What are Numbers Made of? | Infinite Series

Summary

Highlights

• Exploring the basic makeup of numbers using mathematical principles akin to how particles make up atoms. 🔍
• Understanding Peano axioms helps avoid circular definitions in mathematics. ♻️
• Zermelo's Russian doll model illustrates numbers as nested sets. 🪆
• Von Neumann's suitcase model shows numbers growing by adding layers to sets. 📦
• The episode suggests numbers might be constructed from 'nothing' through set theory. 🚫
• Mathematical debates are healthy; they help us probe deeper into theoretical concepts. 🎓

Key Takeaways

• Numbers, like atoms and molecules, can be broken down into even simpler elements in mathematics. 🔢
• Peano axioms allow us to describe natural numbers using a starting point and a function to find the 'next' number without directly referencing numbers. 🌱
• Zermelo's method uses nested sets like Russian dolls, starting from an empty set. 🎎
• Von Neumann's method constructs numbers by expanding sets, starting with nothing and adding everything contained previously. 🎒
• The philosophical question of what numbers are made of is explored, suggesting that numbers might be made of 'nothing.' 🤔
• Discussion is encouraged on different mathematical constructs and methods, highlighting their theoretical rather than absolute nature. 💬

Overview

Chapters

• 00:00 - 00:30: Introduction to the Basics of Numbers This chapter explores the foundational concept of numbers, both in the abstract and in the physical world. It compares the construction of physical objects from basic components, such as molecules and atoms, to the construction of numbers. The natural numbers are identified as the building blocks of the entire number system. The chapter also poses a philosophical question regarding the essence or fundamental nature of natural numbers.
• 00:30 - 01:00: Peano Axioms and Natural Numbers The chapter introduces the concept of the Peano Axioms, which provide a foundation for defining natural numbers without circular references to numerical concepts. It starts with specifying a beginning point, termed as 0, and proceeds to define the 'next' element iteratively. The challenge lies in defining 'next' without inadvertently referencing numbers, which can be addressed by employing the Peano axioms. The chapter encourages revisiting a previous episode for a better understanding of related foundational concepts.
• 01:00 - 02:00: Defining Function s and Set n with Peano Axioms In this chapter, the concept of defining a function 's' and a set 'n' through the Peano axioms is explored. The function 's' must have both its inputs and outputs derived from a non-empty set 'n'. The chapter outlines three primary conditions for this setup: 1) The function 's' must be one-to-one, meaning different inputs must always result in different outputs. 2) There must be at least one element in the set 'n' that is not an output of the function 's'. 3) The set 'n' should be the minimal or smallest possible non-empty set that satisfies these conditions. These foundational concepts are crucial for understanding more complex mathematical structures and theories.
• 02:00 - 03:00: Sets and Peano Axioms in Describing Numbers The chapter discusses the use of Sets and Peano Axioms to describe numbers. It explains that with any set n and a function s that satisfies certain conditions, you can mimic the characteristics of natural numbers and the successor operation. Furthermore, it is possible to define operations that imitate standard addition and multiplication based on any function s, regardless of its internal workings. This demonstrates how the Peano axioms serve to distill the essence of what it means to be a number.
• 03:00 - 04:30: Zero to Numbers using Zermelo's Construction This chapter begins with a recap of previous discussions on the efforts during the 19th and 20th centuries to describe all mathematics purely in terms of sets, as per Zermelo-Fraenkel set theory. The primary implication of this is that natural numbers can be constructed from sets rather than being seen as indivisible entities. The chapter sets the stage to demonstrate several methods for constructing natural numbers using set theory, sharing a logical structure throughout.
• 04:30 - 06:00: Von Neumann's Construction of Numbers The chapter delves into the foundational concepts of number construction as proposed by Von Neumann. It highlights the Peano axioms which remain neutral about the nature of numbers and the principles of the successor function, s. By utilizing set theory, Von Neumann's approach attempts to construct a comprehensive set, denoted as n, whose elements and the successor function s are both sets. The overall aim is to ensure that n and s jointly adhere to all the Peano axioms, providing a robust framework for understanding numbers within set theory.
• 06:00 - 08:30: Comparison of Zermelo and Von Neumann Constructions This chapter focuses on comparing the Zermelo and Von Neumann constructions of natural numbers. The intention is to demonstrate the feasibility of constructing natural numbers using set theory, specifically ZF set theory, without delving into certain technical nuances. The author explicitly opts to bypass discussions related to specific axioms of ZF set theory or the logical formulations of Peano axioms. The main goal is to present the concept to those unfamiliar with the possibility of reducing complex mathematical ideas into simpler constructs.
• 08:30 - 10:00: Mathematical Reductionism and Importance of Existence The chapter introduces the concept of mathematical reductionism, specifically focusing on simplifying complex mathematical concepts into more basic ones, using Zermelo's construction as an example. The Zermelo's construction, or the 'Russian matryoshka model of the numbers,' is discussed. This model involves a function S that transforms a set X into a new set Y, where Y consists of X as its only element. This process can be visualized like Russian dolls, where each doll contains another doll inside it.
• 10:00 - 11:00: Corrections from Previous Episodes This chapter explains the concept of nested arrays or lists in programming by using the analogy of a Russian doll. When an input doll has items inside, a new hollow doll is created as output with the input doll inside it, but not its contents (i.e., candies). This illustrates how nested structures work by showing that only the outermost layer is accessed or 'peeled off', not the contents inside.
• 11:00 - 12:00: Torus Clock Challenge Winners Announcement In this chapter, the winners of the Torus Clock Challenge are announced. The discussion delves into the mathematical concept introduced by Zermelo, which begins with the analogy of an empty set represented as an empty Russian doll. By applying a process referred to as 's', each iteration produces a new doll encapsulating the previous one, analogous to layering sets within sets. This recursive approach is used to compile a collection referred to as 'n'.
• 12:00 - 14:00: Discussion and Viewer Comments on Torus Clock Challenge The chapter explores the properties of a hypothetical 'dollify' function, framed in the context of a collection that satisfies Peano's axioms. The 'dollify' function, when applied to any object in the collection, results in another object within the same collection. The process of dollification ensures a one-to-one correspondence where two distinct starting dolls yield two distinct final dolls. This abstract concept parallels the basic Peano axioms, where the 'empty doll' represents the number 0, embodying the empty set, and the doll containing '0' represents the number 1, hence developing a foundational numerical system using these principles.

