The Riemann Hypothesis and the Call for Maps in Place of Proofs
Modern mathematics still has with it a sense of Holy Grail, a sense that a possible solution is available for a given conjecture, and if it is available, such a solution is best as a proof in the absolute sense. But should such proof should be pursued as an unqualified good in itself?
Indeed, this sense of proof is, in terms of history, much like the Greek predilection for rational numbers at the expense of the irrational, a belief which had both religious implication and deadly consequence.
Such belief does not diminish the greatness of Greek mathematics, but it somewhat limited Greek mathematicians from further developments (as death is wont to do).
Mapping out the range of applicability for a given solution, an act which is often amazing in itself, may one day win out over the absolute quest for proof.
And note that maps in place of proofs serves to qualify proofs as absolutely descriptive and maps as not necessarily fully determinant in the sense of an absolute proof, but highly useful as descriptions.
Note too that maps in place of proofs is also an argument for rigor.
The mapping goal is to define mathematics, in part, as the cataloging of mathematical discoveries, i.e., as a collection of descriptions, and in seeing mathematics as such, to shift the emphasis from the endpoint of the work, the proof as true discovery, to the work itself.
That is to say, we should catalogue the results of our various approaches not as failure to find a proof so much as the discovery of limitation.
After all, limitation tends to be for us what irrational was for the Greeks.
That is to say, we know this much.
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On Beauty ~ On Utility (Deep Ecology)
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The Riemann Hypothesis (n-Lab)
The Riemann Hypothesis (Milne)
The Riemann Hypothesis, Part 1 (n-Category Cafe, John Baez)
The Riemann Hypothesis, Part 2 (n-Category Cafe, John Baez)
The Riemann Hypothesis, Part 3 (n-Category Cafe, John Baez)
Riemann Zeta Function (Wikipedia)
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