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19: Tune of the Hickory Stick (Beginning to Intermediate Math Education)
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The art of mathematics has proven, over the millennia, to be a practical as well as beautiful pursuit. This has required us to use results from math in our daily lives, and there's one thing that has always been true of humanity: we like to do things as easily as possible. Therefore, some very peculiar and interesting mental connections have been developed for the proliferation of this sort of paramathematical skill. What we're talking about when we say "mental connections" is the cerebral process of doing arithmetic and algebra. So who invented arithmetic? How are algebra and arithmetic related? And how have they changed over the years?
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Transcript
Introduction to Mathematics: Practical and Beautiful
00:00:00
Speaker
The art of mathematics has proven, over the millennia, to be a practical as well as beautiful pursuit. This has required us to use results from math in our daily lives, and there's one thing that has always been true of humanity. We like to do things as easily as possible.
Arithmetic and Algebra: Inventions and Historical Perspectives
00:00:13
Speaker
Therefore, some very peculiar and interesting mental connections have developed for the proliferation of this sort of paramathematical skill.
00:00:20
Speaker
What we're talking about when we say mental connections is a cerebral process of doing arithmetic and algebra. So who invented arithmetic? How are algebra and arithmetic related? And how have they changed over the years? All this and more on this episode of Breaking Math.
Exploring Daily Mathematical Applications
00:00:41
Speaker
Episode 19, Tune of the Hickory Stick.
00:00:45
Speaker
I'm Jonathan Baca, and you're listening to Breaking Math. Gabriel's not here because he has other duties. With me, I have on. Hi, I'm Gideon. And hi, I'm John. And we're going to talk about, obviously, math in our daily lives. So I'm going to pose a question to both of you. Gideon, first, when did you use math today? Well, today I did my math homework. You're not in elementary school, obviously, so you're not being trained to use math that we use every day.
00:01:11
Speaker
It's important that we're teaching math, and can you give an example of something that you did in your math homework today that wasn't pure math, that was maybe a little bit tedious? Tedious? Mmm, that's a good question. Like symbolic manipulation, things like that, where you just have to copy something. Well, we're doing quadratics now, and I find that a little bit tedious due to the algorithmic nature of the problems. Algorithmic nature, can you explain?
Math's Symbolic Power: Emotion and Music
00:01:35
Speaker
Well, it's just that with quadratic equations, you just follow a very simple formula. Oh, I see what you mean. Yeah. The algorithmic way of solving it. Exactly. The proof in a box kind of thing. Exactly. And it's just that you have to repeat these problems in a rote way over and over again until you have the formulas memorized, basically. Yeah. And how about you, John? Same question.
00:01:57
Speaker
First what I did was I compared my gas mileage actually and the amount of miles I had driven on my car to actually see the amount that I had gone over my oil change today. That on top of rough estimating my gas, my general gas tank to see how much more miles I could get out of it.
00:02:16
Speaker
So you did at least division and subtraction? At least those two, and then I mean, I also went into my wallet to just count how much money I had today if that counts as mathematics. Yeah, of course it does. I mean, it's not really philosophy. Yeah, exactly. Even stopping at the convenience store today required some simple mathematics.
00:02:34
Speaker
Yeah, completely. So there's math that we use all the time and we're gonna just rattle off a few examples. So in finances, we talked about before the show started, we use bills, we calculate checks and tips, we do multiplication for that. We do budgets where we do estimations and we have to extrapolate. Recreationally, we do things like ELO scoring and chess and every form of gambling like poker or blackjack,
00:03:02
Speaker
Yeah, they all require counting at least, if not more complicated mathematics if you get into the meat of the game. I mean, craps alone, it takes a lot of statistics work. Yeah, and we use markers like time. What kind of time markers do we have? The hours through the day, of course, is like communicating to someone on the East Coast or the West Coast from here in New Mexico requires some civil edition of subtraction. Yeah, the military time system with the 24 hour time clock.
00:03:28
Speaker
as well as age and date, any calendar system really, distance, we have miles, we have volume, we have temperature, all these things just mark concepts. So it's obvious to me and tell me if you agree that mathematics, or at least the representation of numbers,
00:03:44
Speaker
can somehow symbolically evoke emotions or responses? Oh, of course. I completely agree. I mean, you can even see it in a musical, how they use mathematics and numbers in music. Oh yeah, completely. The first thing we're going to talk about, actually good thing you brought up the algorithmic approach to get in,
Understanding Mathematical Algorithms
00:04:03
Speaker
because we're going to talk about addition as an algorithm. We're going to use the word algorithm a lot in this episode, and an algorithm is just a set of instructions. It's worth noting that it is impossible to design an algorithm that invents all of mathematics. It was proven by Godel and his incompleteness theorem.
