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7: QED? Prove it. (Proofs)
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Proofs are sometimes seen as an exercise in tedium, other times as a pure form of beauty, and often as both. But from time immemorial, people have been using mathematics to demonstrate new theorems, and advance the state of the art of mathematics. However, it is only relatively recently, within the last 3,000 years, that the art of mathematical proof has been considered essential to the study of mathematics. Mathematicians constantly fight over what constitutes a proof, and even what makes a proof valid, partially because proof requires delicate insight. So what is the art of mathematical proof? How has it changed? And who can do it?
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Transcript
Introduction and New Blog
00:00:00
Speaker
I'm Jonathan. And I'm Gabriel. And we're coming at you from KUNM Studios. Um, we're brought to you by Generation Listen. And if you haven't checked out the Facebook page yet, you can check us out at facebook.com slash breaking math podcast or breaking math podcast.com. And right now we actually have a new blog feature.
00:00:17
Speaker
Yes, actually, I wanted to take a minute and tell all of our listeners about a new thing that we're doing. So, when we make a Breaking Math episode, obviously we take some time, we put together an outline and we do a lot of research, and then of course we present it in this audio format. However, it's done on us that we really should put this into a written format.
00:00:34
Speaker
and something that our listeners can interact with. We're actually always very curious what our listeners think about the episodes. Starting with this episode, we are actually going to make a blog that will include an episode executive summary. It'll include the main bullet points that we discussed on the episode.
00:00:50
Speaker
We're also going to, if it's fitting, we will elaborate on a few things, maybe include some visual diagrams as well. We will also include a comment section. So if you've got a Facebook profile, you can just comment at the bottom and ask us questions. We would certainly love to interact with our listeners. And you can find that at breakingmathpodcast.com slash blog.html.
The Beauty of Mathematical Proofs
00:01:14
Speaker
And now onto the show.
00:01:16
Speaker
Proofs are sometimes seen as an exercise in tedium, other times as a pure form of beauty, and often as both. But from time immemorial, people have been using mathematics to demonstrate new theorems and advance the state of the art of mathematics. However, it is only relatively recently, within the last three thousand years, that the art of mathematical proof has been considered essential to the study of mathematics.
00:01:39
Speaker
Mathematicians constantly fight over what constitutes a proof and even what makes a proof valid, partially because proof requires delicate insight. So what is the art of mathematical proof? How has it changed? And who can do it? All this and more on this episode of Breaking Math. Episode 7, QED, Prove It. I'm Jonathan. And I'm Gabriel.
00:02:05
Speaker
And we're coming at you from KUNM Studios as part of Generation Listen. And today we have returning guest, Amy. Hi, everybody. And today we're going to be talking about mathematical proofs. Now, Amy, as an educator, what is your opinion on having to teach proof? Well, I don't really get to teach proofs anymore. And so that kind of reveals my opinion of them is that I would love to teach them and I don't really have the opportunity.
00:02:29
Speaker
I wanted to mention real quickly, here we have a very unique opportunity. In my conversations, I know a lot of people who have little to no interest in mathematical proofs, and a lot of people who are absolutely all about them, they're almost, we'll use the term, mathematical evangelicals.
00:02:45
Speaker
Recently, I've seen some really cool proofs and I've actually, I'd say I'm not traditionally one who likes mathematical proofs. So the idea of doing a podcast episode about mathematical proofs, originally I was apprehensive about it. However, I actually think it's a chance to communicate the beauty of mathematical proofs so that more listeners can really appreciate them.
00:03:05
Speaker
When I was a child and I first saw the proof that the square root of two was irrational, it was a proof by contradiction, which we're going to be talking about later. I consider myself to be born that day. That was the epitome of seeing beauty in its purest form. Seeing a proof that was done in such an indirect way.
00:03:25
Speaker
So there's proof that motivation, I'm sorry, I didn't mean to call that proof. I'm being symbolic here. There's proof that there's motivation and there's beauty in the pursuit of math itself. That's pretty cool.
00:03:36
Speaker
No, Amy, you have a favorite proof? My favorite proof is the counter set, the proof of the uncountability of the counter set. And we've talked about it before, and really what was beautiful to me about that proof is it was not intuitive that something that is, you know, it's bounded from zero to one, and you take away essentially all of it, and there's still infinitely much left. The concept of infinity really kind of solidified in that moment for me, as much as the concept of infinity can solidify.
00:04:06
Speaker
And that was really just an enlightening and beautiful moment for me. And that really kind of, I guess, solidified my love of proofs. Now, one thing you mentioned a few times was infinity. And have you heard about the infinity hotel? I have not. It's a hotel that has an infinite number of rooms and a guest shows up, but all the rooms are full. So the guy gets on the loudspeaker and says, everybody move up one room.
00:04:29
Speaker
And so he makes room for that one guy. But then an infinite number of buses show up. So he has everybody look at their room number, double it, and then there's an infinite number of vacancies.
