I enjoyed reading your opinion # 133, and have some comments to add.
One does not need to go all the way back to Babylon and Egypt to find the
"serious" mathematics of the day done empirically:
I quote from Wikipedia's entry on
Oliver Heaviside (1850-1925),
He [Heaviside] famously said, "Mathematics is an experimental science, and definitions do not come first, but later on." He was replying to criticism over his use of operators that were not clearly defined. On another occasion he stated somewhat more defensively, "I do not refuse my dinner simply because I do not understand the process of digestion."
As recently as 1940, G. H. Hardy published the following comment (*Ramanujan: Twelve Lectures Suggested by His Life and Work*, Cambridge UniversityPress, 1940)
"All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for a proof. . . . But the opinion which I have attributed to him, and with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a a mathematician ought to have some reply."
Of course, anyone who knows of Hardy will realize that he found it abhorrent that "a good many quite respectable mathematicians are contemptuous about proof." In a letter to Ramanujan dated 8 February 1913, Hardy wrote (*Ramanujan: Letters and Commentary*, ed. Berndt and Rankin, AMS, 1995, p. 46):
"[T]here are result which appear to be new and important, but in which almost everything depends on the precise rigour of the methods of proof which you have used" and later in the same letter (p. 49): "Of course in all these questions everything depends on absolute rigour" (emphasis his).
So, it might seem that Hardy was ready to dismiss Ramanujan's brilliant contributions if they were not backed up by "absolute rigour." Yet even Hardy, who might be considered among the most conservative mathematicians of all time with regard to the necessity for rigorous proof, conceded that he feared "if I insisted unduly on matters which Ramanujan found irksome *(presumably this included "absolute rigour")*, I might destroy his confidence or break the spell of his inspiration." So at the end of the day, even Hardy seems to admit that the proofs are ultimately of less importance than the results themselves.
In fact, I would argue, mathematics has *always *been practiced as an experimental science if only due to the simple fact that until very recently in history, to be a mathematician automatically meant that one was also a physicist, astronomer, etc. Many great mathematicians (e.g. Archimedes, Newton, Euler, Gauss, Poincar=E9) made fundamental contributions to physics and other sciences as well as mathematics. One vestige of this is that the holder of Cambridge's Lucasian Professor of Mathematics Chair is invariably a physicist! But even in this present age of ultra specialization, where we no longer have even have "mathematicians" but rather algebraists, topologists, combinatorialists, etc., new mathematics is still discovered by experiment (whether paper-and-pencil experiment, computer experiment, or a combination thereof). It's just that when we publish our results, we are expected to suppress the experimentation that led to the discovery, and only reveal the "rigorous proof" of the result!
Nonetheless, I must admit that I cannot go so far as to agree with your statement "'high brow' 'Fields-medals-gathering-subjects' will be soon forgotten, and the Euclidean 'rigorous proof' paradigm will be abandoned." (After all, I was mentored by George Andrews as well as by you!) "Absolute Rigour" in the Hardy sense may well be an ideal that does not exist in the real world (if only due to the facts that "rigorous" proofs are dreamed up by humans, written down by humans, reviewed by human referees, and typeset/published by humans). A purported theorem accompanied by a purportedly rigorous proof is true with probability 1 minus epsilon, where epsilon is hopefully very small (but not zero, Professor Hardy!). But that does not mean we should not *strive* for such idealism. After all, the quest for rigorous proofs (even the failed attempts) can lead to deeper insight, new problems, and even new branches of mathematics (as was the case with the many failed attempts to prove FLT over the centuries). At the individual level, the challenge of seeking proofs is where a good deal (but certainly not all) of the fun of doing mathematics lies.
However, I also believe that we the mathematical community should be more open minded about assessing "truth." When a rigorous proof cannot be found, wouldn't it be nice to be able to apply some deterministic and transparent statistical procedure and announce, e.g., The null hypothesis that "RH is false" can be rejected in favor of the alternative hypothesis that "RH is true" with a P-value<0.001 (say).
This would at least attempt quantify how sure we are about a given statement, and admit that we are not *absolutely* certain, as we pretend to be in the case of "rigorous" proof. If we later find out we committed a Type I error, i.e. rejected the null hypothesis when it was in fact true, then we have an exciting new announcement, and we revise the textbooks. So be it.