00:04:20
Speaker
So the math that we do from day to day is usually algorithmic math where we have proven that an algorithm works in certain situations. And so we're going to talk about addition as an algorithm real quick. Of course, everybody here knows how to add and I'll just describe the process as though it were an algorithm, which it is.
00:04:37
Speaker
You start at the very right hand side of the number. You take the two numerals that you see at the right hand side of that number or the number that you're currently on. You add those together using a table that you have in your head. So six plus three is zero nine or nine. Seven plus six is one three. You write the first number above the next one over, carry down the next one, and then you repeat the process but add three numbers each time.
00:05:02
Speaker
and you just go left, left, left, left, left, until there's no left to go, and then you're done. That sounds like something a computer can and obviously does do, right? Yeah, definitely. Yeah. That's an important thing. So now that I've kind of warmed it up talking about addition, I'd like as a mental exercise for the guests, Gideon and John, and also to sort of describe how difficult algorithms can be to come up with,
00:05:26
Speaker
We're going to describe multiplication as an algorithm in the manner that I just described. All right. Let's figure this out. Okay. All right. Let's figure this out. All right. So you would start on the far right of the problem again. You would be looking at the answer and then taking the two numerals or more that are directly to the left of that, right? And look at the two numerals being multiplied. And then you would mentally put an image of your mind.
00:05:51
Speaker
of what those objects were being multiplied. How would you describe the next part? I don't know. This is an interesting problem to think about. Like just the concept of multiplication is actually quite a jump to make. It is actually. It takes a lot of imagery and you have to translate that into words. It's very difficult innately, I would believe.
00:06:13
Speaker
Yeah. And I mean, one thing that I do when I come up with algorithms as part of my job is I think, what would I tell if I were teaching somebody this? Like, what would I tell them to do? That's essentially the same thing. Because you want to make it the simplest process you can in a way in order to make it understandable.
00:06:32
Speaker
Yeah, but that's the difficulty of algorithms is that teaching is much more difficult than being taught to. That makes sense. And you learn the information much better, obviously that way. If you teach versus you're taught. With that in mind, would you guys like to finish the multiplication algorithm? Okay.
00:06:51
Speaker
um where had we left off we took the two objects to the left of it and you multiply them in your mind you make a mental image of the first object on the left that's the object you're multiplying by then the object on the right next to that you're going to be multiplying by the original number in your mind you're going to separate
00:07:14
Speaker
it's a tough one yeah it is a tough one right and of course for the listeners we're not going to finish this exercise it would probably take quite a while but um it's it's it really is very difficult to uh put into words how is it that we can do these algorithms in our minds so easily
00:07:32
Speaker
Not just in our minds, I'm going to describe real quick the Sotoban, the Japanese abacus basically. What's interesting about this abacus is that masters of the craft of the abacus can often use an invisible abacus instead of a real abacus. Okay. I've heard of this. So masters of the abacus can use basically an invisible imaginary abacus. Oh yeah. Is that because they have just the physical representation of the abacus so they can add that imagery into their mind?
00:08:02
Speaker
Well, really it's because the difference between coming up with an algorithm and running an algorithm is vast. The human mind can come up with algorithms, for example. However, we don't have any computer that can simulate a human mind yet. The human mind is so much more complicated just in structure.
00:08:19
Speaker
So it shows that there's a difference between coming up with an algorithm and running an algorithm. And my personal belief that it's muscle memory and things like that, that actually do the running when these people use the invisible abacus. Oh, it's amazing how well the human nervous system has evolved to run algorithms, like basically without any conscious awareness at all. And you can see algorithms in every facet of our daily life. Even speech patterns could technically be an algorithm. Is that correct?
00:08:48
Speaker
They can be dissected using algorithms and they maybe are constructed using algorithms. So yeah, they have algorithms attached to them in a very deep way. And with the soda bun, you can have algorithms to calculate things like square and cubed roots, for example, just using pieces of wood on dowels.