00:04:38
Speaker
And what you're talking about is a case where for every bus that shows up, there's an infinite number of people in the bus, basically, and you cannot count everybody. And this is some of the non-intuition that goes into mathematical proofs. Yeah, I think that actually, by and large, I think that Cantor's proofs are, I would say, considered among the most, what's the word I'm looking for, the most astonishing proofs. They're really clever, especially the diagonalization argument.
Visual Proofs and Axioms
00:05:07
Speaker
Wow, I I am actually not familiar with that. I look forward to finding more about that basically what this is is that you write down every number write down a bunch of numbers and You pick the first digit from the first number the second digit from the second number and so on and you have a number that is not in that set of numbers and using this Brilliant arguments you can argue some really cool stuff including the fact that the rationals meaning numbers like 2.3 and pi Can't be counted
00:05:36
Speaker
Wow. You know, actually, it just dawned on me real quick. I think that for a lot of people, the first time that they see beauty in proof, and they really appreciate that, I think it actually may involve some of the artistic math. You know, there's a group I follow on Facebook called the Art of Mathematics, and they've got some
00:05:54
Speaker
beautiful proofs that don't use anything written at all. All they have is pictures, in fact. And I think, again, for our listeners who still may be grasping what exactly is interesting and amazing about proofs, I think some visuals would be really helpful for them.
00:06:13
Speaker
and we'll have those on the blog. But what you need to touch on real quick is the difference between a demonstration and a proof. The difference between a demonstration and a proof is something that we kind of came up with on this, but what we really mean is if you have a bunch of right triangles, you can have a three, four, five triangle. There's five, 12, 13. Five, 12, 13. So let's say you have a bunch of triangles. You know that there's a three, four, five triangle. There's a five, 12, 13 triangle, and so on.
00:06:43
Speaker
You can have all these triangles and show that a squared plus b squared equals c squared, but unless you have a proof, all you have is a bunch of examples. There's a famous problem about the Zeta-Riemann function, which is too complicated for us to go into right now, but it has been proven for billions and billions of cases, but it's not considered a proof. I have an example, if I may. Yes. That I once, when I was in college, won a beer for proving that an odd number squared minus another odd number squared will always be divisible by four.
00:07:10
Speaker
So basically all you do, so if you think of an example, right? So an odd number squared minus another odd number squared. If I do 5 squared, that's 25. If I do 3 squared, that's 9. 25 minus 9 is 16. 16 is divisible by 4. That's an example, right? We could come up with infinitely many examples. Think of your own. Think of an odd number squared and subtract any other odd number. Or it could be the same one because 0 is divisible by 4. But think of another odd number and square it. Subtract them. That number will be divisible by 4.
00:07:38
Speaker
the way that we prove it and we'll go ahead and we can put that up on the website. It's a pretty simple proof. Win yourself a beer. Yeah, exactly. Write it on a napkin. Find yourself a tech school and bet somebody that you can prove it. Write it on a napkin and you'll get yourself a beer.
00:07:52
Speaker
Also, never trust a mathematician when they say, I bet I can prove it, because they always will be able to. They wouldn't be betting otherwise. Is that a little bit of resentment? No, I'm just kidding. So really why I brought that up is that it shows pretty clearly the difference between proving something and having an example of it being true. That having an example of it being true is not in and itself a proof. Right, because all you need is one counter example for the demonstration, and it's no longer valid.
00:08:19
Speaker
Exactly. One interesting thing about proofs is that, for example, adding two and three
00:08:26
Speaker
is kind of proving that two plus three is five, but not really unless you go into, um, into, into analysis. And we're going to have an example of that on the blog, but the difference between a proof and just doing a math problem is that a proof requires some insight into the problem.
Methods of Proof
00:08:44
Speaker
It requires you to set things equal to each other that you might not have thought to set equal to each other, but that are equal.
00:08:52
Speaker
Well, that leads into discussion of what a proof really is that we, so do you want to talk about axioms? Yeah, totally. An axiom is a thing that you assume is true, uh, before you go into a math problem. So, um, for example, and we're going to talk a little bit about this later, an axiom in Euclidean geometry, just normal geometry.
00:09:14
Speaker
is that all right angles are the same. It's an obvious statement that you need to prove something. And there's a whole field of mathematics that is based on studying the consequences of axioms.
00:09:30
Speaker
Okay, so just in brief, so now we once had a discussion on Facebook with one of our listeners who happened to be related to me in fact. And he said, aren't axioms in fact just, dare I say, assumptions? How would we address that?
00:09:46
Speaker
They are. There are assumptions and there's parts of math that you are going to have to kind of like be at the bottom of. For example, there's no real reason why we should maybe care about the and logic proposition, but we do. Okay. The and logic proposition. What is that?
