On 1/30/2013 11:19 AM, WM wrote: > On 30 Jan., 12:53, fom <fomJ...@nyms.net> wrote: >> On 1/30/2013 5:29 AM, WM wrote: >> >>> On 30 Jan., 12:02, fom <fomJ...@nyms.net> wrote: >> >>>> As for those "logical considerations," I mean that >>>> one can develop a hierarchy of definitions that >>>> depend on actual infinity. To say that mathematics >>>> is "logical" is to concede to such a framework. I >>>> do not believe that mathematics is logical at all. >> >>> That is a very surprising statement. Why do you think so? >> >> In his papers on algebraic logic, Paul Halmos made >> the observation that logicians are concerned with >> provability while mathematicians are concerned more >> with falsifiability. > > Same is true for physicists. But I had the impression that > mathematicians are more concerned with proving. I, as a physicist, am > more concerned with showing counter examples.
Most mathematicians would agree with that statement on the surface. But, if you think about how people use mathematics arbitrarily, you can interpret the activity of mathematics as establishing the scientific coherence of these uses. By "scientific," I mean precisely the epistemological justification for a demonstrative science given by Aristotle.
What has been lost to the teaching of mathematics because of the mathematical realism of the late nineteenth and early twentieth centuries is the relationship of analysis and synthesis in the practice of mathematics. Analysis is "reverse engineering" and synthesis is proof.
There is an inherent circularity in mathematics that actually belongs there. The best account of why it belongs there is in Emile Borel's book "Space and Time." Space has a sense that can only be made analytic, for example, through the sign of a determinant.
Analytic philosophy takes that geometric distinction and turns it into "the True" and "the False".
To see this, get a book on threshold logic. Of the sixteen basic Boolean functions that represent the semantics of propositional logic, exactly two are not linearly separable. Those two are logical equivalence and exclusive disjunction.
When "logic" finally gets around to a parallel development with geometry in the cylindrical algebras of Tarski and Monk, informative identity, namely
is equated with two-dimensional subspace through the origin. For three dimensions, this is a hyperplane.
Correspondingly, this is exactly why logical equivalence is not a linearly separable switching function. Its truth-set and its falsity-set cannot be separated by a hyperplane.
Analytical philosophy conflates these geometric relations in its assertions of logical priority.
"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. Only on this view does mathematics present itself as completely homogeneous in nature. Counting, which arose psychologically out of the demands of practical life, has lead the learned astray."
> > All that sounds interesting but is a bit above my level.
Sorry about that.
If you can find Robinson's work ("On Constrained Denotation") you will appreciate it. In keeping with Frege's original observation that truth values depend on statements completed with NAMES, Robinson explains that the diagonal of a model -- that is, the (x,x) corresponding to x=x -- is defined by NAMES.
This is largely what you have been saying in your posts.