00:09:07
Speaker
Or an imaginary piece of wood within your body. Yeah. Right. Exactly. Then they actually have competitions between soda-bun masters and people who use desk calculators because they got that efficient. The craft of the soda-bun got so advanced. Who won? I believe that they were basically tied until electronic calculators. Oh, that makes sense.
The Concept of Zero: Historical and Philosophical Impact
00:09:29
Speaker
In India, around 1500 years ago, a new system of numerical representation was invented. This system of representation came to be known as the Hindu-Arabic numerals, which include 1 through 9, but most importantly 0. 0 was used to mean nothingness in a place-value system, which was a radical idea which would change the world. What are Arabic numerals?
00:09:52
Speaker
Well, there's one, two, three. What's interesting about that is, yes, you're correct, but you started with one. Yeah, I guess actually zero would be correct, wouldn't it? It might be correct, but it's just showing that if you're doing an increasing sequence, you started with one, which is the natural instinct. That's why zero was such a difficult concept. Even you don't have zero cemented in your head because nobody does. That's right.
00:10:17
Speaker
Okay, so we're gonna back up a little bit, go to ancient Babylon. They had a place value system, but they did not have a zero. They just left a blank space. And that's a crucial distinction. It's obviously up to debate by anybody. So what are you guys views on this? It's just interesting to me how it took so long for humanity to move from a blank space to the concept of blankness.
00:10:42
Speaker
Well, not only that, we kind of lost the Babylonian numerals in a way. We used the Roman numerals and Greek numerals, which are basically Roman numerals in a different alphabet. That's really interesting. It's like the lack of a number became itself an object. Like it went from being a lack to being the symbol for a lack of
00:11:04
Speaker
anything but we almost need to reach a point to actually have the idea of nothingness an idea of zero before anything else into that we had to have an original counting system we would have to have the number one right
00:11:18
Speaker
Oh yeah, that's definitely essential. The Mayans also came up with zero. But one thing about zero that's interesting is that Mayans used like a dot to mean one, three dots and two bars to mean 13, because each bar was five. And they grouped things in 20s, so it's not like we grouped things in 10, so we do one 10, 100 now, but the Mayans did one 20, 400. If they weren't doing religious work, that's a completely different number system.
00:11:43
Speaker
So why do you think that humans took so long to create the idea of zero? Did it just take that long to have the philosophical idea of nothing? I mean, if you think about it, the idea of marking nothingness mark is always a discovery in itself. For example, the equal sign was an invention of the 1600s and it was based off of two parallel lines because he said they know two things could be more equal.
00:12:09
Speaker
So inequality is zero difference. The Galois theory is seen where polynomials are zero and treating that as a separate thing. And that was revolutionary. So every zero that we find really is revolutionary. And maybe, I mean, even in philosophy, when Descartes was doubting everything, that's the zero of the self. I mean, zeros show up everywhere.
00:12:34
Speaker
Right. It's interesting. It seems to me that the point at which humanity developed the idea of zero could have been the same point that humanity developed mathematics into an abstraction because the idea of zero itself is an abstraction. Well, does it inherently lead into abstraction, the idea of nothing?
00:12:54
Speaker
Well, you have to be careful with different levels of abstraction. For example, a mark on a bone to mean I killed a buffalo is very abstract. What is the similarity between a mark on a bone and a gigantic animal? Right. Definitely. So this is, and that's one level of abstraction and then a whole other level of abstraction would be to have pure mathematics, only numbers. They don't represent buffalo. They don't represent anything. Only numbers for the sake of numbers.
00:13:24
Speaker
Exactly. And zero by necessity is an abstraction. And then you have extreme abstraction such as in gambling, where there's no meaning to the numbers. And the entire point of the game is to have the lack of meaning to the numbers.
00:13:37
Speaker
Yeah, to, to introduce chaos basically. So the point being that zero was a revolutionary invention for the numbers because it made a place value system strong and cemented.
Arabic Numerals vs. Abacus: A Historical Competition
00:13:51
Speaker
It wasn't a bunch of numbers floating around anymore with spaces between them. They were solid numbers. It gave the line between a number and nothing.
00:13:59
Speaker
Yeah, and so Fibonacci brought the Arabic numerals, or at least popularized them in, I mean, there was Gilbert of Orlock, who I think used them, but Fibonacci really popularized them, and he wrote his Libra Bachi in 1202, so that's about 700 years after the invention in India, so it took 700 years to travel over there. Tax men in some cities even feared these numbers, because they thought that people were gonna use them to do financial sorcery, basically.