00:10:07
Speaker
It's true and true is true true and anything else like false is false It's used a lot in symbolic logic. Okay, so that that's our starting axiom or if I'll use that word that is our starting assumption Well, it's one of our starting assumptions
00:10:25
Speaker
So there is a difference between an axiom and an assumption that you use an assumption in a proof and then you can either prove or disprove that assumption and that may change what result you get out of it. An axiom is something that you are not attempting to prove or disprove. An axiom is something that we assume to be true and we use that information to prove or disprove something else. So colloquially they still are assumptions but assumptions are something distinct. I would say so an axiom is something that we are not questioning whereas an assumption maybe we do. Fair point.
00:10:56
Speaker
so my brother has a degree in philosophy and we have his version of logic and my version of logic were very different that the the mathematical logic i learned and the the philosophical logic he learned are very different and when he took a mathematical logic class he took great umbrage with it specifically with axiomatic proofs and with inductive proofs that he didn't care for either of those that an axiomatic proof you have to
00:11:23
Speaker
say like we assume these axioms to be true without question and then based on that foundation we prove other things he's like well that in and of itself is flawed and as a philosopher i think he's right as a mathematician i don't really see an alternative we need to be able to assume things are true i think we need to be able to assume that numbers work a certain way and yes we invented those numbers but that's that's another argument altogether
00:11:44
Speaker
I mean, there's always assumptions in any type of argument. Even of political arguments, people are assuming for some times that what we should do should be for the benefit of the herd. Other times we say what we should do should be for the benefit of the future. But we're always making assumptions, no matter what, that we do not question or that we think that we do not question.
00:12:08
Speaker
Now, one thing that we also want to talk about is the different types of proofs.
Euclid's Influence on Mathematics
00:12:13
Speaker
And you mentioned inductive. Now, what is this issue with inductive proof?
00:12:17
Speaker
Well, so what an inductive proof does is it's basically, here's a bunch of examples of this working. I'm going to assume it works for some finite number of times and then prove that it works. So you assume it works n times and then you can prove that it works n plus one times and therefore it works infinitely. And he basically equates that to having infinitely many examples instead of actually proving it.
00:12:43
Speaker
Is that more or less a common sentiment among other philosophers as well, or is your brother unique? I don't know. Well, he's definitely unique. I don't know if that sentiment he shares, but he's definitely unique. Interesting. Well, I know that the constructivists have problems with axiomatic proofs, but also they have problems with proof by contradiction. So, I mean, they are not to be trusted. So, I know earlier we spoke a bit about axioms, and then also there's different types of proofs, right?
00:13:10
Speaker
Oh yeah, so inductive, and another way of looking at inductive proof is this is the domino argument. You prove that the first domino falls down, then you prove that a domino knocks over another domino, and therefore all the dominoes fall down.
00:13:23
Speaker
Are there any issues with that kind of proof? Depends on who you ask. Okay. And then so in addition to inductive proofs, then the other ones, let's see, there is by way of contradiction and then there is also a contrapositive. Just for listeners, can we say again, by way of contradiction is that simply providing an example that proves that your proof is false?
00:13:45
Speaker
So basically what you do is you assume the opposite of what you're trying to prove and then show that that can't happen. So by reaching that paradox, you prove that your original assumption was false. Okay. And then how is that different from contra-positive? Now what contra-positive is, is that you're trying to get from point A to point B in your proof. So what you do is you show that
00:14:08
Speaker
if not point B, then not point A. It's basically the reverse, right? That if A leads to B, then not having B leads to not having A. Oh, okay. It's the same thing as saying Socrates is a fish, therefore Socrates can swim and Socrates cannot swim, therefore Socrates is not a fish. And of course there's direct proof. That's just where you go from A to B directly, no fanciness. That's generally the hardest.
00:14:33
Speaker
Alrighty, so in talking about proofs, obviously proofs go back quite a long ways, and if I'm not mistaken, I think that one of the original founders who codified a method was Euclid, and he's best known for geometry as we currently study it in high school. Yeah, and he's also one of the original algorithm inventors, but he wrote a whole book about proofs called Euclid's Elements.
00:14:56
Speaker
And I think that that is still considered among the finest piece of writing produced by man of all time, arguably, of course. Because there's been a lot of tomes from the same era, from Chinese, Indian and other cultures trying to do proofs, but they're all just lists of demonstrations.
00:15:17
Speaker
Okay, interesting. So I think the way I once heard it described is that what Euclid did is Euclid established the rigor of mathematical proofs. Yes, he set the tone for what it means to really prove something. He was extremely gifted philosophically in that aspect.
00:15:37
Speaker
So I'd like to comment that geometry was really foundational in the development of mathematics, that when people think about math and Euclid, they're thinking about geometry. And that's really the foundation of physics in a lot of ways, was the geometry of the stars. And I find it very frustrating as a math educator that we don't teach mathematics that way. We do fill in the blank proofs, you know, it's the side-side-side angle, you know, triangle postulate.
