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What Even Is a Number?

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Re: What Even Is a Number?

#51

Post by throwawaymath »

zazen wrote: post #46
throwawaymath wrote: post #29

Earlier quoted context omitted.

{} is the empty set, which as mentioned is natural 0.

GP understands the notation; he means to cast aspersions about the axiomatic project.

Fair enough, though there weren't really any "aspersions" cast. I'm aware you can unravel set theory if you care enough about that particular exercise. But, in context, you could also interpret that comment as being written by someone who isn't aware that the empty set is defined as {}.

Responding to someone's definition that natural 0 = {} that mathematics is a "castle built on sand" isn't exactly a cogent criticism of set theory.

Re: What Even Is a Number?

#52

Post by throwawaymath »

andrewla wrote: post #50
throwawaymath wrote: post #47

Earlier quoted context omitted.

> The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent. > Type theory and category theory give us a much better way of…

The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of all sets that are not members of themselves") is avoided simply because the objects produced are not sets, and sets have no exalted position within the mechanics of category theory. Talking about concepts like "the category of categories that don't contain thems…

I see what you're getting at. My (similarly hand wavey) perspective is that constructions using category theory end up defining natural numbers as categories of sets with cardinality n. What I was really getting at is, how do you define your relation for the category in such a way that isn't pathological ("n is the category of sets with cardinality n except all such sets containing n")?

Alternatively, what are you selecting as the objects for your category if not sets of a given cardinality?

To be clear, I consider a lot of discussion about the foundations of set theory (and paradoxes thereof) to be pretty navel-gazey.

Re: What Even Is a Number?

#53

Post by 12elephant »

xamuel wrote: post #18
12elephant wrote: post #12

Earlier quoted context omitted.

Your castle is built on sand. What, then, is {}?

One of the axioms of set theory is the axiom of the empty set, which states that there exists at least one set which has no elements. Another axiom of set theory is the axiom of extensionality, which states that two sets are equal if they have the exact same elements: from which it follows that all sets without elements are identical, i.e., there is only one set without elements. We call that the emptyset. Other axio…

An axiom is simply something made up.

At the end of the day, all of our Mathematics rests on foundations that are made up.

It's difficult to say what the empty set is. Because it isn't really anything at all.

Re: What Even Is a Number?

#54

Post by throwawaymath »

12elephant wrote: post #53
xamuel wrote: post #18

Earlier quoted context omitted.

One of the axioms of set theory is the axiom of the empty set, which states that there exists at least one set which has no elements. Another axiom of set theory is the axiom of extensionality, which states that two sets are equal if they have the exact same elements: from which it follows that all sets without elements are identical, i.e., there is only one set without elements. We call that the emptyset. Other axio…

An axiom is simply something made up . At the end of the day, all of our Mathematics rests on foundations that are made up. It's difficult to say what the empty set is. Because it isn't really anything at all.

There are legitimate criticisms you can levy against set theory, but I'm starting to lose you here. I'm not really following your point anymore - this seems like arguing about whether or not mathematics is invented or discovered.

Are you trying to argue coming up with new definitions isn't worthwhile if the axioms don't have a foundation in reality? If so, why? If not, what are you saying?

Re: What Even Is a Number?

#55

Post by dahfizz »

zazen wrote: post #45
dahfizz wrote: post #19

Earlier quoted context omitted.

> the natural 0 is the empty set.

That is a definition of "natural 0", not a definition of "empty set". GP was pointing out that GGP was using the concept of "empty set" to provide a definition for the natural numbers, without having first provided a definition for the empty set. Replies that consist solely of throwing a quote at someone are kind of rude even if you're in the right.

I don't see why GGP should have to reinvent set theory to use it in his construction of numbers. Set theory and the idea of {} is well defined, and hardly "built on sand".

Re: What Even Is a Number?

#56

Post by stcredzero »

kazinator wrote: post #49
stcredzero wrote: post #35

Earlier quoted context omitted.

