Infinity, Paradoxes, Gödel Incompleteness & the Mathematical Multiverse | Lex Fridman Podcast #488
Lex Fridman
0:00 The following is a conversation with Joel David Hamkins,
0:03 a mathematician and philosopher specializing in set theory,
0:07 the foundation of mathematics and the nature of infinity.
0:10 He is the number one highest rated user on MathOverflow,
0:15 which I think is a legendary accomplishment.
0:17 MathOverflow, by the way, is like StackOverflow but for research mathematicians.
0:23 He is also the author of several books,
0:27 including Proof in the Art of Mathematics
0:31 and Lectures on the Philosophy of Mathematics.
0:34 And he has a great blog, infinitelymore.xyz.
0:38 This is a super technical and super fun conversation about
0:46 the foundation of modern mathematics and some mind-bending ideas about infinity,
0:52 nature of reality, truth, and the mathematical paradoxes that challenged some
0:57 of the greatest minds of the 20th century.
1:02 I have been hiding from the world a bit, reading, thinking,
1:09 writing, soul-searching, as we all do every once in a while.
1:12 But mostly, just deeply focused on work and preparing mentally for some
1:17 challenging travel I plan to take on in the new year.
1:21 Through all of it, a recurring thought comes to me,
1:25 how damn lucky I am to be alive and to get
1:28 to experience so much love from folks across the world.
1:33 I want to take this moment to say thank
1:36 you from the bottom of my heart for everything,
1:40 for your support, for the many amazing
1:43 conversations I've had with people across the world.
1:46 I got a little bit of hate and a whole lot of love,
1:52 and I wouldn't have it any other way.
1:55 I'm grateful for all of it.
1:58 This is the Lex Fridman Podcast.
1:59 To support it, please check out our sponsors in the description,
2:03 where you can also find ways to contact me,
2:08 ask questions, give feedback, and so on.
2:11 And now, dear friends, here's Joel David Hamkins.
2:17 Some infinities are bigger than others.
2:20 This idea from Cantor at the end of the 19th century,
2:23 I think it's fair to say, broke mathematics before rebuilding it.
2:28 And I also read that this was
2:31 a devastating and transformative discovery for several reasons.
2:33 So one, it created a theological crisis.
2:36 Because infinity is associated with God, how could there be multiple infinities?
2:40 And also, Cantor was deeply religious himself.
2:43 Second, there's a kind of mathematical civil war.
2:47 The leading German mathematician Kronecker called Cantor a corrupter
2:52 of youth and tried to block his career.
2:57 Third, many fascinating paradoxes emerged from this, like Russell's paradox,
3:02 about the set of all sets that don't contain themselves,
3:06 and those threatened to make all of mathematics inconsistent.
3:09 And finally, on the psychological side and the personal side,
3:13 Cantor's own breakdown.
3:15 He literally went mad, spending his final years in and out of sanatoriums,
3:20 obsessed with proving the continuum hypothesis.
3:22 So, laying that all out on the table, can you explain the idea of infinity,
3:28 that some infinities are larger than others,
3:32 and why was this so transformative to mathematics?
3:36 Well, that's a really great question.
3:38 I would want to start talking about infinity
3:41 and telling the story much earlier than Cantor, actually, because, I mean,
3:45 you can go all the way back to Ancient Greek times when
3:49 Aristotle emphasized the potential aspect
3:53 of infinity as opposed to the impossibility,
3:56 according to him, of achieving an actual infinity.
3:59 And Archimedes' method of exhaustion where he is trying to understand the area
4:05 of a region by carving it into more and more triangles, say,
4:10 and sort of exhausting the area
4:12 and thereby understanding the total area in terms
4:14 of the sum of the areas of the pieces that he put into it.
4:18 And it proceeded on this kind of potential understanding
4:20 of infinity for hundreds of years, thousands of years.
4:23 Almost all mathematicians were almost all
4:27 mathematicians were potentialists only and thought
4:30 that it was incoherent to speak of an actual infinity at all.
4:35 Galileo is an extremely prominent exception to this, though he argued against
4:42 this sort of potentialist orthodoxy in The Dialogue of Two New Sciences.
4:47 Really lovely account there that he gave.
4:51 And that the...
4:53 In many ways, Galileo was anticipating Cantor's developments,
4:57 except he couldn't quite push it all the way through
5:00 and ended up throwing up his hands in confusion in a sense.
5:07 I mean, the Galileo paradox is the idea
5:09 or the observation that if you think about the natural numbers,
5:14 I would start with zero but I think maybe he would start with one.
5:17 The numbers one, two, three, four,
5:19 and so on, and you think about which of those numbers are perfect squares.
5:23 So zero squared is zero and one squared is one and two squared is four,
5:29 three squared is nine, 16, 25, and so on.
5:32 And Galileo observed that, that the perfect squares can be
5:37 put into a one-to-one correspondence with all of the numbers.
5:41 I mean, we just did it.
5:43 I associated every number with its square.
5:46 And so it seems like on the basis of this one-to-one
5:51 correspondence that there should be exactly the same number of squares,
5:56 perfect squares as there are numbers,
5:59 and yet there's all the gaps in between the perfect squares, right?
6:03 And, and this suggests that there should be fewer perfect squares,
6:09 more numbers than squares because the numbers include all
6:12 the squares plus a lot more in between them, right?
6:15 And Galileo was quite troubled by this observation because he took it
6:22 to cause a kind of incoherence in the comparison of infinite quantities, right?
6:27 And another example is, if you take two line segments of different lengths,
6:33 and you can imagine drawing a kind of foliation,
6:37 a fan of lines that connect them.
6:39 So the endpoints are matched from the shorter to the longer segment,
6:43 and the midpoints are matched and so on.
6:45 So spreading out the lines as you go.
6:47 And so every point on the shorter line would be associated
6:51 with a, a unique distinct point on the longer line in a one-to-one way.
6:56 And so it seems like the two line segments have the same number
7:01 of points on them because of that, even though the longer one is longer.
7:06 And so it makes, again, a kind of confusion over our ideas about infinity.
7:10 And also with two circles,
7:12 if you just place them concentrically and draw the rays from the center,
7:17 then every point on the smaller circle is
7:20 associated with a corresponding point on the larger circle,
7:23 you know, in a one-to-one way.
7:25 And, and again, that seems to show that the smaller circle
7:29 has the same number of points on it as the larger one,
7:32 precisely precisely because they can be put into this one-to-one correspondence.
7:36 Of course, the contemporary attitude about this situation
7:39 is that those two infinities are exactly the same,
7:42 and that Galileo was right in those observations about the equinumerosity.
7:45 We would talk about it now by appealing
7:48 to what I call the Cantor-Hume principle,
7:50 or some people just call it Hume's principle,
7:53 which is the idea that if you have two collections,
7:55 whether they're finite or infinite,
7:57 then we want to say that those two collections have the same size.
8:02 They're equinumerous if and only if
8:05 there's a one-to-one correspondence between those collections.
8:09 Galileo was observing that line segments of different lengths are equinumerous,
8:14 and the perfect squares are equinumerous with all of the natural numbers,
8:19 and any two circles are equinumerous, and so on.
8:22 The tension between the Cantor-Hume principle
8:26 and what could be called Euclid's principle,
8:28 which is that the whole is always greater than the part,
8:32 is a principle that Euclid appealed to in the Elements
8:35 many times when he's calculating area and so on.
8:40 It's a basic idea that if something is just a part of another thing,
8:45 then the whole is greater than the part.
8:48 So what Galileo was troubled by was this tension
8:53 between what we call the Cantor-Hume principle and Euclid's principle.
8:58 It wasn't fully resolved, I think, until Cantor.
9:02 He's the one who really explained so clearly about these different sizes
9:07 of infinity and so on in a way that was so compelling.
9:12 He exhibited two different infinite sets
9:15 and proved that they're not equinumerous;
9:18 they can't be put into one-to-one correspondence.
9:20 It's traditional to talk about the uncountability of the real numbers.
9:25 Cantor's big result was that the set of all real numbers is an uncountable set.
9:30 Maybe if we're going to talk about countable sets,
9:33 then I would suggest that we talk about Hilbert's Hotel,
9:36 which really makes that idea perfectly clear.
9:40 Yeah, let's talk about Hilbert's Hotel.
9:42 Hilbert's Hotel is a hotel with infinitely many rooms.
9:45 Each room is a full floor suite.
9:47 So there's floor zero...
9:49 I always start with zero because for me, the natural numbers start with zero,
9:53 although that's maybe a point of contention for some mathematicians.
9:56 The other mathematicians are wrong.
9:58 Like I mentioned, I'm a programmer,
9:59 so starting at zero is a wonderful place to start.
10:02 Exactly.
10:02 So there's floor zero, floor one, floor two, or room zero,
10:05 one, two, three, and so on, just like the natural numbers.
10:08 So Hilbert's Hotel has a room for every natural number,
10:13 and it's completely full.
10:14 There's a person occupying room N for every N.
10:17 But meanwhile, a new guest comes up to the desk and wants a room.
10:22 "Can I have a room, please?" The manager says,
10:24 "Hang on a second, just give me a moment." You see,
10:28 when the other guests had checked in, they had to sign an agreement
10:31 with the hotel that maybe there would be
10:36 some changing of the rooms during this stay.
10:39 So the manager sent a message up
10:42 to all the current occupants and told every person,
10:45 "Hey, can you move up one room,
10:48 please?" So the person in room five would move to room six,
10:51 and the person in room six would move to room seven and so on.
10:54 And everyone moved at the same time.
10:56 And of course, we never want to be
10:58 placing two different guests in the same room,
10:59 and we want everyone to have their own private room and...
11:02 But when you move everyone up one room,
11:05 then the bottom room, room zero, becomes available, of course.
11:07 And so he can put the new guest in that room.
11:10 So even when you have infinitely many things,
11:13 then the new guest can be accommodated.
11:16 And that's a way of showing how
11:18 the particular infinity of the occupants of Hilbert's Hotel,
11:22 it violates Euclid's principle.
11:25 I mean, it exactly illustrates this idea because adding
11:29 one more element to a set didn't make it larger,
11:32 because we can still have a one-to-one correspondence between the total
11:35 new guests and the old guests by the room number, right?
11:40 So, to just say one more time, the hotel is full.
11:45 The hotel is full.
11:46 And then you could still squeeze in one more,
11:49 and that breaks the traditional notion of mathematics and breaks
11:54 people's brains when they try to think about infinity, I suppose.
11:58 This is a property of infinity.
12:00 It's a property of infinity that sometimes when you add an element to a set,
12:05 it doesn't get larger.
12:06 That's what this example shows.
12:08 But one can go on with Hilbert's Hotel, for example.
12:12 I mean, maybe the next day, you know, 20 people show up all at once.
12:17 We can easily do the same trick again, just move everybody up 20 rooms.
12:20 And then we would have 20 empty rooms at the bottom,
12:24 and those new 20 guests could go in.
12:27 But on the following weekend, a giant bus pulled up, Hilbert's bus.
12:33 And Hilbert's bus has, of course, infinitely many seats.
12:37 There's Seat Zero, Seat One, Seat Two, Seat Three, and so on.
12:41 And so one wants to...
12:42 You know, all the people on the bus want to check into the hotel,
12:46 but the hotel is completely full.
12:47 So what is the manager going to do?
12:49 And when I talk about Hilbert's Hotel, when I teach Hilbert's Hotel in class,
12:55 I always demand that the students provide, you know,
12:58 the explanation of- of how to do it.
13:00 So maybe I'll ask you.
13:02 Can you tell me, yeah, what is your idea about how to fit them all in the hotel,
13:06 everyone on the bus, and also the current occupants?
13:09 You separate the hotel into even and odd rooms,
13:12 and you squeeze in the new Hilbert bus people into the odd
13:17 rooms and the previous occupants go into the even rooms.
13:20 That's exactly right.
13:21 That's a very easy way to do it.
13:23 If you just tell all the current guests to double their room number,
13:27 so in Room N, you move to Room 2 times N.
13:29 So they're all going to get their own private room, the new room,
13:32 and it will always be an even number 'cause 2 times N is always an even number.
13:36 And so all the odd rooms become empty that way.
13:38 And now we can put the bus occupants into the odd-numbered rooms.
13:43 And by doing so, you have now shoved an infinity into another infinity.
13:47 That's right.
13:48 So what it really shows...
13:49 I mean, another way of thinking about it is that, well,
13:52 we can define that a set is countable if
13:55 it is equinumerous with a set of natural numbers.
13:58 And a kind of easy way to understand what that's saying in terms
14:02 of Hilbert's Hotel is that a set is countable if it fits into Hilbert's Hotel,
14:07 'cause Hilbert's Hotel basically is the set
14:09 of natural numbers in terms of the room numbers.
14:11 So to be equinumerous with a set of natural numbers
14:14 is just the same thing as to fit into Hilbert's Hotel.
14:16 And so what we've shown is that if you have two countably infinite sets,
14:23 then their union is also countably infinite.
14:25 If you put them together and form a new
14:27 set with all of the elements of either of them,
14:30 then that union set is still only countably infinite.
14:34 It didn't get bigger.
14:36 And that's a remarkable property for a notion of infinity to have, I suppose.
14:42 But if you thought that there was only one kind of infinity,
14:45 then it wouldn't be surprising at all,
14:46 because if you take two infinite sets and put them together,
14:49 then it's still infinite.
14:50 And so if there were only one kind of infinity,
14:51 then it shouldn't be surprising...
14:53 that the union of two countable sets is countable.
14:55 So there's another way to push this a bit harder,
14:59 and that is when Hilbert's train arrives,
15:03 and Hilbert's train has infinitely many train cars...
15:08 ...and each train car has infinitely many seats.
15:13 And so we have an infinity of infinities of the train
15:16 passengers together with the current occupants of the hotel,
15:20 and everybody on the train wants to check in to Hilbert's Hotel.
15:26 So the manager can, again, of course, send a message up to all the rooms
15:30 telling every person to double their room number again.
15:35 And so that will occupy all the even-numbered rooms again,
15:38 and but free up again the odd-numbered rooms.
15:41 So somehow, we want to put the train passengers into the odd-numbered rooms.
15:47 And so while every train passenger is on some car,
15:51 let's say Car C and Seat S, so somehow, we have to take these two coordinates,
15:58 you know, C, S, the car number and the seat number,
16:02 and produce from it an odd number in a one-to-one way, you know?
16:07 And that's actually not very difficult.
16:10 In fact, one can just use, say...
16:12 An easy way to do it is to just use the number 3 to the C times 5 to the S.
16:20 3 to the C, 3 to the car number, so 3 x 3 x 3, you know, the number of the car.
16:27 You multiply 3 by itself, the number of the train car,
16:31 and then you multiply 5 by itself the seat number of times,
16:34 and then you multiply those two numbers together.
16:36 So 3 to the C times 5 to the S.
16:40 That's always an odd number,
16:42 'cause the prime factorization has only 3s and 5s in it.
16:46 There's no 2 there.
16:47 So therefore, it's definitely an odd number,
16:50 and it's always different because of the uniqueness of prime factorization.
16:55 So every number can be factored uniquely into primes.
16:58 So if you have a number of that form, then you can just factor it,
17:02 and that tells you the exponent on 3 and the exponent on 5.
17:05 And so you know exactly which person it was,
17:08 which car they came from, and which seat they came from.
17:10 And prime factorization is every single number
17:14 can be decomposed into the atoms of mathematics, which is the prime numbers.
17:19 You can multiply them together to achieve that number.
17:23 And that's prime factorization.
17:24 You're showing 3 and 5 are both prime numbers, odd.
17:29 So through this magical formula, you can deal with this train,
17:35 infinite number of cars, with each car having infinite number of seats.
17:41 Exactly right.
17:42 We've proved that if you have countably many countable sets,
17:46 then the union of those sets,
17:48 putting all those sets together into one giant set, is still countable.
17:52 You know, because the train cars are each countable, plus the current hotel.
17:56 It's sort of like another train car, if you want to think about it that way.
17:59 The current occupants of the hotel could, you know,
18:02 have the same number as any of the train cars.
18:05 So putting countably many countable sets together
18:08 to make one big union set is still countable.
18:12 It's quite remarkable, I think.
18:14 I mean when I first learned this many, many years ago,
18:17 I was completely shocked by it and transfixed by it.
18:20 It was quite amazing to me
18:22 that this notion of countable infinity could be closed
18:25 under this process of infinitely many infinities
18:28 adding up still to the very same infinity, which is a strong instance,
18:33 a strong violation of Euclid's principle once again, right?
18:37 So, the new set that we built is...
18:40 has many more elements than the old set
18:42 in the sense that there are additional elements,
18:45 but it doesn't have many more elements in terms of its size
18:48 because it's still just a countable infinity and it fits into Hilbert's Hotel.
18:53 Have you been able to sort
18:55 of internalize a good intuition about countable infinity?
18:58 Because that is a pretty weird thing.
19:02 You can have a countably infinite set of countably infinite sets,
19:06 and you can shove it all in and it still is a countable infinite set.
19:11 Yeah, that's exactly right.
19:13 I mean, I guess, of course, when you work with these notions,
19:18 the argument of of Hilbert's Hotel becomes kind of clear.
19:21 There are many other ways to talk about it too.
19:25 For example, let's think about, say, the integer lattice,
19:28 the grid of points that you get by taking pairs of natural numbers,
19:33 say, so the upper right quadrant of the integer lattice, yeah?
19:38 So there's the, you know, row zero, row one,
19:40 row two and so on, column zero, column one,
19:41 column two and so on, and each row and column
19:45 has a countable infinity of points on it, right?
19:50 So those dots, if you think about them as dots,
19:54 are really the same as the train cars if you think about each column of...
19:58 in that integer lattice, it's a countable infinity.
20:00 It's like one train car and then there's the next train car next to it,
20:04 and then the next column next to that, the next train car.
20:08 And so, but if we think about it in this grid manner,
20:11 then I can imagine a kind of winding path winding through these grid points,
20:16 like up and down the diagonals.
20:19 Winding back and forth.
20:20 So I start at the corner point and then I go down,
20:23 up and to the left, and then down and to the right,
20:25 up and to the left, down and to the right,
20:26 and so on, in such a way that I'm going to hit every grid point on this path.
20:34 So, this gives me a way of assigning room numbers to the points.
20:38 Because every grid point is going to be the Nth point on that path for some N.
20:45 And that that gives a correspondence between
20:47 the grid points and the natural numbers themselves.
20:50 So it's a kind of different picture.
20:52 Before, we used this 3 to the C, 5 times 5 to the S,
20:55 which is a kind of, you know, overly arithmetic way to think about it.
20:59 But there's a kind of direct way to understand that it's
21:04 still a countable infinity when you have countably many countable sets,
21:07 because you can just start putting them on this list.
21:10 And as long as you give each of the infinite
21:12 collections a chance to add one more person to the list,
21:16 then you're going to accommodate everyone in any of the sets in one list.
21:21 Yeah, it's a really nice visual way to think about it.
21:23 You just zigzag your way across the grid to make sure everybody's included.
21:27 That gives you an algorithm for including everybody.
21:30 So, can you speak to the uncountable infinities?
21:33 Yeah, absolutely.
21:34 What are the integers and the real numbers-- Correct-
21:36 and what is the line that Cantor was able to find?
21:38 Maybe there's one more step I want to insert before doing that.
21:43 Which is the rational numbers.
21:46 So we did pairs of natural numbers.
21:50 Right?
21:50 That's the train car, basically.
21:52 But maybe it's a little bit informative to think about the rational,
21:56 the fractions, the set of fractions, or rational numbers,
22:00 because a lot of people maybe have an expectation that maybe
22:03 this is a bigger infinity because the rational numbers are densely ordered.
22:08 Between any two fractions, you can find another fraction, right?
22:12 The average of two fractions is another fraction.
22:16 And so, sometimes people,
22:18 it seems to be a different character than the integers,
22:22 which are discretely ordered, right?
22:24 From any integer, there's a next one and a previous one, and so on.
22:27 But that's not true in the rational numbers.
22:29 And yet, the rational numbers are also still only a countable infinity.
22:35 And the way to see that is actually
22:40 it's just exactly the same as Hilbert's train again,
22:42 because every fraction consists of two integers:
22:45 the numerator and the denominator.
22:47 And so if I tell you two natural numbers,
22:51 then you know what fraction I'm talking about.
22:53 I mean, plus the sign issue, I mean if it's positive or negative.
22:57 But if you just think about the positive fractions, then, you know,
23:00 you have the numbers of the form P over Q, where Q is not zero.
23:05 So you can still do 3 to the P times 5 to the Q.
23:09 The same idea works.
23:11 with the rational numbers.
23:12 So this is still a countable set.
23:14 And you might think, "Well,
23:16 every set is going to be countable because there's only one infinity." I mean,
23:21 if that's a kind of perspective maybe that you're adopting,
23:24 but it's not true, and that's the profound achievement that Cantor made
23:28 is proving that the set of real numbers is not a countable infinity.
23:32 It's a strictly larger infinity,
23:34 and therefore there's more than one concept of infinity,
23:38 more than one size of infinity.
23:41 So let's talk about the real numbers.
23:42 What are the real numbers?
23:43 Why do they break infinity?
23:44 The countable infinity.
23:45 Looking it up on Perplexity,
23:48 real numbers include all the numbers that can be represented on the number line,
23:53 encompassing both rational and irrational numbers.
23:55 We've spoken about the rational numbers, and the rational numbers, by the way,
23:59 are by definition, the numbers that can
24:00 be represented as a fraction of two integers.
24:05 That's right.
24:06 So with the real numbers, we have the algebraic numbers.
24:08 We have, of course, all the rational numbers.
24:10 The integers and the rationals are all part of the real number system.
24:13 But then also, we have the algebraic numbers like the square
24:16 root of 2 or the cube root of 5 and so on.
24:19 Numbers that solve an algebraic equation over the integers,
24:22 those are known as algebraic numbers.
24:24 It was an open question for a long time whether that was all
24:28 of the real numbers or whether there
24:31 would exist numbers that are the transcendental numbers.
24:35 The transcendental numbers are real numbers that are not algebraic.
24:38 And we won't even go to the surreal
24:40 numbers about which you have a wonderful blog post.
24:42 We'll talk about that a little bit later.
24:44 Oh, great.
24:44 So it was Liouville who first proved that there are transcendental numbers,
24:49 and he exhibited a very specific number
24:51 that's now known as the Liouville constant, which is a transcendental number.
24:56 Cantor also famously proved that there are many, many transcendental numbers.
25:00 In fact, it follows from his argument on the uncountability
25:04 of the real numbers that there are uncountably many transcendental numbers.
25:09 So most real numbers are transcendental.
25:12 And again, going to Perplexity,
25:13 "Transcendental numbers are 'real' or 'complex' numbers;
25:15 they are not the root of any
25:18 nonzero polynomial with integer or rational coefficients.
25:21 This means they cannot be expressed
25:24 as solutions to algebraic equations with integer coefficients,
25:27 setting them apart from algebraic numbers."- So some
25:32 of the famous transcendental numbers would include the number pi,
25:35 you know, the 3.14159265 and so on.
25:41 So that's a transcendental number.
25:42 Also, Euler's constant, the e, like e to the x, the exponential function.
25:47 So you could say that some of the sexiest
25:50 numbers in mathematics are all transcendental numbers?
25:52 Absolutely.
25:52 That's true.
25:52 Yeah, yeah.
25:53 Although, you know, I don't know, square root of two is pretty- Square root.
25:57 All right.
25:57 So it depends.
25:58 Let's not...
25:59 Beauty can be found in in all the different kinds of sets, but yeah.
26:02 That's right.
26:02 And if you have a kind of simplicity attitude,
26:04 then zero and one are looking pretty good too, so...
26:06 And they're definitely not- Sorry to take that tangent,
26:08 but what is your favorite number?
26:10 Do you have one?
26:11 Oh, gosh.
26:11 You know-- Is it zero?
26:13 Did you know there's a proof that every number is interesting?
26:18 You can prove it, because...
26:22 Yeah?
26:22 What's that proof look like?
26:23 Yeah, okay.
26:23 How do you even begin?
26:24 I'm gonna prove to you-- Okay- ...that every natural number is interesting.
26:28 Okay.
26:29 Yeah.
26:29 I mean, zero's interesting because it's the additive identity, right?
26:32 That's pretty interesting.
26:33 And one is the multiplicative identity,
26:35 so when you multiply it by any other number,
26:38 you just get that number back, right?
26:40 And two is, you know, the first prime number.
26:43 That's super interesting, right?
26:45 And- Okay.
26:46 So one can go on this way and give specific reasons,
26:49 but I wanna prove as a general principle that every number is interesting.
26:53 And this is the proof.
26:56 Suppose, toward contradiction, that there were some boring numbers.
27:02 Okay?
27:04 Oh, okay.
27:04 But if there was an uninteresting number-- Yes- ...then
27:08 there would have to be a smallest uninteresting number.
27:11 Mm-hmm.
27:12 Yes.
27:13 But that's a contradiction, because the smallest uninteresting number is
27:16 a super interesting property to have.
27:21 So therefore-- Ah, that's good.
27:24 ...there cannot be any boring numbers.
27:26 I'm gonna have to try to find a hole in that proof-
27:29 because there's a lot of baked in in the word interesting,
27:32 but yeah, that's a beautiful.
27:33 Right.
27:34 That doesn't say anything about the transcendental numbers,
27:36 about the real numbers that you just
27:38 proved from just-- That's right- ...four natural numbers.
27:40 Yeah.
27:40 Okay, should we get back to Cantor's argument, or-- Sure.
27:43 You've masterfully avoided the question.
27:44 Well, you basically said, "I love all numbers."- Yeah, basically.
27:48 Is that what you said?
27:49 Yeah.
27:49 That was my intention.
27:50 Back to Cantor's argument.
27:51 Let's go.
27:52 Okay, so Cantor wants to prove that the infinity of the real
27:57 numbers is different and strictly larger
28:00 than the infinity of the natural numbers.
28:02 So the natural numbers are the numbers
28:04 that start with zero and add one successively,
28:07 so zero, one, two, three, and so on.
28:09 And the real numbers, as we said,
28:11 are the numbers that come from the number line,
28:14 including all the integers and the rationals and the algebraic
28:17 numbers and the transcendental numbers and all of those numbers altogether.
28:21 Now, obviously, since the natural numbers are included in the real numbers,
28:25 we know that the real numbers are at least as large as the natural numbers.
28:30 And so the claim that we want to prove is that it's strictly larger.
28:35 So suppose that it wasn't strictly larger.
28:39 So then they would have the same size.
28:42 But to have the same size, remember,
28:44 means by definition that there's a one-to-one correspondence between them.
28:49 So we suppose that the real numbers can
28:53 be put into one-to-one correspondence with the natural numbers.
28:57 So therefore, for every natural number N,
28:59 we have a real number, let's call it R sub N.
29:01 R sub N is the Nth real number on the list.
29:05 Basically, our assumption allows us to think of the real numbers
29:09 as having been placed on a list, R1, R2, and so on.
29:13 Okay, and now I'm going to define the number Z, and it's going to be...
29:16 The integer part is going to be a zero,
29:18 and then I'm going to put a decimal place,
29:21 and then I'm going to start specifying the digits of this number Z.
29:24 D1, D2, D3, and so on.
29:27 And what I'm going to make sure is that the Nth digit after the decimal
29:32 point of Z is different from the Nth digit of the Nth number on the list.
29:40 Okay?
29:40 So, to specify the Nth digit of Z, I go to the Nth number on the list,
29:45 R sub N, and I look at its Nth digit after the decimal point.
29:49 And whatever that digit is, I make sure that my digit is different from it.
29:56 Okay?
29:57 And then I want to do something a little bit more,
30:00 and that is I'm going to make it different
30:03 in a way that I'm never using the digits zero or nine.
30:06 I'm just always using the other digits and not zero and it
30:14 would form a kind of diagonal going down and to the right,
30:37 and that, for that reason,
30:41 this argument is called the diagonal argument because we're
30:44 looking at the Nth digit of the Nth number,
30:46 and those exist on a kind of diagonal going down.