00:16:01
Speaker
that we don't actually teach the logic, which is what makes this so beautiful. We don't teach the going from something that is known to something that is unknown, which is what Euclid really nailed down, was that I know this, and with this information, I can prove all of these other things. It's almost the same sort of intrigue, in my opinion, as reading a really good Sherlock Holmes story. You have this tiny bit of information, then you get more information just magically.
00:16:32
Speaker
Of course, I know that later on in the episode we have a segment specifically on what Euclid does do and what he doesn't do, so I don't want to get ahead of ourselves, but knowing him is certainly a very good introduction to the beauty and rigor of mathematical proofs.
00:16:48
Speaker
Is it a good time, do you guys think, to talk about some of the methods to attack a proof? I think that actually Jonathan and Amy specifically were discussing just a few general things you can do when you're trying to solve or when you're trying to formulate a proof for something.
00:17:04
Speaker
Well, I know one big thing is if you have an example of a case that you could label zero, like, you know, where zero is just an important part, and a case that you could label one, the next logical place to look is infinity. Would you agree with that assessment a little bit?
00:17:22
Speaker
Well, I think using the word logical when talking about infinities is being pretty generous to our listeners. I think that's the next logical place for a mathematician to look. Yes. Um, that I think everyone else that's listening was expecting you to say too. Oh yeah. Um, but yeah, two is too complicated. Um, we want, yeah, somehow infinity is not, that's what is so brilliant about math.
00:17:50
Speaker
Yeah, you're not doing real math if you use numbers greater than seven. I remember one time I took a homework assignment home that had been graded and my mother looked at it and this was probably a senior level analysis course. So there's no numbers on this page. Like the only numbers on the page were the numbers of the problems. Like this is not math homework. There's no numbers on this page. And I was like, mom, mom, that symbol right there, that represents every number. And she was done. But I was excited.
00:18:19
Speaker
Proofs are not always without controversy, especially when they seemingly overthrow a previous branch of mathematics. Although this is rare in mathematics as it is in science, it does happen,
Controversial Proofs and Challenges
00:18:28
Speaker
and studying what results when it does happen is essential to round witness knowledge of mathematics. So before we talk about real controversy, we're going to talk about apparent controversy or pop controversy. Amy had an example of this.
00:18:41
Speaker
So I was trying to explain to my students that it's about the nature of infinity, which is admittedly difficult to wrap your head around, quite literally. I was trying to tell my students that 0.999 infinity, that if you repeat 0.999 infinitely, on and on and on, that it equals 1. It does not approach 1.
00:19:03
Speaker
It's not rounded to one, it is one. And they looked at me like I was crazy. And so the kind of simplest proof of this, what is one third? One divided by three is point three, three, three, three, three, three, infinitely, right? Two thirds, point six, six, six, six, six, six, six, on and on, infinitely. If I add point three infinite plus point six infinite, what do I get? I get point nine.
00:19:27
Speaker
Because three plus six is nine. Exactly, right. So every single tenth place, hundreds place, thousands place will become nine infinitely. And what is one third plus two thirds? Well, that's three thirds, which equals one. Yeah. And it's a very simple proof. And to a lot of people, it seems that it's not very intuitive. It's just difficult to wrap your head around it, that something that is infinite can also be something very finite.
00:19:52
Speaker
And of course, another controversial proof or pseudo controversial proof is the Monty Hall problem. Oh, I love that Monty Hall. I love the Monty Hall problem. That one drove me nuts for a very long time until I finally took the time and proved it step by step. And actually, I think that we had mentioned earlier the idea of winning bets.
00:20:12
Speaker
So I know that in this day and age everyone's got their smartphone and everyone can just Google something So there's really no excuse for this I'm sure that as we're talking about this somebody could very very easily You know Google on their phone at Monty Hall problem, but but let's let's say you don't do that Let's say you decide not to do it and see if you can solve it in your head
00:20:28
Speaker
Now, this problem is named after a popular game show host who was, I believe he was popular in 1975. He had a show called Let's Make a Deal, and it sometimes would involve doors. Now, the Monty Hall problem itself is just a problem that was proposed. So in this problem, there is a game show, and there are three doors, and the contestant is made aware that behind two doors there are goats, and behind one door is a new car,
00:20:58
Speaker
and the doors are numbered, door number one, door number two, and door number three. So the host would then tell the contestant to just pick one of the doors, and the contestant would then pick a door. Now, before opening that door, the host would then go to one of the other two doors and open it and reveal a goat. They would then say to the contestant, you now have a choice.
00:21:24
Speaker
You can either stick with the first door that you chose, or if you would like to, you can now switch to the other door. Which one would you like to do? So the problem is solving whether or not it is best to either switch doors or whether it's best to stay with the first door that you chose.
