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. This is the difference between a filter and an ideal.
It is also the exact question discussed by Aristotle when speaking of the relation between definitions and identity in Topics.
Logical identity, in the modern parlance, is ontological "self-identity" arising from a combination of Russell's description theory and Wittgenstein's rejection of Leibniz' principle of identity of indiscernibles.
Aristotle points out that one can never prove an assertion of sameness, although one can destroy such an assertion. The modern logic negates this entire relationship between identity and definition.
Given the choice, it is better to side with Halmos and Aristotle (and Frege).
applies simultaneously to ontology and semantics and cannot simply be interpreted ontologically as one must do with Russell and Wittgenstein.
Along similar lines, note that Tarski's paper on truth in formalized languages specifically excludes scientific languages built upon definition whereas Robinson's paper on constrained denotation specifically includes the relationship between descriptively-defined names, identity in models, and truth.
And, in Kant, logic is a *negative criterion of truth*.
In other words, one ought not be proving beliefs in mathematics.
Analysis with synthesis is a circular investigation of structure. Synthesis without analysis is something else altogether. When combined with realism, it is religion.