The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof.

A very interesting article this week by Jonathan Borwen called "The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof."

The first paragraph of this article had me hooked.  The premise:  computers are becoming so powerful that proving by "quasi-inductive methods" is becoming a viable way for proof.  Where does this leave the mathematician?  Borwen explores the "relationship between proof (deduction) and experiment (induction)."  He carries on to show a large number of examples where hand-deduction, and computer-induction proved successful methods in helping to illuminate a problem.  

          His discussion of large and small numbers I found to be very engaging.  Statistics relies on the ‘law of large numbers,’ where correlations, connections, and relations are forged between quantities of data that humans could not find (potentially).  As mathematics ventures into more abstract domains, it too ventures into large-number-territory, however, not with the quantity of data, but rather simply because often you are dealing with a very large or hard-to-find number (like the trillionth digit of pi).  In both of these cases, computers offer a reliable way to access these kinds of numbers.  Small numbers, however, is where mathematicians can shine.  Small numbers, Borwen argues, are important numbers of which a computer cannot grasp the significance of.  He cites wonderfully concise proofs of √2’s irrationality.

          Concluding with one thought of “there are different versions of proof or rigor” was a rewarding way to finish the article, as Borwen points out that historical proofs have since been reneged and renewed on the basis that the standards of rigor change over time.  Computer-assisted proofs may offer a current ‘standard of rigor,’ but if it will stay the test of time is to be determined.