What are Numbers Made of? | Infinite Series Transcription

• 00:00 - 00:30 In the physical world, many seemingly basic things turn out to be built from even more basic things, like molecules being made of atoms, or atoms being made of protons, neutrons, and electrons. But what about numbers? What are they made of? The natural numbers form the basis of the rest of the number system. Last time, we asked whether you can describe the essence of the natural numbers
• 00:30 - 01:00 without circularly evoking numerical concepts in that description. Now, such a description might read as follows, specify a starting point, or a seed, that we arbitrarily call 0, and then include the next thing, and the next thing after that, and so on. That sounds innocuous, but the tricky part is capturing what next means without inadvertently or indirectly referring to numbers. Tricky, mind you, but not impossible via something called the Peano axioms. Now, if you haven't seen that last episode, you should pause me now and go catch up on how that works,
• 01:00 - 01:30 or today's episode may be hard to follow. But here's the executive summary, just in case. Imagine a function s, any function, whose inputs and outputs both come from some non-empty set, n, any set, as long as n and s are subject to the following conditions. 1, different inputs to s always yield different outputs, i.e. the function s is 1 to 1. Second, there's an element in n that is never the output of s. And third, n is the leanest, or minimal, non-empty set
• 01:30 - 02:00 on which you can even define a function s that works like this. Any set n and function s that meet these conditions together will behave, respectively, like the natural numbers and the operation next, or successor. You can even define operations that fully mimic run of the mill addition and multiplication, in terms of any suitable function s, regardless of the details of how s works under the hood. In this sense, the Peano axioms distill number hood
• 02:00 - 02:30 down to its bare essentials. But let's remember Kelsey's earlier episode about the 19th and 20th century efforts to describe all of math in terms only of sets and operations on sets, as codified by the axioms of Zermelo-Fraenkel set theory. The implication there was that the natural numbers don't have to be viewed as indivisible entities, that even they can somehow be synthesized from sets. Well today, we'll demonstrate a couple of different ways to do exactly that, both of which will follo wa very similar logical arc.
• 02:30 - 03:00 Remember, the Peano axioms are agnostic about both the nature of the objects that your calling numbers and about the mechanical details of the successor function s. Now, in set theory, the objects we have available to us to play with are sets. So we're going to assemble a suitable mega set n, whose elements are themselves sets, along with the suitable function s that eats and spits out sets, such that n and s together satisfy all the Peano axioms.
• 03:00 - 03:30 In doing so, we will have built something that we can legitimately call natural numbers. Now, before we start, some disclaimers. I will not address nuances, like which aspects of these constructions rely on which specific axioms of ZF set theory. And I won't even mention technical issues associated with so-called first order versus second order logical formulations of the Peano axioms or of these constructions. That's not my point. My objective today is simply to show, particularly for those who have never seen it before, that it's even possible to reduce something
• 03:30 - 04:00 as seemingly basic as the counting numbers to even more basic ingredients. Let's start with Zermelo's construction, or what I like to call the Russian matryoshka model of the numbers. Here's the operation that Zermelo used as his function, s. You take a set x as input and then you output a set y that contains that set x as its only element. Or if you want to visualize every set as a Russian doll that might have some stuff inside, then s is the operation, stick the input
• 04:00 - 04:30 doll inside of a new hollow doll. Not the contents of the input doll, mind you, just the input doll itself. In other words, if the input were a doll that has three pieces of candy in it, then the output doll wouldn't have three pieces of candy inside of it. It would only have one thing inside it, namely the original doll. This is analogous to how nested arrays, or lists, work in most programming languages. Asking what's inside a given container means peeling off only the outermost layer of its onion,
• 04:30 - 05:00 without peeling any layers off any additional onions that you happen to find inside that onion. Anyway, now let's unmask the elements of n. Zermelo starts with the empty set, or if you will, an empty Russian doll. If you apply s to that doll, you get a new doll whose only element is the original empty doll. If you apply s again, you get a doll that contains the immediately prior doll, and so forth. OK, now let's aggregate all of these successively more nested dolls into a single collection, n.
