Become a Creator today!Start creating today - Share your story with the world!
Start for free
00:00:00
00:00:01
P1: Peano Addition
407 Plays5 years ago
On this problem episode, join Sofía and guest Diane Baca to learn about what an early attempt to formalize the natural numbers has to say about whether or not m+n equals n+m.
This episode is distributed under a CC BY-SA 4.0 license (https://creativecommons.org/licenses/by-sa/4.0/)
---
This episode is sponsored by
· Anchor: The easiest way to make a podcast. https://anchor.fm/app
Support this podcast: https://anchor.fm/breakingmathpodcast/support
This episode is distributed under a CC BY-SA 4.0 license (https://creativecommons.org/licenses/by-sa/4.0/)
---
This episode is sponsored by
· Anchor: The easiest way to make a podcast. https://anchor.fm/app
Support this podcast: https://anchor.fm/breakingmathpodcast/support
Recommended
Transcript
Introduction to Chumba Casino as a Social Escape
00:00:00
Speaker
Judy was boring. Hello. Then, Judy discovered Chumbacasino.com. It's my little escape. Now, Judy's the life of the party. Oh, baby. Mama's bringing home the bacon. Whoa. Take it easy, Judy. The Chumbalife is for everybody. So go to Chumbacasino.com and play over 100 casino-style games. Join today and play for free for your chance to redeem some serious prizes. Chumbacasino.com.
00:00:26
Speaker
No purchase necessary for you or prohibited by law. 18 plus terms and conditions apply. See website for details.
00:00:30
Speaker
Hello, it is Ryan and we could all use an extra bright spot in our day, couldn't we? Just to make up for things like sitting in traffic, doing the dishes, counting your steps, you know, all the mundane stuff. That is why I'm such a big fan of Chumba Casino. Chumba Casino has all your favorite social casino style games that you can play for free anytime, anywhere with daily bonuses. That should brighten your day a lot. Actually a lot. So sign up now at ChumbaCasino.com. That's ChumbaCasino.com.
00:00:59
Speaker
This episode is distributed under a Creative Commons Attribution Share-alike 4.0 international license. For more information, visit creativecommons.org.
Announcement of a New YouTube Show on Science
00:01:10
Speaker
This is Gabrielle Hesh from the Breaking Math podcast. I am thrilled to announce that we have teamed up with a particle physicist and a science fiction author named Dr. Alex Alaniz to create a brand new science show on YouTube. Our new show is called the Touring Rabbit Holes podcast. It's named after the mathematician and father of modern computer science, Alan Turing. You can find us at youtube.com slash touring rabbit holes podcast.
00:01:35
Speaker
and that's spelled T-U-R-I-N-G rabbit holes podcast. Be sure to like and subscribe for great visual storytelling about the history of mathematics, physics, and computer science and the impact that they have had on society. Again, that's youtube.com slash touring rabbit holes podcast. And now back to the breaking math podcast.
Breaking Math Podcast Updates and Platform Tools
00:01:56
Speaker
Somebody stole our website. Oh no, whatever shall we do?
00:02:01
Speaker
I mean, I guess you could go to the new website, http colon slash slash breaking math podcast dot a P P with no www for all you old timers. So breaking me up podcast dot app. I mean, if you're into that sort of thing.
00:02:22
Speaker
Hey, Breaking Math fans. First, I want to thank you for listening. I have an important message for everyone. You can start your own podcast right now with Anchor. Anchor lets you create and distribute your own podcast. Just get an idea, record, and upload. It's just that easy. Anyone can do it. I'm on my way to accomplishing my dream, and you can too. Just get on your device's app store and download Anchor. It contains everything you need to make a podcast. With Anchor, you can put your podcast on all the big platforms.
00:02:50
Speaker
Apple Podcast, Spotify, Google Podcast, Amazon, and more. Reach the whole world with Anchor. Best of all, Anchor is free. You have nothing to lose with a free platform. Download the Anchor app or go to anchor.fm to get started.
