Date: Dec 8, 2012 1:16 PM Author: fom Subject: Re: fom - 01 - preface 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 space-filling

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 two-dimensional 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 3-simplex

(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 semi-modular lattice with

the atomic covering property is an entirely

different beast.

It is the semi-modular 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

context-dependent 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.