• 05:00 - 05:30 This collection n, together with the dollify function s, satisfy Peano's axioms together. If you dollify any object in this collection, you always get something else that's in the collection. If you dollify two different starting dolls, you get two different final dolls, and the empty doll isn't of dollifying anything. So the empty doll, which is just a visualization of the empty set, plays the role for us of 0, and the doll containing 0 plays the role of 1.
• 05:30 - 06:00 And the doll containing 1 plays the role of 2, and so on. Now for Von Neumann's construction, or as I like to think of it, the expanding suitcase model of the numbers. The operation s here is a bit different. The output set will be the union of the input set and a set that contains the input set. In case you're not too fluent with set operations, here's another visual aid. Picture a suitcase, x, that's full of items. That will be the input to s.
• 06:00 - 06:30 Now, take a bigger suitcase, y, and fill it as follows. First, clone every item that's in x and put it into suitcase y. Then take suitcase x itself along with all its contents and put that into suitcase y. Suitcase y is the output of s, and it contains everything that suitcase x contains plus a copy of the entire suitcase x itself. In other words, the output suitcase always has exactly one more element than the input suitcase. Now to build our collection n, let's start again
• 06:30 - 07:00 with the empty set, which this time we'll picture as an empty suitcase. If you feed that into s, you get a new suitcase that contains clones of the contents of the input suitcase, except there aren't any in this case, because it's empty, plus the empty input suitcase itself. Fine. Now, let's feed that into s. You get a new suitcase that has two items in it, the original empty suitcase and a suitcase containing the empty suitcase. Keep going like this. Just as before, you'll end up with an infinite collection
• 07:00 - 07:30 of items that satisfy the Peano axiom. The empty suitcase plays the role of 0. The suitcase containing the empty suitcase, i.e. Containing 0, is 1. The suitcase contains 0 and 1 plays the role of 2. The suitcase containing 0 and 1 and 2 is 3, and so on. But Neumann's model has another neat characteristic. Every number in that construction is a set, each of whose elements is also a subset of that set.
• 07:30 - 08:00 Chew on that for a bit and verify for yourself that it's accurate. And now the punch line. Notice that both the Zermelo and Von Neumann constructions begin with the empty set, and using different set operations, they then synthesize every other number from the empty set, which means when we ask, what are numbers made of, at least in set theory, the answer seems to be nothing, provided, at least, that you stipulate nothing exists.
• 08:00 - 08:30 That's cute. But now, which of these constructions is really the natural numbers? That's a philosophically heavy question and I'll try to answer it in two ways. First, the pragmatic answer. The Von Neumann construction is the de facto standard set theory for a few reasons. Relations like less than are less cumbersome to define with Von Neumann. It's also somewhat simpler to define cardinal numbers, i.e. answers to the question, how many, in terms of the Von Neumann naturals, which, if you think about it, have really so far been
• 08:30 - 09:00 a model of ordinal numbers that only specify position along a sequence. And finally, the Von Neumann ordinals lend themselves better to generalizing to transfinite arithmetic. If you want a nice primer on what transfinite arithmetic means, Vsauce has a very popular video on how to count past infinity that you should go check out. Of course, who's to say that someone tomorrow won't find a different set theoretic model of the naturals that's equally good for these purposes? And that leads me to my second answer, which is just my personal view of the situation,
• 09:00 - 09:30 namely, that asking what anything in math really is somewhat beside the point. In the case of the naturals specifically, it can be shown, and I'm pretty sure Dedikam was the first to do so, that all realizations of the Peano axioms are isomorphic to one another, meaning that you can put any two of them into one-to-one correspondence, so that whatever the s function does in model a winds up with what the s function in model B does to corresponding elements in model B.
• 09:30 - 10:00 But more important, the details of one versus another construction matter a lot less than their actual existence. I've always felt that the whole point of the reductionist enterprise of mathematics is less about uncovering some unique truth and more about minimizing the number of unavoidable assumptions. In that sense, the existence of constructions, in terms of more basic ingredients, matters a lot more than the details. There are, for instance, numerous ways to synthesize your way up to the real numbers starting from the natural.
• 10:00 - 10:30 And preferring one to another of those constructions is less a matter of correctness than a matter of taste. I'm sure a lot of you will disagree with me, and that's good, because we love discussion and debate here in Infinite Series. So have at it down in the comments and I'll see you again soon. Hey, everyone, I'll be responding to comments from today's episode and the previous numbers episode next time. Today, I'm going to respond to comments from the Torus Clock Challenge episode and announce the winners of that challenge question. But first, a quick bit of housekeeping.
