What was Euclid really doing? | Guest video by Ben Syversen

0:00 [Submit subtitle corrections at criblate.com] This is the fifth and final

0:01 in a series of guest videos that I've been putting up on the channel.

0:04 The context, in case any of you missed the first one,

0:07 is that I have been away on leave this summer and I

0:09 decided to use the Patreon funds that were coming into the channel

0:12 during that time to commission a few guest videos from creators

0:15 who I am very excited to share with this particular audience.

0:19 By the way, I am back to making my own videos now,

0:21 currently working on a trilogy about the Laplace Transform,

0:24 the first of which is available as an early view for channel supporters channel.

0:27 Just, you know, throwing that out there.

0:30 This last guest video is probably

0:31 the most different from the channel's usual vibe

0:34 and yet I'm as confident as ever

0:36 that viewers of this channel will thoroughly enjoy it.

0:38 It comes from Ben Syverson,

0:39 who is a very talented and I think underappreciated creator who

0:43 makes these great mini documentaries about the history of math and science.

0:47 The content here changed the way I

0:49 think about Euclid and the history of geometry,

0:51 and perhaps it will for you as well.

0:54 This is the most influential math book in human history.

0:58 Euclid's elements.

1:00 For 2,000 years, it was the ultimate authority for absolute mathematical truth,

1:06 an enduring symbol of exact reasoning from the ancient Greeks.

1:10 But the Greeks didn't approach math the way we do today.

1:14 Their methods fly in the face of some

1:16 of the most important rules of modern practice.

1:18 But yet the approach that Euclid used in the Elements was

1:22 the most successful way to discover mathematical truth until the 17th century.

1:29 Ruler and Compass Constructions.

1:31 But, how could physical tools like these actually

1:36 be a reliable way to do abstract math?

1:40 Understanding what the Greeks were really doing

1:42 starts with the question that anybody who's ever had to use a ruler and compass

1:46 in geometry class has probably asked, what's the point?

1:50 What are these constructions actually for?

1:53 Most people would probably assume that the ruler and compass

1:56 are basically a way of drawing nicer looking shapes.

1:59 Like they're a quaint way of doing it if you don't have a computer.

2:02 And while that's not entirely wrong to the ancient Greeks,

2:06 it's far from the whole story.

2:09 To them, diagrams weren't just incidental schematic illustrations.

2:13 They were part of the reasoning of the proof itself.

2:17 This is not how we do math today.

2:19 Today, every single logical assumption needs to be explicitly stated,

2:23 like a line of code, and a proof isn't considered valid

2:27 until it's completely independent from any diagram.

2:29 But in Euclid's Elements, the compilation of Greek math from around 300 B.C.

2:34 things are different.

2:36 In Euclid's very first proof,

2:38 he makes what modern mathematicians would consider to be an unstated assumption,

2:42 basically a mistake.

2:44 To try to spot what Euclid missed in proposition one of the Elements,

2:48 you have to also know what his stated assumptions were.

2:51 Before he writes any proofs, he lays out some definitions.

2:54 Five logical assumptions called common notions,

2:57 and five assumptions about geometry called postulates.

3:00 Whenever one of these is used in a proof, it'll show up on screen.

3:03 We can read this proposition as a recipe for constructing

3:07 an equilateral triangle using a ruler and a compass.

3:10 You start with a line segment ab.

3:12 When you draw these two circles centered

3:14 at A and B that each have AB as a radius,

3:18 like this, then you get this intersection point.

3:21 Connecting the dots gives you another radius of each circle.

3:24 And of course, we know that the radius

3:26 of a circle is the same length everywhere.

3:29 So this is equal to this and this is equal to this, since

3:32 two things that are equal to the same thing are also equal.

3:35 Euclid proves that he's made an equilateral triangle.

3:40 Did you catch the unstated assumption?

3:42 You never proved that the two circles overlap each other.

3:45 This should have been some other axiom.

3:47 There should have been something else that said how we

3:49 know when the two circles overlap and when they don't.

3:52 But there's another way of looking at this.

