The nature of number

[jonb]

You said this-

The torus would be topologically distinguishable from a sphere--any loop enclosing the hole could not be shrunk to a point while remaining on the torus, whereas every loop on a sphere can be shrunk to a point.

What shape is a point?

[Angrboda]

Geometric shapes are made up of points, which effectively have no dimension. Two points on a geometric object either are the same point or they are not. They either are a part of the line, shape or solid (or hyper-object) or they are not. There is no "infinitely close". (Think of them existing in a space without dimensional units or scale. In such a context, "infinitely close" is the same as "far apart" for some scale.)



(At least as far as I can remember. Been ages since and I was a bad math student then. Still, fun topic. Thank you for starting it.)

[CliveStaples]

You said this-

The torus would be topologically distinguishable from a sphere--any loop enclosing the hole could not be shrunk to a point while remaining on the torus, whereas every loop on a sphere can be shrunk to a point.

What shape is a point?

I don't even know how to answer that. The notion of "shape" is incredibly general. Are you asking for an equation defining a point? What do you mean by "shape"?

[Categories+Sheaves]

Thanks for your help, but before we move into other dimensional use’s of this form. I think it would be nice to examine a possibility if this graph stands.



We seem to have a range of zeros or a stretched zero which is not a point, but that has length.
And although the value does not change we can see a definite progression along the line.

Only in the sense that the preimage of 0 is a set with those properties. Or that you've taken something with these properties and called each of its constituents "0". It seems like we could call each point on the line "fish" or "paisley" and achieve the same result. Is there some concrete sense in which these points actually walk or talk like 0?

My thinking is this way; I have a tours. I press the middle until it is almost connected, but is not actually connected. The central hole exists, but it occupies Zero space. If I categorise the shape by the flow over it, it will fulfil all the criteria of being a sphere, however if I categorise the same shape by the internal side of this zero space skin, it is in all effect a torus. We can do the same the other way round.
Now how could you examine a shape to find holes of zero size and get any meaningful result? So is it not so that Grigori Perelman’s theorem is only proof to a given resolution and not an absolute?

1. This mapping of the torus i) fails to be differentiable ii) fails to be an embedding (even in the weakest topological sense) iii) does not seem to be well-defined (holes of zero width are o.k. if we're just deleting a point from a plane, but have you thought about what it means for a cylinder to have zero width?)
2. The type of vector fields/flows topologists love to put on manifolds are nonvanishing ones. If I understand what you're saying, the flow on the torus in your example has to vanish on the "inside" of your torus. So this doesn't conflict with the established math.
3. Even if this was problematic, Perelman's proof applies to 3-folds, not 2-folds. And if I understand what you mean when you say 'proof to a given resolution'.... yes: Perelman's proof is a technical result about a class of objects that operate under a very rigorous and particular set of rules.
4. A reprise of #1: Topological properties aren't invariant under the stuff you're doing to torii/spheres here, so you and Perelman are playing different games in yet another sense...

Geometric shapes are made up of points, which effectively have no dimension.

This is absolutely correct, but it's still worth noting that the "Geometric object A is a collection of points in space" perspective is surprisingly modern. It isn't too hard to learn/imagine what a straight line is, and we can intuitively understand what "a point on a line" is. But do these points themselves constitute the line, or does the line just happen to have points lying along it? That issue isn't quite so clear. While the former line of thinking is far superior when you have a robust theory of sets + calculus/analysis lying around, Euclid (and a whole bunch of other dead mathematicians) originally approached geometry from the other angle.

Something to think about. (Because: why should the heterodox guy have all the fun here?)

[CliveStaples]

1. This mapping of the torus i) fails to be differentiable ii) fails to be an embedding (even in the weakest topological sense) iii) does not seem to be well-defined (holes of zero width are o.k. if we're just deleting a point from a plane, but have you thought about what it means for a cylinder to have zero width?)

i) Proof? It seems like you're actually claiming that every mapping of the torus with these particular endpoints fails to be differentiable (since no particular map was given).

ii) Proof? This result would actually imply (i), since differentiable functions are necessarily continuous.

iii) I suspect there's actually a bijection from the surface he described (a 'torus' with a hole consisting of one point) to the unit torus, which would mean that the mapping is well-defined (in the sense of being a function). Or did you mean that the resulting object actually fails to be a torus?