30:49 And we've made our number Z so that the Nth digit
30:53 of Z is different from the Nth digit of the Nth number.
30:58 But now it follows that Z is not
31:01 on the list because Z is different from R1 because, well,
31:08 the first digit after the decimal point of Z is
31:11 different from the first digit of R1 after the decimal point.
31:14 That's exactly how we built it.
31:15 And the second digit of Z is different from the second digit of R2 and so on.
31:20 The Nth digit of Z is different from the Nth digit of R sub N for every N.
31:26 So therefore, Z is not equal to any of these numbers R sub N.
31:30 And but that's a contradiction because we had assumed
31:34 that we had every real number on the list,
31:37 but yet here is a real number Z that's not on the list, okay?
31:40 And so that's the main contradiction.
31:43 And so it's a kind of proof by construction.
31:45 Exactly.
31:45 So given a list of numbers, Cantor's proving...
31:48 It's interesting that you say that actually,
31:50 because there's a kind of philosophical controversy that occurs in connection
31:55 with this observation about whether
31:57 Cantor's construction is constructive or not.
32:00 Given a list of numbers, Cantor gives us a specific means of constructing
32:04 a real number that's not on the list, is a way of thinking about it.
32:09 There's this one aspect, which I alluded to earlier,
32:13 but some real numbers have more than one decimal representation,
32:17 and it causes this slight problem in the argument.
32:22 For example, the number one, you can write it as 1.0000 forever,
32:28 but you can also write it as 0.999 forever.
32:33 Those are two different decimal representations of exactly the same number.
32:38 You beautifully got rid of the zeros and the nines.
32:40 Therefore, we don't need to even consider that, and the proof still works.
32:44 Exactly, because the only kind of case where that phenomenon
32:47 occurs is when the number is eventually zero or eventually nine.
32:50 And so since our number Z never had any zeros or nines in it,
32:54 it wasn't one of those numbers.
32:56 And so actually, in those cases,
32:57 we didn't need to do anything special to diagonalize.
33:00 Just the mere fact that our number has a unique
33:03 representation already means that it's not equal to those numbers.
33:06 So maybe it was controversial in Cantor's day more than 100 years ago,
33:10 but I think it's most commonly looked at today as, you know,
33:14 one of the initial main results in set theory,
33:17 and it's profound and amazing and insightful
33:21 and the beginning point of so many later arguments.
33:25 And this diagonalization idea has proved
33:28 to be an extremely fruitful proof method,
33:31 and almost every major result in mathematical logic is
33:35 using in an abstract way the idea of diagonalization.
33:39 It was really the start of so many other observations that were made,
33:45 including Russell's paradox and the halting problem and the recursion theorem,
33:50 and so many other principles are using diagonalization at their core.
33:56 So...
33:56 Can we just step back a little bit?
33:58 Sure.
33:59 This infinity crisis led to a kind of rebuilding of mathematics.
34:04 So it'd be nice if you lay out the things it resulted in.
34:09 So one is set theory became the foundation of mathematics.
34:12 All mathematics could now be built from sets,
34:15 giving math its first truly rigorous foundation.
34:18 The axiomatization of mathematics,
34:22 the paradoxes forced mathematicians to develop ZFC and other axiomatic systems,
34:26 and mathematical logic emerged.
34:29 Gödel, Turing, and others created entire new fields.
34:33 So can you explain what set theory is and, how does it serve
34:39 as a foundation of modern mathematics and maybe even the foundation of truth?
34:43 That's a great question.
34:44 Set theory really has two roles that it's serving.
34:49 There's kind of two ways that set theory emerges.
34:53 On the one hand, set theory is its own subject of mathematics,
34:59 with its own problems and questions and answers and proof methods.
35:04 And so really, from this point of view, set theory is about the transfinite
35:09 recursive constructions or well-founded definitions and constructions.
35:15 And those ideas have been enormously fruitful and set theorists have
35:20 looked into them and developed so many ideas coming out of that.
35:25 But set theory has also happened to serve in this other foundational role.
35:30 It's very common to hear things said about set theory that really
35:34 aren't taking account of this distinction
35:36 between the two roles that it's serving.
35:38 It's its own subject, but it's also serving as a foundation of mathematics.
35:42 So in its foundational role,
35:44 set theory provides a way to think of a collection of things as one thing.
35:49 That's the central idea of set theory.
35:51 A set is a collection of things,
35:53 but you think of the set itself as one abstract thing.
35:58 So when you form the set of real numbers, then that is a set.
36:02 It's one thing.
36:03 It's a set, and it has elements inside of it.
36:06 So it's sort of like a bag of objects.
36:08 A set is kind of like a bag of objects.
36:10 And so we have a lot of different axioms that describe the nature of this idea
36:15 of thinking of a collection of things as one thing itself, one abstract thing.
36:20 And axioms are, I guess, facts that we assume are true,
36:25 based on which we then build the ideas of mathematics.
36:29 So there's a bunch of facts,
36:32 axioms about sets that we can put together, and if they're sufficiently......
36:37 powerful, we can then build on top
36:39 of that a lot of really interesting mathematics.
36:42 Yeah, I think that's right.
36:43 So, I mean, the history of the current set theory axioms,
36:47 known as the Zermelo-Fraenkel axioms,
36:48 came out in the early 20th century with Zermelo's idea.
36:52 I mean, the history is quite fascinating because Zermelo in 1904 offered
36:58 a proof that what's called the axiom of choice implies the well-order principle.
37:05 So he described his proof, and that was extremely controversial at the time.
37:10 And there was no theory, there weren't any axioms there.
37:14 Cantor was not working in an axiomatic framework.
37:16 He didn't have a list of axioms in the way that we have for set theory now,
37:20 and Zermelo didn't either.
37:23 And his ideas were challenged so much with regard to the well-order theorem—
37:29 —that he was pressed to produce the theory
37:33 in which his argument could be formalized,
37:35 and that was the origin of what's known as Zermelo set theory.
37:39 And going to perplexity, the axiom of choice is a fundamental principle in set
37:43 theory which states that for any collection of non-empty sets,
37:46 it is possible to select exactly one element from each set,
37:50 even if no explicit rule to make the choices given.
37:54 This axiom allows the construction of a new
37:56 set containing one element from each original set,
37:59 even in cases where the collection is infinite or where
38:02 there is no natural way to specify a selection rule.
38:06 So this was controversial and this was described
38:10 before there's even a language for axiomatic systems.
38:14 That's right.
38:15 So on the one hand, I mean,
38:17 the axiom of choice principle is completely obvious that we
38:21 want this to be true, that it is true.
38:24 I mean, a lot of people take it as a law of logic.
38:26 If you have a bunch of sets,
38:29 then there's a way of picking an element from each of them.
38:31 picking an element from each of them.
38:34 There's a function.
38:35 If I have a bunch of sets,
38:37 then there's a function that, when you apply it to any one of those sets,
38:41 gives you an element of that set.
38:43 It's- it's a completely natural principle.
38:46 I mean, it's called the axiom of choice,
38:48 which is a way of sort of anthropomorphizing the mathematical idea.
38:51 It's not like the function is choosing something.
38:54 I mean, it's just that if you were to make such choices,
38:58 there would be a function that consisted of the choices that you made.
39:02 And the difficulty is that when you- when you can't
39:05 specify a rule or a procedure by which you're making choices,
39:07 then it's difficult to say choices,
39:09 then it's difficult to say what the function is that you're asserting exists.
39:13 function is that you're asserting exists.
39:15 You know, you- you want to have the view that, well, there is a way of choosing.
39:20 I don't have an easy way to say what the function is,
39:23 but there definitely is one.
39:25 Yeah, this is the way of thinking about the axiom of choice.
39:28 So we're going to say the- the- the three
39:30 letters of ZFC may be a lot on this conversation.
39:32 You already mentioned-- Right- ...Zermelo-Fraenkel set theory,
39:34 that's the Z and the F and the C in that is this...
39:37 and the C in that is this...
39:39 Comes from this axiom of choice.
39:42 That's right.
39:42 So ZFC sounds like a super technical thing,
39:45 but it is the set of axioms that's the foundation of modern mathematics.
39:50 Yeah, absolutely.
39:50 So one should be aware also that there's huge parts of mathematics that don't...
39:54 That pay attention to whether the axiom of choice is being
39:57 used and they don't want to use the axiom of choice,
39:59 so they work out the consequences that's- that are possible without the axiom
40:03 of choice or with weakened forms of- of Zermelo-Fraenkel set theory and so on.
40:08 And that's quite a- there's quite a vibrant amount of work in that area.
40:11 I mean, but going back to the axiom of choice for a bit,
40:16 it's maybe interesting to- to give Russell's description
40:21 of how to think about the axiom of choice.
40:23 So Russell describes, this rich person who has a- an infinite closet.
40:29 And in that closet, he has infinitely many pairs
40:34 of shoes and he tells his butler to "Please go
40:41 and give me one shoe from each pair."
40:43 And- and the butler can do this easily because he can...
40:46 For any pair of shoes, he can just always pick the left shoe.
40:50 I mean, there's a way of picking that we can describe.
40:53 We always take the left one or always take the right one,
40:56 or take the left one if it's a red shoe
40:58 and the right one if it's a brown shoe or, you know.
41:01 We can invent rules that would result in these kind of choice functions,
41:05 so we can describe explicit choice functions.
41:08 And for those cases, you don't need the axiom of choice you don't need
41:12 the Axiom of Choice to know that there's a choice function.
41:15 When you can describe a specific way of choosing,
41:17 then you don't need to appeal to the axiom to know
41:20 that there's a choice But the problematic case occurs when
41:23 you think about the infinite collection when you think about
41:27 the infinite collection of socks that the person has in their closet.
41:31 And if we assume that socks are sort of indistinguishable within each pair,
41:35 you know, they match each other,
41:37 indiscernible, then the- the butler wouldn't have any kind
41:42 of rule for which sock in each pair to pick.
41:46 And so it's not so clear that he has
41:50 a way of- of producing one sock from each pair because...
41:55 Right?
41:56 So that's what's at stake, is the question of whether you can specify
42:00 a rule by which the choice function, you know,
42:04 a rule that it obeys that defines the choice function,
42:08 or whether there's sort of this arbitrary choosing aspect to it.
42:11 That's when you need the axiom of choice to know that there is such a function.
42:16 But of course, as a matter of mathematical ontology,
42:19 we might find attractive the idea that, well,
42:22 look, I mean, I don't- not every way
42:25 of choosing the socks has to be defined by rule.
42:28 Why should everything that exists in mathematical reality
42:31 follow a rule or a procedure of that sort?
42:35 If I have the idea that my mathematical ontology is rich with objects,
42:40 then I think that, that there are all kinds of functions and ways of choosing.
42:45 Those are all part of the mathematical reality that I wanna be talking about....
42:49 and so I don't have any problem asserting the Axiom of Choice.
42:53 Yes, there is a way of choosing, But I can't t- necessarily tell you what it is.
42:59 But in a mathematical argument,
43:01 I can assume that I fix the choice function because I know that there is one.
43:05 So it's a...
43:06 The philosophical difference between working when you have the Axiom of Choice
43:10 and when you don't is the question of this constructive nature of the argument.
43:14 So if you make an argument and you appeal to the Axiom of Choice,
43:18 then maybe you're admitting that the objects that you're
43:22 producing in the proof are not gonna be constructive.
43:25 You're not gonna be able to necessarily say specific things about them.
43:30 But if you're just claiming to make an existence claim, that's totally fine.
43:33 Whereas if you have a constructive attitude about ma- the nature of mathematics,
43:37 and you think that mathematical claims maybe are only
43:40 warranted when you can provide an explicit procedure for producing
43:44 the mathematical objects that you're dealing with then you're probably
43:47 gonna wanna deny the axiom of choice and maybe much more.
43:52 Can we maybe speak to the axioms that underlie ZFC?
43:56 So cone of perplexity, ZFC,
43:58 or Zermelo-Fraenkel set theory with the Axiom of Choice,
44:01 as we mentioned, is the standard foundation for most modern mathematics.
44:04 It consists of the following main axioms: Axiom of Extensionality,
44:08 Axiom of Empty Set, Axiom of Pairing, Axiom of Union,
44:12 Axiom of Power Set, Axiom of Infinity, Axiom of Separation,
44:15 Axiom of Replacement, Axiom of Regularity, and Axiom of Choice.
44:20 Some of these are quite basic, but it would be nice to kinda give people
44:27 a sense- ...of what it means to be an axiom.
44:30 Like, what kind of basic facts we can lay
44:34 on the table on which we can build some beautiful mathematics.
44:37 Yeah, so the history of it is really quite fascinating.
44:40 So, Zermelo introduced most of these axioms, I mean,
44:43 as part of what's now called Zermelo set theory,
44:46 to formalize his proof from the Axiom of Choice to the Well-Order Principle,
44:51 which was an extremely controversial result.
44:53 So in 1904, he gave the proof without the theory,
44:55 and then he was challenged to provide the theory.
44:58 And so in 1908, he produced the Zermelo set
45:02 theory and gave the proof that in that theory,
45:06 you can prove that every set admits a well ordering.
45:09 And so the axioms on the list, these things like extensionality,
45:12 express the most fundamental principles of the understanding
45:16 of sets that he wanted to be talking about.
45:19 So for example, extensionality says if two
45:22 sets have the same members, then they're equal.
45:25 So it's this idea that the sets consist
45:29 of the collection of their members, and that's it.
45:32 There's nothing else that's going on in the set.
45:34 So it's just if two sets have the same members, then they are the same set.
45:38 So it's maybe the most primitive axiom in some respect.
45:45 Well, there's also, just to give a flavor,
45:47 there exists a set with no elements, called the empty set.
45:50 For any two sets, there's a set
45:53 that contains exactly those two sets as elements.
45:56 For any set, there's a set that contains exactly the elements
45:59 of the elements of that set, so the union set.
46:02 And then there's the power set.
46:04 For any set, there's a set whose elements are
46:06 exactly the subsets of the original set, the power set.
46:09 In the axiom of infinity, there exists an infinite set,
46:13 typically a set that contains the empty set and is
46:16 closed under the operation of adding one more element.
46:19 Back to our hotel example.
46:22 That's right.
46:23 And there's more, but this is kind of fascinating.
46:26 Just put yourself in the mindset of people
46:29 at the beginning of this, of trying to formalize set theory.
46:34 It's fascinating that humans can do that.
46:37 I read some historical accounts by historians about that time period,
46:42 specifically about Zermelo's axioms and his proof of the well-order theorem.
46:47 And the historians were saying never before
46:50 in the history of mathematics has a mathematical
46:54 theorem been argued about so publicly
46:57 and so vociferously as that theorem of Zermelo's.
47:02 And it's fascinating also because the axiom of choice
47:07 was widely regarded as a kind of, you know, basic principle at first,
47:12 but then when, but people were very suspicious of the well-order
47:14 theorem because no one could imagine a well ordering, say, of the real numbers.
47:19 And so this was a case when Zermelo seemed to be,
47:23 from principles that seemed quite reasonable, proving this obvious untruth.
47:28 And so people were, mathematicians were objecting.
47:31 But then Zermelo and others actually looked into the mathematical papers and so
47:36 on of some of the people who had been objecting so vociferously,
47:40 and found, in many cases,
47:43 that they were implicitly using the axiom of choice in their own arguments,
47:47 even though they would argue publicly against it.
47:50 Because it's so natural to use it because
47:54 it's such an obvious principle in a way.
47:56 I mean, it's easy to just use it by accident if you're not
48:00 critical enough and you don't even realize
48:02 that you're using the axiom of choice.
48:04 That's true now, even.
48:05 People like to pay attention to when the axiom of choice
48:07 is used or not used in mathematical arguments, up until this day.
48:11 It used to be more important.
48:13 In the early 20th century it was very important because
48:15 people didn't know if it was a consistent theory or not,
48:18 and there were these antinomies arising.
48:20 and so there was a worry about consistency of the axioms.
48:24 but then, of course, eventually,
48:26 with the result of, of Godel and Cohen and so on, they...
48:30 this consistency question specifically about the axiom
48:32 of choice sort of falls away.
48:34 We know that the axiom of choice itself choice itself
48:37 will never be the source of inconsistency in set theory.
48:40 If there's inconsistency with the axiom of choice,
48:43 then it's, it's already inconsistent without the axiom of choice.
48:45 So it's not the cause of inconsistency.
48:47 And so in that...
48:49 from that point of view, the need to pay attention to whether you're using it
48:52 or not from a consistency point of view is somehow less important.
48:55 But still there's this reason to pay attention to it
49:00 on the grounds of these constructivist ideas that I had mentioned earlier.
49:05 And we should say, in set theory,
49:07 consistency means that it is impossible to derive
49:09 a contradiction from the axioms of the theory.
49:12 It means that there are no contradictions.
49:15 That's a...
49:15 That's right- A consistent axiomatic system is that there are no contradictions.
49:19 A consistent theory is one for which
49:21 you cannot prove a contradiction from that theory.
49:24 Maybe a quick pause, a quick break, a quick bathroom break.
49:28 You mentioned to me offline we were talking
49:30 about Russell's paradox and that there's a- a nice,
49:34 another kind of anthropomorphizable proof of uncountability.
49:38 I was wondering if you can lay that out.
49:41 Oh yeah, sure.
49:42 Absolutely.
49:42 Both Russell's paradox and the proof.
49:44 Right.
49:45 So we talked about Cantor's proof that the real numbers,
49:50 the set of real numbers is an uncountable infinity,
49:53 it's a strictly larger infinity than the natural numbers.
49:56 But Cantor actually proved a- a much more general fact,
50:00 namely that for any set whatsoever,
50:03 the power set of that set is a strictly larger set.
50:07 So the power set is the set containing all the subsets of the original set.
50:12 So if you have a set and you look at the collection of all of its subsets,
50:17 then Cantor proved that this is- this is a bigger set.
50:21 They're not equinumerous.
50:22 Of course, there's always at least as many subsets
50:25 as elements because for any element you can make,
50:28 the- the singleton subset that has only that guy as a member, right?
50:33 So there's always at least as many subsets as elements.
50:36 But the question is whether they, whether it's strictly more or not.
50:40 And so Cantor reasoned like this.
50:43 It's very simple.
50:44 It's a kind of distilling the abstract diagonalization idea
50:48 without being encumbered by the complexity of the real numbers.
50:52 So, we have a set X, and we're looking at all of its subsets.
50:58 That's the power set of X.
51:00 Suppose that X and the power set of X have the same size.
51:04 Suppose, towards contradiction, they have the same size.
51:07 So that means we can associate to every individual of X a subset.
51:14 And so now let me define a new set.
51:17 Another set, I'm going to define it.
51:19 Let's call it D.
51:20 And D is the subset of X that contains
51:24 all the individuals that are not in their set.
51:28 Every individual was associated with a subset of X,
51:32 and I'm looking at the individuals that are not in their set.
51:37 Maybe nobody's like that.
51:38 Maybe there's no element of X that's like that, or maybe they're all
51:41 like that, or maybe some of them are and some of them aren't.
51:43 It doesn't really matter for the argument.
51:46 I defined a subset D consisting of the individuals
51:49 that are not in the set that's attached to them,
51:52 but that's a perfectly good subset.
51:54 And so because of the equinumerosity,
51:56 it would have to be attached to a particular individual, you know?
52:00 And- ...but that...
52:02 let's call that person, it should be a name starting with D, so Diana.
52:10 And now we ask, is Diana an element of D or not?
52:15 But if Diana is an element of D, then she is in her set.
52:19 So she shouldn't be because the set D was
52:24 the set of individuals that are not in their set.
52:28 So if Diana is in D, then she shouldn't be.
52:30 But if she isn't in D, then she wouldn't be in her set.
52:33 And so she should be in D.
52:35 That's a contradiction.
52:37 So therefore, the number of subsets is always
52:41 greater than the number of elements for any set.
52:45 And the anthropomorphizing idea is the following.
52:49 I'd like to talk about it this way.
52:51 For any collection of people,
52:53 you can form more committees from them than there are people,
52:59 even if you have infinetely many people.
53:03 Suppose you have an infinite set of people.
53:05 And what's a committee?
53:07 Well, a committee is just a list of who's on the committee, basically.
53:10 The members of the committee.
53:12 So there's all the two-person committees
53:14 and there's all the one-person committees,
53:16 and there's the universal, the worst committee, the one that everyone is on.
53:20 Okay.
53:21 The best committee is the empty committee.
53:23 With no members and never meets and so on.
53:25 Or is the empty committee meeting all the time?
53:28 I'm not sure.
53:30 Yeah.
53:30 That's...
53:30 wow, that's a profound question.
53:32 And does a committee with just one member meet also as a-- Yeah,
53:35 maybe it's always in session.
53:37 I don't know.
53:38 So the claim is that there are more committees than people.
53:44 Okay.
53:45 Suppose not.
53:45 Well, then we could make an association between the people and the committees.
53:50 So we would have every committee could be
53:53 named after a person in a one-to-one way.
53:56 And I'm not saying that the person is on the committee that's named after them,
54:00 or not on it, whatever.
54:01 Maybe sometimes that happens, sometimes it doesn't.
54:03 I don't know.
54:04 It doesn't matter.
54:05 But let's form what I call committee D,
54:08 which consists of all the people that are
54:11 not on the committee that's named after them.
54:16 Okay.
54:16 Maybe that's everyone, maybe it's no one, maybe it's half the people.
54:19 It doesn't matter.
54:20 That's a committee.
54:21 It's a set of people.
54:23 And so it has to be named after someone.
54:27 Let's call that person Daniella.
54:30 So now we ask, is Daniella on the committee that's named after her?
54:37 Well, if she is, then she shouldn't be because it
54:40 was the committee of people who aren't on their own committee.
54:45 And if she isn't, then she should be.
54:47 So again, it's a contradiction.
54:48 So when I was teaching at Oxford, one of my students came up
54:54 with the following different anthropomorphization of Cantor's argument.
55:00 Let's consider all possible fruit salads.
55:03 We have a given collection of fruits.
55:07 You know, apples and oranges and grapes, whatever.
55:09 And a fruit salad consists of some collection of those fruits.
55:13 So there's the banana, pear, grape salad and so on.
55:16 There are a lot of different kinds of salad.
55:18 Every set of fruits makes a salad, a fruit salad.
55:21 Okay.
55:21 And we want to prove that for any collection of fruits,
55:25 even if there are infinitely many different kinds of fruit,
55:29 for any collection of fruits,
55:32 there are more possible fruit salads than there are fruits.
55:38 So if not, then you can put a one-to-one
55:40 correspondence between the fruits and the fruit salads,
55:43 so you could name every fruit salad after a fruit.
55:47 That fruit might not be in that salad, it doesn't matter.
55:50 We're just...
55:51 it's a naming, a one-to-one correspondence.
55:53 And then, of course, we form the diagonal salad,
55:57 which consists of all the fruits that are
56:00 not in the salad that's named after them.
56:04 And that's a perfectly good salad.
56:06 It might be the kind of diet salad, if it was the empty salad,
56:10 or it might be the universal salad which had all fruits in it,
56:13 if all the fruits were in it.
56:14 Or it might have just some and not all.
56:16 So that diagonal salad would have to be named after some fruit.
56:20 So let's suppose it's named after durian,
56:23 meaning that it was associated with durian in the one-to-one correspondence.
56:27 And then we ask, well, is durian in the salad that it's named after?
56:32 And if it is, then it shouldn't be.
56:35 And if it isn't, then it should be.
56:37 And so it's again the same contradiction.
56:39 So all of those arguments are just the same as Cantor's proof
56:43 that the power set of any set is bigger than the set.
56:47 And this is exactly the same logic that comes up in Russell's paradox,
56:52 because Russell is arguing that the class of all sets can't be a set,
56:59 because if it were, then we could form the the set
57:04 of all sets that are not elements of themselves.
57:09 So basically, what Russell is proving is
57:11 that there are more collections of sets than elements.
57:15 Because we can form the diagonal class, you know,
57:20 the class of all sets that are not elements of themselves.
57:23 If that were a set, then it would be an element
57:26 of itself if and only if it was not an element of itself.
57:30 It's exactly the same logic in all four of those arguments.
57:34 Yeah.
57:34 So there can't be a class of all sets, because if there were,
57:36 then there would have to be a class
57:37 of all sets that aren't elements of themselves.
57:40 But that set would be an element of itself
57:43 if and only if it's not an element of itself, which is a contradiction.
57:46 So this is the essence of the Russell paradox.
57:49 I don't call it the Russell paradox.
57:51 Actually, when I teach it, I call it Russell's theorem.
57:54 There's no universal set.
57:56 And it's not really confusing anymore.
57:59 At the time, it was very confusing, but now we've absorbed this nature of set
58:06 theory into our fundamental understanding of how sets are,
58:10 and it's not confusing anymore.
58:11 I mean, the history is fascinating though about the Russell paradox,
58:15 because before that time, Frege was working on his monumental work undertaking,
58:22 implementing the philosophy of logicism,
58:24 which is the attempt to reduce all of mathematics to logic.
58:30 So Frege wanted to give an account
58:32 of all of mathematics in terms of logical notions,
58:36 and he was writing this monumental work and had formulated his basic principles.
58:42 And those principles happened to imply that for any property whatsoever,
58:47 you could form the set of objects with that property.
58:52 This is known as the general comprehension principle.
58:58 And he was appealing to the principles that support that axiom,
59:03 throughout his work.
59:04 I mean, it wasn't just an incidental thing.
59:08 He was really using this principle.
59:10 And Russell wrote him a letter when he observed the work in progress,
59:16 that there was this problem,
59:18 because if you accept the principle that for any property whatsoever,
59:20 you can make a set of objects with that property,
59:23 then you could form the set of all sets that are not members of themselves.
59:28 That's just an instance of the general comprehension principle.
59:33 And the set of all sets that aren't elements
59:38 of themselves can't be a set, because if it were,
59:40 then it would be an element of itself if
59:42 and only if it's not a member of itself, and that's a contradiction.
59:46 And so Russell wrote this letter to Frege,
59:48 and it was just at the moment when Frege was finishing his work.
59:52 It was already at the publishers and, you know, in press basically.
59:56 But it's completely devastating.
59:58 I mean, it must have been such a horrible situation for Frege
1:00:02 to be placed in because he's finished this monumental work, you know,
1:00:10 years of his life dedicated
1:00:12 to this, and Russell finds this basically one-line proof
1:00:16 of a contradiction in the fundamental principles
1:00:20 of the thesis that completely destroys the whole system.
1:00:25 And Frege had put in the appendix of his work
1:00:30 a response to Russell's letter in which he explained what happened,
1:00:34 and he wrote very gracefully,
1:00:35 "Hardly anything more unwelcome can befall a scientific writer than to have
1:00:40 one of the foundations of his edifice shaken after the work is finished.
1:00:43 This is the position into which I was put by a letter from Mr.
1:00:46 Bertrand Russell as the printing of this volume was nearing
1:00:49 completion." And then he goes on to explain the matter,
1:00:51 it concerns his basic law five, and so on, and...
1:00:54 It's heartbreaking.
1:00:55 I mean, there's nothing more traumatic to a person
1:00:58 who dreams of constructing mathematics all from logic.
1:01:02 to get a very clean, simple contradiction.
1:01:06 I mean, that's just...
1:01:08 You devote your life to this work,
1:01:10 and then it's shown to be contradictory, and that must have been heartbreaking.
1:01:16 What do you think about the Frege project,
1:01:19 the philosophy of logic, the dream of the power of logic-- Right-...
1:01:23 to construct a mathematical universe?
1:01:24 So, of course, the project of logicism did not die with Frege,
1:01:28 and it was continued, and, you know, there's a whole movement,
1:01:32 the neologicists and so on, in contemporary times even.
1:01:35 But my view of the matter is that, really,
1:01:38 we should view the main goals of logicism are
1:01:43 basically completely fulfilled in the rise of set-theoretic foundationalism.
1:01:48 I mean, When you view ZFC as the foundation of mathematics, and in my view,
1:01:55 the principles of ZFC are fundamentally illogical
1:01:58 in character including the axiom of choice,
1:02:01 as I mentioned, as a principle of logic.
