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

71–80 of 173 posts

Re: What Even Is a Number?

#71

Post by zazen »

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.

Well, yeah. Axioms are not deduced, they are the starting points for the various theories that are deduced from them. (There is not much you can deduce ex nihilo.)

Did you have some further point?

If you were hoping that mathematics would have - or thinking that it _should_ have - some ontological foundation that is not in this sense "made up", I'm afraid it simply doesn't.

Re: What Even Is a Number?

#72

Post by tgarv »

snarfy wrote: post #66
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.

Where is "one" then, if it's still "there"? :P

Re: What Even Is a Number?

#73

Post by xamuel »

12elephant wrote: post #69
xamuel wrote: post #58

Earlier quoted context omitted.

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 wor…

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

Yep, one of my favorite math quotes and highly under-rated.

One thing I've found is that if you find yourself using the wrong "types", you're almost certainly going in the wrong direction. For example, if you're trying to solve a complex analysis exercise and you find yourself thinking about how the complex numbers = the real-coefficient polynomials modulo the ideal (x^2+1), STOP: You're getting nowhere! :)

Re: What Even Is a Number?

#74

Post by mzs »

dahart wrote: post #62
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?"

Historically true, but glad it's there so ℤ is a group. The way to think about 0 for a child is what you add to anything so it stays the same, ie nothing.

Re: What Even Is a Number?

#75

Post by throwawaymath »

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…

What you're saying isn't demonstrating any inconsistency. A thing is only true in mathematics if it follows from the definitions. If you change your definitions, you should expect that theorems built upon those definitions will no longer hold.

Stated another way, consistency is only a coherent mathematical concept from the perspective of a specific set of definitions. There's no problem here.

Re: What Even Is a Number?

#76

Post by kirkules »

cobbzilla wrote: post #24
mjburgess wrote: post #15

Earlier quoted context omitted.

That's just "Exist as physical objects exist" Ie., you're asking someone to find you an object with spatio-temporal properties which, as an object, does not have spatio-temporal properties. Likewise you could ask, "find me a smile?!!" and then deny every example was a smile because "it was only a face smiling!". "Existence" is defined relative to a context in which it can be evaluated, and it -- in general -- does no…

Yes, I could have really tortured my kids and said, you can't even show me a "ball", because it will always be a specific ball and not the archetypal concept of "ball". I probably should, it'd be fun and they're a bit older than when I pulled the "one" game.

Make sure you say "show me ball" instead of "a ball"...

Re: What Even Is a Number?

#77

Post by throwawaymath »

12elephant wrote: post #68
throwawaymath wrote: post #54

Earlier quoted context omitted.

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…

In that case I'd agree with you. I'm not particularly keen on rehashing foundations of mathematics either.

Re: What Even Is a Number?

#78

Post by kirkules »

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 should keep in mind that being unable to show or find an example doesn't disprove the category. You'll never show me a fork yesterday, but that doesn't mean there were no forks yesterday.

Re: What Even Is a Number?

#79

Post by andrewla »

throwawaymath wrote: post #75
andrewla wrote: post #57

Earlier quoted context omitted.

> 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…

What you're saying isn't demonstrating any inconsistency. A thing is only true in mathematics if it follows from the definitions. If you change your definitions, you should expect that theorems built upon those definitions will no longer hold. Stated another way, consistency is only a coherent mathematical concept from the perspective of a specific set of definitions. There's no problem here.

I don't claim that this makes it inconsistent. Just that it admits meaningless concepts. You can attribute this, if you like, to the fact that I am a computer scientist, and see things through that lens.

If you give me a statement saying "For two sets, A and B, is 'A union intersection B empty", the statement would not be well-formed, so you could just say that it is not a valid proposition without asserting anything about its truthiness. Similarly, a good foundation of mathematics should be able to reject a statement like "2 is a member of 4" as poorly formed, and make no statement about its truthiness.

I guess what I'm saying is that set theory is a bad basis for mathematics and should just be regarded as an interesting relic from previous attempts to formalize mathematics so that we can move on to more powerful systems.

Re: What Even Is a Number?

#80

Post by lisper »

mjburgess wrote: post #15
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…

That's just "Exist as physical objects exist" Ie., you're asking someone to find you an object with spatio-temporal properties which, as an object, does not have spatio-temporal properties. Likewise you could ask, "find me a smile?!!" and then deny every example was a smile because "it was only a face smiling!". "Existence" is defined relative to a context in which it can be evaluated, and it -- in general -- does no…

The "technical" term amongst philosophers is "ontological category":

http://blog.rongarret.info/2015/02/31-flavors-of-ontology.ht...

Post reply on HN