3:55 Rather than making a mistake by relying on the diagram,

3:58 the Greeks were arguably using this series of steps

4:02 to build the equilateral triangle as part of the proof itself.

4:06 This is the perspective of Viktor Blåsjö,

4:09 who hosts the excellent podcast Opinionated History of Mathematics,

4:12 which features an 18 episode season about

4:15 Euclid's Elements and the history of geometry.

4:18 When people say there is a gap in the proof, you know,

4:20 I think it is like an anachronistic way of thinking about it,

4:23 that the existence needs to be ensured by a new axiom.

4:26 That's the way modern, you know, logic nerds think it's supposed to be right.

4:31 But in the spirit of the times,

4:33 we should start by thinking of mathematical proofs as something

4:37 that arose alongside the Greek tradition of antagonistic debate.

4:41 You have somebody trying to prove a claim and they're

4:44 confronted by a skeptic who wants to prove them wrong.

4:46 The prover is saying, now carry out these steps.

4:49 The skeptic is then asked to perform these steps and had

4:52 literally just drawn this in front of their very own eyes.

4:55 And then the skeptic is supposed to say, oh, no, I don't think they intersect.

4:59 There's no credibility.

5:00 You know, you just threw it, it's right there.

5:02 Anybody can see, you know, you just look, make yourself look ridiculous.

5:05 In the modern mathematics, you think of the skeptic as if it was

5:08 like a logic machine who is just like crunching axioms,

5:12 like a machine doing calculations, you know, and then, oh,

5:15 there's no axiom here, error, you know, but that's not this.

5:18 The context we must visualize, you know,

5:20 we must think of the, the degrees we're doing this.

5:23 If the skeptic wants to say, I think maybe they don't intersect,

5:27 then the burden is on him to give

5:28 a credible way that you could doubt that these are.

5:30 But if you literally just drew it in front

5:33 of you and it's obviously cutting right through it,

5:35 there's nothing you can say that has any credibility

5:38 to try to get around the fact that obviously they intersect,

5:41 you know, like any normal person would accept.

5:43 By following the steps in the elements,

5:45 our construction undeniably has two circles that intersect.

5:49 It's impossible to create this construction otherwise.

5:51 But how do we decide what's obvious in a diagram and what's not?

5:55 I mean, we can't just look at this diagram and say

5:58 the triangle is obviously equilateral because that's not a proof at all.

6:02 So why is it that we can use

6:04 the picture to know for sure that the circles intersect.

6:06 That is an inexact property because the drawing doesn't

6:09 need to be exact for the two to cross.

6:11 Obviously, however, what you want to prove

6:13 is that this is an equilateral triangle,

6:16 that all the three sides are exactly the same.

6:18 Well, that's different.

6:19 Well, maybe one of them is 99% the length of the other.

6:23 I don't know.

6:24 That's not self evident.

6:25 Obviously that's something that needs to be established with rational argument.

6:29 And like Euclid does.

6:30 This is a key distinction for this interpretation of Greek geometry,

6:34 which is going to come up again later.

6:37 Because Euclid never establishes exact

6:40 properties like equality based on diagrams,

6:42 but he does allow diagrams to be used for non exact or topological properties.

6:48 Whether a point is inside or outside of a figure

6:51 or the order in which points appear along a line.

6:54 This approach to geometry allows the written proof and the diagram

6:58 to go hand in hand to share the work.

7:01 The parts of the proof that were

7:03 impractical to demonstrate verbally are instead demonstrated diagrammatically.

7:08 So these diagrams, they're not just

7:10 a supplemental illustration to an otherwise logical argument.

7:14 They're actually part of the proof itself, which a skeptic would need to refute.

7:21 But for this to work,

7:23 we can't just draw any picture and start doing geometry with it.

7:26 Our diagrams need to follow rules.

7:29 So for every object that's specified in the elements,

7:32 its construction is given by a precise set of steps.

7:36 If the book says, let the equilateral triangle have been constructed,

7:40 then it's referring you back to Proposition 1,

7:43 where the steps for constructing an equilateral triangle are given.

7:48 From this point of view, an equilateral triangle isn't some perfect entity

7:53 that floats in the eternal Platonic realm.

