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What Even Is a Number?

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Re: What Even Is a Number?

#81

Post by throwawaymath »

edflsafoiewq wrote: post #63
throwawaymath wrote: post #37

Earlier quoted context omitted.

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.

If you've defined the natural number 2 to be an arbitrary set with cardinality 2, you're including sets which contain the number 2. That's the basic form of Russell's paradox.

If you define the natural number 2 to be the category of pairs, your objects are the sets with cardinality 2, and your relations between objects are equivalence relations. As a consequence your category 2 will contain sets which contain itself.

Re: What Even Is a Number?

#82

Post by btilly »

msla wrote: post #70
btilly wrote: post #26

Earlier quoted context omitted.

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?

Can you expand on this?

Yes, maybe more than you want. :-)

Abraham Robinson's version of non-standard analysis went like this.

You start with the standard model of the real numbers and associated concepts like sets, functions, and so on. Using something called the ultrafilter construction, you construct a new, larger model of the real numbers+stuff, and a mapping from the standard model to the nonstandard model. So we have nonstandard numbers, nonstandard sets, nonstandard functions and so on. Some of which are just mappings of the standard ones, and others of which are new.

Thanks to something called the transfer principle, all statements in first order logic about things involving real numbers remain true about the nonstandard versions of the same.

In other words within the nonstandard model, the world looks the same as within the standard model. But there is a key feature. From the construction we know that the nonstandard model has numbers in it that are closer to 0 than any real number except 0. That set is the infinitesmals. The infinitesmals are a set identifiable from the construction, but are NOT a nonstandard set. And 1/infinitesmal gives you infinite numbers whose absolute value is larger than any standard real.

Now here is the point of the whole construction. Suppose that (f(x + dx) - f(x))/dx only varies by an infinitesmal from a non-standard version of a real across all possible infinitesmals. Then it turns out that that real is the derivative in the usual notation. Similarly if a Riemann sum that breaks an area into N pieces always gives the same answer to within an infinitesmal for all infinite integers N, then it turns out that the function is Riemann integral, and that answer is the actual integral. And with these two insights, most of the handwavy arguments that people used to use can be rescued.

Some mathematicians have found that the infinitesmal notation helps them think through problems, and a some important new theorems were proven using this approach. However those proofs can be translated back to more standard notation. Many students intuitively prefer the infinitesmal notation over limits. But when you unpack it, does it make sense to invoke the axiom of choice to define the derivative? (The ultrafilter construction uses the axiom of choice. Alternatives exist, but they use complex model theoretic mechanics under the hood. Really understanding them requires a lot of machinery.)

Re: What Even Is a Number?

#83

Post by prmph »

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.

But if there is no one around to appreciate or think of "one", does it still exist?

Re: What Even Is a Number?

#84

Post by ball_of_lint »

throwawaymath wrote: post #81
edflsafoiewq wrote: post #63

Earlier quoted context omitted.

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

If you've defined the natural number 2 to be an arbitrary set with cardinality 2, you're including sets which contain the number 2. That's the basic form of Russell's paradox. If you define the natural number 2 to be the category of pairs, your objects are the sets with cardinality 2, and your relations between objects are equivalence relations. As a consequence your category 2 will contain sets which contain itself.

Why is that a problem?

Re: What Even Is a Number?

#85

Post by kgwxd »

kirkules wrote: post #76
cobbzilla wrote: post #24

Earlier quoted context omitted.

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"...

And be sure no one is overhearing it out of context.

Re: What Even Is a Number?

#86

Post by johnfn »

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…

Does an apple exist? You can show me an apple, but I would wag my finger in exactly the same way you are to people trying to show you “one”. The apple you’re showing me is not the same thing as the word “apple” which is an abstract category that encompasses all types of apples - honeycrisp, Granny Smith, etc. You could say a honeycrisp is like an instance of apple which is a base class, so then it is indeed an apple. But then youd have to say an apple is also a “fruit” and also a “food” even though those words also encompass broad categories of things.

If you buy the above, you’d have to say that a single apple is also “one”. After all, it’s just another broad category that a single apple fits in to. If not, I think you’d have to reject the fact that a honeycrisp is an apple, which seems untenable.

Re: What Even Is a Number?

#87

Post by cicero »

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…

This is why Aristotle said that the intellect cannot be material. The human intellect is able to receive and understand abstractions. However, whenever you try to represent an abstraction as a physical object or in a physical medium, it is no longer abstract. It is true that you can write about abstractions using physical ink on physical paper, but those are just symbols; they are not the abstraction. It is only when a person reads the symbols, understands them, and forms the abstraction in his mind that the abstraction comes into existence.

So, we can create programs that appear to do intelligent things, but it is the human observer that brings meaning to what the program is doing. The program might output the letters "dog" when a dog pass in front of its camera, but its the human that sees the "dog" that knows what a dog actually is.

Re: What Even Is a Number?

#88

Post by Retra »

kazinator wrote: post #25
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…

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.

That other "universe" is just patterns embedded in our brains. That's all you need.

You could likewise argue that sounds exist in a different universe, but are only "embedded" into the air using pressure waves, but why would you ever think you needed a different universe of sounds when you've already got pressure waves in air? I mean... what's "not enough" about that for you?

>What we call "physical existence" could is also probably just be "math all the way down".

This is a needless diversion. Mathematics is the study of structure. Physical reality has structure, and very well could "be structure" in some equivalent sense, but to say it is math is just abject poetic nonsense. This change of language would literally make no significant difference to anybody about anything. You may as well be declaring the world to be flat and asking everybody else to go and change all our language to make it true.

Re: What Even Is a Number?

#89

Post by TheOtherHobbes »

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 don't exist - unless you're a Platonist.

The experience of numbers certainly exists. The reason you recognise one-ness and two-ness in objects is because your brain experiences a difference between one-object and a-pair-of-those-objects.

The experience is subjective. If you have two objects that are identical except for the fact that one is red and the other is blue, you can parse them as one-each-of-red-and-blue or as two-identical-things. Both interpretations are logically consistent, but the most useful interpretation depends on the context, and your specific needs as defined by the context.

It's easy to underestimate how subjective math is.

I've long suspected that this is why math eventually dissolves into incompleteness theorems. You can't prove a subjective experience, and eventually all mathematical reasoning reduces to subjectivity - even something basic like true/false, which is essentially just an interpretation of experience. (And can be very unreliable.)

Math is really an introspective map of experiences and processes that we find collectively consistent. We like to tell ourselves it's external and objective. But how can we tell the difference between true objectivity and subjective consistency when we only have shared human subjectivity to use as a reference?

Re: What Even Is a Number?

#90

Post by oh_sigh »

mabbo wrote: post #5

I read this as "What is an even number" and spent the entire article stoked for when he got to the part about a number being even. Still, lovely article.

Don't even numbers have a pretty easy definition? If prime factorizartion of the number contains at least a 2
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