Exploration of Arithmetic Proofs and Peano's Axioms
00:03:17
Speaker
This is Sophia, and you're listening to a Breaking Math Problem episode. You may have noticed that we have the P1 label. This episode is called P1 Piano Edition, and it's an experimental format. Most of our stuff, we work on a whole topic. For example, for vintage formulas, we talk about elitism in math.
00:03:35
Speaker
But today we're going to talk specifically about piano's axioms and more specifically how to prove that N plus M equals M plus N. Before we get into all that, we have our guest on and it is my mom again, Diane Baca. Hello. You read the Wikipedia page and you said it was kind of dense.
00:03:53
Speaker
Yeah because it looked like it was calculus based or set based or something and that just was a little intimidating for somebody like me who doesn't have a whole lot of math in their background. Oh yeah and set theory is something that's really simple. It's just a collection of things or like in a way it's a collection of things. It generalizes the idea of a collection of things.
00:04:15
Speaker
Can I just say what I think is funny? Sure. It's mathy. It's so mathy to say n plus m equals m plus n because it feels like everybody knows that n plus m equals m plus n. Okay, which I think is always funny that math people need to prove things like, you know, n plus m equals n plus n because it seems so obvious.
00:04:35
Speaker
Yeah, I mean all of math is kind of like and science was an attempt to Say that the world is maybe not so obvious I mean ideas are tested by experiment and I met this kind of like ideas are tested by ideas But first was do some plugs you could go to patreon and patreon.com slash breaking math and you can Order and you could join our poster tier on there and we'll give you a poster if you don't want to Support us monthly and you just want a poster you could go to facebook.com slash breaking math podcast
00:05:05
Speaker
And on our store you can buy our poster for $22.46 and that's pie to the E dollars plus shipping and handling. I think it's $12.50 right now but we're working on getting that down. And our Twitter is at BreakingMathPod and our website is breakingmathpodcast.com.
00:05:22
Speaker
So yeah, the piano axioms, all of math isn't exactly like that. This is specifically formalism that we're talking about today. And it was in the early 1800s, it was an attempt to reduce math to a series of symbols and a series of rules that govern those symbols. And it's been shown by Godel
00:05:42
Speaker
that it is impossible to do this with the integers. So pianos axioms don't actually describe all the integers, like they don't describe all the things that integers can do. There are things that are true about pianos numbers that cannot be proven in first order logic or second order logic. Oh, okay. Okay. So what's first order logic and second order logic?
00:06:03
Speaker
First order logic is like if then that kind of thing second order logic is like given a collection of things so for example first order logic might be like for all sandwiches for all sandwiches the middle of the sandwich is not the bread of the sandwich or something dumb like that how about if it rains then you're wet if your rains if it rains and you're outside then you're wet
00:06:26
Speaker
Okay, that works. But a second-order logic is like saying, given all the rainy days that can exist, there are perfect umbrellas for each one of those rainy things. So it's like talking about a whole set instead of just the individual variables. It's like running instead of standing.
00:06:48
Speaker
Okay, I think I get it then so it'll become more obvious as stuff goes on Okay, so basically like set notation like we're talking about a group of things but the individuals within the group is that kind of Yeah, like well, for example the axiom of induction which we'll get into is second order logic. Okay
00:07:07
Speaker
Are you going to put this up on the website to like have like, you should have like a step by step thing on the website to just show this particular proof just alone. Yeah. Actually, if you go to breakingmathpodcast.com slash papers at HTML, you can find all this there. So we're going to start with the axioms and the first one is a very simple and you're going to be like, what is a zero is a natural number. Okay. That's the only one that we actually define is zero.
00:07:37
Speaker
simply because we need to say that nothing exists and we need a symbol for that. Well we have to start somewhere because the natural numbers before actually we should probably talk about those the natural numbers are zero one two all the way to infinity. I thought natural numbers were one to infinity and whole numbers it contained zero. Um that is a really old definition I think it's like 70 80 years old. Ah really? Like I mean I
Understanding Induction and Logical Proofs in Math
00:08:00
Speaker
just taught it last week.