7:56 It doesn't exist at all until you

7:58 follow the steps to make an equilateral triangle,

8:01 almost like the output from some lines of code.

8:08 But given that Greek mathematicians didn't write about their thoughts

8:11 and feelings and were making inferences from their technical work,

8:15 this is kind of a lot of inference

8:17 to take away from one little proposition about equilateral triangles.

8:20 So let's take a look at what Euclid does next,

8:23 because it's about to become very clear that there's more going on.

8:28 If the point of the ruler and compass was just to make accurate diagrams,

8:32 then the very next proposition wouldn't make any sense.

8:36 Proposition 2.

8:37 To place a straight line equal to a given

8:40 straight line at a given point at an extremity,

8:42 somebody has drawn a line segment on a piece of paper,

8:45 and now you want to draw an equally

8:47 long line segment somewhere else on the paper.

8:49 Euclid accomplishes this by a very elaborate construction.

8:54 That's Proposition one.

8:55 It involves drawing several circles and an equilateral triangle.

8:59 It's very elegantly indeed.

9:02 This leads exactly to what you need.

9:04 The given segment has been reproduced in a new position.

9:06 With exact mathematical precision, you can prove everything you know.

9:09 The equilateral triangle,

9:10 the radii and the circle are equal as all kinds of logical steps,

9:15 showing that this is correct.

9:17 This is something that I can literally do by picking

9:19 up this compass and putting it down in another spot.

9:22 So Euclid acts as if this is not possible.

9:26 One might say that Euclid behaves as if his compass is collapsible.

9:31 The compass stays at a fixed opening while drawing a particular circle,

9:35 but as soon as it's lifted from the paper, it collapses or closes up.

9:39 Of course, there are no collapsible compasses.

9:41 It's not a real thing.

9:43 Thus, AL is also equal to BC.

9:46 the moral of this is that this straight line here,

9:48 which was the one that I started with, is equal to this straight line here.

9:51 It's all very Neat.

9:53 It's also very weird, isn't it?

9:55 Seems totally out of touch with reality.

9:57 If a craftsman or an engineer or an architect would need to transfer a length,

10:03 surely they would not use Euclid's absolutely baroque procedure.

10:10 So what are we doing here?

10:12 If we followed these steps every time we wanted to copy a length,

10:15 even the simplest diagram would be incredibly complicated to make.

10:19 But if Euclid's constructions are about serving a theoretical purpose,

10:23 then things start to come into focus.

10:26 Proposition 2 becomes another verified operation

10:28 that you can add to your toolkit.

10:31 So each construction is like a little module or a little subroutine.

10:35 It's a bit of proof that signs off on the fact that this object

10:39 can legitimately be constructed using nothing

10:41 but the initial postulates and common notions.

10:45 Euclid's version of this construction works perfectly

10:47 for any situation involving copying a length.

10:50 So it's not saying that you have to do this every time,

10:53 but whenever we do want to copy a length,

10:55 we can invoke Proposition 2 to show that it is,

10:58 in fact constructible from the postulates.

11:00 But why did Euclid need to be this pedantic?

11:04 What was it about the Greeks that made them

11:06 so fixated on proving things in the first place?

11:09 Civilizations like the Egyptians and the Mesopotamians had been

11:13 doing math in service of things like surveying land,

11:17 collecting taxes, and constructing the pyramids for centuries,

11:20 but they never wrote proofs.

11:23 The first civilization to write proofs were the Greeks,

11:26 with the first proof coming around 600 BC.

11:30 For the Greeks, it wasn't enough for math to just be useful.

11:33 They wanted geometry to address deeply philosophical questions,

11:37 because at the time, philosophy was where the action was.

11:41 And the Greeks had a culture of antagonistic debate,

11:44 where the very nature of absolute truth was hotly contested.

11:47 While there were people arguing for all kinds of wild ideas,

11:51 two main camps emerged that we still think about today.

11:56 The first were the rationalists, epitomized by Plato,

11:59 who said that the only thing that can be trusted is intuition.

12:03 In this view, things like shapes exist in the perfect realm of ideas,

12:08 untarnished by worldly imperfections.