00:14:29
Speaker
Huh. It almost adds mysticism to the history of mathematics.
00:14:34
Speaker
Oh yeah, I mean, mathematics arguably was invented by mystics who wanted to divine things or just predict the future, and were successful when they did the thing for flooding, which is basically just counting. Right. I guess we could describe anybody today as an Algorist. What an Algorist is, is someone proficient in using Hindu-Arabic numerals versus an Abbasist, which at the time meant somebody who moved stones around on a sandboard.
00:15:03
Speaker
the abacus kind of devolved in the middle ages. These abacus and algorists would have competitions, and the algorists would win every time because they were better numerals. Even Gilbert of Orlock, when he used the numerals, people thought that he was magical because he was able to multiply like 56 by 13 in his head. It's such a simple task by today's standards, but... Yeah, because we came up with these beautiful new algorithms. That's right. And so it's almost an evolution in our minds of how we use these algorithms.
00:15:32
Speaker
Oh, absolutely, yeah. It's interesting that the meme that is the Arabic numeral system has one out in almost a genetic way. Oh yeah, it's used everywhere. Even the Roman alphabet isn't used nearly everywhere at all, which shows that these numbers are efficient.
00:15:50
Speaker
Are there more efficient ways? Maybe, but probably not. From what I understand from information theory, probably not. Information theory, if you listen to the episode entitled TMI, you'll hear about how information represents the universe and represents numbers. The gist of it being that information in the place value system are very one-to-one.
00:16:11
Speaker
And a place value system actually predates what we call the place value system. And we talked about it earlier, it's the abacus. Do you want to explain why the abacus is a place value system? Well, I've used an abacus before, but it's been a long time. What I remember is that the furthest to the left represents one unit. The left or right, it doesn't matter. Okay. Yeah, it wouldn't matter really. The next one would represent the unit of two. Is that right?
00:16:41
Speaker
No, it's 10s after that. Yeah. And then hundreds. Okay. As you go down the rows, huh? Oh, I mean, a soda bond has two pieces on one side of each line and then five on the other or one in four, um, when the small beads represent five each.
00:16:56
Speaker
But it's all the same concept. You have some beads that represent a digit, and then some beads that represent another digit. And these abacuses have been in use since the Egyptian times. But nobody bothered to actually make a place value system based on it. We decided to have ones represented by I's, and 100 represented by C, and things like that, which is absolutely stupid.
00:17:19
Speaker
Definitely. You would think that they would have like latched on to that, the abacus and developed a place value system out of that. And I mean, once you compare it to the Arabic numeral system, I mean, it almost defies logical thinking at a certain point, but we can only have that through the perspective of having the Arabic numeral system from the first place, I suppose. Oh yeah. We're looking at it from like a mountain. Exactly. We stand on the shoulders of giants.
00:17:46
Speaker
Numbers are pure objects. A 6 is not the symbol that we write down when we mean 6. However, we can comprise a representation of numbers with numerals. This shows how abstract numerals really are. However, there's a further level of abstraction that has been discovered, that of algebra.
Algebra and its Evolution
00:18:05
Speaker
Algebra is treating arbitrary symbols, usually as numbers, though we shall see. This extends to any mathematical concept. A symbol to represent symbols. It's little wonder it took a long time to develop.
00:18:19
Speaker
So obviously everybody here is familiar with algebra. Of course. Yeah. I mean, we all go through it in high school, right? In case we had some really young listeners, uh, do you want to describe algebra to a fifth grader? Okay. So the first thing that comes to mind for me would be the substitution of a variable for a number.
00:18:38
Speaker
Yeah, so you can say like X equals five. Exactly. And if X plus one equals four means X equals three, because if X is three, then three plus one equals four. That's algebra, right? Exactly. It took until the ninth century for somebody to make a book that would basically turn into the algebra that you know today, the symbolic algebra. And it was by Muhammad Ibn Moussa Al Khourizmi. Al Khourizmi was a polymath. And one of the books that he wrote,
00:19:06
Speaker
was called, I'm going to try to pronounce this, Al-Kitaab Al-Maktasar Fihisab Al-Jabbar Wal-Makabala. And if you heard in the middle that you heard Al-Jabbar, which is where algebra comes from. Interesting. And that means a compendious book on calculation by completion and balancing. Very lofty title.