1:02:03 This is a highly disputed point of view, though,
1:02:06 cause a lot of people take even the axiom
1:02:08 of infinity as inherently mathematical and not logical and so on.
1:02:14 But I think if you adopt the view that the principles
1:02:17 of ZFC have to do with the principles of abstract,
1:02:21 you know, set formation, which is fundamentally logical in character,
1:02:25 then it's complete success for logicism.
1:02:28 So the fact that set theory is able to serve
1:02:31 as a foundation means that mathematics can be founded on logic.
1:02:36 I think this is a good moment to talk about Gödel's incompleteness theorems.
1:02:40 So, can you explain them and what do
1:02:43 they teach us about the nature of mathematical truth?
1:02:47 Absolutely.
1:02:47 It's one of the most profound developments in mathematical logic.
1:02:51 I mean, the incompleteness theorems is when mathematical logic,
1:02:55 in my view, first became sophisticated.
1:02:59 It's a kind of birth of the subject of mathematical logic.
1:03:03 But to understand the theorems,
1:03:05 you really have to start a little bit earlier with Hilbert's program.
1:03:10 because at that time, you know, with the Russell Paradox and so
1:03:13 on, there were these various contradictions popping
1:03:14 up in various parts of set theory and the Burali-Forti paradox and so on.
1:03:20 And, and Hilbert was famously supportive of set theory.
1:03:24 I mean, there's this quote of him saying,
1:03:28 "No one shall cast us from the paradise that Cantor has created for us."
1:03:33 And what I take him to mean by that is he was so captured
1:03:37 by the idea of using set theory as a foundation of mathematics and it
1:03:42 was so powerful and convenient and unifying
1:03:44 in a way that was extremely important.
1:03:46 And he wasn't, he didn't wanna give that up,
1:03:49 despite the danger of these paradoxes, these contradictions,
1:03:54 basically, is how some people viewed them.
1:03:57 And so,- this minefield of paradoxes.
1:03:59 Yeah.
1:04:00 Right.
1:04:00 A minefield.
1:04:01 That's a really good way of describing the situation.
1:04:03 And so Hilbert said, "Well, look, we have to fix this problem, you know.
1:04:08 We wanna use the set theory foundations,
1:04:10 but we want to do it in a way that is trustworthy and reliable.
1:04:14 We can't allow that the foundations of mathematics are in question.
1:04:19 You know, this is a kind of attitude,
1:04:21 I think, that underlies Hilbert and the Hilbert program.
1:04:26 And so he proposed, "Look, we're going to have this strong theory,
1:04:30 this set theory that we want to be proving our theorems in but I I mean,
1:04:37 on the one hand, we want it to be as strong as possible.
1:04:40 We would like it to answer all the questions." There's
1:04:44 another famous quote of Hilbert in his retirement address where,
1:04:48 he proclaims, "Wir müssen wissen, wir werden wissen." So,
1:04:53 "We must know, we will know," in which he's very optimistic about the ability
1:04:59 of mathematics to answer all of the questions of mathematics that we have posed.
1:05:05 We have all these problems we want to solve and he is saying,
1:05:09 "We're going to do it.
1:05:10 We're going to solve all these problems." So we
1:05:11 want to propose this strong theory and one has
1:05:14 the sense that he had in mind set theory
1:05:16 in which all the questions are going to be answered.
1:05:20 Okay?
1:05:21 But secondly, we want to combine that with, in a very weak arithmetic,
1:05:28 purely finitistic theory,
1:05:29 we want to prove that the reasoning process of the strong theory is safe.
1:05:36 Okay?
1:05:36 So in order to make sense of that point of view,
1:05:39 you basically have to invent the philosophy of formalism,
1:05:43 where we can look at what is a proof,
1:05:46 what is the nature of mathematical reasoning.
1:05:49 And on Hilbert's way of thinking about this, a proof
1:05:54 is basically itself a finitistic kind of object.
1:05:58 It's a sequence of...
1:06:00 If you think about the nature of what a proof is,
1:06:03 it's a sequence of assertions which can be viewed as sort
1:06:06 of sequences of symbols that conform with certain rules of logical reasoning.
1:06:12 And this is a formalist way of understanding the nature of proof.
1:06:16 So we think about a proof in a kind of syntactic, formal way.
1:06:19 Even though the contents of those statements
1:06:22 might be referring to infinite uncountable objects,
1:06:27 the statements themselves are not infinite uncountable objects.
1:06:29 The statements themselves are just finite sequences of symbols.
1:06:33 So when you think of proof as, maybe it's fair to say,
1:06:36 almost like a outside of math.
1:06:37 It's like, tools operating on math.
1:06:39 And then for Hilbert, he thought proof is inside the axiomatic system.
1:06:44 Something like this.
1:06:46 Yeah, that's helpful.
1:06:47 That's wild.
1:06:48 The main thing about formalism is that you
1:06:52 think of the process of doing mathematics.
1:06:54 You divorce it from the meaning of the mathematical assertions, right?
1:06:59 So the meaning of the mathematical assertions that you make in this infinitary
1:07:03 theory has to do with these huge uncountable infinities and so on possibly.
1:07:08 And that's a very sort of uncertain realm maybe and the source
1:07:14 of the paradoxes and so on in some people's minds.
1:07:18 But the reasoning process itself consists of writing down
1:07:22 sequences of symbols on your page and, you know,
1:07:26 undertaking or, an argument with them which is following these finitary rules.
1:07:32 And so, if we divorce the meaning of the symbols
1:07:36 from just the process of manipulating the symbols,
1:07:38 it's a way of looking at the nature of mathematics as a kind
1:07:41 of formal game in which- the meaning may be totally absent.
1:07:45 I don't think it's necessarily part of the formalist
1:07:48 view that there is no meaning behind,
1:07:50 but rather it's emphasizing that we can divorce the meaning
1:07:54 of the sentences from the process of manipulating those sentences.
1:07:59 And then Hilbert wanted to prove in this purely finitary
1:08:02 theory that if we follow the rules of that game,
1:08:06 we're never going to get a contradiction.
1:08:09 So those were the two aims of the Hilbert program,
1:08:12 is to found the strong infinitary theory,
1:08:15 probably set theory, which is going to answer all the questions.
1:08:20 And then secondly, prove in the finitary theory that the strong theory is safe.
1:08:28 In other words, consistent, yeah?
1:08:31 What does the word "finitary" in finitary theory mean?
1:08:34 Yeah.
1:08:34 Well, this is, of course, philosophically contentious,
1:08:37 and people have different ideas about what exactly it should mean.
1:08:41 So there's hundreds of papers on exactly that question.
1:08:43 But I like to take it just kind of informally.
1:08:45 I mean, it means that we're talking about finite sequences of symbols,
1:08:49 and we're going to have a theory, you know, finite strings of symbols.
1:08:54 A finitary theory would be one whose subject matter is about those kinds
1:08:58 of things so that we can conceivably
1:09:01 argue about the nature of these finite strings.
1:09:05 A proof is just a finite sequence of statements,
1:09:08 so that every statement is either one of the axioms or follows
1:09:12 by the laws of logic from the earlier statements in some specified manner,
1:09:16 like using modus ponens or some other law of logic like that.
1:09:21 And such that the last line on the list is,
1:09:24 you know, the theorem that you're proving.
1:09:26 So that's what a proof is in this kind of way of thinking.
1:09:29 To take a specific example, I mean,
1:09:31 I always conceive of the, perhaps the most natural finitary theory
1:09:35 that one would be called upon to exhibit would be Peano arithmetic,
1:09:39 the theory of Peano arithmetic,
1:09:41 which- which is a first order theory of the nature of arithmetic.
1:09:45 But okay, so some people say, "Well,
1:09:48 Peano arithmetic has these strong first order induction axioms,
1:09:51 and there's much, much weaker versions of arithmetic,
1:09:54 like I-sigma-naught or I-sigma-1 and so on, which are even more finitary
1:09:59 than Peano arithmetic." So different philosophical
1:10:01 positions take different attitudes about what is,
1:10:04 what does it take to be finitary?
1:10:06 How finitary do you have to be to be truly finitary?
1:10:10 So according to Perplexity, Peano arithmetic is a foundational system
1:10:13 for formalizing the properties and operations
1:10:15 of natural numbers using a set of axioms called the Peano axioms.
1:10:19 Peano arithmetic provides a formal
1:10:22 language and axioms for arithmetic operations,
1:10:24 such as addition and multiplication over the natural numbers.
1:10:27 The axioms define the existence of a first natural number, usually zero or one,
1:10:33 the concept of a successor function, which generates the next natural number,
1:10:38 rules for addition and multiplication built from these concepts,
1:10:41 the principle of induction allowing proofs around
1:10:44 all natural numbers, and it goes on.
1:10:46 So it's a very particular kind of arithmetic that is finitary.
1:10:51 You know, in my...
1:10:52 I view it as finitary, but this is a contentious view.
1:10:54 Not everyone agrees with that.
1:10:55 That's what I was trying to hint at.
1:10:58 Okay.
1:10:58 I got it.
1:10:58 All right.
1:10:58 Peano arithmetic is one of the hugely successful
1:11:03 theories of the natural numbers and elementary number theory.
1:11:07 Essentially, all of classical number theory,
1:11:11 so whatever kind of theorems you want to be proving about the prime
1:11:14 numbers or factorization or any kind
1:11:17 of finitary reasoning about finite combinatorial objects,
1:11:20 all of it can be formalized in Peano arithmetic.
1:11:23 I mean, that's the basic situation.
1:11:26 Of course, one has to qualify those statements
1:11:28 in light of the Gödel incompleteness theorem, but for the most part,
1:11:33 the classical number theoretic analysis of the finite
1:11:38 numbers is almost entirely developable inside Peano arithmetic.
1:11:45 So if we go back to the Hilbert program, so Hilbert has these two goals:
1:11:50 produce the strong theory which is going to answer all the questions,
1:11:53 and then prove by purely finitary means
1:11:57 that that theory will never lead into contradiction.
1:12:00 And one can think about, well, the incompleteness theorem should be viewed
1:12:04 as a decisive refutation of the Hilbert program.
1:12:08 It defeats both of those goals decisively, completely.
1:12:13 But before explaining that, maybe one should think about,
1:12:17 you know, what if Hilbert had been right?
1:12:19 What would be the nature of mathematics in the world
1:12:22 that Hilbert is telling us to search for?
1:12:25 And if I may, going to Perplexity's definition of Hilbert's program,
1:12:28 it was David Hilbert's early 20th century project to give
1:12:32 all of classical mathematics a completely secure finitary foundation.
1:12:36 In essence, the goal was to formalize all
1:12:39 of mathematics in precise axiomatic systems and then
1:12:44 prove using only very elementary finitary reasoning about
1:12:48 symbols that these systems are free of contradiction.
1:12:51 Right.
1:12:52 Exactly right.
1:12:52 Let's imagine what it would be like if he had been right.
1:12:55 So we would have this finitary theory,
1:12:59 and it would prove that the strong theory was free of contradiction.
1:13:04 So we could start enumerating proofs from the strong theory.
1:13:08 Right now, we can write a computer program that would
1:13:14 systematically generate all possible proofs from a given theory.
1:13:20 And so we could have, like,
1:13:25 this theorem enumeration machine that would just spit out theorems all day long
1:13:30 in such a manner that every single
1:13:32 theorem would eventually be produced by this device.
1:13:37 And so if you had a mathematical question of any kind,
1:13:42 you could answer it by just waiting for either the answer to come
1:13:47 out yes or from the machine or the answer to come out no.
1:13:51 So the nature of mathematical investigation in Hilbert's world is
1:13:56 one of just turning the crank of the theorem enumeration machine,
1:14:00 devoid of creative thinking or imagination,
1:14:04 it's just getting the answer from this by rote.
1:14:07 procedure.
1:14:08 So Hilbert, in effect, is telling us, I mean,
1:14:12 with his program, that the fundamental
1:14:15 nature of mathematics is rote computation.
1:14:17 I mean, the way I think about the Hilbert program seems
1:14:21 extremely attractive in the historical context
1:14:23 of being worried about the antinomies,
1:14:26 the inconsistencies, and so how can we kind of block them?
1:14:31 It seems natural, first of all,
1:14:33 to have a strong theory that's going to answer all the questions,
1:14:36 because the idea of logical independence and pervasiveness
1:14:40 that we now know exists just wasn't, you know, there was no known.
1:14:47 They didn't know anything like that happening ever.
1:14:50 And so it's natural to think that it wouldn't happen,
1:14:53 and also that they would be able to guard against this inconsistency.
1:14:57 So it seems like the goals of the Hilbert
1:15:00 program are quite natural in that historical context.
1:15:03 But, you know, when you think a little more
1:15:05 about what the nature of it would be like,
1:15:07 it shows you this kind of rote procedure.
1:15:10 And now you're saying, well, that doesn't seem so unlikely maybe, I mean,
1:15:13 in the light of the increasing computer power and so
1:15:17 on, it's actually maybe turning into our everyday experience,
1:15:20 where the machines are calculating more and more
1:15:22 for us in a way that could be alarming.
1:15:26 Okay.
1:15:27 But, okay, so to talk about the alternative to the Hilbert point of view,
1:15:32 I mean, if he's wrong, then what is the nature of mathematical reality?
1:15:36 Well, it would mean that we couldn't ever maybe, for the first goal,
1:15:41 we couldn't ever write down a theory that answered all the questions.
1:15:45 So we would always be in a situation where our best theory,
1:15:51 even the infinitary theories,
1:15:55 would have questions that they stumble with and are unable to answer.
1:15:59 Independence would occur.
1:16:01 But then also, because of the failure of the second goal,
1:16:05 we would also have to be constantly worrying
1:16:07 about whether our theories were consistent or not,
1:16:10 and we wouldn't have any truly convincing means
1:16:14 of saying that they were free from contradiction.
1:16:18 And the fact of Gödel's Incompleteness Theorem shows
1:16:22 that that is exactly the nature of mathematical reality, actually.
1:16:27 Those are the two incompleteness theorems.
1:16:30 So the first incompleteness theorem says you cannot write
1:16:33 down a computably axiomatizable theory that answers all the questions.
1:16:37 Every such theory will be incomplete,
1:16:40 assuming it includes a certain amount of arithmetic.
1:16:43 And secondly, no such theory can ever prove its own consistency.
1:16:47 So not only is it the case that the finitary
1:16:49 theory can't prove the consistency of the strong infinitary theory,
1:16:53 but even the infinitary theory can't prove its own consistency, right?
1:16:57 That's the second incompleteness theorem.
1:16:59 And so it's, in that sense, a decisive takedown of the Hilbert program,
1:17:06 which is really quite remarkable,
1:17:08 the extent to which his theorem just really answered that whole puzzle.
1:17:14 It's quite amazing.
1:17:17 I mean, there's another aspect, kind of easy to think about.
1:17:21 I mean, if you're wondering about
1:17:23 theories that prove their own consistency, then,
1:17:26 I mean, would you trust a theory that proves of itself that it's consistent?
1:17:32 I mean, that's like...
1:17:34 it's like the used car salesman telling you, "Oh, I'm trustworthy." I mean,
1:17:39 it's not a reason to trust the used car salesman, is it?
1:17:43 Just because he says that.
1:17:45 So similarly, if you have a theory that proves its own consistency,
1:17:48 well, even an inconsistent theory would prove its own consistency.
1:17:51 And so it doesn't seem to be a logical reason to believe in the consistency,
1:17:55 if you have a theory that proves itself consistent.
1:17:59 Just for clarification, you used the word theory.
1:18:03 Is it, in this context, synonymous with axiomatic system?
1:18:07 Right.
1:18:08 So in mathematical logic, "theory" is a technical term.
1:18:11 And it means any set of sentences in a formal language.
1:18:14 And so if you say axiomatic system,
1:18:17 it's basically synonymous to my usage with theory.
1:18:20 So a theory means, you know, the consequences of a set of axioms or...
1:18:24 People are sometimes unclear on whether they just mean
1:18:26 the axioms or the consequences of the axioms, but...
1:18:29 So theory includes both the axioms and the consequences of the axioms,
1:18:33 and you use it interchangeably and the context is supposed
1:18:36 to help you figure out which of the two you're talking about?
1:18:40 The axioms or the consequences?
1:18:41 Or maybe to you, they're basically the same?
1:18:44 Yeah, well, they're so closely connected,
1:18:46 although all the features aren't the same.
1:18:47 So if you have a computable list of axioms for a theory,
1:18:54 then you can start enumerating the consequences of the axioms,
1:18:59 but you won't be able to computably decide
1:19:02 whether a given statement is a consequence or not.
1:19:05 You can enumerate the consequences, so you can semi-decide the consequences,
1:19:10 but you won't be able to decide yes or no
1:19:13 whether a given statement is a consequence or not.
1:19:17 So it's the distinction between a problem being
1:19:19 computably decidable and a problem being computably enumerable,
1:19:23 which, was made clear following the work
1:19:26 of Turing and others that came from that.
1:19:29 I mean, so that's one difference between the list
1:19:32 of axioms of the theory and the theory itself.
1:19:35 The axioms could be...
1:19:36 You can decide, maybe computably, whether something is an axiom or not,
1:19:40 but that doesn't mean that you can decide computably
1:19:42 whether or not something is a theorem or not.
1:19:44 Usually, you only get to decide the positive instances.
1:19:48 If something is a theorem,
1:19:49 you will eventually come to recognize that, but if something isn't a theorem,
1:19:52 maybe at no point will you be able to say, "No,
1:19:55 that's not a theorem."- And that's of course connected to the halting problem.
1:19:58 ...and all of these contradictions and paradoxes are all nicely,
1:20:04 beautifully interconnected.
1:20:06 So can we just linger on Gödel's incompleteness theorem?
1:20:09 You mentioned the two components there.
1:20:12 You know, there's so many questions to ask,
1:20:14 like what is the difference between provability and truth?
1:20:17 What is true and what is provable?
1:20:20 Maybe that's a good line to draw.
1:20:22 Yeah, this is a really core distinction that it's fascinating to me to go
1:20:30 back and read even the early 20th century people before Gödel and Tarski,
1:20:38 and they were totally sloppy about this distinction between truth and proof.
1:20:43 It wasn't clear at all until Gödel, basically.
1:20:46 Although even as late as Bourbaki has
1:20:50 the kind of confusion in this foundational work.
1:20:54 So, this standard graduate-level textbook used
1:20:57 in France in the presentation of logic, they are conflating truth and proof.
1:21:04 To be true for them means to be provable.
1:21:07 So, in the early days, maybe it wasn't clear enough that the concept
1:21:10 of truth needed a mathematical investigation or analysis.
1:21:15 Maybe it was already taken to be fully clear.
1:21:19 But because of the incompleteness theorem,
1:21:21 we realized that actually there's quite subtle things happening, right?
1:21:24 And so, why don't we talk about this distinction a bit?
1:21:28 To me, it's absolutely core and fundamental
1:21:30 to our understanding of mathematical logic now.
1:21:34 This distinction between truth and proof.
1:21:38 So, truth is on the semantic side of the syntax-semantics dichotomy.
1:21:44 Truth has to do with the nature of reality.
1:21:48 I mean, okay, when I talk about reality, I'm not talking about physical reality.
1:21:52 I'm talking about mathematical reality.
1:21:54 So, we have a concept of something being true in a structure,
1:21:57 a statement being true in a mathematical structure.
1:22:00 Like maybe you have the real field or something,
1:22:02 and you want to know, does it satisfy this statement or that statement?
1:22:05 Or you have a group of some kind, or maybe you have a graph.
1:22:09 This is a particular kind of mathematical structure
1:22:12 that has a bunch of vertices and edges,
1:22:14 and you want to know does this graph satisfy that statement?
1:22:19 And Tarski gave this absolutely wonderful account of the nature
1:22:24 of truth in what's now known as the disquotational theory of truth.
1:22:29 And what Tarski says is the sentence,
1:22:33 "Snow is white," is true if and only if snow is white.
1:22:40 And what he means by that is, look, to say truth is a property of an assertion,
1:22:48 so we can think of the assertion as it syntactically.
1:22:51 So, the sentence is true if and only if the content of the sentence is the case.
1:23:00 You know?
1:23:01 So, the sentence, "Snow is white," you know,
1:23:05 in quotations is true, that just means that snow is white.
1:23:11 And that's why it's called the disquotational
1:23:13 theory because we remove the quotation marks.
1:23:16 from the assertion, right?
1:23:17 And you can use this idea of disquotation to give a formal definition
1:23:22 of truth in a mathematical structure of a statement in a formal language.
1:23:26 So, for example, if I have a formal language that allows me
1:23:30 to make atomic statements about the objects and relations of the structure,
1:23:35 and I can build up a formal language with, you know,
1:23:38 with the logical connectives of "and" and "or" and "implies"
1:23:42 and "not" and so on, and maybe I have quantifiers, right?
1:23:45 Then, for example, to say that the structure satisfies phi and psi,
1:23:51 that that single statement, phi and psi, I'm thinking of that as one statement,
1:23:57 just means that it satisfies phi and it satisfies psi.
1:24:02 And if you notice what happened there, I...
1:24:06 At first, the 'and' was part of the sentence inside the sentence,
1:24:10 but then in the second part,
1:24:12 I was using the word 'and' to refer to the conjunction of the two conditions.
1:24:17 So-- Yeah, it has the disquotation.
1:24:19 Yeah, it has the disquotation.
1:24:20 And so this idea can be done
1:24:22 for all the logical connectors and quantifiers and everything.
1:24:24 You're applying Tarski's idea of disquotation,
1:24:28 and it allows you to define by induction the truth
1:24:32 of any assertion in a formal language inside any mathematical structure.
1:24:36 And so to say that a sentence is true, first of all,
1:24:40 it's ambiguous unless you tell me which
1:24:42 structure you're talking about it being true in.
1:24:45 And so maybe we have in mind the standard model
1:24:48 of arithmetic or something with the natural
1:24:50 numbers and the arithmetic structure,
1:24:51 and I want to know is a given statement true in that structure.
1:24:54 Then we have a formal definition of what
1:24:56 that means according to the Tarski recursive definition of truth.
1:25:02 Okay, that's truth.
1:25:04 Proof, on the other hand, is, you know,
1:25:07 in this Hilbert way of thinking, we can develop proof theory.
1:25:11 What is a proof for a mathematician, for a mathematical logician?
1:25:15 A proof is a certain sequence or arrangement of sentences in the formal
1:25:21 language that accord with the logical rules of a proof system.
1:25:27 So there are certain modes of reasoning that are allowed.
1:25:30 So if you know A and you know A implies B in the proof,
1:25:34 then at a later step you're allowed to write B as a consequence.
1:25:39 So if you know A and you know A implies B,
1:25:42 those are both two statements that are known,
1:25:44 then you can deduce B as a consequence according to the rule of modus ponens.
1:25:49 This is the rule of modus ponens.
1:25:51 And, you know, there are a lot of other rules.
1:25:53 Some people would call this implication elimination.
1:25:56 There are different kinds of proof systems.
1:25:58 There are a lot of different formal proof systems
1:26:01 that exist that are studied by the proof theorists,
1:26:03 and all of them have the property that they're sound,
1:26:07 which means that if the premises of the argument are all true
1:26:13 in a structure and you have a proof to get a conclusion,
1:26:16 then the conclusion is also true in that structure.
1:26:19 So that's what it means to be sound.
1:26:23 Proofs preserve truth.
1:26:25 They're truth- preserving arguments.
1:26:28 Okay?
1:26:29 But also the proof systems are also generally complete.
1:26:34 They're both sound and complete,
1:26:36 and complete means that whenever a statement is a consequence,
1:26:41 a logical consequence of some other statements,
1:26:45 which means that whenever the assumptions are true,
1:26:48 then then the consequence is also true in the structure.
1:26:52 So whenever you have a logical consequence, then there is a proof of it.
1:26:56 Okay?
1:26:56 And the proof systems generally have both of those properties;
1:27:00 they're sound and complete.
1:27:01 There's a third property a lot of logicians talk about sound and complete,
1:27:05 sound and complete this, sound and complete that.
1:27:07 But actually, there's a hidden third adjective that they
1:27:10 should always be talking about in any such case,
1:27:12 which is that you should be able to recognize whether
1:27:17 or whether or not something is a proof or not.
1:27:20 So there's a computable aspect to the proof systems.
1:27:22 We want to be able to recognize whether something is a proof.
1:27:25 It should be computably decidable whether a given
1:27:28 sequence of statements is a proof or not.
1:27:30 So we don't want a proof system in which someone claims to have a proof,
1:27:35 but we can't check that fact, whether it's a proof or not.
1:27:40 We want to be able to correctly adjudicate all claims to having a proof.
1:27:47 Yeah.
1:27:47 A mathematician comes to mind that said he has a proof,
1:27:49 but the margins are too small- That's right.
1:27:51 to continue.
1:27:52 Exactly.
1:27:53 So- So that doesn't count as a proof.
1:27:55 Yeah.
1:27:55 So generally, all the classical proof systems
1:27:57 that are used are sound and complete and also
1:28:00 computably decidable in the sense that we can
1:28:02 decide whether something is a proof or not.
1:28:04 So what is, again, the tension between truth and proof?
1:28:08 Which is more powerful,
1:28:10 and how do the two interplay with the contradictions that we've been discussing?
1:28:15 Right.
1:28:16 So the incompleteness theorem is the question whether we could,
1:28:19 say, write down a theory for arithmetic.
1:28:23 Say, for the standard model of arithmetic where we
1:28:25 have the natural numbers and plus and times and zero,
1:28:28 one, and less than, and so on.
1:28:30 In that formal language, we can express an enormous number of statements
1:28:34 about the nature not only of arithmetic, but actually by various coding methods,
1:28:39 we can express essentially all of finite mathematics in that structure.
1:28:43 So the question would be, can we write down a computable list of axioms
1:28:48 that will answer all those questions by proof?
1:28:51 In other words, we want to have a complete theory,
1:28:55 a theory of arithmetic that proves all and only the true statements.
1:29:00 That would be the goal.
1:29:01 Hilbert would love that.
1:29:02 I mean, that would be supportive of Hilbert's
1:29:05 program to have such a complete theory of arithmetic,
1:29:09 and Godel proved that this is impossible.
1:29:12 You cannot write down a computable list
1:29:14 of axioms that is complete in that sense.
1:29:16 There will always be statements...
1:29:18 If the theory is consistent,
1:29:19 there will always be statements that you cannot prove and you cannot refute.
1:29:23 So they are independent of that theory.
1:29:26 How traumatic is that, that there are
1:29:28 statements that are independent from the theory?
1:29:31 I mean, my view is that this isn't traumatic at all.
1:29:35 This is rather completely eye-opening in terms
1:29:41 of our understanding of the nature of mathematical reality.
1:29:46 I mean, we're not...
1:29:49 We understand this profound fact about
1:29:52 our situation with regard to mathematical truth.
1:29:57 The incompleteness theorem tells us, look,
1:29:59 we just can't write down a list of axioms that is
1:30:02 going to be consistent and it's going to answer all the questions.
1:30:05 It's impossible.
1:30:05 And so I don't think of it as trauma.
1:30:09 I just think, look, this is the nature
1:30:11 of mathematical reality and it's good that we know it,
1:30:13 and so now we need to move on from that.
1:30:16 And, you know, do what we can in light of that.
1:30:19 Is it fair to say that in general it means if I give you a statement,
1:30:23 you can't know if your axiomatic system would be able to prove it?
1:30:31 That's right.
1:30:32 In general, you cannot.
1:30:34 The provability problem, we can formulate it as a decision problem.
1:30:37 Given a theory and given a statement,
1:30:39 is that statement a consequence of that theory?
1:30:42 Yeah.
1:30:42 This is one of the most famous decision problems.
1:30:45 In fact, the very first one,
1:30:46 because it's equivalent to the Hilbert-Ackermann Entscheidungsproblem,
1:30:50 which is also appearing in the title of Turing's
1:30:54 1936 paper that was so important for computability theory.