• 10:30 - 11:00 If you go back to Tai Danae's "How to Divide by Zero" episode, right around time code 1150, when I'm responding to comments from the metallic ratio episode, I made a mistake. I said the [INAUDIBLE] 128 had said that all A series paper has an aspect ratio that's the silver ratio. That's not right. Actually, all a series paper has an aspect ratio that's root 2:1. But if you cut out the largest square that you can from A4 paper, or any series paper, the remaining rectangle will be in the silver ratio. Sorry I misinterpreted the comment
• 11:00 - 11:30 and didn't do my due diligence. I'll try to do better going forward. And while we're on the topic of housekeeping, Xatnu Rowan and several others correctly pointed out that in the Torus clock episode, I incorrectly identified what the fundamental polygon is for the Klein bottle. The square that I showed that had identified points like this, and then like this on the two edges, actually gives you the projected plan. To get the Klein bottle, you'd identify points that are like this on the top edge and identify points that are directly across each other on the left and right. So I'm glad you all caught that mistake.
• 11:30 - 12:00 I was mortified to have made it on camera, and I wish YouTube would bring back annotations, but what are you going to do? What we did do is pin a comment to the top of the episode that I hope you all have seen, highlighting the error. And the winners of the Torus clock challenge, randomly selected from among the correct responses, are-- and I apologize in advance if I slaughterer your names. The pronunciations, I'm going to try to do my best. yfede, Pelle Kersaan, Trevor Zeffiro, Matan Haller, and Vladamir Jankijevic. I hope I got that close to right. Congratulations, you all win PBS Digital Studios T-shirts.
• 12:00 - 12:30 If you send us an email with your T-shirt size, in US sizing, so small, medium, large, extra large, and your mailing address, we'll get those T-shirts out to you. I made the mistake, by the way, of announcing five winners, just from old space-time habits, not realizing that Infinite Series is a one T-shirt a week kind of show. So going forward, probably one winner a week. Finally, if you want to see the solution written up, or my solution to the problem at least using the Torus geometry, some of you put solutions in comments
• 12:30 - 13:00 that were not quite right, so you might want to take a look. There should be a link flashing on the screen below where you can go download my solution. Finally, a few other comments from the Torus clock challenge episode that I wanted to address. Snilubez commented that, "well, Pac-Man is really more of a cylinder than a Torus. Shouldn't I have used asteroids as a better analogy?" And I just want to say, for the record, that I disagree. I think Pac-Man world actually is a Torus. It happens to be that most editions of Pac-Man, you can only exit through the left and re-enter through the right, or top and bottom, but not usually both. But in some editions of Pac-Man, for example Pac-Man Battle
• 13:00 - 13:30 Royale, you can exit and enter through either the top and bottom or the left and right, making it pretty clear that if the walls weren't there, the Pac-Man world is actually, topologically, a Torus Jimmy Huguet, or Huguet, said that "maybe we shouldn't have made this challenge problem about a Torus. Wouldn't it have been a lot easier to do it algebraically?" And I suppose that depends on your point of view. For some people, I agree, that probably most people natively think to do this problem algebraically, but a lot of people prefer visual solution
• 13:30 - 14:00 just to intuitively understand what's going on. And all I wanted to do is make a visual approach to the problem accessible to people. And finally, Khesed said asked, "well, what if you had a three handed clock? What shape is natural for talking about that kind of problem?" And I'm not sure exactly what you mean because the problem that I posed about swapping hand becomes ambiguous if you have three hands. I mean, which pair do you want to swap, or do you want to do some weird permutation of the hands? And so, I think, naturally, that's still a two dimensional problem. I'm thinking about swapping hour and second hand, or hour and minute hand, or whatever.
• 14:00 - 14:30 But if you want to know, in general, what would be the sort of 3D analog for the coiling helix that goes around the Torus? If you wanted to track a three headed clock, I suppose it would be something going around a three dimensional Torus, or something that would have that equivalent shape, which you could visualize as a three dimensional cube with opposite faces identified so that as the line the tracks the actual valid times of day, goes out one face, it comes in the opposite face at the opposite point, and climbs up the cube
• 14:30 - 15:00 like that, Pac-Manified cube.