fom
Posts:
1,093
Registered:
12/4/12
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Re: fom - 01 - preface
Posted:
Dec 8, 2012 1:16 PM
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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.
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