david petry <firstname.lastname@example.org> writes:
> Platonism, on the other hand, is a perversion of natural logic. It > would be insane to apply platonism to reasoning about the real > world. Falsifiability and platonism are not > compatible. Falsifiability would exclude Cantor from mathematics. > Just as the scientists use falsifiability as a criterion to > distinguish real science from crackpot science, it can serve as a > criterion to distinguish real mathematics from crackpot mathematics.
Regardless of this idea that mathematics should be falsifiable, you must accept the following statement:
It is a theorem of ZF that |R| > |N|,
where cardinality is defined in the usual manner. This fact is trivial to confirm.
So you don't want to call ZF a mathematical theory. To this, what reply can be offered? Have fun trying to enforce your personal re-definition of what mathematics is, while those who are interested in theories such as ZF will continue their research as before.
In the meantime, I'll note that there is nothing in the above statement that comments one to Platonism. It is a simple statement that a particular formula is provable in a particular theory. Personally, I have no particular allegiance to Platonism, but I wouldn't say I have a well-developed mathematical philosophy at all.
Let me ask you about this notion of falsifiability. I presume that you'd agree that Fermat's theorem,
(An > 2)NOT(E a,b,c)( a^n + b^n = c^n )
is falsifiable, since if it is false, we can show that it is false by producing n, a, b and c such that
n > 2 and a^n + b^n = c^n .
So that is a proper mathematical statement (or whatever), right?
Is the negation of Fermat's theorem a falsifiable statement? If so, how might one show that it is false?
Is it the case that if a given mathematical statement is "falsifiable", then so is its negation? Or are there statements which satisfy your view of what mathematics should be, though their negations do not?
-- Jesse F. Hughes "And I will dream that I live underground and people will say, 'How did you get there?' "'I just live there,' I will tell them." -- Quincy P. Hughes, Age 4