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What Even Is a Number?

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Re: What Even Is a Number?

#151

Post by r00fus »

sound1 wrote: post #39
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…

By extension, does God exist? I don't know the answer but he seems to be very useful (at least to some members of the society!)

Sure (s)he does - inside all of us. This works for both spiritual as well as a human nature points of view.

Re: What Even Is a Number?

#152

Post by kazinator »

Retra wrote: post #125
kazinator wrote: post #109

Earlier quoted context omitted.

> 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…

>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? Math is just words. Any description of anything can be reduced to math if you assume the proper math will be developed to describe it. But the same could be said of any subject. You could say the universe…

> You're just confusing the map for the territory.

I am not at all! We do math with symbols and diagrams, but those symbols and diagrams are about something: they are maps, and there is a territory that they are about. When I say "math", I include the concepts, not just the symbolic, graphical and other representations.

So what I'm saying is that if/when mathematics provides a complete map to that territory which is "the world", that territory must then be made of exactly the same stuff as other described-by-math territory-entities like "ellipse", "pi", "finite field", "Hilbert space", ... the map of mathematics doesn't describe any other kind of territory.

I'm not saying that the symbols and pictures used to carry out math (the "maps") are the world.

> you will never find a complete description of the entire universe that fits in a network of human brains

The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be. E.g. the Mandelbrot set is just iterating on z^2 + c for various c. To describe isn't necessarily to evaluate.

Re: What Even Is a Number?

#153

Post by est31 »

btilly wrote: post #144
FakeComments wrote: post #132

Earlier quoted context omitted.

But other kinds of numbers can be made from that: - Differences of counting numbers (integers) - Ratios of numbers (rationals) - Limits to sequences of numbers (reals) - Solutions to polynomials made from numbers (complex) You can think of each of these as adding a layer of behavior to the basic “each number is nothing or one more than another number”.

I agree with your sequence until the last two. The second to last, that should be Cauchy sequences. And for the last, demonstrating that the algebraic closure of the reals is the complex numbers from first principles is much harder than describing complex numbers as pairs of reals with a multiplication rule that (a,b) * (c,d) = (ac - bd, ad + bc) and then much later proving that it is algebraically closed through com…

A much easier to understand definition for the set of reals is probably to use infinite decimal expansions that don't end in 9999 etc. Equivalence classes of cauchy sequences is tougher to explain I'd say.

Re: What Even Is a Number?

#154

Post by btilly »

est31 wrote: post #153
btilly wrote: post #144

Earlier quoted context omitted.

I agree with your sequence until the last two. The second to last, that should be Cauchy sequences. And for the last, demonstrating that the algebraic closure of the reals is the complex numbers from first principles is much harder than describing complex numbers as pairs of reals with a multiplication rule that (a,b) * (c,d) = (ac - bd, ad + bc) and then much later proving that it is algebraically closed through com…

A much easier to understand definition for the set of reals is probably to use infinite decimal expansions that don't end in 9999 etc. Equivalence classes of cauchy sequences is tougher to explain I'd say.

In that case, good luck coming up with a good definition around multiplication where 3 * 0.33333... works out right. And then proving arithmetic properties like the associative law. And then proving that when you do the reals in decimal, you get the same system as the reals in binary.

It sounds harder, but is actually easier to go through the Cauchy sequence definition and then point out that the decimal representation naturally gives rise to a Cauchy sequence. So, for example, 3.1415926535... gives you (3, 31/10, 314/100, 3141/1000, ...). And as Cauchy sequences, of course, (1, 1, 1, 1,...) is easily proved to be the same as (9/10, 99/100, 999/1000, ...).

Re: What Even Is a Number?

#155

Post by est31 »

btilly wrote: post #154
est31 wrote: post #153

Earlier quoted context omitted.

A much easier to understand definition for the set of reals is probably to use infinite decimal expansions that don't end in 9999 etc. Equivalence classes of cauchy sequences is tougher to explain I'd say.

In that case, good luck coming up with a good definition around multiplication where 3 * 0.33333... works out right. And then proving arithmetic properties like the associative law. And then proving that when you do the reals in decimal, you get the same system as the reals in binary. It sounds harder, but is actually easier to go through the Cauchy sequence definition and then point out that the decimal representati…

Oh right, 3 * 0.33333 is an issue, yeah. Still, you need to define that equivalence relation on the Cauchy sequences and then explain what a set modulo a relation means.