12:10 And then there were the empiricists, often associated with Aristotle,

12:14 who said that observations of the world are the only way to gain knowledge.

12:19 But these conflicting viewpoints were doomed to endless argumentation.

12:23 Their perspectives fundamentally came down to beliefs about how the world works,

12:27 so one camp would never definitively win out over the other.

12:31 But alongside this endless debate, mathematicians found a way to use

12:36 constructions to answer the objections of philosophers.

12:40 It let them build a consistent intellectual framework which

12:44 even outsiders could test and conclusively prove right or wrong,

12:48 and it meant that their field could develop new

12:51 irrefutable knowledge and leave those squabbling philosophers in the dust.

12:57 Math, specifically geometry, would become an indisputable source of absolute

13:02 truth in a world filled with uncertainty.

13:08 But to build knowledge within geometry,

13:10 mathematicians needed some starting assumptions,

13:12 a set of ground rules or demands that you would ask of a skeptic.

13:16 In ancient Greece, these demands were known as postulates or axioms.

13:21 We associate these words with math,

13:23 but their origin is in the crucible of philosophical debate.

13:27 For Euclid and the ancient Greek geometers,

13:30 the first three postulates in the elements

13:33 were basic assumptions about constructions.

13:35 If we agree that the straightedge can draw a segment of a straight line,

13:40 and that we can extend it if we need to, and if we also agree

13:43 that a device like a compass can form

13:46 a circle given a certain center and radius,

13:48 then we've established our first three ground rules.

13:52 It's crucial that any starting assumptions are simple

13:55 enough to be plausible and are free from contradictions.

13:58 Grounding the postulatin physical actions addresses these issues.

14:03 That's what the ruler and compass achieves.

14:05 It shows that Euclid's axioms are not just something some guy made up,

14:09 which is, we know that people can make up inconsistent theories.

14:12 So with the ruler and the compass,

14:14 we are now saying the we have geometries based on these axioms.

14:18 Furthermore, these axioms are instantiated in physical reality.

14:21 Therefore, they must be as consistent as the physical experience itself,

14:25 which has a greater track record than human thought and than just imagination.

14:29 Just making stuff up in your head has a fallible track record.

14:33 Many people have failed at doing that perfectly,

14:36 whereas direct experience is not full of contradiction.

14:39 You know, nature does not contradict itself.

14:40 We know that our first three postulates are consistent

14:43 because we can build them in the real world.

14:46 As long as you only use the straightedge for straight lines,

14:49 no measurement allowed, and the compass for circles,

14:52 then any other necessary assumptions are implied by these constructions.

14:57 Lines and curves are continuous because

14:59 they've been formed by a continuous motion.

15:01 And when they cross each other, they intersect at a point.

15:04 The Elements goes on to build every single geometrical object

15:07 that it ever uses with a ruler and compass construction,

15:10 followed by a proof of its validity.

15:13 So in a sense, the ruler and compass become

15:16 like tools for building geometrical knowledge from the basic postulates.

15:21 But if these constructions are meant to be

15:23 embedded in the philosophy of Euclidean geometry,

15:25 then there was a big problem that these greek

15:28 geometers were going to have to reckon with, because

15:30 it was well known at the time that diagrams

15:33 could have subtle mistakes that would ruin an entire proof.

15:36 If a mistake found its way into a geometric theorem,

15:39 then not only would the proof be wrong,

15:41 but it could potentially ruin the credibility of geometry.

15:45 Because how can you trust a system

15:46 of thought if it's been shown to make mistakes?

15:50 Here's an example.

15:51 I'm going to prove to you that some

15:53 right angles are not equal to other right angles.

15:56 Now, to be clear, this is completely wrong.

15:58 In fact, Euclid's fourth postulate says that all right angles are equal.

16:02 But in this proof, the reasoning behind every single step is flawless,

16:06 except for one subtle, almost invisible mistake which ruins the whole thing.

16:12 Here's a square.

16:13 It's got four equal sides and four right angles.

16:17 I'm going to draw a line over here,

16:18 which I've measured to have the same length as the sides of my square.