00:08:02
Speaker
No, yeah. I mean, it's the unnatural numbers. It really doesn't matter if it starts at zero or one. The original natural numbers, they started at one. Now they are usually taught starting at zero. Okay. So it honestly doesn't matter. Either way, we just needed to define zero. Okay. And it's useful to define addition as we'll see in a second. So for all natural, this is the axiom number two, and we're going to go through all 10. Okay. For all natural numbers X, X equals X.
00:08:30
Speaker
Wait, before we go through this might be helpful. I'm just curious. Did we already say this? We're going to go through all 10 rules, all 10 axioms. Yeah. An axiom is like a rule. Yes. An axiom is a foundational concept. Okay. So for example, these are just things that, that you can't argue that are taken as literal truth. And then you build the system for these acts. Piano wanted to do this because he was trying to explain how to work with integers. Well, not necessarily just how to work with integers, but
00:08:55
Speaker
So in a formal system such as this, a proof is derived from the axioms. The axioms are manipulated together and you derive what's called a sentence from the axioms. And a sentence is the same thing as a proof. So if you arrive at something, then you have proven it.
00:09:18
Speaker
Now, there's two properties that a system can have. In an axiomatic system, something can be complete or it can be consistent, but it can't be both if it's sufficiently complex. So if it's complete, that means that it describes all the properties. Consistent means that you can't prove that false equals true.
00:09:38
Speaker
Because if you could prove false equals true, you could derive any proof. So if for example the piano's axioms could be combined in some way which they wouldn't be consistent, which is false, they were proven to be consistent with logic outside of the axioms by necessity, that means that there are certain theorems that cannot be proven from the axioms.
00:10:04
Speaker
Okay, but they just need the axioms in order to build this system of explanation or proof.
00:10:10
Speaker
Yeah, I mean, one thing I want to get across too is that formalism is no longer a very big, like a very big goal for math. At the time, it was thought that math could be described using just axioms completely. And basically, Godel's incompleteness theorem, which we talked about, is that basically the system can either be complete or it could be consistent, but it can't be both. And of course, an inconsistent system is more useless than an incomplete system.
00:10:39
Speaker
OK. Because then it doesn't prove anything. OK. And also, so now we're going to parse the what I said was obviously for all natural numbers x, x equals x. So just we're going to be using talking like this for a while. So we want to discuss what it means. For all something, there is a condition. Means like you could say something like for all fish, for all fish x, x has gills.
00:11:07
Speaker
Okay. Basically it's just describing it. Uh-huh. And you could, the opposite of for all is a does not exist.
00:11:15
Speaker
So for all fish, wings do not exist? So let's say your statement is for all fish. Mine was not probably a good example. Well, I mean for all fish X, X has wings. The opposite of that is there exists a fish such as the fish does not have wings. This is first-order logic.
00:11:37
Speaker
Okay. And also in second of all we said for all natural numbers x, x equals x. But we can make that even simpler by saying for all x, x is a natural number implies that x equals x. Okay, actually I like that better.
00:11:53
Speaker
Yeah, but it's a lot longer. So you just say, for all natural numbers x, so for all people x, x has a head. For all fish, that's a true statement, right? Yes. For all fish, fish have lungs is an untrue statement. Agreed. But there exists a fish with lungs. Such that a fish has lungs is true.
00:12:15
Speaker
Yes. And those two are not opposite statements because the opposite of a true statement can't be false. So the opposite of the other one would be of the incorrect statement. No, for all fish, fish have lungs. I feel like it's sort of the rectangle square thing. Or it would be that there is not exist a fish that has lungs.
00:12:36
Speaker
okay for me it feels like we're talking about the rectangle square thing like all squares are rectangles but i mean all rectangles are not squares kind of thing it's related to that so i mean just to get the subtle distinction uh there's a true statement there's a false statement for all fish fish have lungs there's the opposite of that statement which is uh there exists a fish such as the fish does not have lungs then there's a uh then there's the but that one's true too
00:13:07
Speaker
That one's true. The first one is false. All fish have lungs. Right. Yeah, that's false. The opposite of that is some fish don't have lungs, which can be written as there exists a fish X such that X has lungs. Yes. And then you have the completely different statements. There exists a fish with lungs. Okay.
