The nature of number

[Categories+Sheaves]

Until we get on to Cantor
Ok I wish to add something. It is my contention that the number is not the generator of the series, but that the number is a fraction of the series. If I am wrong please disabuse me of this. Similarly I have also noticed that a point is afected by the field or dimention it is in.

I mean, I don't really know what you mean by "generator of the series", or "fraction of the series". So nobody's getting disabused today. And yes, the type of stuff that can happen around a point is dictated by the 'bigger world' the point is stuck in. It's precisely this structure that allows us to do math

Any number of projections can be made from a point in two dimentional space and be equaly spaced around that point.
However in 3d:
Only specific numbers of projections can be made from a point in three dimensional space and be distributed equally around that point. Given this; it seems the structure of the field dictates the material within it, rather than the material creating the structure. Please help me through this.

Well, my first reaction is, of course you can do that in 3-space. Take the thing you made in 2-space, and imagine that's occurring within a plane in 3-space.
But you overlooked this example for a very clear reason (or at least, my last paragraph was using a very different interpretation of 'regularly spaced'): all the examples of 'regularly spaced' points correspond to the vertices (or faces, if you're into poincare duality) of platonic solids (you're only missing the dodecahedron/icosahedron example, but I also can't see your second 3D example). If my memory serves me right, you have exactly three polytopes in each higher dimension (5+ iirc) and a whole bunch in dimension four (which is somehow the most interesting dimension for manifolds as well).

To prepare for geometric emergencies, I picked up a good book on polytopes/polyhedra (by Coxeter) a while back, so I guess I'll dig into it over the next week and see if I dredge up anything you might enjoy.
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But let's get this Cantor stuff done. EDIT: You should probably read the wikipedia proof first, my run-through is really sloppy/ugly/a PITA to read.

The shortest form of the argument I know is showing that there doesn't exist a 1-1 corresponding between the natural numbers, {1, 2, 3, 4, ...}, and the interval [0,1] (in the real numbers, which we'll express in their decimal expansions).

Suppose there does exist some function f that maps each natural number to each real number in [0,1], leaving none out. Then we can find a number in the following manner: the nth decimal place is '0' if f(n) is not zero, and the nth decimal place is '2' if f(n) = 0.
For the sake of making sure this map is well-defined: in cases where you have two ways of representing a decimal, like 1 = 0.999..., we pick the nth decimal place out of the representation that doesn't have the unending sequence of 9's (you could do it the other way too, but we have to be consistent). Call the number we generate in this way r.
In this way, we'll pick out a number such that there cannot exist some natural number n for which f(n)=r. If the nth decimal place of f(n)'s decimal expansion (sans infinite 9's) is zero, r will have a 2 in this same decimal place, and thus these numbers cannot be equal. If this nth decimal place is not zero, then the respective decimal place in r will be 0, followed by 0's and 2's (so there's no chance of some 999...'s sneaking in and messing things up).
Hence the notion that these two infinite sets, {1,2,3,...} and [0,1] are in some sense different: I cannot map {1,2,3...} to [0,1] in a way that hits every element of [0,1]. But I CAN map [0,1] to {1,2,3...} in a way that hits every natural number: map both 0 and 1 to 1, and for all other numbers, their decimal representation (sans 999...s, again, because we don't want to be ambiguous) has a 'first nonzero term', that must occur in some decimal place--be it the 1st, 2nd, 3rd... it must occur at the nth place for some natural number n--and then map said decimal to this n. For every natural number n, 10-n is a real number, so we'll cover all the natural numbers in this way.
So there is this asymmetry: [0,1] is 'big enough to fill' {1,2,3,...}, but the reverse relation isn't true. In this sense we can say one infinite set is bigger than another.
It should be clear that you can make a partial order relation on infinite sets in this way: If there exists some onto or surjective map f: A -> B, (that is to say, for every element b in B, there exists some a in A such that f(a)=b, i.e. all elements of b are 'hit by f', 'mapped to by f' or 'in the image of f') and no possible onto/surjective map in the opposite direction (some f-1: B -> A) then we can say A is 'bigger' than B. If we have the same situation that a surjective g: B -> C exists but a surjective h-1: C -> B does not, then the composition of f and g, creates a surjective map f(g()): A -> C so there is some transitivity: A > B and B > C yields A > C.

So I'm leaving out a lot, like,
-If you add the Axiom of Choice, this partial order relation becomes a strict order relation (everything is comparable)
-Expressing Cantor's diagonization argument in terms of powersets is more abstract but also the 'most natural' way of expressing the result. For instance, you don't have to fret over the fact that all nonzero terminating decimals also have a non-terminating (including a 999...) decimal expansion.
-Philosophical notes on how to interpret this result.
-Probably other stuff too

Anyway, now we have concrete stuff to discuss on the Cantor thing.

[jonb]

Anyway, now we have concrete stuff to discuss on the Cantor thing.

Thank you I will start to try to proses it.

Quick thought though, what if individual numbers do not have integrity of themselves, but are only created from the series by the observation. Is that a possible line of enquiry?