16:22 And then I'm going to connect this point to this point.

16:26 So this, this, this, this, and this are all congruent.

16:30 Next, I draw a perpendicular bisector to this side up here.

16:34 And then I draw another perpendicular bisector to this side down here.

16:39 Then I connect these points and I make two triangles here.

16:43 This down here is a perpendicular bisector, which makes these two sides equal.

16:48 Meanwhile, since it's perpendicular, this angle is also equal to this angle.

16:52 And then finally, this side is shared by both of the triangles.

16:56 So side, angle, side, these two triangles are congruent.

16:59 So that means that this length is equal to this length.

17:03 Next, I'm going to draw a line here and a line here.

17:07 If I look at these two triangles once again, this was a perpendicular bisector.

17:12 So this length and this length are equal.

17:14 This line is shared between the two triangles,

17:17 and it's perpendicular, which makes these two angles equal.

17:21 Because of this, these two triangles are congruent.

17:23 And that means that this side and this side are also equal.

17:27 Okay, so far, we have two pairs of triangles which are each

17:31 undeniably equal to each other because

17:33 they came from drawing these perpendicular bisectors.

17:36 And that also means that this third side for each pair is congruent as well.

17:42 So now, this triangle and this triangle also have side,

17:45 side, side congruency with each other.

17:49 Hmm.

17:49 But this is a right angle plus some other angle.

17:52 I'll call that measurement alpha.

17:54 We know it's the same over here because these were congruent triangles.

17:58 So alpha plus a right angle again.

18:00 But this right angle is bigger than

18:02 the other one because of this little extra piece.

18:05 So what happened?

18:08 Well, because my perpendicular bisectors are drawn slightly wrong,

18:11 the diagram made the triangle on the right look

18:14 like it's basically symmetrical to the triangle on the left,

18:17 instead of being completely outside of the square like this.

18:21 Since the right angle of the square is no longer inside of the triangle,

18:24 the proof completely falls apart.

18:26 This is why modern math is so stringent about avoiding diagrams.

18:30 That single hidden assumption about where the triangle is located gave

18:33 us a diagram that allowed us to prove an impossible result.

18:37 But the Greek practice of ruler and compass

18:40 constructions has an answer to this problem.

18:43 Because if you use the recipes from the Elements

18:45 to properly construct the diagram for that false proof,

18:48 you would avoid the subtle mistake that I

18:50 just made when I drew the picture by eye.

18:53 Even if your constructions have a certain margin of error,

18:56 following the steps from the Elements still gives you

18:58 the precision that you need to determine these inexact properties,

19:02 like which point lies inside or outside of which shape.

19:05 Shape.

19:06 In the context of Euclid's Elements,

19:08 if a diagram like this hasn't been properly constructed,

19:11 it effectively doesn't exist, or at least it can't be used in a proof.

19:15 In fact, Euclid is very careful throughout the Elements to only

19:19 ever use shapes that he's already shown how to construct.

19:22 As he proves new theorems,

19:23 he introduces new shapes that these new tools allow him to build.

19:28 But this leads to some surprises.

19:30 Shapes that seem like they should be simple,

19:33 but actually aren't, for example, squares.

19:35 If you asked me to construct a square

19:37 using Euclid's propositions and I didn't know any better,

19:40 I'd probably say the simplest way to do it would be something like this.

19:44 Draw a line segment.

19:45 Construct another line segment equal

19:46 to the first and perpendicular to the first, using Proposition 11,

19:50 and then just do the same thing again and then again one more time,

19:54 and you have a square.

19:58 But when it comes time to prove that this really is a square,

20:01 we've got a problem.

20:02 Because to be sure that the side opposite the first

20:05 one really is the same length as the original,

20:08 we would need to say something about parallel lines.

20:11 Euclid waits until more than halfway through book one before

20:15 he finally uses his most famous and most controversial postulate,

20:19 postulate 5, the parallel postulate.

20:23 Unlike the other postulates, it's not simple or intuitive.

20:27 Even the actual statement is strangely convoluted.

