fom
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Registered:
12/4/12
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fom - 01 - preface
Posted:
Dec 7, 2012 1:45 AM
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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."
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