00:19:29
Speaker
Here at Breaking Math, what we did, we looked for an excerpt from this book to show you guys. We actually found a translation, but the translation was from the 19th century. So you have a Breaking Math exclusive retranslation from the, not from the Arabic original, because we don't know Arabic here. Apologies to our listeners in the Middle East, but we do know English and we retranslated it from English.
00:19:53
Speaker
And what you're about to hear is a description of an algebraic technique. So an excerpt from the book is as follows. If the instance be 10 minus thing to be multiplied by 10 plus thing, then you say 10 times 10 is 100. And negative thing times 10 is 10 things negative. And thing times 10 is 10 things positive.
00:20:27
Speaker
Well, this is about my third or fourth time hearing it and I still get lost by about the third thing.
00:20:32
Speaker
It's not complicated stuff because in modern parlance, that's just 10 minus X times 10 plus X is a hundred minus X squared. That's exactly it. This time I was trying to replace everything with X in my head and seeing what would come out of it.
00:20:48
Speaker
Oh yeah, so actually let's reread that where thing is replaced by x. If the problem is 10 minus x times 10 plus x, then you say 10 times 10 equals 100, and negative x times 10 equals negative 10x, and x times 10 equals 10x, and negative x times x equals negative x to the second power. Therefore, the product is 100 minus x to the second power.
00:21:16
Speaker
Okay, so that's fascinating. He's describing modern algebra there. Yeah, just foiling, right? For the first time, yeah.
00:21:23
Speaker
Well, yeah, I mean, there were like algebra types things before this, like this is not the first time somebody realized that, you know, 10 minus X times X plus X is 100 minus X squared, but it was a revolutionary new way of thinking to think, oh yeah, we have thing in this problem that can be substituted by two. So 10 minus two, which is eight times 10 plus two, which is 12 is 100.
00:21:48
Speaker
minus 2 squared, which is 100 minus 4, which is 96. Definitely. That's like magic. That's a brilliant leap. Yeah, really. Oh yeah, completely. Algebra. I mean, look what algebra led to. Calculus. Calculus is really just symbolic manipulation on the same level as algebra if you're getting into differentiation, you know? See, I haven't gotten to calculus yet. I would completely agree. I'm still a kind of a novice at calculus myself, but I would say that I mean it is a huge mental leap.
00:22:16
Speaker
What's interesting is that I think it was a few centuries later when they actually came up with the X notation, but this is, I mean, the whole book, one leap that it doesn't take is it has all these formulas for like 10 minus X and then two minus X and then all these different things. So then seeing, they didn't say like thing one and then thing two, they weren't Dr. Seuss about it. So they didn't have two separate X and Y like we would have today. Not really in this book as far as, I mean, I might be misspeaking, but I'm just from what I've read.
00:22:46
Speaker
Huh, that's really interesting. So they only had a singular concept of abstraction. Yeah, a formula with one input, really. Wow. Because, I mean, if you looked at lambda calculus, each lambda calculus is a formula with one input. And to have a formula with two inputs, you had to have a formula with one input that outputs another formula with one input, and then input one into that one to get a formula with a number.
00:23:12
Speaker
It really shows how the development of math requires a leap of consciousness for every step, basically. Almost an evolution of consciousness, to use the word again, but like, it really seems like it's an evolution of thought. If you haven't yet, please listen to Consciousness One.
00:23:32
Speaker
If you're going to do anything math-related, you're going to have to be comfortable with algebra. If you're not 100% confident, I want to recommend a place that I think will help you do that easily, and that's brilliant.org. Their Algebra Through Puzzles course makes it fun and engaging for you to learn the intuitions and strategic thinking behind the symbols in algebra. It took 700 years for algebra to be accepted, one I worked through the course in seven days. The chapter that I really want you to check out is called Con Misconceptions, which was released for this episode.
00:23:54
Speaker
and highlight several of the common misconceptions that people make when learning algebra. If you make a mistake in this chapter, review the error and correct your understanding. If you're at a computer, go to brilliant.org slash breakingmath. If you're at a computer, go to brilliant.org slash breakingmath, click on their courses, select algebra through puzzles, and select the last chapter, common misconceptions. So that's brilliant.org slash breakingmath. Not only are you supporting the show, but the first 200 breaking math listeners will get 20% off the annual subscription that we've been using. And now, back to the show.
00:24:23
Speaker
There's a skill that humans and chickens have in common. That thing, amongst others, is the ability to count or at least group objects.
