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.