1:30:59 So, it's a formulation of the Entscheidungsproblem.
1:31:03 Does a given theory have a given statement as a logical consequence?
1:31:09 Which, because of Gödel's completeness theorem,
1:31:11 not his incompleteness theorem, but his earlier completeness theorem,
1:31:14 Gödel had proved that the proof systems that they
1:31:17 studied did have this completeness property that I mentioned.
1:31:20 So provability is the same as logical consequence.
1:31:23 And this is an undecidable decision problem.
1:31:26 Turing proved and we now know it's equivalent to the Halting Problem.
1:31:30 Can you describe the Halting Problem?
1:31:32 Because it's a thing that shows up in a very useful and, again,
1:31:35 traumatic way through a lot of computer science, through a lot of mathematics.
1:31:40 Yeah.
1:31:40 The Halting Problem is expressing
1:31:42 a fundamental property of computational processes.
1:31:46 So, given a program,
1:31:48 or maybe we think of it as a program together with its input,
1:31:51 but let me just call it a program.
1:31:52 So given a program, we could run that program,
1:31:54 but I want to pose it as a decision problem.
1:31:58 Will this program ever complete its task?
1:32:02 Will it ever halt?
1:32:03 And the Halting Problem is the question, given a program, will it halt?
1:32:09 Yes or no?
1:32:10 And, of course, for any one instance, the answer's either yes or no.
1:32:15 That's not what we're talking about.
1:32:17 We're talking about whether there's a computable
1:32:19 procedure to answer all instances of this question.
1:32:22 So, it's a decision problem is given as a scheme
1:32:25 of instances for all possible programs that you could ask about.
1:32:28 What I want to know is,
1:32:30 is there a computable procedure that will answer those questions?
1:32:34 And it turns out the answer's no.
1:32:36 The Halting Problem is computably undecidable.
1:32:38 There is no computable procedure that will correctly answer
1:32:41 all instances of whether a given program will halt.
1:32:45 And of course, we can get half the answers
1:32:49 in the sense that you give me a program and you say,
1:32:54 "Will this halt?" And I could take that program and I could run it.
1:32:59 And I could keep running it, and maybe in a week, it would halt.
1:33:03 And at that time, I could say, "Yes,
1:33:05 it halted." So I can get the yes answers correctly for halting,
1:33:10 all the yes answers.
1:33:13 But the problem is if it didn't halt yet, like maybe I waited,
1:33:19 you know, a thousand years and it still hasn't halted,
1:33:23 I don't seem entitled to say, "No, it's not going to halt." Yet,
1:33:27 because maybe in a thousand and one years, it'll halt.
1:33:29 And so at no point can I seem to say, "No." In order to say, "No,
1:33:33 it won't ever halt," it seems like I would have
1:33:37 to really understand how the program worked and what it was doing.
1:33:41 So giving the "yes" answers was sort of trivial.
1:33:43 You didn't have to understand it.
1:33:45 You just needed to run it, which is a kind of rote task.
1:33:48 But to give the "no" answers,
1:33:49 you need to have a kind of deep insight into the nature of the program and what
1:33:53 it's doing in such a way that you would understand it and be able to see,
1:33:57 "Oh, no, I can see this program is never gonna halt." Because,
1:34:01 you know, it's a much more difficult task to say,
1:34:03 "No, it won't halt," than it is to say, "Yes,
1:34:06 it halted because I ran it and it halted." And it turns out
1:34:10 to be impossible to have a computable
1:34:13 procedure that gives the "no" answers, you know?
1:34:15 And the argument is not very difficult.
1:34:17 Should we do it?
1:34:18 Yes, let's do it.
1:34:19 Okay.
1:34:19 Suppose toward contradiction.
1:34:20 I mean, all these proofs are by contradiction,
1:34:23 and this argument is going to be a diagonal argument in the same style
1:34:27 as the Russell argument and the Cantor argument
1:34:30 and Gödel's argument that we haven't talked about yet.
1:34:33 So many diagonal arguments come in.
1:34:35 So suppose towards contradiction that we had a procedure
1:34:39 for determining whether a given program halted on a given input.
1:34:44 Now, let me describe.
1:34:46 I'm gonna use that procedure as a subroutine in the following process.
1:34:52 And my process, let's call it Q, process Q,
1:34:55 and it takes as input a program P, okay?
1:34:59 And the first thing it does is it asks that subroutine, "Hey,
1:35:04 would P halt if I ran it on P itself?" Okay,
1:35:09 that's the diagonal part because we're applying P to P, right?
1:35:14 Okay, so I'm describing program Q,
1:35:16 and program Q takes as input P, which is itself a program.
1:35:19 And the first thing it does is it asks the halting subroutine program,
1:35:24 "Would P halt on P?" And if the answer comes back from the subroutine,
1:35:29 "Yeah, that would halt," then what I do in program
1:35:33 Q is I immediately jump into an infinite loop.
1:35:36 So I don't halt.
1:35:38 If P halts on P, I don't halt.
1:35:41 But if the answer came back, "No,
1:35:43 P is never gonna halt on P," then I halt immediately.
1:35:47 Okay, so and that's it.
1:35:50 I've described what Q does.
1:35:53 And the thing about Q is that Q's behavior
1:35:57 on P was the opposite of P's behavior on P.
1:36:02 I mean, that's how we designed Q specifically so that Q
1:36:07 on P had the opposite behavior as P on P.
1:36:12 Okay, so now, of course, what do we do?
1:36:15 Well, the same thing that Russell did and so forth, and Cantor, we ask,
1:36:20 "Well, what would Q do on Q?" And because of this opposite behavior,
1:36:28 Q would halt on Q if and only if Q does not halt on Q, which is a contradiction,
1:36:33 because Q has to have the opposite behavior on Q than Q does,
1:36:36 but that's just contradictory.
1:36:38 What a beautiful proof.
1:36:39 Simple.
1:36:39 It's absolutely beautiful.
1:36:40 Yeah, I agree.
1:36:41 And it's following the same logic of Russell and Cantor.
1:36:45 I mean, going back to Cantor basically,
1:36:47 because Russell is also quoting Cantor in Cantor basically,
1:36:51 because Russell is also quoting Cantor in his letter to Frege.
1:36:53 So, therefore, the conclusion is
1:36:55 that the halting problem is not computably decidable.
1:36:58 And now we can immediately prove Godel's decidable.
1:37:02 And now we can immediately prove Gödel's theorem using this, actually.
1:37:04 It's an immediate consequence.
1:37:04 So why don't we just do that?
1:37:06 I view this as the simplest proof I
1:37:09 view this as the simplest proof of Gödel's theorem.
1:37:10 You don't need the Gödel sentence to prove Gödel's theorem.
1:37:13 You can do it with the halting problem.
1:37:17 So, suppose that we could write down a computable axiomatization of all that we
1:37:24 could write down a computable axiomatization of all
1:37:26 of the true facts of elementary mathematics,
1:37:28 meaning arithmetic and finite combinatorial things such as Turing and finite
1:37:33 combinatorial things such as Turing machine computations and so on.
1:37:35 So in fact, all those finite
1:37:37 combinatorial processes are formalizable inside arithmetic
1:37:39 with the finite combinatorial processes are formalizable
1:37:41 inside arithmetic with the standard arithmetization coding process.
1:37:44 But let me just be a little bit informal and say suppose we
1:37:46 could write down a complete me just be a little bit informal and say,
1:37:49 suppose we could write down a complete theory of elementary finite mathematics.
1:37:52 So we have a, an axiomatization of that theory.
1:37:56 So we have an axiomatization of that theory.
1:37:58 Then we could produce all possible theorems from those axioms in the way that I
1:38:02 was describing earlier with axioms in the way
1:38:04 that I was describing earlier with Hilbert's program.
1:38:05 I mean, if we had a complete theory of elementary mathematics,
1:38:07 we could construct a theorem enumeration machine that mathematics,
1:38:11 we could construct a theorem enumeration machine that produced
1:38:13 all the theorems and only the theorems from that theory....
1:38:15 so now, I have this theorem enumeration device on my desk,
1:38:21 and I announce theory.
1:38:23 So now I have this theorem enumeration device on my desk,
1:38:24 and I announce that I'm open for business to solve the halting problem.
1:38:26 So you give me a program and input that you wanna run that program on, and input
1:38:30 that you want to run that program
1:38:31 on, and I'm going to answer the halting problem.
1:38:32 And the way I'm going to do it is I'm just gonna wait for the going
1:38:36 to wait for the statement coming out of the theorem
1:38:38 enumeration device that asserts either that P does
1:38:40 halt on that input or I wait for the statement that P does not P does halt
1:38:44 on that input or I wait for the statement that P does not halt on that input.
1:38:46 But one of them's going to happen because
1:38:46 it was a complete theory that was enumerating all
1:38:49 the true statements of elementary complete theory that was
1:38:51 enumerating all the true statements of elementary mathematics.
1:38:52 So therefore, if I had such a system, I could solve the halting problem,
1:38:55 but we already proved that you cannot solve the halting problem,
1:38:58 problem, but we already proved that you cannot solve the halting problem,
1:39:00 so therefore you cannot have such a complete theory of arithmetic.
1:39:02 So that proves Gödel's theorem.
1:39:05 Maybe to take a little bit of a tangent, can you speak...
1:39:07 You've written a wonderful book about proofs and the art of mathematics.
1:39:09 So what can you say about proving stuff in mathematics?
1:39:11 What is the process of say about proving stuff in mathematics?
1:39:15 What is the process of proof?
1:39:17 What are the tools?
1:39:18 What is the art?
1:39:19 What is the science of proving things in mathematics?
1:39:22 proving things in mathematics?
1:39:23 So this is something that I find so
1:39:25 wonderful to teach young mathematicians who are learning
1:39:28 how to become mathematicians and mathematicians who are
1:39:31 learning how to become mathematicians and learning about proof,
1:39:32 and I wrote that book when I was teaching such a proofwriting class in New York.
1:39:37 proofwriting class in New York.
1:39:38 So many universities have such a course, the proofreading course,
1:39:40 which is usually taken by students who have learned some mathematics.
1:39:45 taken by students who have learned some mathematics.
1:39:46 Usually, they've completed maybe the calculus sequence and are
1:39:47 making the kind of transition to higher mathematics,
1:39:50 which tends to involve much more proof,
1:39:53 and it's a kind of challenging step for them.
1:39:56 Many math departments have this kind of course on proofwriting
1:39:59 where the students would get exposed to how to write proofs.
1:40:03 I wasn't happy with most of the other
1:40:06 books that exist for those kind of courses, and the reason was that they were so
1:40:13 often so dull because they would concentrate on, like,
1:40:18 these- totally uninteresting parts of what it's like to write a proof,
1:40:24 these kind of mechanistic procedures about how to write a proof.
1:40:28 You know, if you're going to prove an implication,
1:40:30 then you assume the hypothesis and argue for the conclusion, and so on.
1:40:34 And all of that is true and fine and that's good to know,
1:40:37 except if that's all that you're saying about the nature of proof,
1:40:40 then I don't think you're really learning very much.
1:40:43 So I felt that it was possible to have a much better kind of book,
1:40:47 one that was much more interesting and that had
1:40:51 interesting theorems in it that still admitted of elementary proof.
1:40:56 So I wrote this book and tried to fill it with all of the compelling
1:41:02 mathematical statements with very elementary proofs that exhibited
1:41:06 lots of different proof styles in it.
1:41:09 And so, I found that the students appreciated it a lot.
1:41:13 We should say, we dedicate the book to my students,
1:41:15 may all their theorems be true,
1:41:17 proved by elegant arguments that flow effortlessly
1:41:20 from hypothesis to conclusion while revealing fantastical mathematical beauty.
1:41:27 Are there some interesting proofs that maybe illustrate,
1:41:33 for people outside of mathematics or for people who just
1:41:36 take math classes-- Right- ...in high school and so on?
1:41:39 Yeah, let's do a proof.
1:41:40 There's one in the book.
1:41:41 We can talk about it.
1:41:42 I think it's a nice problem.
1:41:44 It's in the discrete math, yeah, the 5.1,
1:41:47 that one, more pointed at than pointing.
1:41:50 Okay.
1:41:51 So this is the following problem.
1:41:53 Suppose you're gathered with some friends, you know, in a circle,
1:41:58 and you can point at each other however you want, or yourself, whatever,
1:42:02 it doesn't matter, and you can point at more than one person,
1:42:05 you know, use all your fingers or your feet or whatever you want.
1:42:08 So maybe you point at three of your friends or something
1:42:10 and they point at two or three of their friends or whatever,
1:42:13 and one person is pointing at 10
1:42:15 people and somebody isn't pointing at anybody maybe,
1:42:17 or and various people are pointed at also, right?
1:42:20 So the question is, could we arrange a pattern of pointing so
1:42:25 that everyone was more pointed at than they are pointing at others?
1:42:32 So in other words, maybe there's seven people pointing at me,
1:42:35 but I'm only pointing at five people and maybe there's,
1:42:38 you know, 20 people pointing at you,
1:42:41 but you're only pointing at 15 people or something like that, right?
1:42:44 So I want to know.
1:42:45 There's a similar question on Twitter.
1:42:49 For a group of people on Twitter,
1:42:52 could you arrange that everyone has more followers than following?
1:42:57 Yeah, it's the same question.
1:43:00 Mathematically, it's identical.
1:43:02 Although, I don't know, it's not identical,
1:43:04 because I said you could point at yourself, and I think that's not...
1:43:07 Can you follow yourself?
1:43:09 No, I don't think so, no.
1:43:10 I don't think you can.
1:43:11 Okay.
1:43:12 So can you arrange it so that everyone is more pointed at than pointing?
1:43:16 And in my book, I give a couple of different proofs of this.
1:43:20 I think I give an induction proof and then there's another proof.
1:43:23 I think there's three different proofs in there.
1:43:25 But why don't we just talk about my favorite proof?
1:43:28 Suppose it were possible to arrange that we're
1:43:30 all more pointed at than pointing, okay?
1:43:32 Now what we're going to do, we're going to agree,
1:43:36 we're going to give a dollar to everyone that we're pointing at.
1:43:40 Okay?
1:43:41 And so what happens?
1:43:43 Everybody made money, because I was pointed at by more people than I'm pointing,
1:43:49 so I got $10, but I only paid out $7.
1:43:52 And similarly, you got paid $20, but you only paid out $15.
1:43:57 So, if everyone is more pointed at than pointing, then everyone makes money.
1:44:01 But it's obviously impossible for us to make money
1:44:04 as a group by just trading money with ourselves.
1:44:07 And therefore, it can't be possible
1:44:09 that we're all more pointed at than pointing.
1:44:13 And this proof illustrates something.
1:44:14 It's one of my habits that I suggest in the book:
1:44:19 to anthropomorphize your mathematical ideas.
1:44:22 You should imagine that the mathematical objects that are
1:44:26 playing a role in your question are people, or active, somehow, animals,
1:44:32 or something that maybe have a will and a goal and so on.
1:44:36 This is this process of anthropomorphizing.
1:44:40 And it often makes the problems easier to understand because we
1:44:44 all are familiar with the fact that it's difficult to make money,
1:44:48 and the proof is totally convincing because of our knowledge that we
1:44:52 can't make money as a group by trading dollars between us,
1:44:57 you know, without any new money coming into the group.
1:45:01 But that by itself is actually a difficult mathematical claim.
1:45:05 I mean, if someone had to prove that you
1:45:09 can't make money by trading within a group, you know,
1:45:12 it can't be that everyone in the group makes
1:45:15 money just by shifting money around in the group.
1:45:18 Maybe you think that's obvious, and it is obvious if you think about money.
1:45:21 But if you had asked the question, you know,
1:45:23 about mathematical functions of a certain kind and so on, then maybe it
1:45:27 wouldn't be as clear as it is when you're talking about this money thing,
1:45:31 because of we can build on our human experience about
1:45:36 the difficulty of getting money and, you know, or other resources.
1:45:39 It doesn't have to be money, it could be candy, whatever.
1:45:41 You know, we just know that you can't easily get
1:45:45 more things in that kind just by trading within a group.
1:45:49 And we should say that sometimes the power
1:45:51 of proof is such that the non-obvious can be shown,
1:45:53 and then over time that becomes obvious.
1:45:56 So in the context of money, or social systems,
1:45:58 there's a bunch of things that are non-obvious.
1:46:01 And the whole point is that proof can guide us to the truth,
1:46:07 to the accurate description of reality.
1:46:10 We just proved a property of money.
1:46:14 It's interesting to think about, well,
1:46:16 what if there were infinitely many people in your, in your group?
1:46:20 Then it's not true anymore.
1:46:22 The theorem fails.
1:46:23 In fact, you can arrange that everyone is strictly more pointed than pointing.
1:46:28 And also, you can if everyone has even just one dollar bill—
1:46:35 ...then you can arrange that afterwards
1:46:37 everyone has infinitely many dollar bills.
1:46:39 Cause in terms of cardinality, that's the same.
1:46:41 It's just, say, countable infinity in each case.
1:46:43 If you had countably many friends and everyone has one dollar bill,
1:46:46 then you can arrange a pattern of passing those dollar bills
1:46:49 amongst each other so that afterwards everyone has infinitely many dollar bills.
1:46:52 What you need is for each person to be attached to, you know,
1:46:56 one of the train cars or something.
1:46:58 So, think of everyone as coming from Hilbert's train,
1:47:03 but also think of them as fitting into Hilbert's Hotel.
1:47:05 So just have everyone on the Nth car give all
1:47:10 their money to the person who ends up in the Nth room.
1:47:13 So they each give one dollar to that person.
1:47:16 So afterwards, that person has infinitely many dollars,
1:47:18 but everyone only paid out one dollar.
1:47:20 So it's a way of making it happen.
1:47:24 To what degree, sticking on the topic of infinity,
1:47:27 should we think of infinity as something real?
1:47:33 That's an excellent question.
1:47:34 I mean, a huge part of the philosophy
1:47:36 of mathematics is about this kind of question:
1:47:39 what is the nature of the existence of mathematical objects, including infinity?
1:47:45 But I think asking about infinity specifically is
1:47:49 not that different than asking about the number five.
1:47:52 What is— ...what does it mean for the number five to exist?
1:47:56 What are the numbers really, right?
1:47:58 This is maybe one of the fundamental questions of mathematical ontology.
1:48:02 I mean, there's many different positions to take on the question
1:48:05 of the nature of the existence
1:48:07 of mathematical objects or abstract objects in general.
1:48:10 And there's a certain kind of conversation that sometimes happens
1:48:14 when you do that, and it goes something like this.
1:48:19 Sometimes people find it problematic to talk about
1:48:22 the existence of abstract objects such as numbers,
1:48:25 and there seems to be a kind of wish that we could give an account
1:48:30 of the existence of numbers or other mathematical
1:48:33 objects or abstract objects that was more like,
1:48:36 you know, the existence of tables and chairs and rocks and so on.
1:48:42 And so there seems to be
1:48:45 this desire to reduce mathematical existence to something,
1:48:49 you know, that we can experience physically in the real world.
1:48:54 But my attitude about this attempt is that it's very backward, I think,
1:49:03 because I don't think we have such a clear
1:49:09 understanding of the nature of physical objects, actually.
1:49:11 I mean, we all have experience about existing in the physical world, as we must,
1:49:17 because we do exist in the physical world, but I don't know of any satisfactory
1:49:24 account of what it means to exist physically.
1:49:27 I mean, if I ask you, say,
1:49:31 "Imagine a certain kind of steam locomotive," you know,
1:49:37 and I describe the engineering of it and the weight
1:49:40 of it and the nature of the gear linkages and you know,
1:49:45 and I show you schematic drawings of the whole design and so on and, you know,
1:49:50 we talk in detail about every single detailed aspect of this steam locomotive.
1:49:55 But then suppose after all that conversation, I say, "Okay,
1:49:59 now I would like you to tell me what would it mean for it to exist physically,
1:50:03 I mean, as opposed to just being an imaginary steam locomotive?" Then what,
1:50:08 what could you possibly say about it?
1:50:11 I mean, except by saying, "Oh,
1:50:13 I just mean that it exists in the physical world." But what does that mean?
1:50:16 That's the question, right?
1:50:17 It's not an answer to the question.
1:50:18 That is the question.
1:50:20 So I don't think that there's anything sensible
1:50:23 that we can say about the nature of physical existence.
1:50:26 It is a profound mystery.
1:50:28 In fact, it becomes more and more mysterious the more physics we know.
1:50:33 I mean, back in, say, Newtonian physics,
1:50:36 one had a picture of the nature of physical objects as, you know,
1:50:40 little billiard balls or something,
1:50:41 or maybe they're infinitely divisible or something like that.
1:50:43 Okay, but then this picture is upset with the atomic theory of matter.
1:50:48 But then that picture's upset when we realize that the atoms actually can
1:50:53 be split and consist of electrons and protons and neutrons and so on.
1:50:56 But then that picture's upset when we realize that those things themselves are
1:51:00 built out of quarks and leptons and so on, and who knows what's coming.
1:51:03 Furthermore, all of those things,
1:51:05 the nature of their existence is actually as wave functions in you know,
1:51:10 some cloud of probability and so on.
1:51:13 So it just becomes more and more and more mysterious the more we learn,
1:51:17 and not at all clarifying.
1:51:18 So the nature of what it means to say that, you know,
1:51:23 there's an apple on my desk,
1:51:25 and to give an account of what that physical existence really is at bottom,
1:51:30 I think, is totally absent.
1:51:32 Whereas we do seem to have a much
1:51:35 more satisfactory account of the nature of abstract existence.
1:51:39 I mean, I can talk about the nature of the empty set.
1:51:42 You know, this is the predicate which is never true or something like that.
1:51:47 I can talk about those kind of logical properties
1:51:51 or the singleton of the empty set and so on.
1:51:53 I mean, of course, it's very difficult if you go very far with it,
1:51:56 but the point is that it doesn't get more and more mysterious.
1:51:59 The more that you say, it becomes only more and more clear.
1:52:04 So it seems to me that, we don't really have any understanding
1:52:10 of what the physical world is as opposed to the abstract world,
1:52:14 and it's the abstract world where existence is much more clear.
1:52:19 It is very true that we don't know anything about the soda
1:52:23 bottle or the steam locomotive just because we can poke at it.
1:52:26 Again, we anthropomorphize,
1:52:27 and that actually gets us into trouble sometimes because
1:52:30 I'm not feeling the quantum mechanics when I'm touching it.
1:52:34 That's right.
1:52:34 And therefore, it's easy to forget and feel
1:52:38 like this is real and mathematical objects are not,
1:52:42 but you're making the opposite argument.
1:52:44 When you draw a distinction between numerals and numbers,
1:52:47 which numerals are the representation of the number on the page,
1:52:50 and so on, but could you say that a number is real?
1:52:56 Do numbers exist?
1:52:58 I happen to think so.
1:52:59 I mean, I'm on the side of realism in mathematics,
1:53:01 and I think that these abstract objects do have a real existence in a way
1:53:06 that we can give an account of, in a way I just tried to describe.
1:53:10 So, you would describe it as the size of a set with four elements in it?
1:53:14 Well, there are different ways to understand the nature of four.
1:53:16 I mean, actually, this gets into the question of structuralism,
1:53:19 which is maybe a good place to talk about it.
1:53:24 What is structuralism?
1:53:26 Structuralism is a philosophical position in mathematics,
1:53:29 or the philosophy of mathematics,
1:53:31 by which one emphasizes that what's important about mathematical objects is not
1:53:35 what they're made out of or what their substance or essence is,
1:53:38 but rather how they function in a mathematical structure.
1:53:41 And so, what I call the structuralist attitude in mathematics is
1:53:46 that we should only care about our mathematical structures up to isomorphism.
1:53:53 If I have a mathematical structure of a certain kind,
1:53:56 and I make an exact copy of it using
1:53:59 different individuals to form the elements of that structure,
1:54:03 then the isomorphic copy is just as good mathematically,
1:54:06 and there's no important mathematical difference that would ever arise
1:54:11 from working with this isomorphic copy instead of the original structure.
1:54:15 And so, therefore, that's another way of saying
1:54:18 that the substance of individuals, you know,
1:54:21 in a mathematical structure is irrelevant with regard
1:54:25 to any mathematical property of that structure.
1:54:28 And so, to ask a question like, "What is the number four really?"
1:54:36 is an anti-structuralist thing because, you know,
1:54:40 if you have if you have a structure, say,
1:54:42 the natural numbers, you know, with all the numbers in it:
1:54:45 0, 1, 2, 3, 4, and so on, then I
1:54:48 could replace the number four with something else, like, you know,
1:54:53 this bottle of water could play the role of the number four in that structure,
1:54:58 and it would be isomorphic.
1:54:59 And it wouldn't matter at all for any mathematical
1:55:03 purpose to use this alternative mathematical system, you know?
1:55:07 That's to say that we don't care what the number four is really.
1:55:11 That is irrelevant.
1:55:12 The only thing that matters is what are the properties
1:55:15 of the number four in a given mathematical system,
1:55:18 you know, and recognizing that there are other isomorphic copies of that system,
1:55:22 and the properties of that other system's number
1:55:25 four are going to be identical to the properties
1:55:28 of this system's number four with regard
1:55:31 to any question that's important about the number four.
1:55:33 But those questions won't be about essence.
1:55:36 So, in a sense, structuralism is an anti-essential in mathematics.
1:55:43 So, is it fair to think of numbers
1:55:45 as a kind of pointer to a deep underlying structure?
1:55:48 Yeah, I think so, because I guess part of the point of structuralism
1:55:52 is that it doesn't make sense
1:55:54 to consider mathematical objects or individuals in isolation.
1:55:59 What's interesting and important about mathematical objects is how they
1:56:03 interact with each other and how they behave in a system,
1:56:06 and so maybe one wants to think about the structural role that the objects play,
1:56:10 you know, in a larger system, a larger structure.
1:56:13 There's a famous question that Frege had asked actually
1:56:16 when he was looking into the nature of numbers,
1:56:19 because in his logicist program, right,
1:56:21 he was trying to reduce all of mathematics to logic.
1:56:25 And in that process,
1:56:27 he was referring to the Cantor-Hume principle that, you know,
1:56:32 whenever two sets are equinumerous, then they have the same number of elements,
1:56:36 I mean, if and only if.
1:56:37 And he founded his theory of number on this principle,
1:56:41 but he recognized that there was
1:56:44 something that dissatisfied him about that situation,
1:56:47 which is that the Cantor-Hume principle does not seem
1:56:51 to give you criteria for which things are numbers.
1:56:55 It only tells you a kind of identity criteria
1:56:58 for when two numbers are equal to each other.
1:57:01 Well, two numbers are equal just in case
1:57:03 the sets of those sizes are equinumerous,
1:57:05 so that's the criteria for number identity.
1:57:08 But it is not a criteria for what is a number.
1:57:11 And so this problem has become known as the Julius
1:57:14 Caesar problem because Frege said we don't seem
1:57:17 to have any way of telling from the Hume
1:57:21 principle whether Julius Caesar is a number or not.
1:57:25 So he's asking about the essence of number and whether...
1:57:28 Of course, one has a sense that he picked maybe
1:57:32 what he was trying to present as a ridiculous example,
1:57:36 because maybe you have the idea that well,
1:57:39 obviously Julius Caesar is not a number,
1:57:40 and there's a lot of philosophical writing that seems to take that line also,
1:57:44 that obviously the answer is that Julius Caesar is not a number.
1:57:48 But the structuralists disagree with that position.
1:57:51 The structuralist attitude is, "Look, you give me a number system.
1:57:57 If Julius Caesar isn't a number, then I can just...
1:58:01 let's take the number 17 out of that system
1:58:03 and plug in Julius Caesar for that role, and now I've got a new number system,
1:58:07 and now Julius Caesar happens to be
1:58:09 the number 17." And that's totally fine, you know.
1:58:12 So the point of structuralism is is that the question of whether
1:58:18 Julius Caesar is a number or not is irrelevant to mathematics.
1:58:22 It is irrelevant because it is not about structure,
1:58:25 it's about this essence of the mathematical objects.
