In article <36cf1197.0@calwebnnrp>, John M Price PhD <email@example.com> wrote: >SNIP< >But seriously, a more sensible approach might be to propose that different >disciplines lie on a continuum of verifiability. Mathematics and logic, >involving the manipulation of relationships between symbols which don't >have "real-world" referents, are so to speak self-verifying. They occupy >the high end of the continuum; they have theorems, not experiments. At the >low end we have, say, economics--or, even worse, history, where the notion >of reevaluating and correcting "past mistakes" occupies such a large >percentage of the discipline's resources that no-one even talks in terms >of mistakes; they have schools of thought, tendencies, and so forth.
But this attitude itself represents a paradigm shift in mathematics that occurred around the beginning of this century.
If you had suggested to a mathematician in the 19th Century that geometry has no real-world refererent, he would have looked at you as if you were some sort of idiot. The axiomatic approach to mathematics, as it is understood in the 20th Century, primarily goes back to Hilbert (maybe about 1910). Hilbert was the first mathematician to draw up a complete set of axioms for geometry. Proofs in Euclidean geometry generally depended on the use of figures (pictures) and involved taking certain facts for granted, since these facts were obvious from the figures. There are several well known fallacious proofs in Euclidean plane geometry which do not violate any of Euclid's axioms but work because of a figure which looks correct but is actually slightly inaccurate.
Around the beginning of the 20th Century, a lot of things in calculus were taken for granted because they seemed intuitively obvious. To ask for proofs of these things would have seemed a nitpicking waste of time. And yet mathematicians like Lesbesgue gave examples showing that these so-call facts were not actually true.
When Lesbesgue first published examples such as functions which are nowhere differentiable (or a function f(x) which is continuous at all irrational points x and discontinuous whenever x is rational), Poicare said, "Well, even if such functions do exist, mathematicians should just ignore them." Nowadays we look on this statement by Poincare as completely idiotic, but I think that in a way, there was some merit to what he said. Namely, I think that the sort of functions Lesbesgue constructed do not occur in most applications of mathematics and are not of great importance to those who use mathematics as a tool for studying the reality-based sciences.
The other big paradigm shift at the end of the 19th Century was the development of set theory by Cantor. The idea that it could make sense to say that some infinite sets are larger than others, or that there are exactly as many rational numbers as there are integers, but far fewer than there are real numbers, was quite scandalous to most mathematicians.
Cantor was a manic-depressive, by the way, and apparently felt that he needed the enormous energy that came in his manic phases in order to fight the ongoing battles with his mathematical enemies.
Then Goedel came along in the middle of the 20th Century and proved that any formal mathematical language and set of axioms which is reasonably powerful will necessarily contain statements which cannot be proved within that system.
-- Trying to understand learning by studying schooling is rather like trying to understand sexuality by studying bordellos. -- Mary Catherine Bateson, Peripheral Visions lady@Hawaii.Edu