4. A reprise of #1: Topological properties aren't invariant under the stuff you're doing to torii/spheres here, so you and Perelman are playing different games in yet another sense..

This is basically just another claim that any mapping with the endpoints he gave must fail to be homeomorphic. Can you prove this?

[Categories+Sheaves]

i) Proof? It seems like you're actually claiming that every mapping of the torus with these particular endpoints fails to be differentiable (since no particular map was given).

ii) Proof? This result would actually imply (i), since differentiable functions are necessarily continuous.

iii) I suspect there's actually a bijection from the surface he described (a 'torus' with a hole consisting of one point) to the unit torus, which would mean that the mapping is well-defined (in the sense of being a function). Or did you mean that the resulting object actually fails to be a torus?

i) If we're skewing a torus by making the space in the center look like an arbitrarily thin cylinder... (yes yes this becomes a sphere with a line connecting two poles. This was my interpretation of what he said and I ran with it) you can't define a pushforward of the tangent space (of the torus being mapped into 3-space, when you treat it as a manifold) for any of the points that get mapped to a pole.
ii) Failure to be injective. Or at the very least: if he's insisting that the hole have "width zero", that doesn't bode well for the hausdorff-ness/distinctness of the points lying 'across' this hole/gap.
iii) Under that interpretation of jonb's construction (hrm, this looks like the particular shape he was actually talking about... but you still get the shape I was talking about by mapping the torus and then putting it through [some charitable attempt at] the inverse mapping of the sphere) what you end up with is neither a torus nor a sphere, since that 'compressed' shape won't be compact (surely the one-point hole is a limit point?).

This is basically just another claim that any mapping with the endpoints he gave must fail to be homeomorphic. Can you prove this?

Either:
a) there is some 'inner ring' that gets mapped to the point that's supposed to be a hole, and this fails to be injective.
b) If the torus' "hole" is genuinely a point: That compactness thing from earlier.

Although the first part of the 'playing different games' comment was probably too overstated/harsh...

[jonb]

Dear Categories+Sheaves
Thank you for continuing with me. My aproach I know is clumsy, as I am coming at this subject from a wildly different angle. Also I am not as farmiliar with using the computer as you would expect, so I have fcuked up how to show what you have said and give my replies but I hope my solution to that works for you. My problem is that what I am seeing from my end does not seem to tally up with how maths is explained to me.




Only in the sense that the preimage of 0 is a set with those properties. Or that you've taken something with these properties and called each of its constituents "0". It seems like we could call each point on the line "fish" or "paisley" and achieve the same result. Is there some concrete sense in which these points actually walk or talk like 0?

But if the '0' in what I have termed the result is the projection from the origin through the function, does that not satisfy you that we can call it '0'?

Your next explanation is excellent and I think I understand. So I would like to walk away from all the twaddle I was talking about that. But I would like to take up the conversation from this part, because it is central to my theme

Geometric shapes are made up of points, which effectively have no dimension.

This is absolutely correct, but it's still worth noting that the "Geometric object A is a collection of points in space" perspective is surprisingly modern. It isn't too hard to learn/imagine what a straight line is, and we can intuitively understand what "a point on a line" is. But do these points themselves constitute the line, or does the line just happen to have points lying along it? That issue isn't quite so clear. While the former line of thinking is far superior when you have a robust theory of sets + calculus/analysis lying around, Euclid (and a whole bunch of other dead mathematicians) originally approached geometry from the other angle.

This is where my problem is: I would contend that Euclid's books are not about maths. Rather they are help books for Artists etc, on how to use maths. As such the definitions are to enable us to make constructions, not definitions that should be used to define the subject itself. I might use points to map out a shape in space, but vectored space has no points.
As such the point is a tool we use not a thing itself. Where I am going with this is that it seems to me a number is the same it has no integrity, but it is created out of the series, any one number is only given its value by the other numbers in the set or series.

