Date: Dec 7, 2012 1:45 AM Author: fom Subject: fom - 01 - preface

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"

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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."

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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.

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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."