1:58:30 So that's the structuralist criticism of Frege's point.
1:58:35 You've kind of made the case that you can say more concrete things
1:58:39 about the existence of objects in mathematics
1:58:42 than you can in our physical reality,
1:58:44 about which to us human brains, things are obvious or not.
1:58:49 So what's more real?
1:58:51 The reality we see with our eyes
1:58:54 or the reality we can express in mathematical theorems?
1:58:59 I'm not quite sure.
1:59:01 I mean, I live entirely in the Platonic realm,
1:59:06 and I don't really understand the physical universe at all.
1:59:11 So I don't have strong views.
1:59:14 Let's talk about the Platonic realm.
1:59:16 Is it- because you live there, is it real?
1:59:19 Or-- Oh yeah, totally, yeah.
1:59:21 This is the realist position in mathematics
1:59:23 is that abstract objects have a real existence.
1:59:25 And okay, what's meant by that is that there's some sense
1:59:29 of existence in which those objects can be regarded as real.
1:59:32 How should we think about that?
1:59:34 How should we try to visualize that?
1:59:35 What does it mean to live amongst abstract objects?
1:59:41 Right.
1:59:42 Because life is finite.
1:59:43 We're all afraid of death.
1:59:45 We f- fall in love with other physical manifestations of objects.
1:59:52 And you're telling me that maybe reality actually exists elsewhere,
1:59:56 and this is all just a projection-- Well, I mean--...
2:00:00 from the abstract realm.
2:00:02 Do abstract objects exist in a place and at a time?
2:00:06 That's very debatable, I think.
2:00:07 Right.
2:00:07 And what does place and time mean?
2:00:09 Uh, all time, yeah, so...
2:00:10 So what's more real, physics or the mathematical Platonic space?
2:00:16 Well, the mathematical Platonic realm is...
2:00:19 I'm not sure I would say it's more real,
2:00:22 but I'm saying we understand the reality of it
2:00:25 in a much deeper and more- ...a more convincing way.
2:00:28 I don't think we understand the nature of physical reality very well at all.
2:00:32 And I think most people aren't even scratching the surface
2:00:37 of the question as I intend to be asking it.
2:00:40 So, you know, obviously we understand physical reality.
2:00:42 I mean, I knock on the table- ...and so
2:00:45 on, and we know all about what it's like to, you know,
2:00:47 have a birthday party or to drink a martini or whatever.
2:00:50 And so we, we, we have a deep understanding of existing in the physical world.
2:00:56 But maybe understanding is the wrong word.
2:00:58 We have an experience of living in the world-- Yeah, experience.
2:01:01 ...and riding bicycles and all those things,
2:01:03 but I don't think we actually have an understanding at all.
2:01:07 I mean, very, very little of the nature of physical existence.
2:01:11 I think it's a profound mystery.
2:01:13 Whereas I think that we do have something a little better
2:01:18 of an understanding of the nature
2:01:20 of mathematical existence and abstract existence.
2:01:23 So that's how I would describe the point.
2:01:26 Somehow it feels like we're approaching
2:01:30 some deep truth from different directions,
2:01:34 and we just haven't traveled as far in the physics
2:01:38 world as we have in the mathematical world.
2:01:41 Maybe I could hope that someone will give, you know,
2:01:44 the convincing account, but it seems to be a profound mystery to me.
2:01:48 I can't even imagine what it would be
2:01:50 like to give an account of physical existence.
2:01:53 Yeah, I wonder, like a thousand years from now as physics progresses...
2:01:56 Right- ...what this same conversation would look like.
2:01:59 Right.
2:01:59 That would be quite interesting.
2:02:01 Do you think there's breakthroughs a thousand
2:02:02 years from now on the mathematics side?
2:02:04 Because we've just discussed,
2:02:06 and we'll return to, a lot of turmoil a century ago.
2:02:11 Right.
2:02:12 Do you think there's more turmoil to be had?
2:02:15 It's interesting to me because I have
2:02:17 my feet in two worlds of mathematics and philosophy,
2:02:20 and to compare the differences between these subjects.
2:02:24 And one of the, one of the big...
2:02:27 There's many cultural differences, but one of the big cultural differences is
2:02:30 towards the idea of progress in the subject.
2:02:34 Because mathematics has huge progress.
2:02:37 We simply understand the mathematical ideas much, much better,
2:02:41 you're continually improving our understanding and there's growth in knowledge.
2:02:46 We understand the nature of infinity now better than they did 100 years ago.
2:02:51 I mean, definitely better.
2:02:52 And they understood it better 100 years ago than they did,
2:02:55 you know, for the previous thousands of years and so on.
2:02:57 So, in almost every part of mathematics,
2:03:00 there's improved understanding of the core issues so much so that, you know,
2:03:06 the questions at hand become totally different
2:03:08 and the field sort of moves on to more difficult, interesting questions.
2:03:13 Whereas in philosophy, there's...
2:03:16 That's a little bit true that there's progress.
2:03:19 But meanwhile, it's also true that there are these eternal questions
2:03:23 that have been with us for thousands of years and in fact,
2:03:27 so much so that you can find
2:03:29 a lot of philosophers arguing the important contribution
2:03:32 of philosophy is in asking the questions rather
2:03:36 than answering them because it's hopeless to answer them.
2:03:39 I mean, the nature of these deep philosophical questions is so difficult.
2:03:44 Less of a sense of progress is what I'm trying to say.
2:03:48 I don't see any reason to think that the progress in mathematics,
2:03:52 in the growth in our mathematical
2:03:54 understanding and knowledge won't simply continue.
2:03:58 And so, a thousand years from now, maybe the mathematics that they will be doing
2:04:03 at that time would probably be completely unrecognizable to me.
2:04:07 I maybe wouldn't even begin to understand what they're talking about,
2:04:11 even without sort of witnessing, you know, the intervening developments.
2:04:16 So if you bring someone from ancient times to today,
2:04:20 they maybe wouldn't even understand what we're
2:04:22 talking about with some of the questions.
2:04:25 But I feel that, you know, if Archimedes came and we were able to communicate,
2:04:31 I think I would be able to tell him, you know, about some of the things that are
2:04:36 going on in mathematics now and, and maybe, you know...
2:04:43 Or, or anyone from that time, I mean.
2:04:46 So I think it is possible to have this kind of progress even when the subject
2:04:52 kind of shifts away from the earlier
2:04:54 concerns as a result of the progress, basically.
2:04:58 To take a tangent on a tangent since you mentioned philosophy,
2:05:01 maybe potentially more about the questions
2:05:04 and maybe mathematics is about the answers,
2:05:06 I have to say you are a legend on MathOverflow,
2:05:09 which is like Stack Overflow but for math.
2:05:12 You're ranked number one all time
2:05:14 on there with currently over 246,000 reputation points.
2:05:18 How do you approach answering difficult questions on there?
2:05:24 Well, MathOverflow has really been one of the great pleasures of my, of my life.
2:05:29 I've really enjoyed it.
2:05:30 I mean...
2:05:31 And I've learned so much from interacting on MathOverflow.
2:05:36 I've been on there since 2009, which was shortly after it started.
2:05:40 I mean, it wasn't exactly at the start, but a little bit later.
2:05:45 And and I think it gives you the stats for how many
2:05:51 characters I typed and I don't know how many million it is,
2:05:56 but uh, this enormous amount of um,
2:06:01 time that I've spent thinking about those questions
2:06:03 and it has really just been amazing to me.
2:06:06 Uh-- How do you find the questions that grab
2:06:08 you and how do you go about-- Right-...
2:06:10 answering them?
2:06:11 So, I'm interested in any question that I find interesting.
2:06:15 So...
2:06:15 And it's not all questions.
2:06:17 Sometimes certain kinds of questions just don't appeal to me that much.
2:06:22 So you go outside of set theory as well?
2:06:24 So, I think when I first joined MathOverflow, I was,
2:06:28 I was basically one of the only,
2:06:30 one of the few people in logic who was answering.
2:06:33 I mean, there were other people who know some logic,
2:06:36 particularly from category theory and other parts of mathematics
2:06:39 that aren't in the most traditional parts of logic,
2:06:42 but they were answering some of the logic questions.
2:06:45 So I really found myself able to make a contribution
2:06:48 in those very early days by engaging with the logic-related questions.
2:06:53 But there weren't many logic people asking questions either.
2:06:56 But what I found was that there was
2:06:59 an enormous amount of interest in topics that were logic-adjacent.
2:07:03 So a question would arise, you know, in group theory,
2:07:07 but it had a logic aspect or an analysis or whatever,
2:07:10 and there would be some logic angle on it.
2:07:13 And what I found was that I was often able to figure
2:07:17 out an answer by learning enough about that other subject matter.
2:07:21 This is what was so rewarding for me, because basically I had to learn enough.
2:07:25 had to learn enough.
2:07:26 My expertise, my main expertise was logic,
2:07:29 but someone would ask a question, you know, that, that was about, say,
2:07:33 the axiom of choice in this other subject matter or the continuum
2:07:36 hypothesis or something like that in an, in the other subject matter.
2:07:40 And I would have to learn enough about that other subject and the context
2:07:44 of the question in order to answer and I was often able to do that.
2:07:47 And so I was quite happy to do that.
2:07:50 And, and also I learned a lot by doing
2:07:52 that because I had to learn about these other problem areas.
2:07:56 And so it really allowed me to grow enormously as a mathematician.
2:08:01 To give some examples of questions you've answered,
2:08:03 what are some reasonable sounding statements that are independent of ZFC?
2:08:07 What are the most misleading alternate definitions in taught mathematics?
2:08:12 Is the analysis as taught in universities
2:08:14 in fact the analysis of definable numbers?
2:08:17 Solutions to the continuum hypothesis?
2:08:20 Most unintuitive application of the axiom of choice?
2:08:24 Non-trivial theorems with trivial proofs?
2:08:26 Reductio ad absurdum or the contrapositive?
2:08:30 What is a chess piece mathematically?
2:08:33 We should say you worked quite a bit on infinite chess,
2:08:36 which we should definitely talk about.
2:08:38 It's awesome.
2:08:39 You've worked on so many fascinating things.
2:08:41 Has philosophy ever clarified mathematics?...
2:08:44 why do we have two theorems when one implies the other?
2:08:48 And, of course, just as an example you've given a really,
2:08:52 a great, almost historical answer on the topic of the continuum hypothesis.
2:08:56 Maybe that's a good place to go.
2:08:58 We've touched on it a little bit, but it would be nice to lay out
2:09:02 what is the continuum hypothesis that Cantor struggled with.
2:09:05 And I would love to also speak to the psychology of his- his own life story,
2:09:09 his own struggle with it.
2:09:11 It's the human side of mathematics is also fascinating.
2:09:14 So what is the continuum hypothesis?
2:09:16 So the continuum hypothesis is the question that arises so naturally
2:09:20 whenever you prove that there's more than one size of infinity.
2:09:24 So Cantor proved that the infinity of the real numbers
2:09:28 is strictly larger than the infinity of the natural numbers.
2:09:32 But immediately when you prove that, one wants to know,
2:09:37 well, is there anything in between?
2:09:38 I mean, what could be a more natural question to ask immediately after that?
2:09:43 And so Cantor did ask it,
2:09:45 and he spent his whole life thinking about this question.
2:09:49 And so the continuum hypothesis is the assertion that there is
2:09:52 no infinity in between the natural numbers and the real numbers.
2:09:56 And, of course, Cantor knew many sets of real numbers.
2:09:59 Everything in between...
2:10:00 I mean, everything that's in that interval would
2:10:03 be equinumerous with some set of real numbers.
2:10:06 But we know lots of sets of real numbers.
2:10:08 I mean, there's all these various closed sets, Cantor sets, and so on.
2:10:11 There's Vitali sets.
2:10:12 We have all kinds of sets of real numbers.
2:10:14 And so you might think, well, if the continuum hypothesis is false,
2:10:17 then we've probably seen the- the set already.
2:10:20 We just have to prove, you know, that it's strictly in between.
2:10:23 But it turned out that for all the sets
2:10:26 that anyone ever could define or pick out or observe,
2:10:29 for all the sets of real numbers,
2:10:31 it was always the case either that they were countable,
2:10:34 in which case they're equinumerous with the natural numbers or else finite.
2:10:39 Or they were fully equinumerous with the whole real line.
2:10:43 And so they were never strictly in between.
2:10:46 I mean, you're in this situation and you have 100,
2:10:49 thousands of sets that are candidates to be in between,
2:10:54 but in every single case,
2:10:56 you can prove it's on one side or the other and not strictly in between.
2:11:01 And so in every situation where you're able
2:11:03 to figure out whether it's in between or not,
2:11:06 it's always never strictly in between.
2:11:09 Now, Cantor was obsessed with this.
2:11:12 I think he was.
2:11:12 Yeah, I'm not a historian, so I don't know the exact history.
2:11:15 Well, everything I've seen, it seems to be the question that broke him, huh?
2:11:18 Um, I mean, just struggling with different
2:11:21 opinions on the hypothesis within himself and...
2:11:25 ...Desperately chasing, trying to prove it.
2:11:29 So he had a program for proving it,
2:11:31 which has been affirmed in a certain respect.
2:11:35 Of course, the continuum hypothesis holds for open sets.
2:11:38 That's easy to see.
2:11:39 If you have an open interval,
2:11:40 then this is fully equinumerous with the whole real line.
2:11:44 Any interval is equinumerous with the whole line
2:11:47 because all you would need is a function,
2:11:50 you know, like the arctangent function or something
2:11:52 that maps the whole real line into an interval.
2:11:55 And that's a one-to-one function.
2:11:57 So we know the open sets have the property that their non-trivial
2:12:02 open sets are all fully equinumerous with the whole real line.
2:12:05 So, never strictly in between.
2:12:07 But remarkably, Cantor proved it also for the closed sets,
2:12:11 and that is using what's called the Cantor-Bendixson theorem.
2:12:15 So, it's quite a remarkable result.
2:12:17 It's definitely not obvious.
2:12:20 And in this theorem actually was the origin of the ordinals.
2:12:23 Cantor had to invent the ordinals in order
2:12:28 to make sense of his Cantor-Bendixson process.
2:12:32 Can you define the open and the closed set in this context?
2:12:34 Oh, yeah.
2:12:35 Sure.
2:12:35 So a set of reals is open if every point
2:12:38 that it contains is surrounded by a little interval of points,
2:12:42 the whole tiny little interval.
2:12:45 But that tiny little interval is already
2:12:47 just by itself equinumerous with the whole line.
2:12:49 So that's why that question is sort of easy for open sets.
2:12:52 A closed set is a complement of an open set,
2:12:55 and there's a lot of closed sets that are really complicated of varying sizes.
2:13:01 So of course, any closed interval is a closed set, but it's not only those.
2:13:04 There's also things like the Cantor set,
2:13:06 which you get by omitting middle thirds.
2:13:08 Maybe some people have seen this construction.
2:13:11 Or you can imagine sort of randomly taking a lot of little tiny open intervals,
2:13:16 you know, all over the line and so on.
2:13:18 So that altogether would be an open set,
2:13:20 and the complement of it would be a closed set.
2:13:22 So you can imagine just kind of tossing down these open intervals,
2:13:26 and what's left over is the closed set.
2:13:29 Those sets can be quite complicated,
2:13:30 and they can have isolated points, for example,
2:13:33 if the two open intervals were just kissing
2:13:36 and leaving only the one point between them.
2:13:38 But also you could have sequences that are converging to a point,
2:13:42 that would also be a closed set,
2:13:45 or convergent sequences of convergent sequences and so on.
2:13:48 That would be a closed set also.
2:13:50 The Cantor set is constructed by iteratively removing open intervals,
2:13:53 middle thirds, like you mentioned, from the interval, and trying to see,
2:13:57 can we do a thing that that goes in between?
2:14:00 Right.
2:14:01 So the question would be, can you produce a set that has an intermediate size?
2:14:07 an intermediate cardinality, right?
2:14:09 And Cantor proved, with the closed set, "No,
2:14:12 it's impossible." Every closed set is either
2:14:14 countable or equinumerous with the whole real line.
2:14:18 And the Cantor program for solving the Continuum Hypothesis was,
2:14:25 a sort of working up.
2:14:26 So you did it for open sets and for closed sets, and you sort of work up.
2:14:30 Maybe he wants to go into what are called the Borel sets,
2:14:32 which are sort of combinations of open and closed sets.
2:14:35 And there's a vast hierarchy of Borel complexity.
2:14:39 And it turns out that the Continuum Hypothesis has
2:14:43 been proved also for the Borel sets in this hierarchy.
2:14:46 But then one wants to go beyond.
2:14:48 What about more complicated sets?
2:14:50 So there's this hierarchy of complexity for sets of real numbers.
2:14:53 And Cantor's idea was to sort of work your way up the hierarchy by proving
2:14:58 that the Continuum Hypothesis was more and more
2:15:01 true for those more and more complicated sets,
2:15:03 based on our understanding of the earlier cases.
2:15:06 And that has been carried out to a remarkable degree.
2:15:11 It turns out that one begins to need large cardinal assumptions,
2:15:16 though, in order to get to the higher realms,
2:15:20 even at the level of projective hierarchy, which are sets that you can define
2:15:25 by using quantifiers over the real numbers themselves.
2:15:27 So you get this hierarchy on top of the Borel hierarchy,
2:15:32 the hierarchy of projectively definable sets.
2:15:34 And it turns out that if you have enough large cardinals,
2:15:39 then the projective sets also are always either
2:15:43 countable or equinumerous with the whole real line.
2:15:46 And then one can try to go beyond this and so on.
2:15:50 So I view all of those results which came, you know,
2:15:52 in the past 50 years, the later ones,
2:15:56 as fulfilling this Cantor idea that goes back,
2:16:01 you know, 120 years to his idea that we would prove the Continuum
2:16:05 Hypothesis by establishing more and more instances
2:16:08 for greater and greater complexity of sets.
2:16:11 But of course, even with what we know now,
2:16:15 it hasn't fully succeeded and it can't because the hierarchy
2:16:19 of complexity doesn't include all sets of real numbers.
2:16:22 Some of them are, sort of, transcending this hierarchy completely, in a way.
2:16:27 And so the program can't ever fully be successful,
2:16:32 especially in light of the independence result.
2:16:35 Yeah.
2:16:35 Well, spoiler alert, can you go to the independence result?
2:16:39 Sure.
2:16:39 So what does that mean?
2:16:41 So the Continuum Hypothesis was shown to be
2:16:44 independent from the ZFC axioms of mathematics?
2:16:46 Right.
2:16:47 So the ZFC axioms were the axioms that were put forth first by Zermelo
2:16:52 in 1908 in regard to his proof
2:16:54 of the well-order theorem using the axiom of choice.
2:16:58 That wasn't fully ZFC.
2:16:59 At that time, it was just Zermelo theory because he sort of...
2:17:02 There was a kind of missing axiom,
2:17:04 the replacement axiom, and the foundation axiom were added later,
2:17:07 and that's what makes the Zermelo-Fraenkel axiomatization,
2:17:10 which became, sort of, standard.
2:17:13 Actually, there's another aspect, which is Zermelo's original theory allowed
2:17:17 for the existence of ur-elements, or these atoms,
2:17:21 mathematical objects that are not sets but out
2:17:26 of which we build the set theoretic universe,
2:17:28 whereas set theorists today generally don't use ur-elements at all.
2:17:34 I, I argue that it's really the philosophy
2:17:39 of structuralism that leads them to omit
2:17:41 the ur-elements because it turns out that if
2:17:45 you adopt ZFC axioms with ur-elements, ZFCU it's called,
2:17:50 or ZFA, then any structure that exists, any mathematical structure that exists
2:17:55 in that set theoretic universe with the atoms
2:17:57 is isomorphic to a structure that doesn't use the atoms at all.
2:18:02 And you don't need the atoms if you're a structuralist
2:18:05 because you only care about the structures up to isomorphism anyways,
2:18:08 and the theory is simply more elegant and clear without the atoms.
2:18:12 They're just not needed.
2:18:14 And so that's why today when we talk about set theory,
2:18:17 generally we talk about the atom-free version, and ZFC has no ur-elements.
2:18:21 Okay.
2:18:22 So we formulate the ZFC axioms of set theory.
2:18:26 These are expressing the main principle ideas that we
2:18:30 have about the nature of sets and set existence.
2:18:35 And Canter had asked about the continuum hypothesis in the late 19th century,
2:18:44 and it remained open, totally open until 1938.
2:18:50 We should mention, I apologize,
2:18:52 that it was the number one problem in the Hilbert's
2:18:55 23 set of problems formulated at the beginning of the century.
2:18:59 That's right.
2:19:00 Maybe you can comment on why did he put that as number one.
2:19:03 So...
2:19:03 Right.
2:19:03 So Hilbert had introduced at his famous address
2:19:06 at the turn of the century this list
2:19:08 of problems that he thought could guide or were
2:19:12 important to consider in the coming century of mathematics.
2:19:16 I mean, that's how people talk about it now, although I'm not sure at all...
2:19:19 Of course, I can't really speak for Hilbert at all,
2:19:22 but if you were a very prominent mathematician,
2:19:27 I find it a little hard to believe that Hilbert would have conceived
2:19:30 of his list in the same way that we now take his lists.
2:19:34 I mean, having observed the century unfold,
2:19:37 we know that that list of 23 problems did in fact guide whole research programs,
2:19:43 and it was extremely important and influential.
2:19:45 But at the time, Hilbert would have no reason to think that that would be true,
2:19:50 and he was just giving a lecture and had
2:19:53 a list of problems that he thought were very important.
2:19:56 And so I tend to I would find it more reasonable
2:19:59 to think that he was just making a list of problems that he
2:20:02 thought were extremely interesting and important and fundamental in a way
2:20:06 without the kind of heavy burden of guiding this 20th century research.
2:20:13 Although it turns out that in fact that's exactly what they did.
2:20:18 And we already discussed how Hilbert's views
2:20:20 on the nature of set theory and the fundamental character,
2:20:24 that quote where he said,
2:20:26 "No one will cast us from the paradise that Canter has created for us." So,
2:20:31 so I think Hilbert was convinced by Canter on the importance
2:20:36 and the fundamental nature of the continuum
2:20:38 hypothesis for the foundations of mathematics,
2:20:41 which was a critically important development for the unity of mathematics.
2:20:45 I mean, before set theory emerged as a foundation of mathematics,
2:20:48 you know, there are different subjects in mathematics.
2:20:52 There's algebra and there's analysis, real analysis,
2:20:55 and topology and geometry, and so on.
2:20:57 There are all these disparate subjects with their own axioms,
2:21:02 separate axioms, right?
2:21:03 And sometimes it happens, like when you're proving,
2:21:06 say, the fundamental theorem of algebra, you know,
2:21:09 that the complex numbers are an algebraically closed
2:21:11 field that you can solve any polynomial equation in.
2:21:16 But the proof methods for that theorem come from other parts of mathematics.
2:21:21 You know, those topological proofs and so on.
2:21:24 And so how does that work?
2:21:27 I mean, if you have totally different axiom systems,
2:21:30 but you're using results from one subject in another subject,
2:21:33 it's somehow incoherent unless there's one underlying subject.
2:21:39 So the unity of mathematics was provided
2:21:41 by the existence of a mathematical foundation like set theory.
2:21:44 And at the time, it was set theory.
2:21:47 And so it's critically important to be able
2:21:50 to have a single theory in which one views all
2:21:53 of mathematics as taking place to resolve that kind
2:21:57 of transfer and borrowing phenomenon that was definitely happening.
2:22:01 So that must have been part of Hilbert's thinking
2:22:04 about why it's so important to have a uniform foundation,
2:22:07 and set theory was playing that role at the time.
2:22:09 Now, of course, we have other possible
2:22:11 foundations coming from category theory or type theory,
2:22:15 and there's univalent foundations now.
2:22:18 So there are sort of competing foundations now.
2:22:21 There's no need to just use one foundation, one set theoretic foundation.
2:22:25 Although set theory continues to, in my view,
2:22:28 have an extremely successful meta-mathematical analysis as a foundation,
2:22:31 I think is much more successful than
2:22:34 set theory for any of those other foundations,
2:22:37 but it's much less amenable though to things like computer proof and so
2:22:40 on, which is part of the motivation to find these alternative foundations.
2:22:44 So, yeah, okay, so just to talk about Hilbert though,
2:22:47 I think he was motivated by the need for a unifying foundation of mathematics,
2:22:52 and set theory was playing that role,
2:22:55 and the continuum hypothesis is such a core, fundamental question to ask,
2:22:59 so it seems quite natural that he would put it on the list.
2:23:03 There were other logic-related questions though,
2:23:05 like Hilbert's tenth problem is also related to logic.
2:23:08 This is the question about Diophantine equations,
2:23:11 and he asked to provide an algorithm to decide whether
2:23:15 a given Diophantine equation has a solution in the integers.
2:23:18 So a Diophantine equation is just, I mean,
2:23:21 it's maybe a fancy way of talking about something that's easy to understand,
2:23:25 a polynomial equation, except it's not just one variable, many variables.
2:23:31 So you have polynomials in several variables over the integers,
2:23:35 and you want to know, can you solve it?
2:23:37 So the problem is, as stated by Hilbert,
2:23:40 to provide an algorithm for answering the question whether
2:23:44 a given polynomial equation has a solution in the integers.
2:23:49 So he's sort of presuming that there is an algorithm,
2:23:52 but he wants to know what it is.
2:23:54 What is the algorithm?
2:23:56 But the problem was solved by proving that there is no algorithm.
2:24:01 It's an undecidable problem, like the halting problem.
2:24:05 There is no computable procedure that will correctly decide whether
2:24:08 a given polynomial equation has a solution in the integers.
2:24:12 So that's quite a remarkable development, I think.
2:24:14 There were also a few other logic-related questions on the list.
2:24:20 And so eventually, continuum hypothesis was shown
2:24:22 to be independent from ZFC axioms, as we've mentioned.
2:24:25 So, how does that make you feel?
2:24:29 What is independence, and what does that mean?
2:24:31 But once you tell the story, the historical story...
2:24:32 Yes- ...is really quite dramatic.
2:24:34 Yeah, that's great- I think,
2:24:35 because Cantor poses the question, you know, late 19th century.
2:24:39 And then it's totally open.
2:24:40 Hilbert asks about it, you know, at the turn of the 20th century.
2:24:44 Nobody has any clue.
2:24:45 There's no answer coming until 1938.
2:24:48 This is four decades later, right?
2:24:52 So a long time, and Gödel, Kurt Gödel, proved half of it.
2:24:58 What he proved is that if the axioms of set theory are consistent,
2:25:06 then there is a set theoretic world where both
2:25:11 the axiom of choice and the continuum hypothesis are true.
2:25:17 So what he's doing is showing
2:25:19 this is called the constructible universe, Gödel's L.
2:25:24 So he solved this.
2:25:25 This is the same result where he answers
2:25:29 the safety question of the axiom of choice,
2:25:31 but also for the continuum hypothesis.
2:25:33 They're true in the same set theoretic universe we get.
2:25:37 So if ZF, without the axiom of choice, is consistent,
2:25:40 then so is ZFC plus the continuum hypothesis is the result.
2:25:45 1938.
2:25:45 It's really such a beautiful argument.
2:25:48 It's just incredible, I think,
2:25:50 because he's building an alternative mathematical reality.
2:25:55 That's the structure of the proof, is that, okay,
2:25:58 if there's any mathematical reality, if there's any set theoretic world,
2:26:02 then we're going to build another one,
2:26:04 a separate one, a different one, maybe different.
2:26:06 Maybe it's the same as the original one, it could be.
2:26:09 If we started already in the one that he built, then it would be the same.
2:26:12 But there's no reason to assume it was the same.
2:26:15 So he has this kind of model construction
2:26:18 method to build this alternative set theoretic reality,
2:26:22 the constructible universe.
2:26:23 And then he proves that the axiom of choice is true there,
2:26:26 and also the continuum hypothesis is true there, and it's just amazing.
2:26:30 Really beautiful argument.
