```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?  Theycapitalize on the relation of projective geometryto 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 byadding a point distinct from those whichcame before.  Technically, this is calledgeneral position.  To the extent thatmathematics can inform as to the experienceof space, general position is related totopological dimension.  Everyone knowsthat Peano demonstrated a space-fillingcurve.  But, such functions cannothave continuous derivatives.  So, onepart of the issue involves continuityand this places part of the questioninto understanding topological dimension.But, we can ignore topological dimensionif we understand that topological dimensionmerely correlates the notion of linearindependence with the notion of pointsin general position.However, to do that we must understandthe relationship of points in generalposition to the linear dichotomiesdiscussed in switching functions and,in particular, threshold functions.In order for four points in a planeto be distinct, one must be able tofind seven distinct lines that accountfor all of the partitions of thosepoints.So, one can speak of general positionwithin a two-dimensional plane basedon linear separating surfaces.  Thus,for the next step, one must think asif one is counting dimensions.The ability to represent a 3-simplex(that is, a tetrahedron) on a pieceof paper is also relevant here.4: a cone with 4 points in general position           *          /|\         / | \        /  |  \       /   |   \      /    |    \     /     |     \    /      |      \   * ------|------ *    \      |      /      \    |    /        \  |  /          \|/           *This is called a cone because that is preciselywhat the definition of cone is.  A new pointin general position is added to the systemand then edges are added to connect each of theoriginal points.  It is the achievement oftopology to have demonstrated the role of distinguishingparts of a collection in order to connectlinear independence to general position.  But,in geometry we simply use Peano's trick in reverseto recognize that we do not have to leave thepiece of paper to do this.5: a cone with  points in general position           * ----------------------------      *          /|\                            /         / | \                      /    /  /        /  |  \              /       /   /       /   |   \      /          /    /      /    |    /           /     /     /     /     \     /       /    //     |      \         /   * ------|------ *      /    \      |      /    /      \    |    /   /        \  |  /  /          \|/ /           * From this point you should get the idea.  If youclean up the diagram above, you will recognize apentagram inside of a pentagon.So, one can "count" using cones on pointsin general positionNow, the inconsistency in the claims of the usuallogicist position is the coincidence of claimsconcerning Boolean algebras and counting.  ABoolean lattice is one kind of beast with one setof properties whereas a semi-modular lattice withthe atomic covering property is an entirelydifferent beast.It is the semi-modular lattice with the atomiccovering property.  These lattice are calledmatroid lattices and their theory is the algebraictheory which connects the partitioning ofsets (equivalence classes) with certainquestions about linear dependence.There are certain ongoing investigationsinto the structure of mathematical proofsthat interpret the linguistic usage differentlyfrom "mathematical logic".  You would belooking for various discussions ofcontext-dependent quantification where itis 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 twodifferent domains of discourse.This is just how one would imaginetraversing from the bottom of apartition lattice.
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