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What Even Is a Number?

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Re: What Even Is a Number?

#41

Post by jessermeyer »

carapace wrote: post #23
cobbzilla wrote: post #10

I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course. OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea. You cannot show just "one" unattached to anything else. The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of…

Ah! I wrote the below before seeing your comment: Numbers don't exist. Take the number two. You can have two apples but that's not the number two. You can write the numeral "2", but that's not the number two. It's one line. The word "two" has three letters, it's obviously not the number two. In fact, the number two doesn't exist (or it's existence is not contingent on an arrangement of matter/energy. No pattern of ma…

Numbers do not have a physical existence, but we can talk about them because in some capacity they absolutely do exist: conceptually.

There is the Tao and then there are the 10,000 things.

Re: What Even Is a Number?

#42

Post by andrewla »

xamuel wrote: post #8

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…

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. Not to mention that this is one possible construction of the natural numbers in set theory; in other formulations the theorem "is 2 a member of 4" would not be true, making things even more confusing. For example, we have the other standard approach of 0 = {}, 1 = {{}, 0}, n={{}, n-1}. The untyped lambda calculus and things like Church numerals have the same defect.

If we can start assigning types, then we can reason about these things in a much more concrete manner without these nasty side effects of theorems that only make sense because of definitional shortcuts. We can even derive isometries between different derivations of the same objects to show that two different definitions (like the Peano numerals and binary numbers) of a "natural number" can be used exactly interchangeably, because theorems like "is 2 a member of 4" are simply not expressible given the derivation.

Re: What Even Is a Number?

#43

Post by skh »

12elephant wrote: post #12
xamuel wrote: post #8

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…

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

All of mathematics is built on sand in the sense that there necessarily are going to be undefined terms. There’s no way around this. Think about the English language. Grab a dictionary. Look up any word in it. Look at the definition of that word. Pick a word in that definition. Repeat and eventually you’ll end up with a defintition that contains one of the words you previously looked up. So English is built on sand too and yet we are able to still communicate.

Re: What Even Is a Number?

#44

Post by Someone »

”This means we eventually end up either with an empty sequence (the set is empty) or with a sequence with only + operations in it”

For valid starting points, that is true, but it doesn’t follow from the text. The loop has an error exit (”If the sequence starts with − then something has gone wrong”), and the text doesn’t show that you won’t get there for valid inputs, and showing that isn’t trivial.

For example, if the sequence is ++---+, the first iteration removes the second and third item, leaving +--+, and the second iteration removes the first and second item, yielding -+.

The ‘program’ crashes there because the input is invalid, but proving that it never will crash for valid inputs without resorting to “that’s how integers behave” isn’t trivial.

Re: What Even Is a Number?

#45

Post by zazen »

dahfizz wrote: post #19
12elephant wrote: post #12

Earlier quoted context omitted.

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

> 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.

Re: What Even Is a Number?

#47

Post by throwawaymath »

andrewla wrote: post #42
xamuel wrote: post #8

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…

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 constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects.

Out of curiosity, what is the category theoretic construction that avoids Russell's paradox? I don't know that there isn't one, but I can't think of it off the top of my head. I know there are category theoretic constructions in general (I responded to one someone else posted in this thread).

Re: What Even Is a Number?

#48

Post by lmarinho »

My usual answer is that math isn't too concerned with what things are, but with how they work with each other. So the answer would be: anything that works like the Peano axioms says it should. The "what it is" can be filled in later, and that's what makes math so powerful.

Re: What Even Is a Number?

#49

Post by kazinator »

stcredzero wrote: post #35
kazinator wrote: post #25

Earlier quoted context omitted.

I believe that in fact the numbers do exist as physical objects exist. Just in their own universe. When we reason about numbers, we're embedding a representation of a universe of numbers into our universe. What we call "physical existence" could is also probably just be "math all the way down". Just not in such a way that the number two per se can be an entity for us to behold.

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 such a thing is plausible with genetic engineering in our universe.) As we endow the unicorn with additional properties, then it becomes less and less clear that there exists a universe where such a thing can exists other than as an imaginary being. If the concept of a unicorn could be axiomatized, then it could constitute a universe.

> Or, math is also a number of simplified alternate views of reality.

Well, that's the math that we do; not the math that we (possibly) are.

Re: What Even Is a Number?

#50

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…

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 themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category.

Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.

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