The nature of number

[Cato]

Inovking Heideggar without a mention of Wittgenstein is the same as mentioning Newton without Einstein. Heidegger? Platonic circles?

Are you really prepared to invoke Heideggar?

[Categories+Sheaves]

Inovking Heideggar without a mention of Wittgenstein is the same as mentioning Newton without Einstein. Heidegger? Platonic circles?

Are you really prepared to invoke Heideggar?

Some folks on a philosophy forum I frequent were throwing around of W's criticism of the infinite in math, and that was a bunch of fun.

Also: I'm a fan of Heidegger too. So bring it, sucka. (Maybe start this in a new thread, though?)

[jonb]

Dear Categories+Sheaves

WOW, I think were singing from exactly the same song sheet, The link( http://www.thebigquestions.com/2012/02/08/rock-on/), yes that is what I think I may have found a pointer towards. It will take me a little while to draw up some new graphs to try to make it clearer what I think I have found.
When I mentioned Euclid it was not to say the maths was not real, but it seems evident to me that the definitions he was using were designed to apply it rather than examine the thing itself.

I did not know anybody was even thinking about maths that way, thank you so much!

[jonb]

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').

I am not sure what this means and how it applies to what I think I may have seen, so forgive me an let me try to reiterate what I have done in what I think maybe a clearer way, then If you would pull it apart again I might have a clearer idea of where I am going wrong.

I thought of an equation as a geometric form- for instance 2x3=6





Then thinking of 'x3' as a common focus I could lay out every number that could be multiplied by three and got a consistent result in the order of the numbers which had been multiplied (even though the series that resulted are in the opposite direction to the origin).



It seems to me what I have here is not just a method of comparing individual numbers, but of sets, and series. While at the same time each individual number in the series can be mapped to a number in the resulting series. Thus I am not making a lot of individual calculations, but an infinite set of numbers are all dealt with in one go.

Now when I multiply by '0' I get this result.



The series multiplied by '0' gives me a series of '0' and all the individual numbers in the original series become '0's in just the place where you would expect them to be in the order you would expect them to be in. So is the result a series of distinct '0's? So we have a series of individual parts that do not progress in value, or is the result a single '0' that is not a point, but that has at least some width?

What you have written above I think would be consistent with all the individual lines from origin to result. So is what I have done so far OK?

[Categories+Sheaves]

The link( http://www.thebigquestions.com/2012/02/08/rock-on/), yes that is what I think I may have found a pointer towards.

Well, if you're hungry for more...
This is pretty relevant: it's a discussion of how the natural numbers work, and the axioms we prefer to take concerning them.

Here he spells out a 'strong' realist position concerning mathematics. (I remember throwing this at a currently-absent forumgoer, houseofcantor... I miss that guy)

This collection is a little less relevant to the precise nature of numbers, but still good:
First post in a series on whether math is invented or discovered.
Second post in that series, where he argues explicitly for taking natural numbers as real.
Third in that series where... hey I was taking that stance he's arguing against (in opposition to CliveStaples) in a previous thread...
Here he spells out a 'strong' realist position concerning mathematics.

It will take me a little while to draw up some new graphs to try to make it clearer what I think I have found.

No rush.

When I mentioned Euclid it was not to say the maths was not real, but it seems evident to me that the definitions he was using were designed to apply it rather than examine the thing itself.

Hrm...

By the same token then, Set Theory is designed to apply sets to other problems, rather than examine sets themselves. Euclid's axioms could produce novel statements about lines and how they behaved, but only in regard to the properties declared by those axioms in their application. The difference in cardinalities between the naturals and the reals revealed a great amount of information about how sets can behave and how they can be applied but any questions about what we mean by 'set' (beyond an intuitive "the collection/class of such-and-such stuff") remained in the dark--it's the paradoxes we run into (Cantor's, Russell's, "Skolem's", etc.) that actually struck the foundations. Similarly, the alternative geometries that arose from the investigation of alternatives to Euclid's fifth postulate were the ones that changed the common interpretation of "line".