00:13:29
Speaker
and for all, which is true, and the opposite of that is all fish don't have lungs, which is false. So you can see the subtle distinction between those four. Okay. All right. So X equals X. That seems pretty obvious, but we have to define it. Otherwise we don't know what equals Y. Okay. All right. Next one. Number three, for all our natural numbers X and Y, X equals Y means Y equals X.
00:13:56
Speaker
Okay. So like for, for, so two equals two plus three equals three plus two means three plus two equals two plus three. It's reflexive basically.
00:14:07
Speaker
Exactly. It makes sense. Yeah. And then we have basically the transitive property of equality. So for x, y, and z, x equals y and y equals z means x equals z. Of course. Yeah. So if I am the same age as Taft and Taft is the same age as Leonidas, then I'm the same age as Leonidas. Sure. For all initial numbers x, this is the next one. If y equals x, then y is a natural number.
00:14:37
Speaker
because x is a natural number so therefore y is a natural number yeah yeah all natural everything that's equal to a natural number is a natural number okay then the next one defines the what's called the successor function for all natural numbers x the successor of x is a natural number so the successor function is just plus one exactly so the successor of 23 is 24 successor of zero is one
00:15:01
Speaker
And we're not dealing with negative numbers, right? So we're just in the natural numbers, correct? Yeah, exactly. Okay, either way. I mean, either way it's plus one, but still. So the next one says that if we have two natural numbers, they're equal if and only if the successors are equal. Okay, that makes sense to me. Yeah, so like if we know that like two numbers like are followed by the same number, they're the same number.
00:15:25
Speaker
Yeah, I mean, but basically you're just like five is equal to five, then six is equal to six. Yeah, exactly. And vice versa. For all natural numbers X, the successor of X is not zero. That would make sense because even if X were zero, then the successor would be one.
00:15:41
Speaker
Yeah, because the successor of negative 1 could be said to be 0, but negative 1 is not a natural number. Exactly. And the last one is, and these so far have all been first-order logic. This last one is going to be second-order logic. And it says, for a given set k, if 0 is in k, and for all natural numbers x, if x is in k implies that successor of x is in k, then all natural numbers are in k. So let me break that down a little bit.
00:16:12
Speaker
So we have, we have a, let me explain how induction is used first and then, uh, explain how this, uh, this describes it. So if we want to prove that all dominoes fall over, if you topple the first domino, um, how would we prove that using induction? You would say if the second one falls over, when the first one hits it, then when we start only with the first one,
00:16:38
Speaker
Well, if we tipped, but it has to have an action after it. Yes. Oh, yeah. Well, but we'll see. That's the second part. So. Oh, so if we tip the first domino over and it knocks the second one over. Almost even more elementary than that. OK, go ahead. So the statement is all dominoes fall over if we tip the first one over, right? Oh, so we need to say if we tip the first one over. So does the first domino fall over? Yes or no?
00:17:03
Speaker
Yeah. Cause we tipped it. Exactly. Oh, if we tip the first one over, then it falls. Yeah. Okay. Got it.
00:17:09
Speaker
Now let's say that the nth domino falls over. The action that that has is to knock down the n plus first domino, therefore all dominoes fall. So basically it's saying that if you could show that the first one is true and that for any one that you pick that the next one will be true based on that one being true, then all of them are true.
Examining the Commutative Property and Proof Techniques
00:17:33
Speaker
Dominoes really helped me figure that out.
00:17:35
Speaker
You never really considered the case where the dominoes weren't close enough and so the domino fell and it hit the ground and it did not get over. Well a property of one of the axioms in our domino little universe is that dominoes not go for dominoes invariably.
00:17:50
Speaker
Okay, and that they're equidistant from each other so that you said them the right distance apart They're definitely not gonna have that space to fall. Well, you doubt be Implementing implementing the axiom in something else embedding it. Okay, so that would ensure that the axioms are followed But it's a correlation. It's a homeomorphism basically, in other words, you're just saying if you tip over the domino it will fall and therefore
00:18:16
Speaker
If you tip over the first one, it'll fall because it's tipped over. If the second, if the nth one is tipped over, then the next one will fall over, therefore all of them fall. Got it, okay.