00:21:48
Speaker
or whether it doesn't matter either way. Now the first answer that I gave, I know the first answer I gave, what was, do you all know the first answer that you gave? I thought it was irrelevant. Yes, I actually thought in my mind I was very sure that I could prove, mathematically speaking, that it utterly did not matter whether you changed doors or not.
00:22:10
Speaker
Now, I think I'll go ahead and disclose right now, of course you all can just Google it, but I would encourage you not to Google it yet because we want to talk you through the problem and see if you agree with us and see what it takes to get you all to change your minds. So the answer is not that it doesn't matter, it does in fact matter.
00:22:29
Speaker
Alrighty, so now my first guess was of course that it did not matter at all. One way or the other. In fact, I was almost positive. I spent all class planning out doing scenarios in my head when I first heard this and I arrived at the conclusion that it does not matter. And I think, Amy, you arrived to a similar one, right?
00:22:48
Speaker
I did. I figured eliminating a door changed the probability. Now instead of a one out of three chance, I have a one out of two chance of getting it right. But it didn't seem to matter to me that whichever one I chose, it was still a 50-50 chance. Exactly. Now, the cool thing is we're going to walk you through this. We will tell you that it does, in fact, matter. So it is not correct to say that it is irrelevant. There is a correct choice. What I want to say is before we tell you the answer,
00:23:16
Speaker
Well, we can tell you the answer, but we're not going to walk you through it quite yet. The answer that was published a while back, back in the 1970s, was that it is in your best interest to change guesses, to change doors. So you pick a door,
00:23:35
Speaker
at first then Monty Hall opens up a door and says do you oh sorry then Monty Hall opens up a door and reveals a goat and says now do you want to stay where you're at or change the correct answers you always always want to change now why that's the case we'll get to in just a minute there's one who published this in a paper her name is a Marilyn Svant
00:23:57
Speaker
And she published the answer, and she explained that it is better to change your guess when you're given the choice. And that was met with a lot of resistance. In fact, if I understand it correctly, there were PhD doctors in statistics who wrote that her solution that she provided was absolutely wrong.
00:24:20
Speaker
However, the solution that we're going to give you can be reasoned through step by step, does not involve doing a bunch of trials, which you can do and you could see the probabilities involved very easily. But yeah, we have a direct way of reasoning through this. Okay, cool. Yeah. And I think right now we can we can probably give our listeners the step by step. You know, obviously, if they want to pause it and try to figure it out on their own, they're welcome to but we can provide the answer.
00:24:46
Speaker
Well, one thing that I would like to do is give the slightly more intuitive version of the Monty Hall problem. Let's say there's a million doors and you choose a door and obviously the probability that that's going to be the car is very, very low. Now you open every single door except for two doors. Now we think about the probability that switching doors
00:25:12
Speaker
will change the outcome. It's a lot more obvious that it's gonna be a lot more obvious that the door that you did not choose is more likely to be the one. The reason why it's more obvious is because think about the number of goats that they opened up the doors for.
00:25:31
Speaker
So if you had a million doors, and you get to choose one of them, and you chose that one door out of a million, you have a one out of a million chance of having chosen the winning door. And then Monty Hall opens every other door except one other door. Suddenly that one other door does look way more attractive.
00:25:50
Speaker
Why did he open all of the other ones except that exact one? I want that door, right? It makes a little more sense when you're dealing with much larger numbers than one out of three or two out of three. And I encourage that you work things out on paper because there are still some who will be thinking
00:26:05
Speaker
Well, the probability is still one out of two because now there's just two doors left. Well, and that was so this is one of my favorite mathematicians that I've ever read anything about. His name is Paul Erdős. He was a Hungarian mathematician who worked on something like two or three thousand papers. He published some prolific amount of work. He is more prolific than Euler.
00:26:22
Speaker
Yeah, the man, there's something called an Erdos number that is your degree of separation away from publishing a paper with Erdos. And during his lifetime, there were, you know, hundreds of people who had an Erdos number of one, meaning that hundreds of people, if not more, it might have been thousands.
00:26:38
Speaker
He lived a transient lifestyle. He would sleep on somebody's couch, help improve something, and move on to the next couch. Yeah, he was kind of a mooch in that regard. And he actually got an honorary degree from, I believe, Cambridge? Maybe Oxford? I can't remember which one. But at the time, he was homeless. He was literally, he had no home. And that was pretty much his entire life. One of the most celebrated and renowned mathematicians in history was homeless.
Non-Euclidean Geometry and Set Theory
00:27:03
Speaker
So when he was introduced to the Monty Hall problem, he did not like her proof because her proof was just, here's the three ways that this can happen and look, two out of those three ways you win. And that wasn't an elegant enough proof for him. It really bothered him that there wasn't a more elegant way to do it than by example, showing this is the three cases and two of those three cases are a winner.
