What are Numbers Made of? | Infinite Series

In the physical world, many seemingly basic things turn out to be built from even more basic things. Molecules are made of atoms, atoms are made of protons, neutrons, and electrons. So what are numbers made of?

Check out the previous episode to find out What It Means to be a Number

And to see Gabe's solution to the Torus Clock Challenge check out:

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Previous Episode:
What Does it Mean to Be A Number?

Torus Clock Challenge:

How to Divide by "Zero"

Blog post about the Peano axioms and construction of natural numbers by Robert Low:

Recommended by a viewer for connections to formulation of numbers in computer science:

Any set N and function S that meet these conditions 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 S, regardless of the details of how S works . In this sense, the Peano axioms distill numberhood down to its bare essentials.

Written and Hosted by Gabe Perez-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Linda Huang

Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon.

Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!

And thanks to Mauricio Pacheco who are supporting us at the Lemma level!

[KohuGaly]

well, the peano axioms demonstrate that numbers are not really specific entities. They are more of a property of things within some system, be it sets, apples, money, bits, energy, ... as long as the system satisfies peano axioms, the things within that systems are numbers.

[mebamme]

So in terms of programming concepts, the Peano axioms are the interface, and Zermelo's and von Neumann's models are implementations.

[jameswise]

In the span of 7 hours almost all of my favourite channels uploaded a new video. Today was good.

[Melomathics]

This could be my favourite episode so far.
Great topic, interesting, deep and even philosophical, but set up in a way that makes it accessible to pretty much everyone with an interest in this kind of thing. Great job!
You should definitely make more of these - the really technical stuff, although interesting, isn't always accessible to many of us still busy with our undergraduate degrees : (

[DustinRodriguez_]

Might as well do surreal numbers next! They get way too little attention!

[flamephlegm]

The set theory stuff comparing Zermelo and Von Neumann is really cool. Thank you PBS Digital Studios for all that you do.

[diribigal]

He said "I'm sure a lot of you will disagree", but I agree with his answers to those philosophical questions.

[reishvedaur]

This was my favorite video in this series. Thank you for making it. <3

[zairaner]

If someone wants to understand 9:27, you can try to prove the following: Every countable set can be equipped with a function S so that it satisfies the peano axioms.
Even more exact: If you have a bijective function f from the natural numbers to a set X, then there is exactly one function S' on X so that S' composed with f is the same as f composed with S, and S' turns X into "a set of natural numbers" (fullfills the peano axioms)

[cliffordhodge]

I am very glad PBS, the BBC, and others put educational videos on Youtube.

[HouseofObiwan]

The suitcase concept reminds me of blockchain, in the sense that each new block is built on the hash of all previous blocks. The lesson would even synergize with the videos on cryptography and hashing.

Edit** PBS please do a video on blockchain!!

[alansmithee]

"What are numbers made of? [...] Nothing, provided at least that you stipulate nothing exists."
Incredible statement.

[josealvim]

I've had a very interesting convo with a prof.

He suggested that on any infinite set X (so there are injective functions onto itself that are not surjective, take some such F). You take an element of the set outside the image of F, and take the chain of that element.

It's easy to show that the set we get from the chain (name it C) and some operation "+" -- recursively defined by F restricted to C -- form a commutative monoid isomorphic to ℕ.

Interestingly enough, you can get some ordinal structure by finding other elements outside of F's image and "glueing" those chains together. You get a weird order where every element outside of F's image is made into a ''limit ordinal''.

[FernandoVinny]

When algebraic structures are generalized they become more powerful and you can find more mathematical objects that behave like the original structure you abstracted. Examples: vector spaces, groups, rings, fields...
So, when the structure of natural numbers is abstracted, is natural that somente will find other things that behave like them. That's awesome and beautiful.

[AnarchoAmericium]

+PBS Infinite Series

Since you've done videos on sets, foundations, functors, topology, and logic, would you ever consider doing a video on Topos theory?

[ChurchOfThought]

Gabe, I love the personal wisdom you provide at the end of the video.

Regarding which model is preferred, I think the VN model is beautiful for its ability to convey arithmetic in simple set operations. The less than as a subset you mentioned already. Addition in VN is the size of the disjoint union. But of course we have to be careful saying size so we get around that by saying that we want the "pure set" that is isomorphic to the disjoint union one that is all fuddled with duplicates.

Thanks for another wonderful video Gabe. Have a great weekend!

[JayLikesLasers]

I like the video, and I really respect that you discuss comment points and make clarifications at the end. Good engagement with your audience I'd say.

[legendarybanditmb]

So interesting! This is the first time I've heard a lot of these concepts, but I'm always impressed at how well you explain them. What a fascinating way to think about numbers! Thank you!!

[phpn]

The object of mathematics is not numbers but mathematical language. Or, to state it in linguistics terms, math is a metalanguage that operates on its own concepts (containers). It so happens that the containers/concepts of math are paradoxically the 'content' of Nature. So mathematical language seeks to be the projection (or subsumption) of Nature into an intensional set of concepts.

[WadelDee]

If 3 is really the same as {0; 1; 2} it would mean that you should be able to iterate over numbers in a for loop in programming!
In Python, if you want to iterate over 3, you'll have to write "for i in range(3)". In any other language, you'll have to write "for(i=0;i<3;i++)" or even "i=0; while(i<3) i++;"
But what if you could merely write "for i in 3" or "for(i:3)"? I have always believed that this would be practical and useful and compact and convenient!
And now, you've given me some grounding for this idea! Whoever is reading this, do you agree with this idea? If you ever invent a programming language I hope I just gave you an idea!