00:18:26
Speaker
And so let's look at the definition again. For a given set k, if 0 is in k, and for all natural numbers x, if x is in k implies that the successor of x is in k, then all natural numbers are in k. Yes. So k here is the statement. So let's say if some number is in k, then it means that domino number n falls over. So 0 is in k because the 0th domino does fall over because of the tipping property.
00:18:57
Speaker
And if n is in K, that means that the nth domino fell over, right? Which means that the n plus first domino fell over, which means that n plus 1 is in K, therefore all natural numbers are in K, which translates to all dominoes fell over. Interesting. I feel like it's pretty easy to understand without the domino analogy, personally. Maybe I'm thinking it's too easy.
00:19:20
Speaker
like in other words basically you have a set K, K contains X, X has already been defined as a natural number and we're saying that X plus 1 is also in the set so therefore... Well let's do a little bit more of a complex example then maybe that'll drive it home. Okay. So the sum of 1 to n is n times n plus 1 all over 2, right? Oh my gosh that was a big ol' huge jump but yes I do know that. I gave that to you when you were 7, do you remember that? I do. Okay.
00:19:51
Speaker
I gave that to you to solve. Figure out this formula. Anyway, or create the formula, I think I gave it to you. Anyway. It was done on many paper plates. But let's test out this formula with the first natural number, which is 0, right? Yes. So n times n plus 1 over 2 is 0 times 1 over 2, right? Exactly. Which is 0. Of course. Is 0 the sum of all natural numbers up to 0?
00:20:15
Speaker
Yes. Yeah, so it works for zero, right? Yeah. So now let's suppose that the sum up to n is equal to our formula, right? Okay. So this is the sum of all numbers up to n, right? Okay. So the next term, we would have to add n plus 1 to that. Yes.
00:20:32
Speaker
And if we do that, then the whole thing factors into n plus 1 times n plus 2 over 2, which is equal to n plus 1 times n plus 1 plus 1 over 2, which is our formula for n plus 1. Therefore, it describes the next number. And therefore, we have proven by induction that this formula is valid. Okay, okay.
00:20:57
Speaker
And here k is the set of natural numbers such that one plus dot dot dot two n is equal to our formula. So that is how we define k. But we're not going to define k in all of these, we're just going to use the intuitive way of doing it. I just wanted to drive home that it is possible using second order logic to define induction.
00:21:24
Speaker
So now let's define addition real quick because we're going to prove that n plus n equals n plus m, right? Yes. So we have to define addition. So and these are the only the last two definitions that we have. So for all natural numbers n, n plus zero equals n. Uh-huh. That seems pretty self-explanatory. Sure, that's identity. And for all natural numbers m and n, m plus a successor of n is equal to the successor of quantity m plus n.
00:21:55
Speaker
Okay. So like two plus three is equal to two plus two plus one is what it's saying. Okay. So, and, and, uh, that's useful for addition because let's say we're doing two plus two, right? Wait, wait. Can you say that again? So let me do it with two plus two. Wait, would do the, do the M plus N thing again, plus the successor of N, which is going to be N plus one. No, no. Emma plus a successor of N. Yeah. Isn't the successor of N and plus one.
00:22:23
Speaker
You said the whole thing was n plus 1. Oh, no. No, I meant that I meant like n plus n plus 1 is the successor of n plus n. Oh Okay. Yes, that makes sense. Yeah, so Yeah, so like so in useful for addition because let's say we're doing 2 plus 2, right? Okay 2 is the successor of 1, right? Yes So 2 plus 2 is a successor of 2 plus 1 Now let's do 2 plus 1 2 plus 1 is equals to 2 plus is a successor of 2 plus 0 and
00:22:53
Speaker
Yes, yes, yes. And then so the successor... Go forwards. Go forward. Yeah, and then we're going forward now that we're down to this. Oh, I see. Okay. And then successor that we have the successor of two plus, then two plus zero equals two, right? Two plus zero equals two, yes. And the successor of two is? Three. And the successor of three is? Four. And so we have our answer. Oh my God, that's so weird. We went backward to go forward.
