The nature of number

[Categories+Sheaves]

[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.