I may have gone a little off-track with the paradoxes, so to return to my original point: the applications* are the math. A formal treatment of geometry didn't exist prior to Euclid, and new things (about how lines/angles behave) were discovered via the foundations he laid. Exploring the results or applications of axiomatic system A isn't just an integral part of determining what we mean by A and finding a paradox in A. This is why we axiomatized A in the first place.

*I'm trying to use this in the broadest sense possible: e.g. K-theory is applicable to determining whether two topological spaces are homeomorphic or not. Perhaps the issue, jonb is that in those 'early' stages of math, there wasn't much math to discuss beyond the direct, real-world applications.
-------Update-------
I'll try to respond to your new post later..

[jonb]

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

You and me both

Thought a universe being caused by a flaw, or inconsistency.



Because the potentials don't meet correctly and resolve, new lines are formed, which create all prime numbers.

[jonb]

Further to my post on this thread no 34, I am I know over eager, and want to get this off my desk.

So I will presume there is nothing intrinsically wrong with the notion of a set of '0's as it is mentioned in places in the links mentioned on this thread.
There is a second feature in the graph I am using.


If we line up the origial series (A) and the resulting series (B) there is a field between the two that if we move what I have termed the function about in, we alter which part of series (B) we are examining. So for each position in the field we seem to have a distinct function, yet all of these functions relate to each other in what seems to be a consistent way.



As far as I can see this is logical and consistent, and allows me to examine not just individual numbers, but also sets in the series. So if I take two series of unrestricted length and line them up at their neutral '0' point and multiply a given set in series (A) by '0' we get this graph.



At first sight this seems to break down, with the function seeming to be in the same place as the result. The '0' of the function being the same as the '0' of the result, but as we have I think established that cannot be the same. Also if we think of this in geometric terms the function and the result have to be in different places. Also as mentioned on this thread by apophenia in geometry a distance '0' is still a distance so I would contend that the following graph seems consistent.



Now this I think depicts that the series is not just composed of numbers of gradually increasing/decreasing value, but within it are numbers that are of the same value. I have done lots of speculating about this, but I really need some feed back, even if that is to tell me I am as thick as two short planks, and didn't you notice this?

[Categories+Sheaves]

My apologies for taking so long to respond. Just really unsure what I should be responding to.

You're willing to change a bunch of stuff (e.g. the whole 'the distance between a and b is 0 iff a = b) which means you aren't going after the same stuff most mathematics is reaching for. For the most part, you're looking at a class of affine transformations (subset of the projective transformations...) of a line. And then you look at something that would normally map your line to a single point ("x0", this function isn't injective or surjective like the others so it's a bit odd to think it's going to behave in the same way) and produce a line of zeros (because it came from a line, the image ought to look like a line? This need not be the case...). I could make some arguments about why drawing this line full of zeros seems silly, but I'm not sure it involves math you're familiar with (else you'd probably have thought of them by now ).

So... where are we?

[jonb]

OK, lots of stuff in there, so could we unpick it a bit, and is my way lets jump in at the centre. 'x0' what about the position for the 'function' 'x0+1' etc are these all not part of the same field?

[Categories+Sheaves]

Yes, these 'x0 +n' transformations are, in a sense, part of the same "field". But much in the same way that multiplication by zero isn't the same as multiplying by 3 (I can 'undo' the latter by multiplying with 1/3, but I can't undo the former) these functions aren't carrying a line to a line, but collapsing a line to a point. Here, for example: You have a 'field' of n-by-n matrices, some of which have "rank n"/"nullity 0" and map an n-dimensional euclidean space onto an n-dimensional euclidean space/map only one point in the domain to zero, some of which have "rank n-1"/"nullity 1" and map an n-dimensional euclidean space onto an (n-1)-dimensional euclidean space/map a line in the domain (or 1-dimensional euclidean space) to zero, some of which have "rank n-2"/"nullity 2" and map an n-dimensional euclidean space onto an (n-2)-dimensional euclidean space/map a plane in the domain (or 2-dimensional euclidean space) to zero...
A lot of things have caveats that pop up in the "zero"/"trivial" case