[Categories+Sheaves]

Quick thought though, what if individual numbers do not have integrity of themselves, but are only created from the series by the observation. Is that a possible line of enquiry?

I mean, what do we count "observation" as? There's a very old discussion about whether the mathematical structures we play with exist in their own right, or just through our intuition/visualization/understanding. Finitism/Ultrafinitism is also a thing; if I prove that some formula works when I plug in '1', and then (abstractly) prove "if the formula works for some integer n, then the formula must work for n+1 as well" I've proven this formula for all natural numbers, right? It's common practice in math to say "yes, of course!" but some folks aren't so sure (e.g. Ed Nelson from Princeton tried to prove that the Peano Axioms weren't consistent about a year ago). It's not like I'll ever "observe" all the integers out there (given a finite brain and a finite lifespan, anyway) so how can I intelligently make statements about all the integers? The phenomenology of math is pretty weird.

[jonb]

The phenomenology of math is pretty weird.

I was wondering if the number could in some way be analogues to the electron in that the act of observation affected it, that was why I was trying to observe the actions of numbers through an outside medium geometry.
If you use a fixed point to define a number, then the number can only be a fixed point, but I thought if we examine a series then we might be able to see if the number is fixed or not.

[Categories+Sheaves]

I was wondering if the number could in some way be analogues to the electron in that the act of observation affected it, that was why I was trying to observe the actions of numbers through an outside medium geometry.
If you use a fixed point to define a number, then the number can only be a fixed point, but I thought if we examine a series then we might be able to see if the number is fixed or not.

1. We use numbers to model the things we observe. So once we start talking about the ways we "observe" the model... too meta. Should we model our model? Model our model of our model? Reminds me of an Achilles/tortoise dialogue. Ever read G.E.B.?
2. If 'number blarg' is one thing at one moment and a completely different object at another moment, mathematicians would probably say 'number blarg' isn't well-defined.
3. I mean, mathematical objects exist through their behavior/intelligibility and because of it. There's nothing to really say about "a point" on its own--and I'm not even sure what we should mean by that. But a point in a plane, a point on a torus, a point in the prime spectrum of a ring... these
make sense.

Otherwise: still interested in that geometry stuff?

[jonb]

Otherwise: still interested in that geometry stuff?

Yes, you don't have to ask. The pupil will ask questions, but even if these seem silly, I am paying attention.

PS
G.E.B. Gödel, Escher, Bach; no I hadn't just found it my sort of thing thank you.

[Categories+Sheaves]

Yes, you don't have to ask. The pupil will ask questions, but even if these seem silly, I am paying attention.

PS
G.E.B. Gödel, Escher, Bach; no I hadn't just found it my sort of thing thank you.

And G.E.B. seemed right up your alley...
1. Napoleon put up a really cute video on the diagonalization proof in the 'existence and infinity thread', so feel free to use that video (if the jumble I dumped here is too... jumbly) if you want to keep kicking Cantor around.
2. That Coxeter book has a wonderfully short proof that there are only 5 platonic solids:

A convex polyhedron is said to be regular if its faces are regular and equal, while its vertices are all surrounded alike... ...If its faces are {p}'s [that is, p-gons], [with q such faces] surrounding each vertex, the polyhedron is denoted by {p,q}
The possible values for p and q may be enumerated as follows. The solid angle at a vertex has q face-angles, each (1-2/p)*pi... A familiar theorem states that these q angles must total less than 2*pi. Hence 1-2/p < 2/q

Which we can rewrite as (p-2)(q-2)<4. Because p>=3, (there are no 1-gons or 2-gons) and q>=3 (q=1 just gives you a p-gon in 3-space, and q=2 isn't much better) the only admissible values are {3,3}. {3,4}, {4,3}, {3,5} and {5,3} (tetrahedron, octahedron, cube, icosahedron, and dodecahedron respectively)

Man, geometry... I feel so nostalgic...

[jonb]

With Cantor and the book to digest and read my responses, are going to be slow to say the least on those subjects thanks again for the book, it really looks as though it will be right up my tree.

On the geometry,
Given that only specific numbers of projections from a point will give an even distribution, how about this for a thought?
A finite amount of matter emanating from a singularity, could be to an extent unevenly distributed.
So could a theory covering a finite amount of matter emenating from a singularity, that had within it a formula to explain an uneven distribution of matter from a singularity be affected adversely by the inclusion of that formula?

[Categories+Sheaves]

Given that only specific numbers of projections from a point will give an even distribution, how about this for a thought?
A finite amount of matter emanating from a singularity, could be to an extent unevenly distributed.
So could a theory covering a finite amount of matter emenating from a singularity, that had within it a formula to explain an uneven distribution of matter from a singularity be affected adversely by the inclusion of that formula?

You may have to take that up with the physicists.
As-is, I'm not sure how to parse that.

[jonb]

I am dyslexic, but that is only covers part of it, I am by nature ill defined, I could come up with an alternative way of making the statement, but in all truth it probably would not be any more clear. And anyway I think you understand the gist of it.