Foundations of Mathematics: Counting and Axioms
00:24:31
Speaker
A chicken can tell if an object is missing from a few as can a human. As the number of objects grow, however, this is harder and harder to do. You would not be able to easily tell if there were 183 marbles one moment and 182 the next. Counting is foundational to mathematics, as we will see shortly.
00:24:49
Speaker
This innately reminds me of a term in psychology, which is the chunking theory. It's a memory technique where you can group up to, I believe, seven objects for humans in your memory at one time. And even groups of groups. I mean, it's all about memorization. I mean, you can have a memory hotel if you want. Yeah, exactly. You can view, visualize it and create how knows how many groups of numbers you would need. Seven objects is one phone number. And how many of these, how many of those do each of us have memorized? Exactly.
00:25:17
Speaker
Yeah, I mean, that's a common example, actually. And it varies from person to person, too. Some people take more effort to memorize seven things, and some people take less. But if you take into account the amount of memorization we have, phone numbers, social security numbers, everything we basically use requires memorization in some level, it almost seems like the original counting system is the foundation of actually this abstract style of thinking.
00:25:43
Speaker
Yeah, absolutely. In a very deep way, you could define mathematics using counting, as was done in the 19th century by piano. He came up with the piano's axioms, and we're going to discuss each one of them a little bit. So the first one is, zero is a natural number. Already we have a little bit of stuff going on here. We start at zero, for example, which is, as we talked about, a revolutionary idea.
00:26:09
Speaker
So in order to continue with the style of thinking, we need to have that foundational conception of zero being a natural number.
00:26:17
Speaker
And not only that, we have the concept of belonging here. Zero belongs to something called the natural numbers. So that's a concept in and of itself. Which is the concept of the set, eventually. Yeah, actually, this is based on set theory. Well, I mean, not directly. You could make set theory versions of pianos, axioms. We're not going to go into those. But yeah, completely. It's a set. The set of all natural numbers. Zero, one, two, three, et cetera. Every natural number equals itself.
00:26:46
Speaker
Okay, that one makes sense. That one's obvious. It's obvious, right? So that's why it's an axiom. Definitely. But here we're saying that we're starting to define equality. We literally don't have equality defined yet. Okay, yeah. So in order for something to be equal, we must originally have the object start out with is almost what it's saying.
00:27:03
Speaker
Yeah, almost exactly. So it's almost ontological, but it's a subtle difference. It's a formal distinction. And what we mean by formal isn't that these numbers put on suits and ties. What we mean is that we're talking about the form of the number. That's what formal means. Form plus a suffix, all. All right, ready for the next one? Yeah, that sounds great. Yeah.
00:27:24
Speaker
If x equals y, then y equals x. Okay, so we're going deeper into the concept of equality there. Now we're equating two separate objects, it sounds like. Yeah, we're showing that things can be equal even if they're not formally exactly the same thing.
00:27:40
Speaker
So we're actually conceptualizing the logic during each step and we're building on that logic each step. Yeah, and as you'll see in the next one, if x equals y and y equals z, then x equals z. So we're showing that equality is now a chain. So we can have things equal to each other and have the first thing equal to the last one with this new rule. Right, we create a chain of events now.
00:28:01
Speaker
Yeah, chain of causality almost with equal symbol. So we have this very special symbol called equal symbol. We have a symbol that means to belong to something, a symbol that means the natural numbers, a symbol that means zero. And yeah, that's what we have so far. And also when we say X and Y, they're assumed to be natural numbers. We just left that out. It innately has a very supreme beauty to it, this analytical, logical style of building this process.
00:28:29
Speaker
Oh yeah, actually the breaking math blog, you could see how to prove one plus one equals two using the system. Oh wow. So yeah, the next one is introducing a new symbol and that's it. There exists a successor function. We don't say what this is intuitively. What this is is a successor of zero is one, the successor of 130 is 131, et cetera. But we don't say that we're doing this formally. Right. So we say there's a successor function, which can be applied to natural numbers.
00:28:57
Speaker
And in a way, it's almost algebraic. You can implant any number into that successor place, and then you can immediately follow it with the successor. Yeah, however, is a successor of a natural number a natural number? It would be. You would think it should be, right? Right. However, that's the next axiom. Okay, well, let's hear it. The successor of every natural number is itself a natural number. We have to make that explicit. We're assuming nothing.
