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.
> > 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.
Well in mathematics we can ask whether in a = a the right a can be the same as the left a, because both can be distinguished by their position. One can ask whether the digits of a decimal can be distinguished, which is possible when they are indexed, i.e., when their positions are taken into account, and one can ask whether 1 + 1 = 2 or whether that has to be proved. I deny the latter because we must start with something. And the very best to start in mathematics is to take the natural way namely to count, i.e., to add 1. Therefore it is in priciple nonsense to define the natural numbers by axioms. And every time I do it in class room I try to excuse that superfluous procedure as simply being a convention. > > 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). > > The axiom, > > x=x > > 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*.
All that sounds interesting but is a bit above my level.