Jerry, I suggest that this is a very good question, but I think that we have to consider Hilbert's position as an unfinished product and a moving target. Probably the best indication can be gotten by considering that there is not so much *a* "Hilbert program" as there are "Hilbert programs" (see, e.g. Wilfried Sieg's SIEG, "Hilbert's Programs, 19171922", Bulletin of Symbolic Logic 5 (1999), 1-44).
I would therefore preface my answer by noting that I think it important to remember that Hilbert was a mathematician first and foremost, and that, although interested in philosophical issues in foundations of mathematics, did not systematically develop his formalism. He is better considered an amateur at philosophy. Apart from his handful of brief publications such as "Axiomatische Denken" and "Die logischen Grundlagen der Mathematik", there is, e.g. his correspondence with Frege and his unpublished lectures. The best early articulation of Hilbert's formalism is probably that given by John von Neumann in the round-table discussion in 1930 on foundations, in which Heyting also presented Brouwer's intuitionism and Carnap presented logicism, all published in Erkenntnis in 1931. All of this having been said, the best answer I can give is that, the "points, lines, and planes" and "tables, chairs, and beer mugs" remark aside, Hilbert would give different axiomatizations for different parts of mathematics. That is to say, therwe is one set of axioms and primitives suitable to develop, say, projective geometry, and another for algebraic numbers; there is one suitable for Euclidean geometry and another for metageometry. In the case of the latter, for example, one needs to devise an axiom set that is powerful enough to develop all of the theorems required for the articulation not only required for Euclidean geometry, but also for hyperbolic geometry and elliptical geometry, but which do not also generate superfluous theorems of other theories. Hilbert's axiom system for geometry, then, is not the same athat which he erected for physics. What I think is the correct understanding of Hilbert's "throw-away" remark about points, lines and planes and tables, chairs, and beer mugs, is the more profound -- or perhaps more mundane -- idea that axiom systems are sets of signs which are meaningless unless and until they are interpreted, and by themselves, the only mathematically legitimate characteristic of axiom systems is that they be proof-theoretically sound, that is, the completeness, consistency, and independence of the axiom system, and capable of allowing valid derivation of all, and only those, theorems, required for the piece of mathematics being investigated. 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
