On Nov. 27, I wrote: "... I would have to say that he would agree that there is a strong empiricism underlying Hilbert's work, and that this is the philosophical import of his quote from Kant's K.d.r.V. in the Grundlagen der Geometrie: "So fängt denn alle menschliche Erkenntnis mit Anschauungen an, geht von da zu Begriffen und endigt mit Ideen." I would argue, however, that this is about how we obtain our information, and, assuming Corry is correct, how Hilbert thought we select the elements of our universe of discourse; but I would also argue that it has nothing to do with how axiomatic systems operate, which is to say, having established the axioms, chosen the inference rules for the system, and selected the primitives from which theorems are constructed from the axioms in accordance with the inference rules, is strictly mechanical, and it does not, working within the axiom system, whether what is being manipulated are points, lines and planes, or tables, chairs, and beer mugs, or integers, , or whatever we may require for the axiomatizing task at hand. What matters within the system, while the calculations are occurring, is that complex formulas (theorems) are being constructed on the basis of the formulas that do duty as axioms, in accordance with the rules. (It is this distinction, of having inference rules in place, that renders Hilbert's systems not merely axiomatic systems, but formal deductive systems.) Hilbert's formalism amounts to the mechanization of these manipulations, and for practical purposes, the formulas are combinations of marks, and these marks become signs as soon as an interpretation is give, that is, a universe of discourse - - whether points, lines, and planes, or tables, chairs and beer mugs, or the integers. What concerns me is whether, in considering what (else) or what different Hilbert might have meant by his formalism, and whether or not there was an underlying empiricism behind this, is that we might be demanding too much of Hilbert, who was, I understand, concerned with mathematics and only peripherally with philosophy of mathematics. (Having said this,I have to also confess that I have not seen or read the contents of Hilbert's late, unpublished, lectures on foundations, but I believe that Corry has, and it is on that basis that Corry proposes an empiricist epistemology behind Hilbert's formalism.)
"The only other point I would make w.r.t. Hilbert on physics, is that, at least according to Corry, part of Hilbert's empiricism is exhibited by the requirement that his axiomatization depends upon his axiomatization of geometry, and that the Kantian root of geometry is spatial intuition." Since then, I have come across some preprints (headed for publication in Erkenntnis or Synthese) that stress the empiricist aspect of Hilbert's philosophy, such as Helen De Cruz & Johan De Smedt's "Mathematical Symbols as Epistemic Actions" that takes Hilbert to be a radical empiricist in the style of, or at least very close to, Husserl's pre-phenomenological psychologism, and Soren Stenlund's "Different Senses of Finitude: An Inquiry into Hilbert's Finitism". And then there is Solomon Feferman's "And so on...: Reasoning with Infinite Diagrams", in which, in footnote 10, Sol, who I had known primarily and essentially as a mathematician specializing in recursion-theoretic aspects of proof theory and a disciple of Georg Kreisel, and secondarily as a friend and associate of Jean van Heijenoort and as editor-in-chief of Gödel's Collected Works, straightforwardly and unequivocally asserts that it is a mistake to regard Hilbert as a formalism. (What this all suggests to me is that, *if* correct, everything about Hilbert and twentieth-century formalist foundational philosophy of mathematics that I was -- and probably many of us were -- taught 47 years and more ago ... is just plain *wrong*.) Irving H. Anellis Visiting Research Associate Peirce Edition, Institute for American Thought 902 W. New York St. Indiana University-Purdue University at Indianapolis Indianapolis, IN 46202-5159 USA URL: http://www.irvinganellis.info --------------------------------------------------------------------------------- You are receiving this message because you are subscribed to the PEIRCE-L listserv. To remove yourself from this list, send a message to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of the message. To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU
