Date: Dec 8, 2012 4:02 AM
Subject: Re: fom - 01 - preface

On Dec 7, 9:45 am, fom <> 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.