20:30 "If a straight line falling across two other straight

20:33 lines makes internal angles on the same side of itself,

20:36 whose sum is less than two right angles,

20:38 then the two other straight lines being produced to infinity

20:41 meet on that side of the original straight line,

20:44 that the sum of the internal angles is less than

20:47 two right angles." This is a statement about non parallel lines.

20:52 But taking the contrapositive gives

20:54 logically equivalent information about parallels.

20:57 Anytime two lines don't cross, then the sum of the angles equals 180 degrees.

21:02 Today, we usually say it like for any line and a point not on the line,

21:07 there's exactly one line through the point which is parallel to the first line.

21:12 But in the context of ruler and compass constructions,

21:16 the first one is not the same as the other two,

21:19 because there's no way to physically verify

21:22 that lines continue forever and never intersect.

21:25 How do you check for never?

21:28 Even though Euclid's definitions say that parallel

21:30 lines are two lines that never intersect,

21:33 when it comes time to ask somebody to accept an assumption about them,

21:37 he avoids the problem of checking for never

21:39 by phrasing it in a way that can be constructed.

21:44 Whenever two lines are drawn through a transversal so that they're converging,

21:47 they'll cross eventually.

21:49 It's an assertion that's physically checkable,

21:52 even if doing it in practice requires a bit of extra space.

21:57 But even so, this is not empirical science,

22:00 because this postulate lets Euclid derive theorems

22:02 about parallel lines that are not physically checkable.

22:06 We can't actually make lines that go on forever without touching,

22:10 especially on the spherical surface of the Earth.

22:13 So Euclid is is using physically verifiable principles to develop

22:17 a theory that goes beyond the scope of empirical observation.

22:21 It's an approach that's different

22:23 from both Platonic rationalism and Aristotelian empiricism.

22:27 And once he introduces the parallel postulate

22:29 for the first time in Proposition 29,

22:31 all of the subsequent theorems and constructions require it for their existence.

22:36 This kind of additive structure of the elements reveals hidden

22:39 complexities in shapes that we might have assumed to be simple.

22:43 So Euclid's construction of a square is much more complicated than mine.

22:47 In fact, when I followed his construction

22:49 and I included all the subroutines from earlier propositions,

22:52 it took me close to 10 minutes just to make a square.

22:56 Proposition 46: "To describe a square on a given straight line,

23:01 let AB be the given straight line." "Let AC have been drawn at right angles

23:05 to the straight line AB from the point A on it." To get a right angle,

23:09 I have to go to proposition 11.

23:11 Proposition 11: to draw a straight line

23:12 at right angles to a given straight line.

23:14 Let AB given straight line, and C a given point have been taken at random on it,

23:19 and let CE be made equal.

23:20 And let the equilateral triangle FDE have been constructed on proposition 1.

23:24 Make AD equal.

23:25 Propositions 3, DE have been drawn through point D parallel to...

23:30 31.

23:30 Draw a straight line parallel to a given straight line through a given.

23:33 Gonna have me make alternate.

23:35 Let angle DAE equal to angle ADC have

23:37 been constructed on the straight line DA proposition 23.

23:42 Back to proposition 47.

23:43 Let DE have been drawn through point D parallel to AD.

23:46 And let BE have been drawn through point B parallel to AD.

23:49 This and this are parallel, and this and this are also parallel.

23:52 And this angle is equal to.

23:53 Thus we have a square with four equal sides and four right angles.

23:58 The key steps here were that after drawing the second side

24:01 so that it was perpendicular and equal in length to the first,

24:05 the third side was constructed to be parallel to the first,

24:08 and the fourth side was parallel to the second.

24:11 This allows him to then use

24:13 the already established properties of parallelograms to state

24:16 that the opposite sides and the opposite angles are all equal to each other.

24:21 With my simpler method of constructing the square,

24:24 I wouldn't have been able to prove that it

24:26 was a square at all without invoking the parallel postulate.

24:29 But it's actually even worse than that, because without the parallel postulate,

24:34 squares don't exist.

24:37 If the parallel postulate is false,

24:39 one can instead prove that this configuration does not make a square,

24:44 but rather a weirdly disfigured quadrilateral.

