fom
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fom  01  preface
Posted:
Dec 7, 2012 1:45 AM


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 wellentrenched nonparticulars 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, nonparticulars, 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 nonempirical 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 selfidentity 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."