00:27:28
Speaker
And it really bugged him. And that really appealed to me as a mathematician that I did my graduate work in analysis, in proofs. And I always appreciated the beauty of a perfect proof. And that is something that Erdos spent his entire life pursuing. That is awesome. And Gabriel, do you want to give the answer to the probabilities involved?
00:27:51
Speaker
Yes, now the very, very quick way of proving this problem is when you say when you first select a door, you have a one out of three chance of being correct, however, or I'm sorry, along with that, you have a two out of three chance of being wrong. So what's interesting is since you have a 66.66, since you have a two thirds chance of being wrong,
00:28:15
Speaker
when Monty Hall gives you the option of changing doors, you are betting that you were wrong the first time. So if you switch doors when you're given the option to after originally having chosen a door at random, you have a two out of three chance of being correct. And that is how this problem works out.
00:28:35
Speaker
And I have a little bit of an interesting experience with the Monty Hall problem before we move on to real controversy. Um, in trying to prove it to myself, I wrote a program to play the game over and over again. Then I simplified the program and I kept simplifying it and simplifying it to the point where it didn't need to actually play it over and over again. It just had the probabilities right there laid out before me. And that's what convinced me that it was true. I went from a statistical demonstration to an actual proof.
00:29:01
Speaker
Well, and that's what bothered Paul Erdos as well, is that he was shown a statistical demonstration. He's like, yes, this says what you said it would say, but it's still not a proof and I don't like it. That it was not a sufficient proof for him. And again, it wasn't elegant enough. And what's the difference between this type of controversy, the 0.99 thing, the Mani Hall problem and actual controversy in mathematics?
00:29:21
Speaker
I think if I'll take a stab at this one. So everything that we said right now has had some degree of controversy as far as it being accepted by people. However, every single thing we've said so far has been accepted as mathematically proven at some point.
00:29:36
Speaker
Yeah, and it relies on things like axioms and sets of assumptions that have not been shown to be contradictory. However, there are some cases where assumptions have been shown to stand on pillars of salt. They have been shown to be contradictory, and we're going to give you an example of both of these.
00:29:56
Speaker
So first we're going to talk about the parallel postulate. Now the parallel postulate is one of five postulates, introduced by Euclid in the book The Elements. We're going to go through all of Euclid's postulates, or assumptions, or axioms. So postulate number one, a straight line connects two points.
00:30:15
Speaker
Postulate number two, any straight line segments can be extended indefinitely in a straight line. Number three, given any straight line segment, a circle can be constructed such that the center is at one end and the edge touches the other end. Postulate four, all right angles are congruent. Postulate five, if two lines are drawn which intersect a third such that the inner angles on one side is less than two right angles, then the two lines must intersect one another if extended far enough.
00:30:43
Speaker
Now, one of these things is obviously not like the other, and it's number five. Everything that we said is, I mean, would you agree, Gabriel, it's pretty self-evident? Quite, quite, yes. Except for number five. Number five, like, what did you call it earlier? I said this last one, unlike the other ones, it goes on and on and on. It sounds like, you know, what a five-year-old does when he's telling you about his day and the story just goes on and on. It's not elegant.
00:31:07
Speaker
And Euclid was very bothered by this. He tried to prove the fifth postulate with the other four and he never could. If you look at his book, he actually avoids it as long as he possibly can and whenever he possibly can. And people tried for hundreds of years to try to prove this and nobody was successful until somebody said, well, what if it's not always true? Well, isn't it like the simple version of this is that two parallel lines will never intersect?
00:31:31
Speaker
Oh yeah, but this is the definition of parallelism. Right, but for our listeners, that's the basic, the two lines that are parallel will never intersect is his fifth postulate that we couldn't really prove.
00:31:42
Speaker
Oh yeah, and now think about a big, like a sphere, for example. If two ships start sailing slightly away from each other, think about, think about like a slice on an orange. They'll come back together. So all straight lines, all parallel lines in that case intersect, because there'll be a time where they are parallel. So in that case, you have exactly two instances in which case the parallel lines intersect.
00:32:05
Speaker
And then you have a saddle where an infinite number of different lines do not intersect. So it's like the opposite of that. Again, the progression of zero, one, infinity. So I think the main takeaway here is if we were to really simplify this, is that Euclid's postulates are correct in two dimensions. They are not correct in three dimensions.
00:32:24
Speaker
Well, they are correct in three dimensions, but not two dimensional space embedded in three dimensions. That's to say two parallel lines in three dimensions will be parallel. It's just that on the surface of a sphere, they look like they're not parallel. They look like they're curved, right? Yes.
00:32:39
Speaker
but they are to the surface of the sphere. They're the shortest distance between two points on that sphere. It's like if you were to unwind a globe and turn it into a map, the difference between that three-dimensional space and the two-dimensional space is very marked. The way that it appears is very different. And we actually came out with a poster about tensors that describes this kind of geometry.