24:47 So even though you made sure you had

24:50 right angles at the base of the quadrilateral,

24:52 and that the perpendicular sides were equal,

24:54 the fourth and final side somehow still manages to miss the mark, so to speak.

24:59 It makes non right angles.

25:01 It's as if the sides are sort of bent.

25:04 It's not that Greek mathematicians were doubting whether or not squares existed.

25:08 Or that Euclid somehow anticipated non Euclidean geometry, which he didn't.

25:13 But rather it's that Euclid was using the process of constructions combined

25:18 with axiomatic deductive proofs to pin

25:20 down the very foundations of geometric knowledge.

25:23 To systematically catalogue every assumption that's required

25:27 before we can make something like a square.

25:31 So the Elements isn't just a collection of proofs

25:34 accompanied by subroutines showing how to construct geometric objects.

25:38 It's a taxonomy of the exact components

25:41 that are required for any geometric object.

25:44 A family tree of the universe of geometry.

25:47 And with this type of analysis,

25:49 Euclid is able to identify precisely what fundamental starting

25:53 assumptions are required for the entire Greek system of geometry.

25:58 It's not obvious that the square requires the parallel postulate to be

26:02 true until you pull it apart and put it under a microscope.

26:07 But for centuries, other mathematicians were sure that Euclid

26:11 had made another mistake by calling postulate 5 a postulate.

26:16 This complicated prediction about two lines crossing seems

26:19 more like a theorem that should be proven, not a postulate to be accepted.

26:23 In the Dark Ages when Europe had forgotten about the elements

26:27 and the only people actively studying it were in the Middle east,

26:31 the Arab mathematician Ibn Al Haytham and the Persian scholar Omar

26:35 Khayyam both tried to prove the parallel postulate as a theorem.

26:40 And they weren't the last to try either.

26:43 Many people tried to improve on Euclid in this way.

26:47 Around 1800, people like Lagrange and Legendre

26:50 proposed so called proofs of the parallel postulate.

26:54 These are big name mathematicians.

26:57 Their names are engraved in gold on the Eiffel Tower elite establishment stuff.

27:04 But even these bigwigs were wrong.

27:07 Their proofs contain hidden mistakes.

27:10 It's astonishing that this was more than 2,000 years after Euclid.

27:14 People tried to improve on Euclid for millennia.

27:19 The fact is that Euclid was right all along.

27:22 The parallel postulate really does need to be a separate assumption,

27:25 just as Euclid had made it.

27:27 It cannot be proved from the other axioms,

27:29 as so many mathematicians during those millennia had mistakenly believed.

27:35 The Greeks, you know, those guys were really something else.

27:38 It's so easy to make subtle mistakes in the theory of parallels.

27:42 History shows that there are a hundred ways to make

27:45 tiny invisible mistakes that fool even the best mathematicians.

27:49 Top mathematicians, they were never wrong about anything else.

27:51 They stumbled on this one issue and somehow Euclid got it exactly right.

27:57 He didn't make any of those hundred mistakes that later mathematicians did.

28:01 It's not luck, in my opinion.

28:04 Arguably, the Greeks were genuinely more sophisticated foundational

28:09 geometers than even the Paris elite in 1800.

28:14 Unbelievable, but true.

28:16 Euclid's Elements is not a classic for nothing.

28:19 Euclid is not a symbol of exact

28:22 reasoning because of some lazy Eurocentric birthright.

28:26 No, Euclid's Elements, it really is that good.

28:33 Euclid's construction of the square leads directly

28:37 into the culminating proof of book one,

28:40 the Pythagorean theorem and its converse.

28:42 From there, Euclid has five more books about plane geometry

28:46 before moving on to number theory in books 7 through 10,

28:49 where he gives theories of prime and irrational numbers.

28:53 Even within number theory, Euclid gives every proof a diagram seeming

28:57 to suggest that even numbers are subservient to geometry.

29:02 After all, numerals can't precisely represent an irrational

29:06 number like the square root of 2.

29:08 But a geometric construction can easily show that length.

