Steve Gray wrote: << <<< James Buddenhagen wrote: . . . something that has been bothering me for a long time: when should I accept that something is true? . . .
You raise interesting points. Like it or not, mathematics is partly a matter of consensus. This is dismaying because we were educated, explicitly or implicitly, to think of math as absolute. Instead, it's only the most absolute thing we humans are allowed. There being no "god," the limitation to our possible knowledge (also evident in physics for several reasons) is built into the universe at a very deep level. The most complex proofs must be accepted provisionally, since all advanced science and math is accepted on the basis of approval by experts. As much as I dislike it, even mathematics is therefore subject to authority. But I, for one, plug along, continuing to find and prove some of these maximally absolute things (in my case, rather minor ones) knowing that a theorem once established will very likely never be disestablished.
I would distinguish between mathematics per se, on the one hand, which I think of as the absolute body of things that can be proved (and being a Platonist, I also include facts that may be undecidable, possibly including things like the Twin Prime Conjecture) . . . . . . and the *sociology of mathematics*, on the other hand, which includes the vague hierarchy of authority that determines what each individual mathematician believes to be proved, and the consensi whomof. (It is indeed rare, but there are striking examples of putative proofs of valid theorems that were debunked many years later, like Kempe's 1879 putative proof of the four-color theorem (about 15 years), Lucas's 1875 putative proof that 70^2 is the unique square > 1 that equals 1^1 + 2^2 + . . . + n^2 (43 years?), and Dulac's 1923 putative proof that a polynomial vector field V(x,y) = (P(x,y),Q(x,y)) in the plane has at most a finite number of limit cycles (65 years?).) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele