fom
Posts:
1,968
Registered:
12/4/12


Re: fom  01  preface
Posted:
Dec 8, 2012 1:16 PM


On 12/8/2012 9:08 AM, WM wrote: > On 8 Dez., 10:02, Zuhair <zaljo...@gmail.com> wrote: >> On Dec 7, 9:45 am, fom <fomJ...@nyms.net> wrote: >> >> >> >> >> >>> Although it is not mentioned frequently >>> in the literature, Frege actually >>> retracted his logicism at the end of >>> his career. His actual statement, >>> however, is much stronger. He rejects >>> the historical trend of arithmetization >>> in mathematics as foundational. >> >>> In "Numbers and Arithmetic" he writes: >> >>> "The more I have thought the matter >>> over, the more convinced I have become >>> that arithmetic and geometry have >>> developed on the same basis  a >>> geometrical one in fact  so that >>> mathematics in its entirety is >>> really geometry" >> > >> I agree with Frege. Geometry or more generally thought about structure >> is what mathematics is all about, number is basically nothing but a >> very trivial structure. > > Then everybody should understand that the infinities in the numbers > forming the following triangle and the geometrical aspects have a > common origin: > > 1 > 11 > 111 > ... > > Height and diagonal have lenght aleph_0. What about the basis? > > Regards, WM >
The answer here resolves to simplexes, cones, and linear dichotomies.
Have you ever seen the works of M. C. Escher? They capitalize on the relation of projective geometry to the perception of space.
It is easy to think of the first few steps,
0: a point
*
1: a line segment
**
2: a triangle
*  * \ / \ / *
in each case, a dimension was traversed by adding a point distinct from those which came before. Technically, this is called general position. To the extent that mathematics can inform as to the experience of space, general position is related to topological dimension. Everyone knows that Peano demonstrated a spacefilling curve. But, such functions cannot have continuous derivatives. So, one part of the issue involves continuity and this places part of the question into understanding topological dimension.
But, we can ignore topological dimension if we understand that topological dimension merely correlates the notion of linear independence with the notion of points in general position.
However, to do that we must understand the relationship of points in general position to the linear dichotomies discussed in switching functions and, in particular, threshold functions.
In order for four points in a plane to be distinct, one must be able to find seven distinct lines that account for all of the partitions of those points.
So, one can speak of general position within a twodimensional plane based on linear separating surfaces. Thus, for the next step, one must think as if one is counting dimensions.
The ability to represent a 3simplex (that is, a tetrahedron) on a piece of paper is also relevant here.
4: a cone with 4 points in general position
* /\ /  \ /  \ /  \ /  \ /  \ /  \ *  * \  / \  / \  / \/ *
This is called a cone because that is precisely what the definition of cone is. A new point in general position is added to the system and then edges are added to connect each of the original points. It is the achievement of topology to have demonstrated the role of distinguishing parts of a collection in order to connect linear independence to general position. But, in geometry we simply use Peano's trick in reverse to recognize that we do not have to leave the piece of paper to do this.
5: a cone with points in general position
*  * /\ / /  \ / / / /  \ / / / /  \ / / / /  / / / / / \ / / //  \ / *  * / \  / / \  / / \  / / \/ / *
From this point you should get the idea. If you clean up the diagram above, you will recognize a pentagram inside of a pentagon.
So, one can "count" using cones on points in general position
Now, the inconsistency in the claims of the usual logicist position is the coincidence of claims concerning Boolean algebras and counting. A Boolean lattice is one kind of beast with one set of properties whereas a semimodular lattice with the atomic covering property is an entirely different beast.
It is the semimodular lattice with the atomic covering property. These lattice are called matroid lattices and their theory is the algebraic theory which connects the partitioning of sets (equivalence classes) with certain questions about linear dependence.
There are certain ongoing investigations into the structure of mathematical proofs that interpret the linguistic usage differently from "mathematical logic". You would be looking for various discussions of contextdependent quantification where it is being related to mathematical usage.
You will find that a statment such as
"Fix x"
followed by
"Let y be chosen distinct from x"
is interpreted relative to two different domains of discourse.
This is just how one would imagine traversing from the bottom of a partition lattice.