00:33:01
Speaker
You know, and actually, in planning this episode, I actually really, really enjoyed this discussion. I think previously, I had greatly overlooked it. And in essence, Jonathan, I don't know if you recall, but we were talking about specifically that the corners of a square on a flat plane, of course, are 90 degrees. And of course, if you were to draw a square on the Earth, the Earth is obviously incredibly huge compared to how we would draw a square that's one foot by one foot.
00:33:30
Speaker
we can still say that those angles are basically 90 degrees. But as you were saying, as the size of the square you draw increases, also on a sphere, and as the sphere decreases, the square warps around the edges of the sphere. And I think the example you even said is, you know, if you have a sphere and you've got an equator along the sphere, pick four dots along that equator and connect them through the shortest line possible, right?
00:33:58
Speaker
Yeah, and then you have 720 degrees. And of course, if you're paying attention, you could tell that if we picked five points, there'd be 900 degrees and so on. So this is one of the other weirdnesses about non-Euclidean geometry.
00:34:10
Speaker
Wow, yeah. But this is a whole field of mathematics that was developed over a controversy. Another controversial part of mathematics is set theory. Now there's an axiom called the axiom of choice, which basically says that given a bunch of sets, you could choose one element out of each set. A set is like a bucket full of things, and those things can also be buckets full of other things. If you try to define the set that contains all sets that do not contain themselves,
00:34:39
Speaker
then you run into a contradiction. Gabriel, what is that contradiction? OK, I'm sorry. Let me say it again. So if you try to define the set that contains all sets that do not contain themselves. So obviously that set wouldn't contain itself. Or it would. OK. Both at the same time. So you run into a contradiction. Oh, yeah. You have a paradox. Yeah, yeah, absolutely. So yeah, say that again. There's no way to describe a set that contains all sets that do not contain themselves.
00:35:05
Speaker
Yeah, and so then a new form of set theory called ZFC and other variants were developed in the late 1800s, early 1900s. Well, and this is where we really started challenging the axioms that were the foundation of mathematics that we started thinking pretty literally outside of the box. That's interesting. Can you give any more examples of axioms that were contradicted?
00:35:29
Speaker
Well, I think, again, I go back to my favorite, and I don't know that this is necessarily an axiom, but the kind of foundational shift it took to talk about fractals and chaos that we can't think of dimensions as being
00:35:47
Speaker
one and two and three anymore, that there are dimensions in between mathematically, that that was really a foundational shift, that we had to introduce a whole different kind of dimension, that that was similar in my mind to having to kind of have a different set of axioms. So I don't know if we necessarily had different axioms, but we had to kind of think of a different way of approaching mathematics, which I think started with this kind of axiom of choice, the set that cannot contain itself, but must.
00:36:14
Speaker
So would you say earlier you said you were thinking outside of the set, outside of the set that contains all the things you were, I'm sorry, we can edit that out. Going down a paradoxical hole here. No, is anyone familiar with Benoktarski? That sounds familiar to know that you say it.
00:36:32
Speaker
It is a completely unintuitive proof, but one that must be accepted if you want to accept the axiom of choice, which sometimes you can take, depending on what kind of set theory you're doing.
Fractals, Chaos Theory, and Complex Numbers
00:36:44
Speaker
And that is this, you take a sphere, you cut it up into an infinite number of pieces. You put the pieces back together. You have two spheres now.
00:36:52
Speaker
So, of course, as you're saying this, I'm imagining it in my head. And obviously this is not something that you may be able to imagine in your head. But suffice it to say that it is true and it has been proven or rather what Jonathan just said has not been disproven. I mean, you cannot do this if you use a certain type of set theory like ZFC. However, there are certain people who think that this says something about the way that particles interact. So you could take that for what it was worth.
00:37:22
Speaker
So this is fascinating. We have some people who say that this actually might explain a real phenomenon. That is that you can have one sphere and you divide it into an infinite amount of pieces, then you reassemble them into two spheres that equal the same volume as the first sphere. It sounds like a violation of conservation of something, doesn't it? It does, but the universe has done weirder things.
00:37:46
Speaker
Well, set theory is pretty mind-blowing in and of itself that trying to change the way that, because we're raised thinking mathematics is one thing and then set theory kind of takes you into a whole different world. And there's definitely branches of mathematics that do that and I think chaos is one of those. And so when we were talking about chaos theory that the imaginary numbers, that was a paradigm shift. That was definitely something that we had to kind of introduce a new metric or at least a new
00:38:13
Speaker
Like I said, I don't know that the axioms were different because we just had to kind of define their behavior a little bit differently. But the introduction of complex numbers to our number system was paradigm altering. And I think that set theory has a similar, you have to think in a different way. And it leads you to all of these exciting and different conclusions. That's what I love about proofs. You go from something which is known
00:38:38
Speaker
Whether you've assumed it to be known or whether it's actually known, whether it's an axiom or whether it's just a definition, we go from something that is known and prove something that is unknown. And that is so exciting.