I believe that in fact the numbers do exist as physical objects exist. Just in their own universe. How is it any different than believing that unicorns exist as physical objects, just in their own universe? What we call "physical existence" could...just be "math all the way down". Or, math is also a number of simplified alternate views of reality of varying degrees of "applicability." (Where "applicability" has to do…

> How is it any different than believing that unicorns exist as physical objects, just in their own universe? We have very well-developed representations of the number universe that we can embed into our own universe. These things are governed by precise axioms. By contrast, we don't have such well-developed "unicorn universe". (For starters, what is the definition of "unicorn"? If it's just a horse with a horn, then…

We have very well-developed representations of the number universe that we can embed into our own universe. These things are governed by precise axioms. By contrast, we don't have such well-developed "unicorn universe".

Indeed. The current better known attempts at a "unicorn universe" have problems with power scaling and the square-cube law. https://www.reddit.com/r/mylittlepony/comments/mrs81/accordi...

Do you have a good definition of what constitutes a "well-developed" universe?

Re: What Even Is a Number?

#57

Post by andrewla »

throwawaymath wrote: post #47
andrewla wrote: post #42

Earlier quoted context omitted.

The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects…

> The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent. > Type theory and category theory give us a much better way of…

> Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent.

That's the problem -- it's only within this definition. In other definitions, notable the nesting example that I gave, that theorem is false. And the theorem makes no sense in and of itself, because we're talking about numbers, so it is unexpected that the membership operator would apply at all.

Whereas I can take two definitions of the natural numbers

  PN =
  Peano_One: PN
  Peano_Succ: PN -> PN

  BN =
  Binary_One: BN
  Binary_2x: BN -> BN
  Binary_2xp1: BN -> BN
and I can define a Plus: PN x PN -> PN and a Plus: BN x BN -> BN, and so on, and once I can define Binary_Succ: BN -> BN and Peano_2x and Peano_2xp1 I can prove that these are isometric types, so all theorems derivable from PN apply to BN and vice versa, not just the convenient theorems that don't use any syntax from the meta-language (e.g. set theory).

Re: What Even Is a Number?

#58

Post by xamuel »

12elephant wrote: post #53
xamuel wrote: post #18

Earlier quoted context omitted.

One of the axioms of set theory is the axiom of the empty set, which states that there exists at least one set which has no elements. Another axiom of set theory is the axiom of extensionality, which states that two sets are equal if they have the exact same elements: from which it follows that all sets without elements are identical, i.e., there is only one set without elements. We call that the emptyset. Other axio…

An axiom is simply something made up . At the end of the day, all of our Mathematics rests on foundations that are made up. It's difficult to say what the empty set is. Because it isn't really anything at all.

Do you complain similarly when Euclid states that for any two distinct points, there exists an infinitely long straight line which passes through those two points? We cannot point to that line, we can only point to a small portion of it within our field of vision, and have to extrapolate it to infinity.

Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.

Re: What Even Is a Number?

#59

Post by ajkjk »

ddebernardy wrote: post #33

> What even is a number? It's a category. 1 is the category of singletons, 2 is the category of pairs, etc. As to what numbers are, they're an ordered collection of categories. Edit: To whoever downvoted this comment, kindly explain. Insofar as I'm aware this is textbook maths, psychology, and philosophy.

I downvoted it; it's dismissive and obnoxious to just provide a intuitionless one-liner explanation: "what are numbers? they're X", as though there was no complicated ideas involved, or like it should be obvious or trivial. The whole point of an article about this is to acknowledge and grapple with the subtleties.

Re: What Even Is a Number?

#60

Post by zazen »

dahfizz wrote: post #55
zazen wrote: post #45

Earlier quoted context omitted.

That is a definition of "natural 0", not a definition of "empty set". GP was pointing out that GGP was using the concept of "empty set" to provide a definition for the natural numbers, without having first provided a definition for the empty set. Replies that consist solely of throwing a quote at someone are kind of rude even if you're in the right.

I don't see why GGP should have to reinvent set theory to use it in his construction of numbers. Set theory and the idea of {} is well defined, and hardly "built on sand".

He doesn't necessarily have to. But when he defines "natural zero" as "the empty set", and someone asks "what's the empty set?", and you say "natural zero", you're being circular.
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