This episode from PBS Infinite Series takes a deep dive into the fundamental nature of numbers, specifically natural numbers, using the Peano axioms to describe them in a non-circular, abstract mathematical perspective. Viewers are introduced to two primary set-theoretic constructions: Zermelo's 'Russian matryoshka model' and Von Neumann's 'expanding suitcase model,' both of which synthesize numbers from the empty set. The discussion also delves into the philosophical implications of these constructions, such as the ideas that the essence of numbers might be built from nothing and that any realization of Peano axioms is isomorphic to another. The episode encourages mathematical discussion and debate among its audience.
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
The episode kicks off with a philosophical question: What are numbers made of? This initiates a deep dive into mathematics, where the team draws parallels between physical and numerical worlds, likening numbers to atoms and molecules in their fundamental aspects.
The show creatively explores the Peano axioms, which describe the natural numbers using abstract mathematical functions and starting points. It introduces two intriguing models: Zermelo's Russian doll method and Von Neumann's expanding suitcase method. Both models ingeniously construct natural numbers from the simple concept of nothing, relying heavily on set theory.
Finally, the episode wraps up by engaging its viewers in philosophical and practical discussions, encouraging them to think about the nature of numbers. It emphasizes that while mathematical constructs offer a way to describe numbers, they leave space for philosophical debates on their true essence. The show also invites viewers to share thoughts and corrections, fostering a community of engaged learners.
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