Interactive Proof Problem Challenge
00:38:49
Speaker
It really is. Um, especially because the proofs build on each other and they build and build until you have a cathedral of proof. Cathedral of corollaries. And before we go, we're going to talk about a proof that we would like to leave you with.
00:39:03
Speaker
So this is a fun fact, actually. Before Breaking Math even existed in its current format, while we were still planning for the episodes, Jonathan and I researched a couple of other podcasts that dealt with math and STEM topics, that is science, technology, education, and math. And we came across one podcast in particular that featured a professor from Cornell named Dr. Steven Strogatz.
00:39:28
Speaker
And in this podcast, he introduced the audience to a story that described where his passion for mathematics was in part from. And in the story, he talks about a proof or rather a problem to prove that one of his high school math teachers provided him with. Now, of course, in the podcast, Stephen Storgas talks about wrestling with the proof and taking a long time with it.
00:39:52
Speaker
And then, you know, after that he decided maybe I should study math. He never actually provides the audience with the proof itself.
00:40:00
Speaker
So if Stephen Strogatz attempted it, we thought that we'd attempt it. And I proved the first version of this proof. It only took a little bit, but however, this proof was ugly. It's one of the ugliest proofs I've ever done. And I actually threw away the paper. I was ashamed of it. Then it took me about a week to come up with a beautiful proof. And this is the problem. Take an isosceles triangle, bisect the equal angles, and measure the line that goes to the edge. Those two lines are equal in length.
00:40:30
Speaker
And that's how simple the problem is. It's a very simple problem. However, it will take you down a rabbit hole. Oh, cool. So basically, uh, I think, um, we are going to include that problem, um, at the breaking math website. He'll be in the blog. Okay. So yeah, so search, searched on, on the blog and don't worry, it won't be a spoiler. It'll be, it'll be, you know, um, it'll say following is the solution. So you can attempt it on your own. And when you feel ready, you can take a look at the solution.
Conclusion and Listener Engagement
00:41:00
Speaker
Yes, and if you haven't heard from Amy, it's because she's been attempting the solution on her own. I feel like I have a really simple solution, which probably makes some assumptions that I shouldn't make. So I am bisecting the angle. So I have an isosceles triangle. The opposite sides are congruent, which of those angles. So if those angles are congruent, the opposite sides are congruent. That's a lot of signs, really. I'm bisecting the angle. Does that also bisect the opposite side? No. That ruins my proof.
00:41:29
Speaker
I ran into the same problem. That's an important part of proving something is trying and failing. I'm learning from that mistake. As we've said before on this podcast, a proof is like trying to drive to the store, winding up at the park, then the morgue, then the moon, and then finally the store.
00:41:46
Speaker
Well, we mentioned that earlier, talking about what it takes, like general strategies for trying to prove something. And it's thinking about whatever it is you're doing, every possible way that you can think about it. And I think it was, I want to say it was Einstein who said that you will never solve a problem by thinking about it the same way you came up with the problem. Something along those lines of paraphrasing. Sounds Einstein-y.
00:42:09
Speaker
I think I have a quote, I have that exact, well, the precise quote is up in my classroom, that you can't solve a problem using the same thinking that created the problem, right? And I tell my students that all the time, that there may be a hundred different ways to explain this concept to you, and only one of those ways might make sense, but I'll explain it a hundred different ways until you find that one that makes sense. What I think kind of separates
00:42:36
Speaker
A mathematician is that they understand maybe all hundred ways or they understand 99 of those ways and they're looking for the one way in there that is best. That that's a mathematician. That was what Erdosz did. He wanted the best, most elegant, eloquent solution or explanation rather than just looking for the one that made sense.
00:42:56
Speaker
So you're saying they have 99 problems, but I'm just kidding. We've explored the challenges surrounding proofs, the controversy surrounding them, the difficulties in understanding them and the reasons for their existence. As you go forward, think about the proofs that have built our understanding of the world around us and the people who have struggled to come up with them. I'm Jonathan. And I'm Gabriel. And this has been Breaking Math. Breaking Math is brought to you through KUNM Studios, through KUNM Generation Listen.
00:43:24
Speaker
Visit the website at breakingmathpodcast.com, the Facebook page at facebook.com slash breakingmathpodcast, or show your support by visiting patreon.com slash breakingmathpodcast. Just $17 a month would cover all of our podcasting costs. So if you donate $5, you go a long way towards helping us out. And please don't forget that we now have a new blog that goes with every episode where you're going to hear the bullet points, the main points of the episode, the terminologies, as well as a lively discussion.
00:43:52
Speaker
Is that also where our proofs will be? That is. Yes, you'll find those also on the Breaking Math website. You know what, just for old time's sake, maybe I'll write my proof on a beer napkin.