I do not know if this is standard thinking or not. When as an outsider to maths you get an entire hour long BBC programme talking about Cantor and infinity, and declaring he said you cannot map one to one fractions and decimals, without saying that is an analogy. And I try to look up definitions of number and find:-

Can I take where I am with this is a reasonable starting point?

[CliveStaples]

i) If we're skewing a torus by making the space in the center look like an arbitrarily thin cylinder... (yes yes this becomes a sphere with a line connecting two poles. This was my interpretation of what he said and I ran with it) you can't define a pushforward of the tangent space (of the torus being mapped into 3-space, when you treat it as a manifold) for any of the points that get mapped to a pole.

But, to my understanding, nothing gets mapped to the pole--but rather to every neighborhood around the pole.

ii) Failure to be injective. Or at the very least: if he's insisting that the hole have "width zero", that doesn't bode well for the hausdorff-ness/distinctness of the points lying 'across' this hole/gap.

I don't think I understand what you mean by "lying across this hole". Do you mean points on either side of the hole?

iii) Under that interpretation of jonb's construction (hrm, this looks like the particular shape he was actually talking about... but you still get the shape I was talking about by mapping the torus and then putting it through [some charitable attempt at] the inverse mapping of the sphere) what you end up with is neither a torus nor a sphere, since that 'compressed' shape won't be compact (surely the one-point hole is a limit point?).[/quote]

Hmm. That's an interesting line of attack; I hadn't thought of that. The result would immediately follow, since compactness is preserved by homeomorphisms.

Either:
a) there is some 'inner ring' that gets mapped to the point that's supposed to be a hole, and this fails to be injective.

Which doesn't make sense; the ring lies on the torus, whereas the hole does not lie on the new shape.

b) If the torus' "hole" is genuinely a point: That compactness thing from earlier.

Although the first part of the 'playing different games' comment was probably too overstated/harsh...

I think this is the easiest proof for me to follow.

[Categories+Sheaves]

Dear Categories+Sheaves
Thank you for continuing with me. My aproach I know is clumsy, as I am coming at this subject from a wildly different angle...
...My problem is that what I am seeing from my end does not seem to tally up with how maths is explained to me.

Nothing wrong with being clumsy, as long as we're all able and willing to think about this stuff.

But if the '0' in what I have termed the result is the projection from the origin through the function, does that not satisfy you that we can call it '0'?

Well, what do we mean by zero?
For the most part, we insist that an element is "zero" when we have some sort of addition operation on a set such that 0 + a = a =a+ 0 for all a our addition operation can take as an input. The origin takes this role in vector addition, and the 'zero function' (f(x)= 0 for all inputs x) fills this role when we're adding functions. In this sense, there can only be one 'zero element' for a given addition operation (if we have 0 and 0' both acting as zeros, 0 = 0 + 0' = 0').

Not to say that you have to approach things that way... but if you aren't doing it that way, you may need to go into more detail/think harder about what you mean by '0'.

Your next explanation is excellent and I think I understand. So I would like to walk away from all the twaddle I was talking about that. But I would like to take up the conversation from this part, because it is central to my theme..

...This is where my problem is: I would contend that Euclid's books are not about maths. Rather they are help books for Artists etc, on how to use maths. As such the definitions are to enable us to make constructions, not definitions that should be used to define the subject itself. I might use points to map out a shape in space, but vectored space has no points.
As such the point is a tool we use not a thing itself. Where I am going with this is that it seems to me a number is the same it has no integrity, but it is created out of the series, any one number is only given its value by the other numbers in the set or series.

The whole "Object X doesn't genuinely exist, we just use it as a tool" angle irks me a little. Here are my gripes expressed by a wiser man.

Also: Euclid pioneered the axiomatic approach to mathematics, so calling his geometry 'not the math itself (but applications of the math)' as a general statement is a little iffy (although I think I can feel where you're going...)