00:23:17
Speaker
Yeah, that's the cool thing about axioms is that it's just manipulating symbols around. That's what makes it so attractive because if Godel's incompleteness terms didn't exist, we could just have a math machine basically. That was what Kilbert was working on and hated Godel for. Math wars.
00:23:37
Speaker
Oh yeah, math wars are totally a thing. We should do an episode on that. You actually should. Or do like thievery or plagiarism or whatever. There's probably lots of that too. So we know that n plus zero equals n, right? Yes. How would we prove that zero plus n equals n? Probably with another axiom. No, no more axioms, no more definitions. We have everything we need.
00:24:05
Speaker
induction. And the reason why is because the statement is zero plus n equals n, right?
00:24:12
Speaker
Yes, but we proved that zero plus n equals n just a step before. We proved that n plus zero equals n. We showed that as a definition. Okay, so we proved that n plus zero equals n. And now you're saying y is zero plus n equal to n. Yeah. Okay, so how are you going to induce that? So zero plus zero equals zero, right? That's, we know that because we just, we switched the zero that we're focusing on. Yes, yes, yes. I got it. Okay. And then you're going to just change the order.
00:24:40
Speaker
Yeah. And so, well, I mean, not necessarily change the order because anything plus zero is that thing, right? Yes. So we are now just using the N as the zero that we're adding to the zero that's always there. Yes. So you just substituted zero for N, right? Yeah. So it works for zero. Zero equals zero. So now let's suppose that zero plus N equals N. Okay.
00:25:03
Speaker
Now we know that if x equals y, if and only if successor of x equals successor of y, right? Right. So that means that if we have 0 plus n equals n, then the successor of 0 plus n is equal to the successor of n.
The Impact of Godel's Theorem on Mathematical Formalism
00:25:18
Speaker
OK. And remember that the successor of x plus y is equal to x plus the successor of y. Say that one again.
00:25:26
Speaker
Remember that x plus the successor of y is equal to the successor of x plus y in my first class? Yes. So what would the successor of 0 plus n be? The successor... That's a hard word to say. The successor of n. 0 plus the successor of n. Yes. Which is very important. Sorry, I didn't put the 0 because I figured it was... You know what I mean? Yeah, you have to be very careful with these. Yes, okay. 0 plus the successor... I can't say that successor of n... There you go.
00:25:55
Speaker
And so now we have the statement 0 plus the successor of n equals the successor of n. So we've proven the statement for the successor of n based on n. Exactly. And now we've proven that 0 plus n equals n by induction. Exactly. OK, gotcha.
00:26:10
Speaker
We're going to prove that the successor of M plus N is equal to the successor of M plus N. So let's say M is equal to 10 and N is 5. We're saying that the successor of 10 plus 5 is equal to the successor of 15. Which is the same thing, yes. And remember before we would have been saying that 10 plus 6 is equal to the successor of 15.
00:26:32
Speaker
Okay, so we're just reversed order. We just reverse ordered this thing. Yeah. Okay. Before it was M plus successor of N. Yes. Equals the successor of M plus N, but now it's the opposite. Okay. Okay. I got it. We just switched it around. So basically 11 plus five is equal equal to 16. Yeah. Okay. So let's prove it for zero and let's do induction on N. So the successor of M plus zero is equal to what?
00:26:59
Speaker
M plus one. Yeah, the successor of M. Oh, I need to use the words, not the one. Because remember we're doing informal, informal stuff. Sometimes when you're doing fractions, you use two divided signs because you really want to drive home the point that this isn't division. It is something that symbolizes division. Okay. Okay. Think I got it. You know what I mean? It's all kind of a little bit new for me, but yeah.
00:27:21
Speaker
Yeah, and of course it's equal to the successor of M because by the axiom, N plus zero equals N, right? Sure. So now we've proven it for N. So now let's prove it. And this one is kind of a involved one. So the successor of M plus N equals a successor of M plus N. We're going to prove, we're going to suppose that we're going to assume it's true for a second. Sure. Wouldn't we already prove it? No, we haven't proved it yet. We're proving it now.