2:26:32 Okay, so then for the other part of the independence, that's only half of it,
2:26:36 because Gödel shows basically that you can't refute the continuum hypothesis,
2:26:43 but that's not the same thing as proving that it's true.
2:26:47 He showed that if set theory is consistent...
2:26:52 without the continuum hypothesis,
2:26:53 then it's consistent with the continuum hypothesis.
2:26:55 So that's not the same thing as proving that it's true.
2:26:59 Yeah.
2:26:59 And then it didn't come until 1963,
2:27:02 when Paul Cohen invented the method of forcing.
2:27:06 And proved that if there's a model of set theory,
2:27:10 then there's a model of set theory in which the continuum hypothesis is false.
2:27:15 So Cohen also is giving us this extremely
2:27:21 powerful tool for building alternative mathematical realities,
2:27:24 is how I think about it.
2:27:27 He's explained to us how to take any set theoretic world
2:27:31 and build another different one in which the continuum hypothesis is false.
2:27:35 The forcing extension.
2:27:38 It's just such a fascinating technique, tool of forcing.
2:27:42 Maybe I'm anthropomorphizing it,
2:27:44 but it seems like a way to escape one mathematical universe into another,
2:27:51 or to expand it or to alter it.
2:27:54 So you travel between mathematical universes.
2:27:56 Can you explain the technique of forcing?
2:27:57 Yeah, exactly.
2:27:58 It's all those things.
2:27:59 It's so wonderful.
2:28:00 I mean, that's exactly how I think about it.
2:28:03 I mean...
2:28:03 And we should mention, maybe this is a good place to even give a bigger picture.
2:28:07 One of your more controversial ideas in mathematics as laid out in the paper,
2:28:13 The Set-Theoretic Multiverse,
2:28:14 you describe that there may not be one true mathematics,
2:28:18 but rather multiple mathematical universes,
2:28:21 and forcing is one of the techniques that gets you from one to the other, so...
2:28:25 The-- Can you explain the whole shebang?
2:28:27 The whole...
2:28:27 Yeah, sure.
2:28:28 Let's get into it.
2:28:30 So the lesson of Cohen's result and Gödel's result
2:28:35 and so on, these producing these alternative set theoretic universes.
2:28:39 We've observed that the continuum hypothesis is independent
2:28:42 and the axiom of choice is independent of the other axioms,
2:28:46 but it's not just those two.
2:28:48 We have thousands of independence results.
2:28:50 Practically every non-trivial statement of infinite
2:28:53 combinatorics is independent of ZFC.
2:28:54 I mean, this is the fact.
2:28:57 It's not universally true.
2:28:59 There are some extremely difficult prominent
2:29:02 results where people proved things in ZFC, but for the most part,
2:29:08 if you ask a non-trivial question about infinite cardinalities,
2:29:11 then it's very likely to be independent of ZFC.
2:29:15 And we have these thousands of arguments,
2:29:18 these forcing arguments that are used to establish that.
2:29:22 And so how should we take that?
2:29:24 I mean, on the one hand, if you have a theory and it doesn't
2:29:28 answer any of the questions that you're interested in...
2:29:31 Okay, so what does that mean?
2:29:33 If you're following what I call the universe view or the monist view,
2:29:38 you might naturally say, "Well, look, ZFC is a weak theory,
2:29:43 and there's the true set theoretic reality out there,
2:29:48 and we need a better theory 'cause
2:29:51 the current theory isn't answering the questions.
2:29:53 Everything's independent." And so that seems
2:29:55 like a quite reasonable thing to take.
2:29:57 If you think that there is...
2:29:59 that every set theoretic question has a definite answer and there's
2:30:02 a unique set theoretic truth or a unique fact of the matter,
2:30:06 right, this is the universe view.
2:30:09 And by the way, to reiterate, independent means it cannot be proved or disproved
2:30:14 within this axiomatic system within this theory.
2:30:17 Right.
2:30:17 Exactly.
2:30:17 So to be independent means you can't prove
2:30:19 it and also you can't prove that it's false.
2:30:21 You can't refute it.
2:30:22 And you're saying that's why the statement is so traumatic or sad,
2:30:26 that most of the interesting stuff,
2:30:27 as you said, has been shown to be independent.
2:30:31 of ZFC.
2:30:32 But, but that's an interesting way to put it,
2:30:34 I think, because it reminds me of this, uh...
2:30:36 when I was a graduate student in Berkeley,
2:30:39 there was another graduate student who was working
2:30:42 with a non-logic professor in C-star algebras or something like this.
2:30:48 So it's a part of analysis or functional analysis,
2:30:51 and they were looking at a question,
2:30:53 and it turned out to be independent of ZFC, right?
2:30:57 And the attitude of this other professor was that, "Oh,
2:31:00 I guess I asked the wrong question." But my attitude and the attitude of all
2:31:07 the set theorists was when you ask a question that turns out to be independent,
2:31:11 then you asked exactly the right question because this is the one...
2:31:15 You know, it's carving nature at its joints.
2:31:18 You're adjudicating the nature of set
2:31:21 theoretic reality by finding these two realms.
2:31:23 You find one of these dichotomies.
2:31:25 You know, there's the worlds where it's true and the worlds where it's false.
2:31:28 And so when you ask that question, that's to be celebrated.
2:31:31 It means you asked exactly the right, interesting, fascinating question.
2:31:34 So it's not a kind of bleak thing
2:31:37 that you can't prove it and you can't refute it, and that's such a disaster.
2:31:40 Rather, it means that you found this...
2:31:44 this...
2:31:45 this cleavage in reality, in mathematical reality,
2:31:48 and it's good to know about those when they happen, you know?
2:31:53 Carving nature at its joints.
2:31:54 So what can you do about the things that are shown to be independent from ZFC?
2:32:00 Right.
2:32:00 So...
2:32:00 What are the techniques?
2:32:01 So one thing is that because of the incompleteness theorem,
2:32:04 we know that there's going to be...
2:32:07 For any theory that we can write down,
2:32:10 there's going to be things that we can't prove,
2:32:12 true things we can't prove in it.
2:32:14 So there's those things are gonna be independent.
2:32:17 And so we're already aware of the fact that there will
2:32:22 always be these independent phenomenon for any theory that we write.
2:32:27 And furthermore, some of those theories we won't
2:32:29 even be able to prove that they're consistent,
2:32:31 you know, like the consistency of their own theory.
2:32:33 So that's called the consistency-strength hierarchy.
2:32:37 So it's a direct consequence of Gödel's second incompleteness
2:32:40 theorem that for any theory we can write down,
2:32:44 then towering over it is this incredibly tall tower of consistency strength,
2:32:49 where the strength in theories aren't just adding another axiom,
2:32:53 but they're adding another axiom even whose consistency was
2:32:56 not provable in the previous layers of the hierarchy.
2:33:01 So, and so how lucky we are to find the large
2:33:04 cardinal axioms that instantiate exactly
2:33:08 this feature of increasing consistency strength,
2:33:13 this unending and extremely tall hierarchy of consistency strength of axioms.
2:33:21 And it exactly fulfills the prediction that Gödel's
2:33:24 theorem makes about that kind of thing.
2:33:27 Except, it's the axioms in the large cardinal hierarchy aren't, you know,
2:33:34 metalogical self-referential statements of the form
2:33:36 that sometimes arise in the Gödel analysis,
2:33:39 but rather they're professing existence of big infinities,
2:33:43 these large cardinal axioms.
2:33:45 And so it's such a welcome development,
2:33:49 and yet it's also known that the continuum hypothesis
2:33:54 is independent of all of the known large cardinal axioms.
2:33:59 So none of the large cardinal axioms we can prove,
2:34:04 none of them can settle the continuum hypothesis.
2:34:07 So the independence phenomenon is still there for things like
2:34:10 the continuum hypothesis and the cardinal combinatorics that I mentioned.
2:34:17 So you're building this incredible hierarchy of axiomatic
2:34:22 systems that are more powerful than the ZFC.
2:34:25 More powerful than ZFC and then more powerful
2:34:26 than that, more powerful than that, and so on.
2:34:28 It keeps going forever, and it will never be finished.
2:34:33 And still, to this day, the continuum hypothesis does not...
2:34:37 It's not settled by any of the large cardinal axioms.
2:34:41 Wow.
2:34:42 Wow.
2:34:44 What does that mean?
2:34:44 How does that make you feel?
2:34:46 Will it ever be settled?
2:34:47 Yeah, well, it's part of my multiverse view, I guess.
2:34:50 So which we started by, I was describing the universe view,
2:34:54 which is the view that, look, there are facts of the matter about all
2:34:59 of these questions and that it will turn out, if you're a universe view person,
2:35:03 which I'm not, but if you are, then you will hold that there is
2:35:07 a right answer to the continuum hypothesis question,
2:35:09 and there's a right answer to the large cardinal questions and so on.
2:35:15 And that what we should be aiming to do
2:35:17 is figure out this one true set theory, okay?
2:35:21 In contrast I take the developments of set theory over the past half
2:35:27 century or more as evidence that there
2:35:30 isn't such a unique set theoretic reality.
2:35:33 Rather, what we've been doing for decades
2:35:37 now is producing more and more alternative
2:35:40 set theoretic universes in which the fundamental
2:35:43 truths are differing from one to the other.
2:35:46 And that is the answer to the continuum hypothesis question.
2:35:51 The fact that given any model of set theory,
2:35:55 there's a forcing extension where the continuum hypothesis is true,
2:35:58 and another one where it's false.
2:36:00 You can sort of turn it on and off like a light switch.
2:36:03 And that's the fundamental nature of the continuum
2:36:05 hypothesis is that you can have it or you can have the negation as you
2:36:08 like within a very closely related set theoretic world.
2:36:14 Wherever you happen to be living,
2:36:16 there's a closely related one where CH is true,
2:36:19 where the continuum hypothesis is true, and one where it's false.
2:36:23 And that itself is a kind of answer.
2:36:25 It's not a singularist answer, a universe view answer.
2:36:30 It's a pluralist answer.
2:36:32 And this led me to my views
2:36:34 on the multiverse view of set theory and pluralist truth,
2:36:37 namely the fundamental nature of set theoretic truth has this plural
2:36:43 character in that there isn't a singular meaning to the fundamental terms,
2:36:48 but rather, there's this choice of alternative
2:36:52 set-theoretic universes that have different truths.
2:36:55 So, what does the multiverse view of mathematics enable you to do?
2:36:59 What does it empower you to do and what are the limitations?
2:37:02 What are the things it breaks about mathematics as a field,
2:37:06 as a space of knowledge, and what does it enable?
2:37:11 First of all, I guess one should say that these different
2:37:13 philosophical positions that you might take in the philosophy of set theory,
2:37:17 like the multiverse view or the universe view,
2:37:19 we don't ever disagree about the mathematics.
2:37:22 We're all agreeing on what the theorems are.
2:37:25 It's a question of philosophical perspective
2:37:28 on the underlying meaning or the context,
2:37:31 or really what is a philosophy of mathematics for, right?
2:37:36 And I mean, if you look back in history, for example,
2:37:40 like to the time of calculus with Newton and Leibniz, right?
2:37:44 They famously developed the ideas
2:37:47 of calculus using their concepts of infinitesimals,
2:37:51 and those foundations were roundly mocked by Bishop
2:37:55 Berkeley and so on who talked about,
2:37:57 you know, what are these same evanescent increments,
2:38:00 and shall we not call them the ghosts of departed quantities?
2:38:05 But the foundations really were kind of completely suspect,
2:38:09 I think, at the time.
2:38:12 And that foundation of infinitesimal calculus
2:38:15 really only became rigorous in the 1950s
2:38:18 or so with the development of non-standard analysis and Robinson's work.
2:38:21 Okay, so the point I'm trying to make is that, do you need a robust,
2:38:26 rigorous foundation of mathematics to make enduring insights in mathematics?
2:38:33 And the answer, regrettably,
2:38:36 is apparently not because in calculus, even with that lousy,
2:38:43 creaky foundation of infinitesimals not even
2:38:47 well understood that Newton and Leibniz had,
2:38:50 they proved all the fundamental theorems of calculus and, you know,
2:38:54 they had all the main insights
2:38:57 in those early days with that extremely bad foundation.
2:39:01 And so that shows you something about the relevance of the kind of foundational
2:39:07 views on mathematics and how important they
2:39:10 are for mathematical developments and progress and insight.
2:39:13 I mean, because I view those early mathematical developments
2:39:17 in calculus as genuinely mathematical and extremely important and insightful,
2:39:23 even though the foundations weren't any good,
2:39:27 from, you know, by contemporary perspectives.
2:39:29 Okay.
2:39:30 So, rather...
2:39:30 So when it comes to the philosophy of set theory
2:39:33 and the dispute between the universe view and the pluralism,
2:39:38 my view is that the choice of the philosophical perspective doesn't
2:39:42 actually have to do with the mathematical developments directly at all.
2:39:47 Rather, it tells us, "Where should set theory go?
2:39:52 What kind of set theory should we be looking at?
2:39:55 What kind of questions should we be asking?" So
2:39:58 if you have a universe mentality, the universe view,
2:40:03 then you're gonna be pushed to try to find
2:40:07 and articulate the nature of the one true set-theoretic universe.
2:40:12 And I think that remark is really well
2:40:14 borne out by the developments with Hugh Woodin, who's one of the most prominent
2:40:19 mathematicians and philosophers with the universe
2:40:22 view and his theory of ultimate L and so on.
2:40:25 And he's really striving.
2:40:27 Who was also your advisor.
2:40:28 He was also my supervisor.
2:40:29 My graduate supervisor.
2:40:30 Which is a, a personal story as well.
2:40:33 This, this fundamental dispute, yeah, on this question.
2:40:37 he is uh, has a very strong and successful research program,
2:40:42 sort of trying to give legs to finding
2:40:45 the nature of the one true set theoretic universe.
2:40:49 And it's driving the questions that he's
2:40:51 asking and the mathematical programs that he's pursuing.
2:40:54 Whereas if you have a pluralist view, as I do,
2:40:57 then you're gonna be led and attracted to questions that have
2:41:01 to do with the interaction of different set theoretic universes.
2:41:05 Or maybe you wanna understand the nature of how are the models
2:41:09 of set theory related to their forcing extensions and so on.
2:41:12 And so this led to things, um that I call, say, set theoretic potentialism,
2:41:16 where you think about a set theoretic universe in a potentialist way.
2:41:21 Not in the sense of potential infinity directly,
2:41:24 because all of these universes have infinite sets inside them already.
2:41:27 But they're potentialist in the sense that we could have more sets.
2:41:31 The universe could be wider and taller and so on, you know,
2:41:35 by forcing or by extending upward.
2:41:38 And so we wanna understand the nature of this realm of set theoretic universes.
2:41:45 And, and that's quite some exciting work.
2:41:47 And so with Benedikt Loewe and I, we proved some theorems on the modal logic
2:41:51 of forcing and set theoretic potentialism under end extension.
2:41:55 I've done a bunch of work on this topic.
2:41:58 And, and also I I mounted together with Gunter Fuchs and Jonas Riets,
2:42:01 who was one of my own PhD students,
2:42:03 the topic of set theoretic my own PhD students,
2:42:05 the topic of set theoretic geology, which is studying...
2:42:10 It's taking the metaphor of forcing.
2:42:12 I mean, in forcing, you have the ground model and the forcing extension.
2:42:15 And when I was first working with Jonas he said, "I wanna undo forcing.
2:42:22 I wanna go backwards." And I at first said,
2:42:25 "But Jonas, it doesn't work that way.
2:42:27 You start in the model, in the ground model,
2:42:29 and you go out, you go to the bigger one.
2:42:31 bigger one.
2:42:31 You know, that's how forcing works." And he said, "No, no,
2:42:34 I wanna go backwards." And, and so he was quite persistent, actually.
2:42:38 And um, and so finally, I said, "Okay, let's, let's do it.
2:42:42 Let's take it seriously." And so we sat
2:42:44 down and, and s- and started thinking, you know,
2:42:47 more precisely and carefully and deeply about the nature of taking
2:42:50 a set theoretic universe and seeing where did it come from by forcing,
2:42:54 which was a new way of thinking about forcing at the time.
2:42:58 Like reverse engineering the forcing?
2:43:00 Yeah, something like that.
2:43:01 Forcing is a way of producing a new universe.
2:43:04 And so you could start somewhere and go to that new universe,
2:43:07 or you could look where you are and say, "Well, look,
2:43:09 I got here by doing that already in the past."
2:43:12 So we defined models of the bedrock model and ground,
2:43:17 you know, sort of undoing the forcing.
2:43:19 And, and really, it was quite fruitful.
2:43:21 And I view this as part of the sort of pluralist perspective,
2:43:24 except the difference is that set theoretic
2:43:27 geology is amenable to the universe view.
2:43:30 So even though the work was inspired
2:43:33 by this philosophical view on the multiverse view, nevertheless,
2:43:39 the central ideas of geology have now been picked up
2:43:43 by the people with the research program in the universe view.
2:43:46 Because it turns out that set theoretic geology is helping them or us
2:43:51 to discover the nature of the one true universe relates to its mantle.
2:43:56 There's this concept of the set theoretic mantle that I
2:43:58 had introduced in a way that is extremely interesting.
2:44:02 And so it's historically quite funny, I think,
2:44:05 because this research program that grew entirely out
2:44:08 of the pluralist point of view ended up being picked
2:44:12 up by the universe point of view research program
2:44:16 in a, in a way that is quite important.
2:44:21 Can you prove something in the world that you arrived at through
2:44:26 forcing and then take some of that back to the ground model?
2:44:30 Yeah, absolutely.
2:44:31 And that's a really powerful argument method, actually.
2:44:33 People often want to do that.
2:44:36 Suppose you're in some set theoretic context.
2:44:38 You know, you could think about as living in a set theoretic universe,
2:44:41 and you want to prove something in that universe only.
2:44:45 But maybe one way to do it is to first
2:44:49 construct this forcing extension and then use the features
2:44:53 about this forcing extension to realize that certain things
2:44:55 must have already been true in the ground model.
2:44:58 And then you throw the forcing extensions away and you...
2:45:01 Oh, cool- ...yeah.
2:45:02 So this can happen.
2:45:03 To pick a more elementary example, if you think about the early days of people
2:45:08 reasoning with the complex numbers before they really understood them.
2:45:14 So they would have these algebraic equations
2:45:15 that they're trying to solve, you know,
2:45:17 and they would have the tools and methods of doing it,
2:45:20 but then in the course of, you know,
2:45:22 so they would have to do things to the polynomial and change the factors
2:45:25 and so on, and produce other polynomials and solve them and so on.
2:45:29 And sometimes, they could produce solutions.
2:45:33 in the middle of their construction, they were led to, like,
2:45:37 the square root of minus five or something, you know, in the construction.
2:45:41 And they didn't have any meaning for that, but they
2:45:44 would just do it symbolically, you know.
2:45:47 And, and eventually, it would turn in, you know,
2:45:50 because of the methods that they had,
2:45:52 they would combine and they would cancel and so on, and all
2:45:54 the complex parts would cancel out and they'd end up with this, you know,
2:45:58 actual answer, you know, three plus square root of 17 or whatever.
2:46:01 And, and they could check it and it worked.
2:46:03 It was a solution of the original equation.
2:46:06 And so it must have been bewildering to them because
2:46:09 they would start with this question purely in the real numbers,
2:46:13 an algebraic question,
2:46:14 and they would march on their method and proceed through the land of nonsense,
2:46:19 you know, with these square roots of negative numbers and then end up
2:46:23 with an answer that was real again that they could verify was correct.
2:46:29 And so, I view this kind of forcing argument
2:46:31 that I was just describing in a similar way.
2:46:34 You start in set theory,
2:46:35 and you go to this land of nonsense in the forcing extension,
2:46:40 this imaginary world.
2:46:41 And you argue and you come back.
2:46:42 I mean, you make a consequence in the ground model,
2:46:45 and it's such a beautiful way of arguing.
2:46:47 So, speaking of the land of nonsense, I have to ask you about surreal numbers,
2:46:51 but first, I need another bathroom break.
2:46:54 All right, we're back,
2:46:56 and there's this aforementioned wonderful blog post on the surreal numbers,
2:47:01 and that there's quite a simple surreal number
2:47:07 generation process that can basically construct all numbers.
2:47:11 So, maybe this is a good spot to ask,
2:47:15 what are surreal numbers and what is the way we can generate all numbers?
2:47:20 So, the surreal number system is an amazing,
2:47:24 amazingly beautiful mathematical system that was introduced by John Conway.
2:47:30 Rest in peace, one of the great mathematicians ever on this earth.
2:47:33 Yes, absolutely.
2:47:34 And I really admire his style
2:47:37 of mathematical thinking and working in mathematics,
2:47:40 and the surreal number system is a good instance of this.
2:47:43 So, the way I think about the surreal numbers system is what it's
2:47:47 doing is providing us a number system that unifies all the other number systems.
2:47:52 So, it extends the real numbers.
2:47:54 Well, not only, it extends the integers,
2:47:56 the natural numbers and the integers and the rational numbers,
2:47:59 and the real numbers, but also the ordinals and the infinitesimals.
2:48:03 So, they're all sitting there inside the surreal numbers,
2:48:08 and it's this colossal system of numbers.
2:48:12 It's not a set even.
2:48:14 It's a proper class, it turns out, because it contains all the ordinal numbers.
2:48:19 But it's generated from nothing by a single rule, and the rule is,
2:48:26 so we're gonna generate the numbers in stages,
2:48:30 in transfinite sequence of stages.
2:48:32 And at every stage, we take the numbers
2:48:34 that we have so far and in all possible ways,
2:48:38 we divide them into two sets, a lower set and an upper set,
2:48:41 or a left set and a right set.
2:48:44 So we divide them into these two sets,
2:48:46 so that everything in the left set is less than everything in the right set,
2:48:50 and then at that moment,
2:48:51 we create a new number that fits in the gap between L and R.
2:48:58 Okay?
2:48:58 That's it.
2:48:59 That's all we do.
2:49:01 So, let me say it again.
2:49:05 The rule is, we proceed in stages, and at any stage,
2:49:08 then in all possible ways, we divide the numbers we have into two collections,
2:49:16 the left set and the right set,
2:49:17 so that everything in the left set is less than everything in the right set.
2:49:20 And we create a new number, a new surreal number that will fit in that gap.
2:49:27 Okay.
2:49:27 So, for example, we could start...
2:49:29 At the beginning, we don't have any numbers.
2:49:31 We haven't created anything yet, and so, well, we could take nothing,
2:49:36 and we could divide it into two sets,
2:49:38 the empty lower set and the empty upper set.
2:49:41 I mean, the two empty sets.
2:49:43 And everything in the empty set is less than
2:49:45 everything in the empty set because that's a vacuous statement.
2:49:48 So we satisfy the conditions, and we apply the number generation rule,
2:49:53 which says we should create a new number.
2:49:56 And this is what I call the Big Bang of numbers,
2:49:59 the surreal genesis when the number zero is born.
2:50:03 Zero is the firstborn number that is bigger than everything
2:50:07 in the empty set and less than everything in the empty set.
2:50:10 Okay, but now we have this number zero,
2:50:12 and so therefore, we now can define new gaps.
2:50:17 Because if we put zero into the left set and have an empty right set,
2:50:22 then we should create a new number that's bigger
2:50:25 than zero and less than everything in the empty set,
2:50:27 and that number is called the number one.
2:50:29 And similarly, at that same stage, we could have put zero into the right set,
2:50:36 and so that would be the firstborn number that's less than zero,
2:50:39 which is called minus one.
2:50:40 So now we have three numbers, minus one, zero and one,
2:50:44 and they have four gaps because there could be a number below minus one
2:50:48 or between minus one and zero or between zero and one or above one,
2:50:52 and so we create those four new numbers.
2:50:55 The first number above one is called two.
2:50:57 The first number between zero and one is called 1/2,
2:51:00 and then on the negative side, we have minus 1/2 and minus two and so on.
2:51:04 So now we have, what is that, seven numbers?
2:51:07 seven numbers.
2:51:08 So there's eight gaps between them.
2:51:09 So at the next birthday, they call them,
2:51:12 the next stage will be born all the numbers between those gaps,
2:51:16 and then between those and between those and so on.
2:51:18 And as the days progress, we get more and more numbers,
2:51:22 but those are just the finite birthdays,
2:51:24 because as I said, it's a transfinite process.
2:51:26 So at day omega, that's the first infinite day,
2:51:30 we're going to create a lot of new surreal numbers.
2:51:34 So every real number will be born at that stage because every real number
2:51:40 fills a gap in the previously born
2:51:42 rational numbers that we had just talked about.
2:51:45 It's not all the rationals,
2:51:46 because actually the rational numbers that are born at the finite stages are
2:51:49 just the rationals whose denominator is a power of two, it turns out.
2:51:54 Those are called the dyadic rationals.
2:51:56 So the real numbers are all born on day omega,
2:51:59 but also some other numbers are born on day omega, namely,
2:52:04 the ordinal omega itself is the firstborn
2:52:06 number that's bigger than all those finite numbers,
2:52:09 and minus omega is the firstborn number
2:52:12 that's less than all those finite numbers.
2:52:14 But also, we have the number epsilon,
2:52:17 which is the firstborn number that's strictly bigger than
2:52:20 zero and strictly less than all the positive rational numbers.
2:52:24 So that's going to be an infinitesimal number in that gap, and so on.
2:52:28 On day omega plus one, we get more numbers, and then omega plus two and so on.
2:52:33 And the numbers just keep coming forever.
2:52:35 So, this is how you build the surreal number system.
2:52:38 And then it turns out you can define the arithmetic operations of addition
2:52:43 and multiplication in a natural way
2:52:45 that is engaging with this recursive definition.
2:52:48 So we have sort of recursive definitions
2:52:51 of plus and times for the surreal numbers.
2:52:54 And it turns out you can prove that they
2:52:57 make the surreal numbers into what's called an ordered field.
2:53:01 So they satisfy the field axioms, which means that you have distributivity
2:53:05 and commutativity of addition and multiplication.
2:53:09 multiplication, and also you have reciprocals for every non-zero number.
2:53:13 You can divide by the number.
2:53:15 So you can add and multiply and divide and subtract.
2:53:18 And furthermore, you can take square roots.
2:53:21 And furthermore, every odd degree polynomial has a root,
2:53:27 which is true in the real numbers,
2:53:28 because if you think about, say, a cubic or a fifth degree polynomial,
2:53:32 then you know it's going to cross the axis,
2:53:35 because it has opposite behaviors on the two infinities,
2:53:39 because it's an odd degree polynomial.
2:53:40 So on the positive side, it's going to the positive infinity.
2:53:43 On the negative side, it would be going to minus infinity.
2:53:46 So it has to cross.
2:53:47 So we know in the real numbers, every odd degree polynomial has a root.
2:53:53 And that's also true in the surreal numbers.
2:53:55 So that makes it what's called a real closed field,
2:53:58 which is a very nice mathematical theory.
2:54:02 So it's really quite interesting how we can find copies
2:54:06 of all these other number systems inside the surreal numbers.
2:54:09 But the surreal numbers are fundamentally discontinuous as you're worried about.
2:54:12 What are the consequences of this?
2:54:14 Right.
2:54:14 So the surreal numbers have a property that they
2:54:17 form a non-standard model of the real field,
2:54:21 which means that they provide a notion of infinitesimality that one can use
2:54:27 to develop calculus on the grounds of Robinson's
2:54:30 non-standard theory that I had mentioned earlier.
2:54:33 But they don't have the least upper bound property for subcollections.
2:54:39 There's no set of surreal numbers,
2:54:41 no non-trivial set of surreal numbers has at least upper bound,
2:54:45 and there are no convergent sequences in the surreal numbers.
2:54:48 And so for the sort of ordinary use in calculus based on limits and convergence,
2:54:55 that method does not work in the surreal numbers at all.
2:54:59 So that's what I mean when I
2:55:00 say the surreal numbers are fundamentally discontinuous.
2:55:02 They have a fundamental discontinuity going on.
