david petry <email@example.com> writes:
> On Tuesday, March 26, 2013 12:51:21 PM UTC-7, Dan wrote: > >> The first >> principle that is to be falsified when doing mathematics is the >> principle of falsifiability . > > > I'm not convinced you understand what "falsifiability" means. > > We say that a statement is falsifiable if and only if it makes > predictions about the outcome of well-specified, feasible > experiments. For mathematics, those experiments will be > computational experiments. > > For a simple example, consider Fermat's Last Theorem. We could > write a computer program that methodically searches for > counterexamples to the theorem and then halts when it finds such a > counterexample. Fermat's Last Theorem predicts that the program > will never halt. And, of course, Fermat's Last Theorem would be > "falsified" if the program did halt. > > The principle of falsifiability asserts that statements that make no > predictions are meaningless and should be ignored.
And so, for instance, while Fermat's Last Theorem is kosher, its negation is pseudoscientific claptrap.
And, similarly, Goldbach's conjecture is a good, honest hypothesis, but its negation is twaddle.
Interestingly, the scientific hypothesis, "The universe is infinite," is an upstanding example of falsifiability, while, "The universe is finite," fails the falsifiability test.
And, see "The Logic of Reliable Inquiry" by Kevin Kelly for many examples of apparently scientific hypotheses (matter is finitely divisible, human behavior is computable, etc.) which are *NOT* falsifiable.
> The mathematics that helps us reason about the real world is > necessarily compatible with the principle of falsifiability.
Honestly, I agree with part of what you say. In so far as our measurements are always to some finite precision, it seems plausible that we can use finitist mathematics in our scientific theories.
But I don't see why anyone would want to do so, given the fact that classical real analysis is simple and beautiful. It allows concise statements of theorems. And, in any case, even if we agree that scientific hypotheses must be falsifiable, it does not follow that the math used in science must be "falsifiable".
-- Jesse F. Hughes "Well, I guess that's what a teacher from Oklahoma State University considers proper as Ullrich has said it, and he is, in fact, a teacher at Oklahoma State University." -- James S. Harris presents a syllogism