Re: What Even Is a Number?

#156

Post by btilly »

est31 wrote: post #155
btilly wrote: post #154

Earlier quoted context omitted.

In that case, good luck coming up with a good definition around multiplication where 3 * 0.33333... works out right. And then proving arithmetic properties like the associative law. And then proving that when you do the reals in decimal, you get the same system as the reals in binary. It sounds harder, but is actually easier to go through the Cauchy sequence definition and then point out that the decimal representati…

Oh right, 3 * 0.33333 is an issue, yeah. Still, you need to define that equivalence relation on the Cauchy sequences and then explain what a set modulo a relation means.

This is all standard mathematics.

The equivalence relationship is that the sequence (x_1, x_2, x_3, ...) is equivalent to (y_1, y_2, y_2, ...) if and only if the limit as n goes to infinity of x_n - y_n = 0.

Formally, the real number represented by (x_1, x_2, x_3, ...) is the set of all Cauchy sequences which are equivalent to that one. Since "equivalent to" is transitive, any Cauchy sequence in that set will define the same set.

Addition and multiplication are defined elementwise. Proving that they are well-defined is relatively straightforward. Their algebraic properties follow for free. Any rational number q can be mapped to the Cauchy sequence (q, q, q, ...) which leads to a unique real number that we somewhat sloppily call q again.

I've left some details out, but this construction is well-understood, and is how we define the completion of a metric space.

Re: What Even Is a Number?

#157

Post by ajkjk »

hopler wrote: post #147
ajkjk wrote: post #64

Earlier quoted context omitted.

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 imple…

Those "factor" zeros look like infinitesimal forms, which are well studied.

There is a similarity, but no one claims infinitesimals are _literally 0_, and certainly no one would claim that 2-2 = 2 epsilon.

Re: What Even Is a Number?

#158

Post by mcv »

qubex wrote: post #91
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…

This reminds me of my Italian grammar teacher telling me that nouns can be broken into two categories: ‘concrete’ and ‘abstract’ (for example, ‘chair’ and ‘love’ respectively). Then came a totally mind bending digression for my eight-year-old mind when she mumbled “well actually all nouns are abstract because there’s no such thing as a chair divorced from reality, there’s nobly instantiations of these abstract concep…

  > all nouns are abstract because there’s no such thing as a chair divorced from reality,
  > there’s nobly instantiations of these abstract concepts we consider close enough to be
  > concrete
Very Plato for an 8 year old.

Re: What Even Is a Number?

#159

Post by Ntrails »

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…

A coiuple of my favourites:

- Child is put into perfectly sealed cube. They have a torch, which provides light. Since the light cannot escape the cube, what happens when they turn the torch off? Where does it go?

- If you put a brain in a jar and wired up electrodes which pulsed up every nerve ending etc to simulate reality - would the brain know it was in a jar? How? (The matrix being the modern representation)

-

Re: What Even Is a Number?

#160

Post by Retra »

kazinator wrote: post #152
Retra wrote: post #125

Earlier quoted context omitted.

>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? Math is just words. Any description of anything can be reduced to math if you assume the proper math will be developed to describe it. But the same could be said of any subject. You could say the universe…

> You're just confusing the map for the territory. I am not at all! We do math with symbols and diagrams, but those symbols and diagrams are about something: they are maps, and there is a territory that they are about. When I say "math", I include the concepts, not just the symbolic, graphical and other representations. So what I'm saying is that if/when mathematics provides a complete map to that territory which is…

The concepts are also just symbols, just in different media. They are also a map.

>if/when mathematics provides a complete map to that territory which is "the world",

Then it will encode all the information that exists in the universe. Which will not fit in your head, and thus will not be mathematics. Just because a territory is trivially a map of itself doesn't mean it is useful to call it a map.

>The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be.

Reality isn't just the initial rules. It has state. That state must be described for you to have a complete description of reality. Yes, we could mathematically describe the complete laws of nature. But reality is necessarily more than just that, because the state is really damn relevant. You can't just ignore it because it doesn't fit your argument.

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