> On Wednesday, September 12, 2012 1:30:44 AM UTC-6, Andrzej Kozlowski wrote: >> As I have written before, I don't think this is an appropriate space for >> >> explaining the basic ideas on modern philosophy to people who clearly >> >> have never heard of them. > > Andrzej, > > Of course I've heard of them. But I don't consider mid 20th century "modern" here. Science and mathematics have changed drastically in the last century: philosophy hasn't yet fully digested these developments. However, philosopher/mathematicians like Hersh have made very significant advances in "humanist" mathematical philosophy. > >> http://faculty.unlv.edu/jwood/wm/Quine.pdf > > Quine is an interesting fantasist, but I can't consider him any more reliable as a guide to reality than, say, J. R. R. Tolkien. He utterly misunderstands physics. An ontological commitment to the existence of real numbers is certainly unnecessary to a physicist. Real numbers present serious difficulties to physics. We must carefully step around these problems. This has caused a number of physicists (including one who should be well known here) to seek discrete models of "fundamental" physics. From my perspective, approximate numbers are much truer to physics than mathematical reals. But approximation is toxic to much mathematics, and we also like to tap the power of the exact. So we wander from realistic approximation into the unrealism of mathematical "real" numbers and back. Mathematica is an extremely useful tool for this kind of journey: that's why I'm here. But we are also constantly reminded here of the difficulties of mixing exact mathematical reasoning with realistic approximation.
To my knowledge, - that might be very limited -, mathematics is most effective in physics where differential equations can be found and solved. Also to my knowledge the existence of differential equations and their solutions rely on differentiable functions. Differentiable function has to be continuous. Continuity relies on real numbers and real numbers cannot exist without the continuum hypotheses. The continuum hypotheses cannot be proved or disproved with all the axioms crafted for real numbers.
Once upon a time I had a short discussion of it with my analysis teacher ZoltE1n Dar=F3czy, who made a name for himself in the area of "function equations" . His brother S=E1ndor, was a well known physicist by the way under Alex Szalay in Debrecen, so I am sure there were some brotherly and some not so brotherly philosophical discussions between the two as they advanced in their selected area of interest. It was first semester analysis and in one of the breaks between the two hour long lectures, I told to him, that in a real quantum theory the usual mathematical tools cannot be applied, because as we go down the distance scale, the metric cannot be established preciously in reality because of the Heisenberg relations, neither in the Weierstarss, nor in the Cauchy interpretation. He smiled at me and replied: "then you have to trow out the continuum hypotheses, but with it you loose all the power of mathematics in physics". Then he further said, still smiling: "Show me a physicist who can tell me preciously how the light is bouncing back from a surface even using all the mathematical tools they have."
> > As Hersh points out, part of the difficulty is that there are different kinds of existence. In physics, we frequently "throw out" solutions to the math when they violate informal conditions like causality. Such solutions "exist" in the mathematical sense, but since the correspond to behavior that is never observed in the physical world they do not "exist" in the physical sense. > >> It might (or might not) become clear what I have been hinting at: that >> >> is that the idea of a sharp distinction between "experiment" and >> >> "theory" etc. are illusionary. > > That's very much a theorist's ideology. This is why it is advisable to keep theorists away from dangerous machinery ;-) > > "In theory there is no difference between theory and practice. In practice there is." Regardless of the attribution you choose (Einstein, Berra, ...), this is wisdom. > >> There is no >> point repeating all this stuff on a forum devoted to Mathematica. > > The original issue of this thread was education. A philosophy that cannot distinguish between reality and hallucination is useless here. But, if we can understand mathematics as a human social construct, we can connect it to human social activities like education. >