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What Even Is a Number?

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Re: What Even Is a Number?

#101

Post by qubex »

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…

I’m a mathematician and you seem a bit confused. Natural numbers ({1, 2, 3, ... }) are for counting. Giving a bunch of objects as being prime examples of that which can be counted with natural numbers is as close as one can get (short of just scribbling down the numerals and kind-of circumventing the point). Other classes of numbers (particularly the Reals) are for measuring. Again, short of writing down examples of decimal expansions, a child could point to a (non-integer) mark on a measuring tape.

I get the feeling you’re not quite so well versed on this stuff as you think you are. Want a true example of ‘one’: the number of solutions to the intersection of two linear equations that are not parallel. Want an example of ‘two’: the number of solutions between a quadratic equation and a linear equation (given that certain existence criteria are met).

I think you need to revise your measure theory.

Re: What Even Is a Number?

#102

Post by [deleted] »

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"...

[deleted]

Re: What Even Is a Number?

#103

Post by carapace »

johnfn wrote: post #86
carapace wrote: post #23

Earlier quoted context omitted.

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.…

Reflecting on this stuff is my golf...

Even if I agree that THE number "one" is "just another broad category that a single apple fits in to", I think the epistemological status of the Broad Category is then itself just as problematical, existence-wise, as the original concept of "number", eh? We can at least display a pair of apples and "count" to "two" with them, etc. What hope do we have of enumerating (no pun intended) the Broad Category of "one"? And what do we gain from introducing it?

Re: What Even Is a Number?

#104

Post by roywiggins »

Retra wrote: post #88
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.

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…

The strange thing about mathematics, as opposed to other noncorporeal ideas, is that it is nearly uniquely suited to being transmitted between brains. My idea of the number "three" and your idea the number "three" are probably very similar. This sort sensation that they have an independent existence leads people to the Platonic notion of a mathematical "realm" or "universe".

(personally I think mathematics exists in our brains, but also and crucially in the interactions between our brains, since a proof is never accepted until it is reviewed and so on. A proof that's just in my brain is no proof at all, even if it happens to be correct)

Re: What Even Is a Number?

#105

Post by kazinator »

FabHK wrote: post #98
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? Good question, but I think there might be a difference. Suppose, in a galaxy far, far away, there are aliens. They probably don't have the concept of unicorns. However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc. So, does the fact that something is discovered (rather…

I would sadly have to say no. Because in some completely different universe unlike ours, the aliens have no concept of, say, "hemoglobin" because there isn't matter like ours: no atoms, no iron, no carbon. Unlike numbers, hemoglobin and unicorns contain a kind of arbitrary complexity. Yet one is definitely real. Some real entities are far removed from elementary math (even though they may ultimately rest upon it, many levels down). The question is: are unicorns bunk, or are they like hemoglobin.

Re: What Even Is a Number?

#106

Post by stcredzero »

FabHK wrote: post #98
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? Good question, but I think there might be a difference. Suppose, in a galaxy far, far away, there are aliens. They probably don't have the concept of unicorns. However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc. So, does the fact that something is discovered (rather…

However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc.

We can only be confident of that if they're in this universe. Also, this is only a conjecture. It's just something some of us imagine would be true. You can't elevate such an idea to the level of empirical data.

So, does the fact that something is discovered (rather than invented), and can be discovered many times independently in identical fashion, confer it some sort of elevated ontological status?

Unicorns. Kirin. Narwhals. Rhinos. So you have two independent origins in culture for such a creature, plus two appearances of something similar in nature. Does that confer it some sort of elevated ontological status?

Re: What Even Is a Number?

#107

Post by mabbo »

oh_sigh wrote: post #90
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

Oh sure, but then the article started talking about sheep and rocks and I'm like "Oh boy, I bet I totally don't understand what it means to be even!"

Re: What Even Is a Number?

#108

Post by kansface »

FabHK wrote: post #98
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? Good question, but I think there might be a difference. Suppose, in a galaxy far, far away, there are aliens. They probably don't have the concept of unicorns. However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc. So, does the fact that something is discovered (rather…

> They probably don't have the concept of unicorns.

What would happen to your epistemology if every alien civilization has unicorns but some lack π?

Re: What Even Is a Number?

#109

Post by kazinator »

Retra wrote: post #88
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.

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…

> abject poetic nonsense.

No it isn't! Either it is math all the way down, or else it isn't.

If it isn't, then what is at the bottom?

So this is actually an existential question.

Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes).

Won't that description consist of nothing but math?

Then, if it is complete, how does that description account for the fact that reality isn't math? If there is a difference between reality and math, can't that be described with math, so then it goes away?

If you want math to just be a description of a structure that itself isn't math, then the structure has to have attributes that are not covered by the math. The math is then using a substitute structure with fewer properties as a proxy for that structure. Once all the attributes are rolled into the math, there ceases to be a difference between that target structure and the math; it is that mathematical object. To prove otherwise requires one to name the difference. If two entities are completely described and found to be the same, they are just one entity.

> You may as well be declaring the world to be flat.

It is testable that the world isn't flat.

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