29:13 The final three books cover three dimensional geometry,

29:16 culminating with the construction of the five Platonic solids and proving

29:20 that they're the only five ways to create 3D figures from regular polygons.

29:25 But this wasn't just the work of a single person.

29:28 It was the culmination of the work of generations of Greek mathematicians,

29:32 other elements of mathematics, and existed before Euclid's.

29:36 But his is the one that survives

29:38 because its impeccable structure made the others obsolete.

29:42 This book and the system of ruler and compass constructions at its core,

29:47 Acted as the arbiter of absolute mathematical truth for the next two millennia.

29:52 The middle eastern scholars who built on the elements

29:55 during the dark ages Developed powerful empirical science,

29:58 which laid the groundwork for the modern scientific method.

30:02 Once Europeans started to read Euclid again,

30:04 Ruler and compass constructions were seen

30:07 as the only path to absolute mathematical certainty.

30:10 Even as the powerful Islamic innovation of algebra found its way to Europe,

30:15 the Europeans saw it as less rigorous than geometry,

30:18 A tool for mere practical applications

30:20 or a mathematical trick for finding the right answer.

30:23 They were skeptical of algebra for centuries because it used

30:27 symbols instead of the physically verifiable procedures of Euclidean geometry.

30:32 In the 17th century,

30:34 when Rene Descartes took the bold step of assigning coordinate values to points,

30:39 the beginnings of analytic geometry,

30:41 he still justified his work with constructions.

30:45 To him, the ruler and compass were like a mechanism for trust.

30:48 His philosophy was to doubt everything.

30:50 But even he didn't doubt Euclid's postulates.

30:54 Descartes went so far as to create new compasses,

30:56 Mechanical devices that were more complex than the ruler and compass,

31:00 but which served a similar constructive purpose.

31:03 He wanted to bring absolute certainty

31:05 to the mathematics of higher degree curves,

31:08 which go beyond the scope of what's constructible with a ruler and compass.

31:12 So even as he revealed the power of using algebra to do geometry,

31:17 he still couldn't let go of constructions,

31:19 Even though he has the alternative right there.

31:22 You know, to do it algebraically, to take algebras to fundamental foundation,

31:26 is he's the one who shows that that could easily be done.

31:29 And still he refuses.

31:31 He has the standing at the gates of paradise where he's denying himself,

31:34 you know, saying, no, no, no, that's wrong.

31:36 We should do it this way.

31:37 He insists on this classical stuff eat despite himself.

31:40 But soon the reign of constructions as the philosophical

31:43 bedrock for math would have to come to an end.

31:46 The 19th century realization that there are valid

31:49 forms of geometry where the parallel postulate is false,

31:53 along with other seemingly contradictory discoveries in that century,

31:56 led to a sear formal foundations for math.

32:00 Math increasingly turned to formal logic,

32:02 which today is embodied by computer systems.

32:06 Rather than relying on intuition or abstract thought,

32:09 computerized proof checkers like Lean serve a similar purpose to the one

32:12 that a skeptic with a ruler and compass would have served in ancient Greece.

32:17 Just like in those early days of axiomatic deductive mathematics,

32:22 we still need tools that ground abstract reasoning in in systematic,

32:26 verifiable procedures.

32:29 Instead of constructions.

32:30 Computer systems are built on countless on off switches of transistors,

32:34 following explicit checkable rules.

32:38 As is the case with Euclid,

32:40 smaller operations based on fundamental starting points form

32:43 subroutines that build into ever larger programmatic structures.

32:47 But where Euclid's basic operations were lines and circles,

32:52 ours are ones and zeros.

32:57 There was a very strong parallel between understanding

32:59 what Euclid was really getting at with constructions,

33:02 and like my own process for making videos with programmatic animation,

33:05 that which felt like a very modern thing,

33:08 like wanting to implement it on a computer,

33:10 was actually maybe more reflective of how people's

33:13 relationship was with math through most of history.

33:16 I don't have...

33:16 I'm not a computer coder,

33:18 but there are many similar ways where actually having to instantiate

33:21 something changes it from what you had in your mind.

33:25 And I did find that kind of core idea.