00:27:47
Speaker
OK, I think I just misunderstood where the successor went. That's OK. Well, successor of m plus n is equal to successor of m plus n. Got it. OK. So remember that if x equals y, then the successor of x equals the successor of y? Yes. So that means that we can set the sites equal to one another using the successor function.
00:28:10
Speaker
Oh, okay. So we're going to work both sides of the equal sign basically. Oh, okay. So now we have the successor of the successor of M plus N is equal to the successor of the successor of M plus N. Okay. Basically saying that like one plus 11 plus five is equal to one plus one plus 15. If M is 10 and N is five, for example,
00:28:36
Speaker
I was going to say, okay, I was, I just want to make sure if you said 11, yeah, I got it now. If it, if it's 10 and five, that makes sense. Yeah. Okay. So now let's look at the left-hand side, the successor of quantity, the successor of M plus N end quantity. What we can do with that is because we know that the successor of X plus, uh, Y is equal to X plus the successor of Y, right? Yeah. So if we set X to a successor of M, right? Okay. So now we're substituting in.
00:29:04
Speaker
Yeah. Oh yeah. You always substitute in when you're doing axioms like, oh, this whole time you've been substituting. Right. So we know that. So if the successor of the successor of M plus N, we take that, we know that that's equal to the successor of M plus successor of N, right? Uh-huh. Because, uh, for example, just to give an example of M is 10, the successor of M is what? Uh, well, wait, a real example, 11.
00:29:29
Speaker
11 and then if n is 5 then 11 plus 5 is well 16 and the successor of 16 is 17.
00:29:37
Speaker
Yeah. And so in the next line down, we have the successor of M plus the successor of N.
Modern Applications and Perspectives on Axioms
00:29:44
Speaker
And they're the equal because if we set X to the successor of M and then Y to N, we just move the successor function from the outside of everything to the inside. So we have the successor of M plus the successor of N is equal to what the right-hand side was already, which is the successor of the successor of M plus N.
00:30:05
Speaker
Okay, so basically you're just pulling off that X, the successor of X and successor of Y initially. Yeah, you're substituting in various things in order to get to that point. Yeah, and now we work on the right hand side. On the inside of the successor function, we have successor of M plus N, right? Which we've proven is equal to the successor of M plus N.
00:30:26
Speaker
Yeah, so now we have the statements the successor of M plus the successor of N is equal to the successor of the successor of M plus N. Yeah, this gets a little tedious with all this successor stuff. Oh, it really does. This kind of math is extremely tedious.
00:30:45
Speaker
Yeah, and I would assume notation is incredibly important. It's visually though. It's not only important, it's literally the whole thing. The whole thing, because these can be represented using numbers, for example. You have a number equal to this function. Okay. Are you telling me that before they decided to write all this down, they used to talk, they used to speak math like this? Only, I mean, in a way. Yeah. I mean, yes, absolutely. Like for example, uh, the words.
00:31:12
Speaker
Yeah, this is the same era where they start where I can remember which mathematician tried to formalize Euclid Because Euclid you know his elements like a point is that which has neither length nor breadth Like it's hard to make that into an axiom. I've tried it's pretty fun. Oh, yeah. Mm-hmm. So well, that's geometry though. Oh
00:31:32
Speaker
Oh yeah, that would be axioms for geometry, and these are axioms for the integers. So in the same way that Euclid defined things for his book, this defines things for a huge number of integer problems. Which is funny, I feel like the easier the understanding, the harder the proof is to understand if that makes sense.
00:31:51
Speaker
Yeah, because it's so intuitive that you don't know you don't really know what you don't really know how to make a statement that's falsifiable. Yeah, because it just seems so obvious that the that the opposite seems to be only the lack of that thing. Exactly. Like to say one plus one is two doesn't even seem like there's ever an argument or even a discussion necessary.
00:32:13
Speaker
Yeah, but this allows us to talk about the integers in a very interesting way. There's also axioms for set theory, and some of them compete with each other. For example, there's the axiom of choice, which says that given an infinite number of sets, you can take one item from each set with a certain property.