2:55:06 But you can still do calculus with them,
2:55:08 because you have infinitesimals if you use these non-standard methods,
2:55:13 the infinitesimal-based methods to calculus.
2:55:15 And people do that.
2:55:18 I once organized a conference in New York,
2:55:21 and we had John Conway as a speaker at that conference.
2:55:23 And there was a question session, and someone asked him, I mean,
2:55:28 it's a bit rude, I think, but they asked it and the question was,
2:55:34 "What is your greatest disappointment in life?" I mean,
2:55:37 I would never ask a question like that at a conference in a very public setting.
2:55:41 But Conway was extremely graceful, and he answered by saying
2:55:46 that, "The surreal numbers..." Not the numbers themselves,
2:55:50 but the reception of the surreal numbers,
2:55:52 because he had ambition that the surreal numbers would
2:55:56 become a fundamental number system used throughout mathematics and science,
2:56:01 because it was able to do non-set analysis,
2:56:06 it was able to do calculus, it unified the ordinals and so on.
2:56:08 And it's such a unifying, amazing structure,
2:56:12 beautiful structure with elegant proofs and sophisticated ideas all around it.
2:56:18 And he was disappointed that it never really achieved
2:56:24 that unifying status that he had the ambition for.
2:56:28 And this, he mentioned as his greatest disappointment.
2:56:32 Yeah, Donald Knuth tried to celebrate it.
2:56:34 It never quite took hold.
2:56:36 So I don't want to give the impression
2:56:38 though that the surreal numbers are not widely studied,
2:56:40 because there are thousands of people who are...
2:56:41 Sure- ...studying it.
2:56:42 In fact, Philip Ehrlich, who is one of the world experts on the surreal numbers,
2:56:46 mentioned to me once that Conway was his own
2:56:50 worst enemy with regard to that very issue,
2:56:52 because in the Conway style, everything is a game.
2:56:56 And he treated the surreal numbers as a kind of plaything,
2:57:00 a toy, and maybe that makes people not take it seriously.
2:57:05 Although my view is that it is extremely serious, useful, and profound,
2:57:11 and I've been writing a whole series of essays
2:57:14 on the surreal numbers for my Substack at Infinitely More.
2:57:17 And I just find the whole subject so fascinating and beautiful.
2:57:22 I mean, it's true.
2:57:23 I'm not applying it in engineering,
2:57:26 which maybe was part of this Conway ambition.
2:57:30 And I just wanted to, before I forget,
2:57:32 mention the Conway, turning everything into a game.
2:57:36 It is a fascinating point that I didn't quite think about,
2:57:39 which I think the Game of Life is
2:57:41 just an example of exploration of cellular automata.
2:57:43 I think cellular automata is one of the most incredible,
2:57:46 complicated, fascinating...
2:57:47 It feels like an open door into a world we have not quite yet explored.
2:57:52 And it's such a beautiful illustration of that world,
2:57:55 the Game of Life, but calling it a game,
2:57:57 maybe life balances it, 'cause that's your powerful word,
2:58:01 but it's not quite a game.
2:58:03 It's a fascinating invitation to an incredibly
2:58:06 complicated and fascinating mathematical world.
2:58:08 I think every time I see cellular automata and the fact
2:58:11 that we don't quite have mathematical tools to make sense of that world,
2:58:17 it fills me with awe.
2:58:18 Speaking of a thousand years from now,
2:58:20 it feels like that is a world we might make some progress on.
2:58:24 The Game of Life is a sort
2:58:26 of playground for computably undecidable questions because in fact,
2:58:29 you can prove that the question of whether
2:58:32 a given cell will ever become alive is computably undecidable.
2:58:37 In other words, given a configuration and you ask,
2:58:41 "Will this particular cell ever you know,
2:58:44 be alive?" "...in the evolution?" And you can prove
2:58:48 that that question is equivalent to the Halting Problem.
2:58:52 It's computably undecidable.
2:58:53 It's semi-decidable in the sense that if it will become alive,
2:58:57 then you will know it at a finite stage because you
2:59:00 could just run the Game of Life algorithm and let it run.
2:59:03 And if it ever did come alive, you could say, "Yeah,
2:59:06 it was alive." But if you've run it
2:59:08 for a thousand years and it hasn't come alive yet,
2:59:11 then you don't necessarily seem to have any basis for saying,
2:59:14 "No, it won't ever come alive" if the behavior was very complicated.
2:59:18 Maybe if you have a complete understanding of the evolution of the behavior,
2:59:21 then you can say no,
2:59:22 but you can prove you won't always have that understanding-...
2:59:25 precisely because the problem is equivalent to the halting problem.
2:59:28 And nevertheless, when you sit back and look and visualize the thing,
2:59:31 some little mini cellular automata civilizations are born and die quickly,
2:59:35 and some are very predictable and boring,
2:59:38 but some have this rich incredible complexity.
2:59:41 And maybe that speaks to a thing I
2:59:45 wanted to ask on the halting problem and decidability.
2:59:50 You've mentioned this thing where if you understand the program deeply,
2:59:53 you might be able to say something.
2:59:55 So can we say something interesting about maybe once statistically how many
3:00:01 programs we know something about in terms of whether they halt or not?
3:00:05 Or what does it mean to understand a program
3:00:07 deeply enough to be able to make a prediction?
3:00:11 The main lesson of computability theory in my view
3:00:16 is that it's never the case that you
3:00:20 can have a thorough understanding of the behavior
3:00:23 of a program by looking at the program,
3:00:26 and that the content of what you learn from a program, I mean,
3:00:30 in the most general case,
3:00:32 is always obtained just by running it and looking at the behavior.
3:00:36 And the proof of that is there's a theorem called Rice's Theorem,
3:00:41 which makes that idea completely robust.
3:00:43 But I want to just take a little detour
3:00:47 towards another question riffing on something that you just said.
3:00:52 Namely, one can ask the question, What is the behavior of a random program?
3:00:59 So you have some formal computing language, you know, and you want to, you know,
3:01:03 look at the collection of all programs of a certain size.
3:01:06 Maybe there's only finitely many.
3:01:08 And can you say something about the behavior of a randomly chosen one,
3:01:13 like with a certain likelihood it will have a certain behavior?
3:01:16 And the answer turns out to be extremely interesting.
3:01:20 Once years ago, Alexey Myasnikov asked me a question.
3:01:23 He had this concept of a decision problem with a black hole,
3:01:27 and what that means is it's a decision
3:01:30 problem which is possibly difficult in the worst case,
3:01:33 but the difficulty was concentrated in a very tiny region called the black hole.
3:01:39 And outside of that black hole, it was very easy.
3:01:42 And so, for example, this kind of problem is a terrible problem
3:01:45 to use if you're basing your encryption scheme, you know,
3:01:49 you don't want to use a black hole problem because
3:01:52 if someone can rob the bank 95% of the time,
3:01:55 you know, then that's not what you want,
3:01:58 or even any non-trivial percent of the time is too dangerous.
3:02:01 So you don't want to use problems that are, you know,
3:02:06 almost every case is easily solved as the basis of your encryption.
3:02:10 And the question Alexey asked me was,
3:02:13 "Does the halting problem have a black hole?" You know?
3:02:16 And so if we take, say, the standard model of Turing machines,
3:02:20 it's one-way infinite tape with zeros and ones on the tape
3:02:24 and so on, the head moving back and forth,
3:02:26 and you know, it stops when it gets into the halt state,
3:02:30 then it turns out we proved that there is a black hole.
3:02:34 And what that means is there's a computer procedure
3:02:37 that decides correctly almost every instance of the halting problem.
3:02:41 Even though the halting problem is not decidable,
3:02:43 we can decide almost every instance.
3:02:46 So, more precisely, there's a collection of Turing machine programs such
3:02:51 that we can easily decide whether a program's in that collection or not.
3:02:56 And for the programs in the collection,
3:02:59 we can decide the halting problem for those programs easily.
3:03:03 And furthermore, almost every program is in the collection in the sense
3:03:08 that as the number of states goes to, you know,
3:03:10 becomes large, the proportion of programs in the collection goes to 100%.
3:03:18 So the asymptotic density of the programs is one.
3:03:23 And the proof was quite fascinating because it's one
3:03:25 of these situations where the theorem sounds really surprising,
3:03:28 I think, to many people when I first tell it, I mean, to computability experts.
3:03:32 Then it's sort of intriguing to think that you
3:03:36 can solve almost every instance of a halting problem.
3:03:38 But then when they hear the proof, it's completely a letdown.
3:03:43 Unfortunately, nobody likes the theorem after the proof.
3:03:47 And so the proof is so simple, though.
3:03:51 If you know how a Turing machine operates,
3:03:53 there's this infinite paper tape on which the machine writes zeros and ones,
3:03:56 and the head moves back and forth according to rigid instructions.
3:04:00 And the instructions are all of the form,
3:04:05 if the machine is in such and such a state
3:04:08 and it's reading such and such symbol on the tape,
3:04:10 then it should write this symbol on the tape
3:04:13 and it should change to this new state specified,
3:04:15 and it should either move left or right as specified.
3:04:18 So a program consists of instructions like that.
3:04:22 If you look at, If you look at a program, you know,
3:04:27 one of the states is the halt state and that's when the program halts.
3:04:31 But you can calculate how many programs don't
3:04:36 have any instruction that transitions to the halt state.
3:04:39 You can easily calculate the proportion.
3:04:42 And in the limit, it goes to 1 over E squared, 13 and a half percent.
3:04:47 If you calculate the limit,
3:04:49 the proportion of programs with end states that don't
3:04:54 ever halt because they don't have any instruction saying halt.
3:04:58 Those programs obviously never halt because they can't halt.
3:05:01 They don't have any instruction that says halt.
3:05:04 So 13% of programs, you could say,- 13%, you can say they don't halt,
3:05:08 because you just look at them and you can understand them.
3:05:10 There's no halt state.
3:05:11 There's no...
3:05:11 They never change to the halt state, so they can't halt.
3:05:14 I mean, that nevertheless is beautiful to know.
3:05:17 So that's a kind of trivial reason for non-halting, you know.
3:05:21 And when I first made that observation, I thought,
3:05:23 "Okay, this is the proof strategy." Because we wanted...
3:05:25 I wanted to say at first the goal was, look,
3:05:29 that's a stupid reason for a program not to halt.
3:05:32 And I just want to pile up as many stupid reasons as I can think of...
3:05:37 ...until it gets more than 50%, and then I can say most.
3:05:43 That was brilliant.
3:05:43 Yeah.
3:05:43 That was my goal.
3:05:44 I love this.
3:05:45 Yeah.
3:05:45 So we thought more about it, though, and we hit the jackpot because we found
3:05:50 one gigantic stupid reason that converged to 100%.
3:05:54 I mean, in the limit.
3:05:56 And so, the stupid reason for a program not to halt is that, well,
3:06:02 if you think about the behavior, see, the head is sitting there.
3:06:06 It's on the leftmost cell of the tape at the very beginning.
3:06:09 It's in the start state, and the head is following an instruction.
3:06:13 And the instruction says, "When you're in the start state," which it is,
3:06:17 "and you're reading something on the tape,
3:06:18 then you should write something and you should change to a new state,
3:06:21 and you should either move left or right." But half of them move left.
3:06:27 But if you move left and you are already at the end, then the head falls off.
3:06:31 And so the computation stops because the head fell off the tape.
3:06:37 That's a pretty stupid reason.
3:06:38 Okay, but that's half of them already just like that.
3:06:40 Okay, and then some of them went right, and they changed to a new state.
3:06:44 And amongst those, you know, the new state,
3:06:46 half of those ones are going left and half are going right from that place,
3:06:50 and then most of those are changing to a new state.
3:06:53 When there's a lot of states,
3:06:54 it's very likely that the next state that you transition to is new.
3:06:57 And so you get this random walk behavior, if you know what that means,
3:07:01 where half go left and half go right at each step.
3:07:04 And there's a theorem due to Pólya which is called the Pólya recurrence theorem,
3:07:09 which says when you have a random walk, a one-dimensional random walk,
3:07:13 then it's very likely to come back to where you started.
3:07:17 And when that happens for us,
3:07:19 then half of them from that place fall off on the next step.
3:07:23 And so you can show using this kind
3:07:26 of analysis that the probability one behavior
3:07:29 of a random Turing machine is that the head
3:07:33 falls off the tape before it repeats a state.
3:07:36 And that is the stupid proof that shows how to solve the halting problem.
3:07:40 Because when that happens, we can answer the halting problem saying,
3:07:43 "No, the computation stopped because the machine crashed, not because it halted,
3:07:47 so therefore it doesn't count as halting on some accounts." Or, you know,
3:07:51 if you want to define that as halting, crashing as halting, then...
3:07:54 But in any case, however it is that you set up your formalism,
3:07:58 you're going to be able to answer the question
3:08:00 for the behavior of the machine when the head falls off.
3:08:04 So statistically, in the limit, you solve the halting problem.
3:08:08 Yes, exactly.
3:08:09 Computably solve it.
3:08:11 Yeah.
3:08:12 What do we take from that?
3:08:13 Because you didn't solve the halting problem.
3:08:15 No, it's impossible to fully solve...
3:08:17 ...the halting problem correctly in all cases.
3:08:20 That's pretty cool.
3:08:20 That's kind of...
3:08:21 I mean, I don't know.
3:08:21 This is...
3:08:22 It's a probabilistic way...
3:08:23 I mean, it's probabilistic in the sense
3:08:25 that we're solving almost all instances...
3:08:28 ...computably.
3:08:29 There are versions of this that are maybe more interesting
3:08:32 from the point of view of complexity theory and actually useful.
3:08:35 I mean, there's the whole P-NP problem and so on.
3:08:37 And there's this genre of NP-complete problems,
3:08:40 which are problems that are infeasible.
3:08:43 They would take exponential time to solve them in the ordinary way.
3:08:47 And they're not known to be polynomial time solvable,
3:08:50 although in these cases it's an open
3:08:52 question whether there is a polynomial time algorithm, a feasible algorithm.
3:08:56 And for most for most of the NP-complete problems,
3:09:01 you can prove that there's a polynomial
3:09:05 time approximation that solves almost all instances...
3:09:09 ...in a feasible amount of time.
3:09:10 So like the knapsack problem, you know, packing problems,
3:09:13 and so on, other kinds of problems, satisfaction problems, when...
3:09:16 Depending on how you set up the formalism, you can prove,
3:09:20 and I've proven many instances of this but also
3:09:23 I think it's widespread for almost all the NP-complete problems,
3:09:28 the difficult problems,
3:09:29 and these are important problems for industrial application when
3:09:32 these are problems that we actually want to solve.
3:09:35 We can have feasible algorithms that solve almost every instance of them.
3:09:42 The amount of fields and topics you've worked on is truly incredible.
3:09:46 I have to ask about P versus NP.
3:09:48 This is one of the big open problems in complexity theory.
3:09:51 So for people who don't know,
3:09:52 it's about the relation between computation time and problem complexity.
3:09:55 Do you think it will ever be solved?
3:09:57 And is there any chance the weird counterintuitive
3:10:03 thing might be true that P equals NP?
3:10:06 Yeah, that's an interesting question.
3:10:07 Sometimes people ask about whether it could be independent, which I think is...
3:10:12 ...An interesting question for logicians.
3:10:15 And of course, well,
3:10:16 one has to say if you're entertaining the idea of independence,
3:10:20 you know, over which theory?
3:10:22 Because every statement is going to be
3:10:24 independent over an extremely weak theory.
3:10:26 So that's, you know,
3:10:27 it doesn't make sense to say it's independent all by itself.
3:10:30 You're only independent relative to a theory, right?
3:10:32 So the way I think about P-NP is that...
3:10:35 I mean, of course it's a theoretical
3:10:38 question about the asymptotic behavior of these problems.
3:10:42 I mean, for a problem to be in P means that there, you know,
3:10:47 there is a computable decision procedure
3:10:49 that runs in time bounded by some polynomial.
3:10:52 But the coefficients on that polynomial could be enormous,
3:10:55 and the degree could be incredibly high.
3:10:59 And so for small values of inputs, then it doesn't make sense to talk about
3:11:05 this polynomial time feasibility with respect to, say,
3:11:08 the range of problem inputs that we will ever give it
3:11:12 in our lifetime or in the span of human civilization or whatever.
3:11:16 I mean, because it's an asymptotic property,
3:11:19 it's really in the limit as the size of the inputs goes to infinity,
3:11:24 that's the only time that polynomial or NP becomes relevant.
3:11:28 And so maybe it's important to keep that in mind when.
3:11:31 Sometimes you find kind of overblown remarks made about,
3:11:35 you know, if P equals NP,
3:11:37 then this will be incredibly important for human civilization because it
3:11:41 means that we'll have feasible
3:11:43 algorithms for solving these incredibly important-...
3:11:45 problems in NP.
3:11:47 You know, that it would cause immense wealth for human societies and so
3:11:51 on because we would be able to solve these otherwise intractable problems,
3:11:56 and that would be the basis of new technology and industry and so forth.
3:12:00 I mean, people make these kind of remarks, but-- Of course-...
3:12:04 you have to temper those remarks by the realization
3:12:07 that P and P equal NP or P not
3:12:10 equal NP are not about these practical things at all
3:12:14 because of the asymptotic nature of the question itself.
3:12:19 Okay, that's on the one hand.
3:12:20 But on the second hand, we already have the algorithm,
3:12:22 so we could use it already, except it's a terrible algorithm because it involves
3:12:27 all this incredible amount of coding and so on.
3:12:30 And on the third hand, like you said,
3:12:32 we already have approximation algorithms that-- Yes-...
3:12:35 that from a pragmatic perspective,
3:12:37 solve all the actual real engineering problems of human civilization.
3:12:42 Like the SAT solvers work amazingly well, you know,
3:12:45 in lots and lots of cases, even though we can prove we don't expect...
3:12:48 If P is not equal to NP, then there won't be a polynomial time SAT solver.
3:12:52 But the actually the SAT solver approximations,
3:12:55 you know, are really quite amazing.
3:12:59 Sorry to ask the ridiculous question, but who is the greatest mathematician...
3:13:03 of all time?
3:13:04 Who are the possible candidates?
3:13:06 Euler, Gauss, Newton, Ramanujan, Hilbert.
3:13:09 We mentioned Gödel, Turing, if you throw him into the bucket.
3:13:15 So this is, I think, an incredibly difficult question to answer.
3:13:19 Personally, I don't really think this way
3:13:23 about ranking the mathematicians by greatness.
3:13:28 So you don't have, like...
3:13:29 You know, some people have a Taylor Swift poster in their dorm room.
3:13:31 You don't have it.
3:13:33 I mean, if you forced me to pick someone,
3:13:35 it would probably be Archimedes because-...
3:13:37 he had such incredible achievements in such an early era,
3:13:43 which totally transcended the work of the other people in his era.
3:13:49 But I also have the view that I want to learn mathematics and gain
3:13:54 mathematical insight from whoever can provide it and wherever I can find it.
3:13:59 And this isn't always just coming from the greats.
3:14:02 And sometimes the greats are doing things that are just first and not...
3:14:06 You know, somebody else could have easily been first.
3:14:09 And so there's a kind of luck aspect to it
3:14:11 when you go back and look at, you know, the achievements.
3:14:14 And because of this progress issue in mathematics that we talked about earlier,
3:14:18 namely, we really do understand things much better now than they used to.
3:14:22 And when you look back at the achievements that had been made,
3:14:25 then maybe you can imagine,
3:14:27 you know, thinking, "Well, you know, somebody else could've,
3:14:30 could've had that insight also." And maybe they would have.
3:14:36 It's already a known phenomenon that disparate mathematicians end
3:14:40 up proving essentially similar results at approximately the same time.
3:14:44 But okay, the person who did it first is getting the credit and so on.
3:14:49 What do you make of that?
3:14:50 Because I see that sometimes when mathematicians...
3:14:52 This also applies in physics and science,
3:14:55 where completely separately, discoveries are made...
3:14:58 ...Maybe at a very similar time.
3:15:00 What does that mean?
3:15:01 It's relatively common.
3:15:02 I mean, I think it's certain ideas are
3:15:05 in the air and being thought about but not fully articulated,
3:15:09 and so this is the nature of growth in knowledge.
3:15:14 Do you understand where ideas come from?
3:15:16 Not really.
3:15:17 I mean, what's your own process when you're thinking through a problem?
3:15:22 Yeah, that's another difficult question.
3:15:23 I suppose it has to do with...
3:15:25 I mean, my mathematical style, my style as a mathematician,
3:15:29 is that I don't really like difficult mathematics.
3:15:37 What I love is simple, clear,
3:15:42 easy-to-understand arguments that prove a surprising result.
3:15:46 That's my favorite situation.
3:15:48 And actually, so the question of whether it's a new
3:15:51 result or not is somehow less important to me.
3:15:53 And so that has to do with this question
3:15:56 of the greats and so on, whoever does it first.
3:15:58 Because I think, for example,
3:15:59 if you prove a new result with a bad argument or complicated argument,
3:16:07 that's great because you proved something new.
3:16:09 But I still want to see the beautiful
3:16:13 simple because that's what I can understand.
3:16:16 Also, I mean, I'm kind of naturally skeptical
3:16:21 about any complicated argument because it might be wrong.
3:16:25 And- ...if I can't really understand it fully,
3:16:28 like every single step all at once in my head,
3:16:32 then I'm just worried maybe it's wrong.
3:16:34 And so these different styles, sometimes mathematicians get involved
3:16:38 with these enormous research projects that involve
3:16:41 huge numbers of working parts and- ...different technology coming together.
3:16:45 I mean, mathematical technology, not physical technology.
3:16:49 And sometimes it actually involves now more and more something
3:16:51 like the Lean programming language where some parts are automated,
3:16:53 so you have this gigantic-- Yeah, yeah, I see.
3:16:56 Well, that's another issue because maybe those things are,
3:16:58 you know, less subject to skepticism when it's validated- ...by Lean.
3:17:02 But I'm thinking about the case
3:17:04 where the arguments are just extremely complicated,
3:17:07 and so I sort of worry whether it's right or not,
3:17:11 whereas you know, I like the simple thing.
3:17:13 And so, so I tend to have often worked on things that are a little bit
3:17:18 off the beaten path from what other people
3:17:21 are working on from that point of view.
3:17:23 Your curiosity draws you towards simplicity.
3:17:25 Yeah.
3:17:25 I wanna work on the things that I can understand and that are s- And luckily,
3:17:30 I've found that I've been able to make
3:17:35 contributions that other people seem to like,
3:17:38 you know, in this way, in this style.
3:17:41 And so I've been kind of fortunate from that point of view.
3:17:44 I mean, my process always, though,
3:17:47 and I've recommended this always to my students,
3:17:51 is just a kind of playful curiosity.
3:17:54 So whenever I have...
3:17:55 whenever there's an idea or a topic, then I just play around with it and change
3:18:03 little things or understand a basic case and then make
3:18:07 it more complicated or press things a little bit
3:18:10 on this side or apply the idea to my favorite example,
3:18:15 you know, that's relevant or, and see what happens,
3:18:18 or you just play around with ideas,
3:18:20 and this often leads to insights that then lead to more methods or more,
3:18:25 you know, then pretty soon you're making progress on the problem.
3:18:29 And so this is basically my method, is I just, you know,
3:18:33 fool around with the ideas until I
3:18:35 can see a path through towards something interesting.
3:18:41 And then prove that, and that's worked extremely well for me.
3:18:45 So I'm pretty pleased with that method.
3:18:48 You do like thought experiments where you anthropomorphize, like you mentioned?
3:18:51 Yeah, yeah.
3:18:52 So this is a basic tool.
3:18:53 I mean, I use this all the time.
3:18:54 You know, you imagine a set-theoretic model,
3:18:57 a model of ZFC as like a place where you're living,
3:19:02 and you might travel to distant lands by forcing,
3:19:04 and this is a kind of metaphor for what's going on.
3:19:07 Of course, you know,
3:19:08 the actual arguments aren't anything like that because there's not land,
3:19:11 and you're not traveling and you're not...
3:19:13 But you allow your mind to visualize that kind of thing,
3:19:15 in the natural real world.
3:19:16 And it helps you to understand, particularly when there's parts of the argument
3:19:19 that are in tension with one another,
3:19:21 then you can imagine that people are fighting or something.
3:19:24 And those kind of metaphors, you know,
3:19:26 or you imagine it in terms of a game-theoretic,
3:19:28 you know, two players trying to win.
3:19:30 So that's kind of tension.
3:19:32 And those kind of metaphorical ways of understanding
3:19:34 a mathematical problem often are extremely helpful in realizing,
3:19:38 aha, the enemy is going to pick this thing to be like that because,
3:19:43 you know, it makes it more continuous or whatever,
3:19:46 and then we should do this other thing in order to...
3:19:49 So it makes you realize mathematical strategies
3:19:53 for finding the answer and proving the theorem that you want to prove because
3:19:57 of the ideas that come out of that anthropomorphization.
3:20:02 What do you think of somebody like Andrew Wiles,
3:20:04 who spent seven years grinding at one
3:20:06 of the hardest problems in the history of mathematics?
3:20:09 And maybe contrasting that a little bit
3:20:11 with somebody who's also brilliant, Terence Tao,
3:20:13 who basically says if he hits a wall,
3:20:16 he just switches to a different problem and he comes back and so on.
3:20:20 So it's less of a focused grind for many
3:20:24 years without any guarantee that you'll get there,
3:20:27 which is what Andrew Wiles went through.
3:20:30 Maybe Grigori Perelman did the same.
3:20:32 I mean, Wiles proved an amazing theorem,
3:20:34 the Fermat's Last Theorem result is incredible.
3:20:36 This is a totally different style than my own practice,
3:20:40 though, of working in isolation.
3:20:42 I mean, for me, mathematics is often a kind of social activity.
3:20:47 I have- I counted, I mean, it's pushing towards a hundred collaborators,
3:20:52 co-authors on various papers and so on.
3:20:55 And, you know, anybody has an idea they want to talk about with me,
3:20:58 if I'm interested in it, then I'm gonna wanna collaborate with them and we might
3:21:02 solve the problem and have a joint paper or whatever.
3:21:05 You wanna have a joint paper?
3:21:06 Let me-- Yeah, exactly.
3:21:07 Let's go.
3:21:09 So my approach to, like,
3:21:10 making mathematical progress tends to involve working with other people
3:21:14 quite a lot rather than just working on my- ...own,
3:21:17 and I enjoy that aspect very much.
3:21:20 So I, personally, I couldn't ever do what Wiles did.
3:21:23 Maybe I'm missing out.
3:21:24 Maybe if I locked myself, you know,
3:21:26 in the bedroom and just worked on whatever, then, uh, I would solve it.
3:21:30 But I tend to think that no, actually,
3:21:32 like being on MathOverflow so much and I've gotten so many ideas,
3:21:38 so many papers have grown out
3:21:40 of the MathOverflow conversations and back and forth.
3:21:42 Someone posts an, you know,
3:21:44 someone posts a question and I post an answer on part of it,
3:21:46 and then someone else has an idea and it turns into a full solution,
3:21:49 and then we have a three-way paper coming out of that.
3:21:52 That's happened many times.
3:21:53 And so for me, it's...
3:21:55 I enjoy this kind of social aspect to it.
3:21:58 And it's not just the social part.
3:22:01 Rather, that's the nature of mathematical investigation as I see it,
3:22:06 is putting forth mathematical ideas to other people and they
3:22:12 respond to it in a way that helps me learn, helps them learn, and I think that's
3:22:17 a very productive way of undertaking mathematics.
3:22:21 I think it's when you work solo on mathematics,
3:22:23 from my outsider perspective, it seems terrifyingly lonely.
3:22:28 And because you're, especially if you do stick to a single problem,
3:22:32 especially if that problem has broken many brilliant mathematicians in the past,
3:22:37 that you're really putting all your chips in.
3:22:40 And just the torment- ...the rollercoaster of day to...
3:22:43 that day.
3:22:44 Because I imagine you have these moments of hopeful break, mini breakthroughs,
3:22:51 and then you have to deal with the occasional realization that, no,
3:22:57 it was not a breakthrough, and that disappointment.
