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Discord Invitation
Georg Kreisel, R.I.P.
Georg Kreisel in a photograph by Paul Halmos
One of the great intellects of our time has passed away. Georg Kreisel died today, 01 March 2015, in Salzburg at 2:30 am.
I owe many happy hours to reading Kreisel’s papers on the foundations and philosophy of logic and mathematics. Kreisel had a great facility for presenting difficult ideas in a clear and unambiguous manner, though I doubt anyone would have called him a “popularizer” of the philosophy of mathematics.
Though Kreisel could not accurately have been called a popularizer, he did directly address the issue of popularization in his paper, “Observations on popular discussions of foundations” (1971), from which I learned much and which I occasionally reread.
It has not been particularly easy to get a hold of Kreisel’s various papers, as they are to be found across a range of academic journals and collected conference proceedings. Whenever I have chanced upon a volume of the North-Holland Publishing Company series “Studies in Logic and the Foundations of Mathematics” in a used book store I have always snapped it up, partly in the hope that it might contain a Kreisel paper of which I was previously unaware.
Kreisel was a regular contributor to the international congresses on Logic, Methodology, and Philosophy of Science, and I possess several of these conference proceedings volumes bound in bright yellow with black markings, like a bumblebee, as though to underline Nietzsche’s fanciful characterization of philosophers: “…our treasure is where the beehives of our knowledge stand. We are forever underway toward them, as born winged animals and honey-gatherers of the spirit."
Kreisel’s continuing interest in the relationship between constructive and non-constructive reasoning issued in his "unwinding program” (also called “proof mining”), which research seeks to extract the constructive content from proofs that appear non-constructive–a metaphysical task within the philosophical of mathematics distinguishing the appearance and reality of formal thought.
Kreisel is sometimes quoted as having said, “we can recognize a proof when we see one,” which accords well with Dummett’s claim on the second page of his Elements of Intuitionism: “…mathematics, when correctly carried on, would not need any justification with without, a buttress from the side or a foundation from below: it would wear its justification on its face.”
Kriesel did say this (“Foundations of Intuitionistic Logic,” 1962), but he said it in this context: “…we are adopting the basic intuitionist idealization that we can recognize a proof when we see one.” A clear and also a careful scholar, Kreisel noted the distinction between unproblematic advocacy of constructivism and theoretical engagement with constructivism.