00:32:34
Speaker
But if you take the axiom of choice, you can prove that by just cutting up a sphere into different points and moving the parts that you cut up around and then recombining them, you can make two spheres. You never stretch the pieces, you never change the size.
00:32:50
Speaker
That is bizarre. Yeah, right. Oh goodness. Okay. So I assume that, that obviously wouldn't, I mean, they do this for a reason. It's just kind of funny if it's, if it's not particularly your discipline of study, it just seems funny. You know what I mean? But I get, I get that it has to be done for a purpose.
00:33:10
Speaker
Well, I mean, it was done for a purpose for a while. Again, most of math now isn't trying to create some overarching all of math kind of thing. It's just people solving problems. Probably because people already did that stuff.
00:33:23
Speaker
Well, they tried to, and Goodell proved that it's impossible. Oh, okay, okay. Like this is like the Principia Mathematica, or Principia Mathematica, I don't know how to pronounce it, by Whitehead and Russell. I think on page like 200, they proved that two plus two equals four, one plus one equals two. Yeah. But, and it just builds up to that. And that was an attempt to codify all mathematics. And people thought that it actually did, but then Goodell was like, nope. Oh, okay. Okay. And everybody kind of tore their hats in frustration.
00:33:54
Speaker
Yeah, as you do. They were wearing hats? Yeah. What year was this? I mean, this would have been like 1911. Okay, they'd be wearing hats, but not inside. So anyway, now we're going to finally prove what we set out to prove, that m plus n equals n plus m. Finally. Yeah, right. So m plus zero equals what?
00:34:19
Speaker
Yeah, by the rule n plus 0 equals n. Sure. And then we proved, remember the first proof we did was 0 plus n equals n? Yes. So 0 plus m equals m. And therefore, m plus 0 equals 0 plus m by basically the properties of equality that we defined. I was like, is it transitive property-ish?
00:34:39
Speaker
Yeah, because we, I mean, we proved that M plus zero equals M and zero plus M equals M. So if X equals Y and Y equals Z, then X equals Z. That's what we're using. Transitive property. Yeah. Okay. Gotcha. So now suppose that M plus N equals N plus M. That means that the successors are equal, right? Since the thing would be equal. Yes, of course.
00:35:00
Speaker
So the successor of M plus N equals the successor of N plus M, right? Right. Now we can break this down on both sides. So the left hand side, we could change that to M plus the successor of N because we proved that the successor of M, but because there's a rule that the successor of M plus N is equal to M plus the successor of N.
00:35:21
Speaker
Wow, that's really getting tedious to say, isn't it? Right, but we only have one more step and we're done. Okay. Finally, we have m plus the successor of n equals the successor of mn plus m. So now we're going to look at the right-hand side, the successor of n plus m. We can make the successor function go in the n by the lemma that we proved. Remember we proved successor of m plus n equals successor of n plus n? Yes.
00:35:48
Speaker
So that means that S of n plus m is equal to the successor of n plus m. Yes. And so now, as soon as we've proven both sides, we've done both sides, we have the statement m plus successor of n is equal to the successor of n plus m. Right. And therefore, we have our thing proven by induction. Exactly. Wow. Long induction.
00:36:13
Speaker
Yeah, I mean this was like I did I've had this proof in my math 401 class and I remember doing it over and over and over again. Is this just because like to get it down, right? Cause there's different ways to do some of these parts. Interesting. So they still teach it. Oh yeah. I mean, uh, turns towel has a very good book on analysis. I can't remember the title right now, but he discusses this in there.
00:36:36
Speaker
Interesting. So it's not ancient. It's not like as if, it's not like as if nobody talks about it. It's just. Oh yeah. It's just that we're not trying to actively create rules that describe everything anymore. We're just using rules and understanding them for what they are as human tools. Okay. Got it. So that was the first problem episode of Breaking Math. And of course we had on Diane Baca. Anything you want to prove? Anyone think about a plug?
00:37:06
Speaker
Um, no, I don't, but I do think it's really helpful to see it. Oh yeah. And if you'd like to see it, you can go to breaking math, podcast.com slash papers.html and it'll be up there. Awesome. Signing off.