00:29:22
Speaker
It has to be a separate rule. It has to be a separate logical rule. Is the concept of infinity inherent in this axiom? It's okay. Infinity is a symbol that actually we develop. We'll talk about that in just a second. Okay. It's a symbol that basically means keep going, but it's not part of the natural numbers per se. It's part of the extended natural numbers, I believe, but it's not used all the time. So what do we have next?
00:29:48
Speaker
Next we got up, two natural numbers are equal if and only if their successors be equal. Two numbers are equal if and only if their successors are equal. Again, that seems obvious, but it has to be stated. Because we could have the successor of two not equal the successor of three, for example. We haven't defined what the successor function is at all.
00:30:13
Speaker
And mentally, this divides the line between the equal symbol, for me at least. It really shows that one side of the equal symbol is equal to the other side. This cements that, for sure. And one thing I want to bring up is that we said that we had some symbols earlier and a new symbol is a successor function, but I want to go over some non-obvious symbols. And, for example,
00:30:34
Speaker
Then, if, those are all important symbols in mathematics. That's very interesting, because in analytical philosophy, and if, or therefore, are all very, very important. They're all atomic though, right? Exactly. The next one is interesting. For every natural number x, it is false to say that the successor of x is zero.
00:30:57
Speaker
for every natural, okay, so that means that we start at zero. So pick a natural number. Three. Is a successor of three, zero? Nah. Is any natural number zero, one, two, three, et cetera. The success are going to be zero ever? No. So this creates zero as the starting point. This creates the idea of nothing as a starting point. Definitely. And it gives us the definition of what the set of natural numbers is.
00:31:18
Speaker
Yeah, and this stuff is actually like, I think this is Math 401 for me. So this is pretty high up there, but it's easy to understand. Definitely. Right. It's easy to conceptualize.
00:31:30
Speaker
And here we have a few new symbols, for example, false or negation. So that's important. Now, this next one is a doozy. OK, do you guys know about Euclid's fifth axiom? What was that one? That is, if two parallel lines are intersected by a third line, then the sum of the interior angles is equal to the exterior angles. I might have watched that.
00:31:56
Speaker
All the other ones are very simple postulates and axioms. However, that one is kind of out of the left field. I do remember that one from geometry. And this one has a similar left field axiom. For every set of natural numbers, if zero is in the set, and if for any natural number its existence in the set guarantees its successor's existence in the set, then all natural numbers belong to that set. Okay, I can understand that. That's the set of all natural numbers.
00:32:22
Speaker
Well, that's not the set of all natural numbers. That's a key distinction. We have a set. So the set doesn't have to be the set of all natural numbers. Okay, so for example, let's say we have something we want to test. For example, any natural number times two is even.
00:32:39
Speaker
zero belongs in that set, the set defined by that definition. This is induction, basically. This is how induction is defined in terms of the natural numbers. So let me reread that. For every set of natural numbers, so every thing that you could do, do natural numbers, because like, for example, the odd numbers don't have every natural number in them, right? It's just 1, 3, 5, 7, et cetera. If zero is in the set, so we know that zero is in the set, that's the first thing we prove.
00:33:06
Speaker
Imagine this is dominoes zero is the first domino we topple that one
00:33:11
Speaker
And if for any natural number, so some domino down the road, its existence in the set, so we know this is part of the set, guarantees its successor's existence of the set, so this domino falls and it knocks the next one over, then all natural numbers belong to the set, all dominoes fall. So if dominoes push down dominoes and you push down the first domino, all dominoes fall. Right, that makes perfect sense. This is basically saying that a number will build on itself into the next number.
00:33:37
Speaker
Yeah. And, um, or these existence of the set, it implies the existence of the next one. So the mirror cannot be proven with the other axioms. So the mere existence of a number proves the existence of a successor in this set in this set. So for example, zero times two is zero and zero is even, right? Right.
00:33:55
Speaker
100 times 2 is 200 and 200 is even, right? Right. 200 plus 2 is 202, right? Right. And that's the next one up. And since 202 equals 101 times 2, and 100 guaranteed the existence of 101 in the times 2 equals even set, then we know that any natural number times 2 is even. That makes perfect sense.
