Date: Dec 8, 2012 4:02 AM Author: Zaljohar@gmail.com Subject: Re: fom - 01 - preface 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"

>

> ==============================

>

> Once geometry is no longer precluded from

> the debate, positions such as Strawson's

> become admissible. Specifically,

> linguistic analysis does not suggest

> that abstract objects are treated

> differently in the lexicon from paradigmatic

> material objects whose geometric relations

> are intrinsic to their description.

>

> In his chapter on Logical Subjects and

> Existence from "Individuals" he writes:

>

> "Of course, not all well-entrenched

> non-particulars exhibit this kind of

> relationship to particulars. Numbers

> do not. Nor do propositions. But

> there are other ways in which things

> can exhibit analogies with particulars

> besides being themselves, as it were,

> models of particulars. Particulars

> have their place in the spatiotemporal

> system, or, if they have no place of

> their own there, are identified by

> reference to other particulars which

> do have such a place. But,

> non-particulars, too, may be related

> and ordered among themselves; they

> may form systems; and the structure

> of such a system may acquire a kind

> of autonomy, so that further members

> are essentially identified by their

> position in the system. That these

> non-empirical relationships are often

> conceived on analogy with spatial

> or temporal relationships is

> sufficiently attested by the vocabulary

> in which we describe them."

>

> =================================

>

> Continuing along this line of inquiry,

> the ontological positions that confer

> self-identity to objects are subject

> to the same criticism that Mach

> applied to Kant's treatment of

> spatial intuition without objects.

>

> In "Space and Geometry" Mach writes:

>

> "Today, scarcely anyone doubts that

> sensations of objects and sensations

> of space can enter consciousness only

> in combination with one another: and

> that, vice versa, they can leave

> consciousness only in combination

> with one another. And the same must

> hold true with regard to the concepts

> which correspond to those sensations."

>

> Thus, there is a simultaneity in the

> presentation of objects and the geometric

> relations between objects that should

> be apparent in any intial presentation

> of a system.

>

> ============================================

>

> Looking to a geometric foundation, one

> reverts from Fregean logicism back to

> Kantian intuition. But, the caveat is

> to be found in Russell's "An Essay on

> the Foundations of Modern Geometry".

>

> Russell writes:

>

> "I shall deal first with projective

> geometry. This, I shall maintain,

> is necessarily true of any such

> form of externality, and is, since

> some such form is necessary to

> experience, completely a priori."

>

> "For the present, I wish to point

> out that projective geometry is

> wholly a priori; that it deals

> with an object whose properties

> are logically deduced from its

> definition, not empirically

> discovered from data; that its

> definition, again, is founded on

> the possibility of experiencing

> diversity in relation, or

> multiplicity in unity."

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.

Zuhair