3:23:00 And then you have to go, like, a weekly,
3:23:03 maybe daily disappointment where you hit a wall,
3:23:06 and you have no other person to brainstorm with.
3:23:11 You have no other avenue to pursue.
3:23:14 And it's I don't know.
3:23:17 The mental fortitude it takes to go through that.
3:23:20 But every- Everybody's different.
3:23:21 Some people are recluse and just really find solace in that lone grind.
3:23:28 I have to ask about Grisha Grigori Perelman.
3:23:33 What do you think of him famously
3:23:36 declining the Fields Medal and the Millennial Prize?
3:23:39 So he stated, "I'm not interested in money or fame.
3:23:42 The prize is completely irrelevant to me.
3:23:45 If the proof is correct, then no other recognition is needed." What do
3:23:49 you think of him turning down the prize?
3:23:52 I guess what I think is that mathematics is
3:23:55 full of a lot of different kinds of people.
3:23:58 And my attitude is that, hey, it doesn't matter.
3:24:01 Maybe they have a good math idea,
3:24:02 and so I want to talk to them and interact with them.
3:24:06 And so I think the Perelman case,
3:24:08 you know, is maybe an instance where, you know,
3:24:13 he's such a brilliant mind and he
3:24:17 solved this extremely famous and difficult problem,
3:24:20 and that is a huge achievement.
3:24:23 But he also had these views about, you know, prizes and somehow,
3:24:28 I don't really fully understand why he would turn it down.
3:24:33 I do think I have a similar thing,
3:24:35 just observing Olympic athletes that, in many cases, don't get paid very much,
3:24:40 and they nevertheless dedicate their entire lives for the pursuit...
3:24:42 ...of the gold medal.
3:24:43 I think his case is a reminder that some of the greatest mathematicians,
3:24:47 some of the greatest scientists and human beings do the thing they do,
3:24:51 take on these problems for the love of it,
3:24:56 not for the prizes or the money or any of that.
3:24:59 Now, as you're saying, if the money comes, you could use it for stuff.
3:25:03 If the prizes come, and the fame, and so on, that might be useful.
3:25:06 But the reason fundamentally the greats do it is because of the art itself.
3:25:13 Sure, I totally agree with that.
3:25:15 I mean, I share the view.
3:25:16 That's, you know, that's why I'm a mathematician is because I find
3:25:22 the question so compelling and I've spent
3:25:24 my whole life thinking about these problems.
3:25:27 But, you know, if I won an award-- Yeah, it's great.
3:25:33 It's great.
3:25:33 I mean, I'm pretty sure you don't contribute
3:25:35 to MathOverflow for the wealth, and the power.
3:25:41 That you gain.
3:25:43 I mean, it's genuine curiosity.
3:25:46 Well, you asked who the greatest mathematician is, and of course,
3:25:50 if we want to be truly objective about it,
3:25:52 we would need a kind of an objective criteria.
3:25:55 Criteria, yeah.
3:25:55 About how to evaluate the relative
3:25:57 strength and the reputation of various mathematicians.
3:26:00 And so, of course, we should use MathOverflow score.
3:26:03 ...Because-- That you're definitively.
3:26:07 I mean, nobody's objectively the greatest mathematician of all time.
3:26:11 Yes, that's true.
3:26:11 I've also argued that tenure and promotion decisions should be based...
3:26:15 Based on MathOverflow.
3:26:16 ...Yeah.
3:26:17 So my daughter introduced me to her boyfriend.
3:26:20 and told me that she had a boyfriend.
3:26:24 And I, um-- asked him what his MathOverflow...
3:26:26 I wanted to know, first of all,
3:26:28 what is his chess rating, and secondly, what is his MathOverflow score?
3:26:34 Oh, man.
3:26:34 Well, that's the only way to judge a person, I think.
3:26:37 That's, I think, objectively correct.
3:26:39 Yeah.
3:26:41 I mean, since you bring up chess, I've got to ask you about infinite chess.
3:26:44 I can't let you go.
3:26:45 You've, I mean, you worked on a million things,
3:26:47 but infinite chess is one of them.
3:26:49 Somebody asked on MathOverflow, the mathematical definition of chess.
3:26:54 Right.
3:26:54 So can we talk about the math of chess and the math of infinite chess?
3:26:58 What is infinite chess?
3:26:59 Oh, yeah, absolutely.
3:27:00 Infinite chess is fantastic.
3:27:02 Chess ordinarily is played on this tiny, tiny board.
3:27:04 It's an eight by eight board, right?
3:27:06 So when you play chess, normally it's on the eight by eight board.
3:27:09 But we want to play infinite chess, so on the, on the integer board.
3:27:15 It's infinite in all four directions, you know,
3:27:19 but it still has the chessboard pattern,
3:27:21 and maybe there's pieces on this board, maybe infinitely many pieces we allow.
3:27:25 But one difference from finite ordinary chess, in infinite chess,
3:27:29 we don't play from a standard starting position.
3:27:34 Rather, you...
3:27:36 The interesting situation is that you present a position where there's a lot
3:27:40 of pieces already on the board in a complicated way, and you say,
3:27:44 "What would it be like to start from this position or from that one?" You know,
3:27:47 and we want to produce positions that have interesting features,
3:27:51 meaning mathematically interesting features.
3:27:53 And so I can tell you for example,
3:27:57 probably a lot of people are familiar with, say,
3:28:03 the mate in two genre of chess problem.
3:28:07 You know, you have a chess problem and it's white to mate in two,
3:28:10 which means that white is going to make two moves,
3:28:12 but the second move is going to be a checkmate.
3:28:15 Or maybe mate in three or mate in five or whatever.
3:28:17 We can have mate in N positions for any N.
3:28:21 I mean, in infinite chess,
3:28:23 you can create a position which is not mate in N for any N,
3:28:31 but white has a winning strategy that will win infinitely many moves.
3:28:38 So in other words, let me say it again.
3:28:41 There are positions in infinite chess that white can definitely win.
3:28:45 Infinitely many moves, white is going to make checkmate.
3:28:49 But there's no particular N for which white can guarantee to win in N moves.
3:28:56 There's no N?
3:28:57 No N.
3:28:57 So it's not mate in N for any N, but it's a white win, infinitely many.
3:29:02 The way to think about it is,
3:29:05 white is going to win, but black controls how long it takes.
3:29:09 Ah, got it.
3:29:11 But it's doomed.
3:29:11 Black can say, "Well, I know you're gonna win, but this time it's gonna...
3:29:14 you're gonna take a thousand moves at least."
3:29:17 Or maybe in a different way of playing, black can say, "Well,
3:29:19 I know you're gonna win, but this time you're gonna have to take
3:29:22 a million moves." For any number, black can say that.
3:29:24 So it's these really interesting positions.
3:29:26 There's a position in my first infinite chess paper.
3:29:30 So it's black to play in this position,
3:29:33 and if black doesn't move that rook there,
3:29:38 then white is gonna checkmate pretty quickly.
3:29:41 By the way, can we describe the rules of infinite chess?
3:29:45 Right.
3:29:45 So the rules of infinite chess are there's just the ordinary pieces,
3:29:48 and they move on this infinite board, which is just a chessboard,
3:29:51 but extended in all directions- infinitely, with no edge.
3:29:55 So there's no boundary.
3:29:57 But the pieces move just like you'd expect.
3:29:59 So the knights move just the same and the rooks move,
3:30:02 you know, on the ranks and files, and the bishops move on the same
3:30:05 color diagonals and, just like you would expect,
3:30:07 except they can move as far as they want,
3:30:10 you know, if there's no intervening piece in the way.
3:30:13 The one thing is that, okay, so the white pawns always move upwards
3:30:18 and the black pawns always move downwards,
3:30:20 but when they're capturing, the pawns, you know, capture on the diagonal.
3:30:24 So I think the piece movement is pretty clear.
3:30:27 There's a couple of differences that you
3:30:30 have to pay attention to from ordinary chess.
3:30:32 For example, there's this threefold repetition rule in ordinary chess,
3:30:37 but we just get rid of this for infinite chess because,
3:30:40 of course, threefold repetition is just a proxy for infinite play.
3:30:44 The real rule is infinite play is a draw, not threefold repetition is a draw.
3:30:49 That's just a kind of convenient approximation
3:30:51 to the, what I view as the actual rule, which is that infinite play is a draw.
3:30:56 So the only way to win is to make
3:30:59 checkmate on the board at a finite stage of play.
3:31:01 And if you play infinitely, you haven't done that, and so it's a draw.
3:31:05 And the pawns can't be converted
3:31:06 into-- And there's no promotion 'cause there's no edge.
3:31:09 Right, exactly.
3:31:09 And this position that we were just talking
3:31:12 about is a position with game value omega,
3:31:13 which means that because it has an ordinal value, white is going to win,
3:31:19 but black can play as though counting down from omega.
3:31:23 What is the nature of counting down from omega?
3:31:25 If you're black and you need to count down from omega,
3:31:29 then you have to say a finite number, and then after that, it's gonna be at most
3:31:35 that many numbers afterwards to count down, right?
3:31:38 So the nature of counting down from omega is
3:31:40 that you take this giant step on the first count,
3:31:44 and then after that, you subtract one each time.
3:31:47 You can't subtract one from omega because that's not an ordinal.
3:31:50 So if you count down from omega, you have to go to some finite number,
3:31:54 and then if you just subtract one each time,
3:31:56 then that's how many more moves you get.
3:31:58 So that's the sense in which black can make it take as long
3:32:01 as he wants because he can pick his initial number to be whatever he wants.
3:32:05 By the way, I just noticed that you were citing a MathOverflow question,
3:32:08 which is really cool.
3:32:10 That's right, yeah.
3:32:11 My interest in infinite chess was born
3:32:13 on MathOverflow 'cause someone asked this question.
3:32:16 Noam Elkies asked this question.
3:32:17 That's so cool to see a MathOverflow citation in an arXiv paper.
3:32:21 That's cool.
3:32:23 How do you construct the position-- Right- the position that satisfies this?
3:32:29 Is there an algorithm for construction?
3:32:32 No.
3:32:32 This is an act of mathematical creativity, really, to come up with...
3:32:35 I had a co-author, my co-author, Corey Evans.
3:32:38 He's a US national master chess player.
3:32:42 A very strong chess player.
3:32:45 He's also a philosophy professor of law.
3:32:50 Your collaborations are wonderful.
3:32:52 That's great.
3:32:53 So I met him because he was a grad student
3:32:55 at CUNY where I was at the time in New York.
3:32:57 And also he was my son's chess coach for when
3:33:00 my son was playing chess competitively in elementary school.
3:33:05 Then Corey was the coach.
3:33:06 And so we knew him that way.
3:33:09 And that was right around the time
3:33:10 when I was getting interested in infinite chess,
3:33:13 and I knew I needed a chess-knowledgeable partner.
3:33:18 And so Corey was invaluable for the paper because the proofs
3:33:26 in infinite chess are extremely finicky because you create these positions,
3:33:32 but the details of the argument have
3:33:34 to do with kind of chess reasoning, you know?
3:33:38 My chess reasoning wasn't quite up to it because I would create the positions.
3:33:45 Almost all the positions are ones that I made,
3:33:48 but this is like after many generations,
3:33:50 of being corrected by Corey because Corey would come and say,
3:33:53 "Hey, you know, this pawn is hanging, and it breaks your argument,
3:33:57 and" "or, or, you know, this bishop can leak out" "of the cage," or whatever.
3:34:02 And so...
3:34:03 the process was I knew kind of in terms
3:34:07 of these ordinals what we needed to create with the position,
3:34:11 and I would struggle to do it and create
3:34:13 something that sort of had the features that I wanted,
3:34:16 and then I would show it to Corey and he would say,
3:34:18 "Look, it doesn't work because of this and that," and so on.
3:34:20 And so this kind of back and forth was extremely helpful to me,
3:34:24 and eventually we, you know, converged on arguments that were correct.
3:34:30 So it's...
3:34:30 yeah, it's quite interesting.
3:34:32 Also, maybe another thing to say is the follow-up paper to this one
3:34:36 was a three-way paper with also Corey and myself and my PhD student,
3:34:42 Norman Perlmutter, in which we improved the bound.
3:34:44 So we were aiming to produce more and more
3:34:46 chess positions with higher and higher ordinal values.
3:34:48 chess positions with higher and higher ordinal values.
3:34:52 So the initial position was value omega,
3:34:54 and then we made omega-squared and omega-cubed in the first paper,
3:34:56 omega-squared and omega-cubed in the first paper,
3:34:57 and then in this three-way collaboration, we made omega to the 4th.
3:35:00 then in this three-way collaboration, we made omega to the 4th.
3:35:04 The title of the paper:
3:35:05 The Position in Infinite Chess with Game Value Omega to the 4th.
3:35:09 Right.
3:35:10 And so, at the time, this was the best-known result,
3:35:13 the sort of state of the art,
3:35:15 but since that time, it's been improved now dramatically.
3:35:18 And, in fact, we know now that every countable ordinal
3:35:22 arises as the game value of a position in infinite chess,
3:35:26 so it's a fantastic result.
3:35:29 Before I forget, let me ask about your views on AI
3:35:34 and LLMs that are getting better and better in mathematics.
3:35:38 We've spoken about collaborators, and you have so many collaborators.
3:35:41 Do you see AI as a potential great collaborator to you as a mathematician,
3:35:48 and what do you think the future role of those- kinds of AI systems is?
3:35:52 I guess I would draw a distinction between what
3:35:55 we have currently and what might come in future years.
3:35:59 I've played around with it and I've tried experimenting,
3:36:04 but I haven't found it helpful at all, basically zero.
3:36:08 It's not helpful to me.
3:36:10 helpful to me.
3:36:11 And, you know, I've used various systems and so on, the paid models and so on.
3:36:17 My typical experience is interacting with AI on a mathematical question
3:36:23 is that it gives me garbage answers that are not mathematically correct.
3:36:30 And so I find that not helpful and also frustrating.
3:36:36 Like, if I was interacting with a person,
3:36:39 the frustrating thing is when you have to argue about whether or not,
3:36:43 you know, the argument that they gave you is right,
3:36:45 and you point out exactly the error,
3:36:47 in the AI saying, "Oh, it's totally fine." And, you know,
3:36:53 if I were having such an experience with a person,
3:36:55 I would simply refuse to talk to that person again.
3:36:58 But okay, one has to overlook these kind of flaws.
3:37:02 And so I tend to be a kind of skeptic about the value
3:37:07 of the current AI systems as far as mathematical reasoning is concerned.
3:37:12 It seems not reliable.
3:37:14 Okay, but I know for a fact that many,
3:37:18 that there are several prominent mathematicians
3:37:20 whom I have enormous respect for, who
3:37:23 are saying that they are using it in a way that's helpful,
3:37:28 and I'm often very surprised to hear that based on my own experience,
3:37:33 which is quite the opposite.
3:37:35 And so maybe my process isn't any good,
3:37:39 although, you know, I use it for other things,
3:37:41 like, you know, for programming things or for image generation and so on.
3:37:47 It's amazingly powerful and helpful.
3:37:50 But for mathematical arguments, I haven't found it helpful,
3:37:54 and maybe I'm not interacting with it in the right way.
3:37:59 Yet, or it could be.
3:38:00 And so maybe I just need to improve my skill.
3:38:03 But also maybe I wonder, like, these examples that are provided by other people
3:38:11 maybe involved quite a huge amount of interaction,
3:38:14 and so I wonder if maybe the mathematical
3:38:17 ideas are really coming from the person,
3:38:19 you know, these great mathematicians who are doing it rather than the AI.
3:38:23 And so so I tend to be kind of skeptical.
3:38:27 But also, I'm skeptical for another reason, and that is because of the nature
3:38:35 of the large language model approach to AI doing mathematics.
3:38:41 Um, I recognize that the AI is trying to give me an argument
3:38:49 that sounds like a proof rather than an argument that is a proof.
3:38:55 The motivation is misplaced.
3:38:57 And so I worry that this is a very dangerous source
3:39:02 of error because it often happens in mathematics that, I mean,
3:39:07 if I think back to when I was an undergrad, you know, here at Caltech,
3:39:11 and I was a math major eventually,
3:39:14 and at that time, LaTeX was a pretty new thing, and I was learning LaTeX,
3:39:19 and so I was typing up my homeworks in LaTeX and they looked beautiful.
3:39:24 Actually, they looked like garbage.
3:39:26 From my current standards, I'm sure it was terrible.
3:39:29 Except at the time, you know, I didn't know anything.
3:39:32 I was an undergrad, and LaTeX was sort of unheard of.
3:39:37 And so I was producing these beautifully typeset,
3:39:39 you know, problem sets, solutions, and so on.
3:39:43 And I would print it up and submit it
3:39:45 and so on, and the grades would come back, terrible grades.
3:39:48 And I realized what was happening is that the, you know,
3:39:54 the copy was so beautiful mathematically typeset in this way.
3:39:59 It looked like the kind of mathematics you find in a book, you know?
3:40:02 Because basically that's the only time you saw that kind
3:40:05 of mathematical typesetting was in a, in a professional,
3:40:08 you know, published book.
3:40:10 And those me- that mathematics was almost always correct-...
3:40:14 in a book, right?
3:40:15 And so I had somehow, you know, lost my-...
3:40:21 because it was so beautiful,
3:40:22 and I'm used to only seeing that kind of type setting when an argument was,
3:40:26 you know, totally right-...
3:40:28 I wasn't critical enough,
3:40:29 and making these sort of bonehead mistakes in the proofs.
3:40:32 And, and so, okay, so I, I corrected this, of course.
3:40:36 But this kind of effect is very much real with the modern LLM system.
3:40:39 That's right.
3:40:40 And so I think that the chat programs and so
3:40:44 on are producing these arguments that look really, they look like a...
3:40:48 that's what they're striving to do, that it's what they're designed to do.
3:40:53 They're not designed to make a logically correct argument.
3:40:56 They're designed to make something that looks like a logically correct argument.
3:41:00 And it's easy to get fooled if you're not skeptical.
3:41:03 And so that's why I worry a bit
3:41:05 when people rely on AI for mathematical arguments.
3:41:10 I mean, using...
3:41:11 tying them to Lean in the formal proof,
3:41:13 um verification systems and so on, this is a totally different way of operating.
3:41:18 But for the sort of ordinary person sitting down
3:41:20 and using chat to come up with a mathematical argument,
3:41:24 I think it's a dangerous source of error
3:41:26 if you're not especially attuned to this very
3:41:30 issue that the AI is going to produce
3:41:33 something that's not grounded in mathematical understanding,
3:41:36 but rather something that is trying to look
3:41:40 like something that is grounded in mathematical understanding.
3:41:42 And those are not the same thing at all.
3:41:45 And furthermore, I really wonder if one can make
3:41:48 a kind of system for producing genuine mathematical insight
3:41:52 that isn't based in what I would view
3:41:55 as mathematical understanding as opposed to the text generation systems.
3:42:01 The methods that are used, you know, they don't seem close enough grounded
3:42:06 in understanding of the underlying mathematical concepts,
3:42:09 but rather grounded in the way words appear
3:42:13 on a page in arguments about those concepts, which are not the same.
3:42:18 So there's a couple of things to say there.
3:42:19 So one, I think there is a real skill in providing
3:42:23 the LLM system with enough information to be a good collaborator.
3:42:29 Because you really are dealing with a different...
3:42:30 It's not a human being.
3:42:32 You really have to load in everything you possibly can from your body of work,
3:42:37 from the way you're thinking, and that's a real skill.
3:42:40 And then the other thing is, you know,
3:42:42 for me, if it's at all anything like programming,
3:42:45 because I have a lot of colleagues and friends
3:42:48 who are programmers who kind of feel similarly to you.
3:42:53 And for me, I've gotten better and better and better at giving
3:42:58 as much information as possible to the systems in a really structured way,
3:43:03 maybe because I just like natural language as a way to express my thinking.
3:43:08 And then the benefit comes from the inspiration
3:43:12 that the system can provide by its ability to know a lot of things and make
3:43:19 connections between disparate fields and between disparate concepts.
3:43:22 And in that way, it provides not
3:43:26 the answer but the inspiration, the handholding,
3:43:29 the camaraderie that helps me get to the answer,
3:43:33 because it does know a lot more than me.
3:43:38 Know, like knowledge.
3:43:39 And if you give it a lot of information and ask the broader questions,
3:43:45 it can make some really beautiful connections.
3:43:47 But I do find that I have to be extremely patient, like you said.
3:43:53 The, the amount of times I'll do something dumb where I feel like, "Uh,
3:43:58 you don't get this at all,
3:43:59 do you?" That's a source of a lot of frustration for us humans.
3:44:03 Like, "This...
3:44:04 Wait, this thing doesn't understand at all." If you can have the patience
3:44:07 to look past that, there might be
3:44:11 some brilliant little insights that it can provide.
3:44:16 Right.
3:44:16 At least for me in the realm of programming.
3:44:19 I should say programming, there's just so much training data.
3:44:22 There's so much there.
3:44:23 And at least I see the light at the end
3:44:29 of the tunnel of promising possibilities of it being a good collaborator,
3:44:32 versus like something that gives you really true genius-level insights.
3:44:39 Right.
3:44:39 It's probably true.
3:44:40 Uh, I also find it likely that a lot of the...
3:44:44 As far as mathematical training data is concerned,
3:44:48 I just have to assume that math overflow answers are part of the training data.
3:44:52 Yes, of course.
3:44:53 It's so...
3:44:55 And you're...
3:44:56 So-- I mean, you're talking to yourself, essentially.
3:44:58 Yeah, maybe.
3:45:01 Sorry for the ridiculously big question,
3:45:03 but what idea in mathematics is most beautiful to you?
3:45:08 We've talked about so many.
3:45:11 The most beautiful idea in mathematics is the transfinite ordinals.
3:45:15 These were the number system invented
3:45:18 by Georg Cantor about counting beyond infinity,
3:45:21 just the idea of counting beyond infinity.
3:45:24 I mean, you count through the ordinary numbers,
3:45:27 the natural numbers: zero, one, two, three, and so on.
3:45:30 And then you're not done because after that comes omega,
3:45:34 and then omega plus one, and omega plus two, and so on.
3:45:38 And you can always add one.
3:45:40 And so of course after you count through
3:45:43 all those numbers of the form omega plus N,
3:45:45 then you get to omega plus omega, the first number after all those.
3:45:50 And then comes omega plus omega plus one, and so on.
3:45:54 You can always add one.
3:45:55 And so you can just keep counting through the ordinals.
3:45:58 It never ends.
3:45:59 Eventually, you get to omega times three, omega times four, and so on.
3:46:03 And then the limit of those numbers,
3:46:05 the first number that comes after all those numbers will be omega squared.
3:46:10 And this one is the first compound limit ordinal because it's a limit ordinal,
3:46:15 is one of these numbers, an ordinal that doesn't have an immediate predecessor
3:46:19 like omega and omega times two, omega times three.
3:46:22 Those are all limit ordinals.
3:46:24 But omega squared is a limit ordinal,
3:46:27 but it's also a limit of limit ordinals because the omega times three,
3:46:32 omega times four, and so on, those are
3:46:34 all limit ordinals that limit up to omega squared.
3:46:36 And then, of course, you form omega squared plus one,
3:46:39 and then omega squared plus two, and so on, and it never stops.
3:46:43 And it's just absolutely beautiful and amazing, and furthermore,
3:46:48 forms the foundation for these transfinite
3:46:51 recursive constructions that came later.
3:46:53 I mean starting with the Cantor-Bendixson theorem that I mentioned.
3:47:00 And continuing with, the construction of the V
3:47:06 hierarchy and Gödel's constructible universe is built this way,
3:47:10 and Zermelo's proof of the well-order principle using
3:47:14 the axiom of choice is a transfinite recursive construction.
3:47:17 And, and so the idea of just counting past infinity is so simple and elegant,
3:47:24 and has led to so much fascinating mathematics.
3:47:28 Yeah, the infinity's not the end.
3:47:29 And what about philosophy?
3:47:30 What to you is the most beautiful idea in philosophy?
3:47:35 So I have a foot in both fields: philosophy and mathematics,
3:47:39 and in some contexts I seem to be required
3:47:44 to choose whether I'm a mathematician or a philosopher.
3:47:47 I mean, my training is in mathematics.
3:47:49 My PhD, all my degrees are mathematics.
3:47:51 But somehow I turned myself into a philosopher over the years
3:47:55 because my mathematical work was engaging with these philosophical issues.
3:47:59 And so when I went...
3:48:01 In New York, I had appointments first in mathematics only,
3:48:04 but then eventually I was also joining
3:48:07 the philosophy faculty at the graduate center.
3:48:10 And when I went to Oxford for the first time,
3:48:12 my main appointment was in philosophy, and that's also true now at Notre Dame
3:48:17 although I'm also a concurrent professor in mathematics.
3:48:20 And I have math PhD students still and philosophy PhD students.
3:48:24 And so I don't really care to decide
3:48:28 whether I'm a mathematician or a philosopher.
3:48:32 And my work is engaging with mathematics
3:48:34 and with philosophical issues in mathematics and with plain philosophy,
3:48:39 and there's this ample region between these re- between these two subjects.
3:48:44 So it's not necessary to choose.
3:48:46 I remember when I first went to Oxford and I told
3:48:49 my daughter that I was going to become professor of philosophy in Oxford,
3:48:54 And she looked at me plaintively and said,
3:48:57 "Uh, but, but Papa, you're not a philosopher.
3:49:01 Because in her mind, you know, her father was the mathematician and her mother
3:49:06 was the philosopher 'cause my wife, Barbara, is a philosopher.
3:49:10 Now also at Notre Dame.
3:49:13 We're together there.
3:49:15 And okay, but fortunately, I don't really have to choose between them.
3:49:21 So you ask about the most beautiful idea in philosophy,
3:49:23 and I would have to say that I
3:49:26 think it's the distinction between truth and proof,
3:49:30 the one that we discussed already.
3:49:33 Um, it's, it's so profound and gets
3:49:41 at the heart of so many philosophical issues.
3:49:44 I mean, of course this is a distinction
3:49:46 that's maybe born in mathematics or mathematical logic,
3:49:49 but that's already philosophical to a degree,
3:49:53 and it's ph- you know, fundamentally a philosophical distinction.
3:49:58 The truth is about the, nature of the world and the way things are.
3:50:05 It's about objective reality in a sense.
3:50:08 Whereas proof is about our understanding of the world and about how
3:50:13 we come to know the things that we know about the world.
3:50:18 And so to focus on proof is to focus
3:50:22 on the interaction that we have with the objective reality.
3:50:27 And, okay, I'm talking about the reality of mathematics,
3:50:30 not the physical world, because, as I said, I live in the Platonic
3:50:33 realm and I interact with mathematical reality,
3:50:36 and so proof is about the interaction and how we come
3:50:40 to know the facts that are true in this mathematical reality,
3:50:45 whereas truth is about what's really the case,
3:50:48 sort of apart from our knowledge of it.
3:50:52 And this is, I think, such a core way that I have of understanding
3:50:59 the world and the nature of logic and reasonings.
3:51:03 And the gap between the two is full of fascinating mysteries,
3:51:07 both in the Platonic realm,
3:51:09 but also in the physics realm, and I would even say in the human psychology,
3:51:15 sociology, politics, geopolitics, all of it,
3:51:19 if you think about proof more generally,
3:51:21 which is the process of discovery versus the truth itself.
3:51:27 And that's our journey whatever field we're in.
3:51:32 Well, I for one, am grateful for how marvelous of a philosopher,
3:51:39 mathematician, and human being you are.
3:51:41 It's truly an honor to speak with you today.
3:51:43 Well, thank you so much.
3:51:44 It's such a pleasure to be here, and thank you for inviting me.
3:51:47 Thanks for listening to this conversation with Joel David Hamkins.
3:51:51 To support this podcast, please check out our sponsors in the description
3:51:54 where you can also find links to contact me,
3:51:57 ask questions, get feedback, and so on.
3:52:00 Thank you for listening.
3:52:02 As always, happy New Year.
3:52:05 I love you all.