[FOM] SHIFTING PARADIGMS?

The Vienna Summer of Logic, 9th–24th July 2014, has just concluded. The meeting consisted of twelve large conferences and numerous workshops, attracting at least 2500 researchers from all over the world. More details can be found at http://vsl2014.at/. I had the honor of giving one of the opening talks in welcoming participants. Part of what I said concerned the place of Kurt Gödel in the History of Logic -- as my personal opinion. I went far out on a limb in giving my assessment and my prediction for the future -- as this was a chance of a lifetime (in Logic) to make such a personal statement. It would be very interesting to me to get FOMers reactions to my claims, either on this list or by private e-mail. Roughly speaking, the FORMATIVE PERIOD OF FORMAL LOGIC goes from Cantor, Frege, Dedekind, Peano, Russell-Whitehead, C.S. Pierce, to the Hilbert School (and of course through many others). We can briefly sum up what was done in this period as: LOGIC IS APPIED TO MATHEMATICS, in that axiomatics is developed along with how to make formal proofs and rigorous definitions. We could cite Hilbert for Geometry, and Zermelo for Set Theory, as examples. After Hilbert states the basic problems of logic as Completeness and Decidability, Gödel settles them, much to the distress of Russell and Hilbert, as we know. I wish to assert that Gödel can be credited with performing a major Paradigm Shift in Logic by showing how MATHEMATICS IS APPLIED TO LOGIC. König’s Infinity Lemma is needed for the completeness proof, arithmetization is used to represent syntax and proof steps. Gödel thus opens the MODERN PERIOD OF FORMAL LOGIC, a period we are still in. And quite soon after Gödel’s proofs we have Tarski’s Truth Definition and the beginnings of Model Theory, Marshall Stone’s Boolean Representation Theorem (where expertise in Functional Analysis is used for Logic), and Kleene’s seeing how Descriptive Set Theory relates to Higher-Type Recursion (to name some high points). What, then, is going to be the next Paradigm Shift? When will we enter the POST-MODERN PERIOD OF FORMAL LOGIC? Some would like to say that it will happen when Constructive Logic and Mathematics becomes the norm. George Polya and Hermann Weyl made a bet on that back in 1918, and we are still waiting. Perhaps the new insights by Vladimir Voevodsky on Univalent Foundations for Homotopy Type Theory will bring about a very productive JOIN of Classical and Constructive Mathematics, but it seems likely to me that more development is needed before we can see clearly that a Paradigm Shift actually took place. But here is my prediction today: Big Proofs will soon show that computers and logic have to be used TOGETHER to make progress in certain areas of mathematics. That is, we need to show convincingly how COMPUTER-ASSISTED PROOFS APPLY TO MATHEMATICS. We are almost there, and I think I shall live to see this as a Paradigm Shift long before I will ever see practical Quantum Computing! And Voevodsky himself wants to show how computer proofs are helpful. The proofs, for example, of the Four-Color Theorem and the Kepler’s Conjecture are excellent, but -- and forgive me for saying this -- the FACTS are not so surprising. The surprise is that the proofs remain so lengthy. What is needed -- in my personal view -- are mathematical DISCOVERIES made while doing computer- aided proofs -- where the new facts ARE surprising. The hard reality seems to be that the mathematics community will not pay very much attention to new tools in logic unless they can see them as essential for making new discoveries. Big Data has already become essential in many areas of science: Biology, Genetics, Physics, Chemistry, Astronomy, for example. And we have have many uses of Big SAT Solvers for Model Checking and other basic problems. Big Proofs have to come next! Do folks agree? or disagree?