00:34:18
Speaker
And then you go on to define all of, like, not all of mathematics, but a lot of mathematics using this. For example, addition can be defined using these, like, similar axioms in piano arithmetic. Any natural number plus zero is equal to that natural number. Okay, that makes sense, right? Right. And for any two natural numbers, x and y,
00:34:37
Speaker
x plus the successor of y is equal to the successor of x plus y. So like 2 plus 3 is equal to 2 plus 2 plus 1, which is 1 plus 2 plus 2. That's all that's saying. Right. This is completely algebraic, this whole thing. Propositional logic is what it's called. And multiplication, any natural number times 0 is 0. And for any two natural numbers, x and y, x times the successor of y is equal to x plus x times y.
00:35:05
Speaker
So you keep building these rules and you go from addition multiplication, for example, I want you to take a stab at this. How do you define an integer? An integer is what happens when there is a separation between two natural objects, two natural numbers per se.
00:35:21
Speaker
Yeah, that's exactly correct, actually. We define them as a pair of natural numbers separated by a minus sign. And that minus sign, we never give it any meaning. It has inherent meaning because of the axioms. We don't think minus, we write line. So the minus sign is almost created inductively through the axioms.
00:35:43
Speaker
More deductively. And then you have the rational numbers, which is any integer divided by another integer. Okay, so then we get fractions and decimals.
00:35:54
Speaker
Yeah, and then for the real numbers, which include things that are not fractions, like the square root of 2, what you do is you define a sequence. So, for example, for square root of 2, it could be 1, then 1.4, then 1.41, then 1.414, et cetera, until like forever.
00:36:13
Speaker
And if you define the sequence, and you can prove that the sequence is getting closer and closer using what's called Cauchy limits, using the Cauchy inequality, you can do this. And the point being that counting this very simple skill can be used to do a wonderful amount of mathematics, but all of this is algorithmic in nature.
Algorithms: Cognition and Universality
00:36:38
Speaker
It's as if our minds naturally run on algorithms, which seems to be the case.
00:36:43
Speaker
And something that I was wondering about while we were having this discussion is whether algorithms or something that has come from the way that human consciousness functions, or whether they exist independently of us. Almost a cause and effect style. Yeah, exactly. Yeah. I mean, the mind is not necessarily discrete, but the nature of the mind is still not understood.
00:37:04
Speaker
In the neural net episode, artificial thought, we covered that you can use weights in a neural net that connects back to itself to simulate any mathematical function. In a sense, the human brain does run on algorithms because it can simulate any function. Proof of that is that we know the sentence, I can simulate any natural function. The fact that we can conceive of that means we can do it.
00:37:26
Speaker
Definitely that makes sense. I don't almost feel like arguing that algorithms are Inherent to the universe rather than just to human consciousness like for example chemistry runs on algorithms Oh, yeah, part of that is it from bits and also there's a lot of philosophy that has to do with computability and
00:37:47
Speaker
For example, the N-body system, I read a paper that said that if you don't apply Einsteinian mechanics, then it's not seemly to believe your computer, but when you do, it can be. So there's bizarre things in the universe, but it's all based on algorithms. These things that we use every single day, day to day, are based on something as simple an animal as counting, but all we need is counting to bootstrap our way into the stratosphere.
00:38:14
Speaker
It's the foundation of all thought it almost seems like. Counting? Counting. It almost seems like a representation of what it actually is to think. There is some study in Gestalt theory. There's grouping, there's inferring, there's all sorts of things.
Mathematics as a Universal Exploration Tool
00:38:31
Speaker
And Gestalt is really the way that we group things and see things and hear things, really. I think that has to do a lot with what you're talking about.
00:38:38
Speaker
Any more thoughts guys? Do you think that humans now are naturally born with the counting system ingrained in them just evolutionarily? I think that it's developed way too quickly to be evolutionary right now. That makes sense. So do you think that it's just taught from parent to child innately?
00:38:56
Speaker
Yeah, I mean, we have the evolution of ideas, memetics. The theme of this episode really has been mental connections and the power of mental connections to create and use algorithms interchangeably.
00:39:09
Speaker
Arithmetic and algorithms have been used since time immemorial to let mathematics breathe life into human endeavor. As we explore the universe, mathematics will prove to be our faithful companion, not only in helping us decipher the world around us, but to provide a mental framework to accept the ideas that are emanated therefrom. I'm Jonathan, and this has been Breaking Math. Today we head on to Gideon and John. If you enjoyed Tune to the Hickory Stick, then why not buy us a coffee?
00:39:37
Speaker
go to patreon.com slash breaking math podcast to support the show. Thank you. And until next time.