I do not know if this is standard thinking or not. When as an outsider to maths you get an entire hour long BBC programme talking about Cantor and infinity, and declaring he said you cannot map one to one fractions and decimals, without saying that is an analogy. And I try to look up definitions of number and find:-

Can I take where I am with this is a reasonable starting point?

Hrm... I'll try to find a juicy bone to throw your way...

I don't think I understand what you mean by "lying across this hole". Do you mean points on either side of the hole?

Yeah... I would have used 'antipodes' but that can also mean a different pairing of points on the torus... (since it's a direct product of circles...)

[Angrboda]

I'm getting lost because my maths is weak and my brain is old and gray. Still, I enjoy trying. So please continue.


I have this.... let's call it a notion, as the word "idea" is perhaps too strong. My idea is that if you look at the set of all possible universes, varying on whatever variables are possible for a universe, some subset of those universes will have regularities in them; in our universe, we call such regularities the laws of physics. And inside that subset, arrangements of "stuff" will occur that are able to capitalize on the inherent predictability and order implicit in such regularities. Call that life. Now I so lack the necessary math and physics to prosecute this statement, but my hunch is that in the phase space of all possible universes, these subsets span a significant portion of the entire space. In my view, it's possible that life and intelligent life aren't rare chance occurrences, but likely built into the relationship between order and the possible. I rather suspect that the potential universes without some forms of regularity and order may be a minority. (This dips into questions of the meaning of causality, questions which I'm going to side step for the moment.)

Now, given this understanding, one can view life forms, and especially intelligent life forms, as having evolved mechanisms for extracting the utilitarian value of the predictatability of order by mental mechanisms (evolution itself being an algorithmic process which capitalizes on the ordered regularities and "builds them into" machines that behave in ways that are useful if the regularities are assumed to hold true [they have predictive value]). (Just a side example, a fish doesn't view the minnow it wants to eat as existing or moving in more or less than 3 dimensions, nor as if it might disappear here and appear there. It doesn't "know" these things in the sense of having thought them, but the regularities are built into how its brain thinks about the possible, so that what it thinks is often quite probable, and following the probable is profitable — that profitability being the ratchet that drives evolution.)

Now I'm going to make a gross simplification for the sake of example. When we look at brains, and human brains specifically, they appear to be composed of neural networks arranged in a connectionist model sort of way, with a bunch of nodes at the input or perception side, a bunch of nodes at the output side, and a bunch of nodes in between that are doing the work. (See ANN, artificial neural networks.) Now this isn't an invariant property of such networks, but many can be characterized as having the nodes at the input side dealing with the entire input in all its particularity, there are no generalizations or inferences made at this level. However, as you go farther back into the network, you encounter things which can be described as abstractions: expressions of properties and aspects of the particulars but which aren't themselves particulars. Example.... humma. We don't see a circle, what we see are a bunch of points that cause one of those higher order abstraction levels to gravitate both to shape and the particular shape, but neither shape nor that particular shape "exist" in the real world, nor in our particulars.

Now to connect the dots. As life forms that capitalize on the order and regularity in our environment, some of that order and regularity can be expressed as abstractions. Math probably being the important one. If my view is correct, then math and number are just pure forms of what is happening in our neural networks; they express awareness of the output of layers that output abstractions of the real as real things themselves (e.g. square, triangle, circle are not real, but thinking about them in relation to perception or idea is real, because our thoughts are the categories which our brains create for containing what we think; one layer, then, gravitates towards idealizations of shape as idealizations — as idealizations qua idealizations; these aren't truly abstractions or generalizations in the pure sense, they are more "creations"). However, given what gave rise to these mental forms, they have an intimate connection to the real that can never be completely severed.

It is in this latter sense I would call to mind Heidegger's notion that "there is sameness in difference and difference in sameness". In that same sense, the particulars of real world circles in some sense "infect" our mental, platonic circles, and yet they are also quite distinct; the circle inside us is not the circle outside us, it is just a useful way for brains to model regularities outside, regularities that the circle inside shares with them.


Anyway, I hope this hasn't been an inchoate mess. For whatever it's worth, which is surprisingly little in today's economy.