Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

What Even Is a Number?

notebook.drmaciver.com

Post reply on HN
Print view View original

61–70 of 173 posts

Re: What Even Is a Number?

#61

Post by xamuel »

andrewla wrote: post #57
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…

> 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…

It's similar to the way in C you can malloc memory for two data structures, and then check which pointer is smaller. Say, one pointer points at the data for Mario, and the other pointer points at the data for Luigi. You look at the raw pointers and notice that the Mario pointer is a smaller 64-bit number than the Luigi pointer. That seems absurd or arbitrary because why should "Mario" be "smaller" than "Luigi"? But it doesn't really matter: the game still runs fine and nothing breaks as a result of the arbitrariness.

Re: What Even Is a Number?

#62

Post by dahart »

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…

> You cannot show just "one" unattached to anything else.

Also interesting to try to show "zero" of something. Zero is fun because it took longer to exist as a number than one & two, and there was some debate over whether it should be a number. "How can nothing be something?"

Re: What Even Is a Number?

#63

Post by edflsafoiewq »

throwawaymath wrote: post #37
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.

If you define the natural numbers this way you'll run into Russell's paradox. For practical purposes that's fine and this is an elegant, modern restatement of Frege. But if you need to avoid Russell's paradox the sets defining each natural n can't contain n as an element. The easiest such construction was outlined elsewhere in this thread by xamael. Under your category theoretic construction, every natural number n i…

I don't understand how Russell's paradox comes in. The set of all eg. pairs does not contain itself.

Re: What Even Is a Number?

#64

Post by ajkjk »

cobbzilla wrote: post #11
msla wrote: post #4

Earlier quoted context omitted.

> was there ever an attempt to do math with something like y=1/0 or another illegal construct? Wheels are a type of algebra where division by 0 is defined. https://en.wikipedia.org/wiki/Wheel_theory https://math.stackexchange.com/questions/994508/wheel-theory... To call wheels "obscure" is to make them out to be better-known than they are, however.

I had not heard of Wheel theory, quite fascinating. But eek! From the wikipedia article: 0x ≠ 0 in the general case x − x ≠ 0 in the general case Like, you get the ability to divide by zero, but at what cost? Multiplying by zero is also not zero! Seems bonkers. Are there any practical uses for this kind of algebra?

Likely no (the peculiar distributive laws seem unusable to me). But I have come across similar rules that I think may have a real usage. It sometimes seems to me that it might make more sense to keep 'factors' attached to 0s, such that x-x = (1-1)x = 0x might be 'sound', and to keep track of 'powers' of 0s by having factors of 0 attached to 0s: (0)0 = 0^2 != 0, etc.

If you keep factors like this, then you could implement L'Hopital's rule without the result only being true if considered under a limit: say, lim(x->0) (5x^2/ 3x^2) = 5/3 could be computed as (5)0^2 / (3)0^2 = 5/3.

This is not like Wheels though; it requires that any power of 0 be a distinct number. I of course have no idea if it is sound or meaningful, but I do find myself thinking about it a lot.

Re: What Even Is a Number?

#65

Post by andrewla »

dandare wrote: post #3

> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…

Infinitesimals [1][2][3] are probably the best example. Though there's a whole range of theories, the basic version is just a magic number, d, such that d^2 == 0. So (x+d)^3 = x^3 + 3dx^2 + 0, giving us a more "natural" path to get to the notion of derivatives without resorting to limits. They get hard to manage, though, so they haven't really made it into the mainstream.

[1] https://en.wikipedia.org/wiki/Non-standard_analysis

[2] https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis

[3] https://en.wikipedia.org/wiki/Dual_number

Re: What Even Is a Number?

#66

Post by snarfy »

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…

You might even say the unattached "one" is more real than the fork. At the end of the universe, that fork is long gone, but "one" is still there.

Re: What Even Is a Number?

#67

Post by Nomentatus »

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…

"Reification" is the term to search for if you wish to delve into the extensive Philosophical bibliography on this topic that's accumulated over the last few centuries. Within mathematics "Intuitionist Mathematics" is the rubric for the anti-reification view. This relies on computability as a substitute for "existence." The Intuitionist view has pretty much triumphed.

Re: What Even Is a Number?

#68

Post by 12elephant »

throwawaymath wrote: post #54
12elephant wrote: post #53

Earlier quoted context omitted.

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?

The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism".

What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down.

If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement:

> In mathematics you don't understand things. You just get used to them.

Re: What Even Is a Number?

#69

Post by 12elephant »

xamuel wrote: post #58
12elephant wrote: post #53

Earlier quoted context omitted.

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 se…

This is the same response I gave to throwawaymath:

The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism".

What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down.

If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement:

> In mathematics you don't understand things. You just get used to them.

Re: What Even Is a Number?

#70

Post by msla »

btilly wrote: post #26
dandare wrote: post #3

> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…

The original version of Calculus by Newton used "fluxions". That doesn't correspond to any number system that we use today. Leibniz's (re?)invention of Calculus used infinitesmals. Infinitesmals as understood by mathematicians then do not correspond to any numbers we use today. (Yes, yes, something else called infinitesmals do show up in nonstandard analysis and the notation deliberately looks the same. But the under…

> But the underlying concepts are more..complicated.

Can you expand on this?

Post reply on HN