Dear Irving, I did not intend to draw the association between Peirce and Hilbert that concerns you. My use of Hilbert's well-known line, thought now to be a matter of some embarrassment after Goedel's result, is only a reflection of my own view and not that of Peirce. I acknowledge the context of Hilbert's original statement. In the years after that context I have come to believe that the statement gained a broader context, that of what was widely described as "Hilbert's Program" - in which all science is tractable to mathematics.
I cannot tell if Peirce would have agreed with Hilbert. I personally believe that Hilbert was right but too optimistic in 1900, and this should caution us. I certainly do not think we should give up on the program, though clearly new ways of thinking about mathematics and epistemology are required and I personally believe that Peirce and Semeiotic Theory more generally can eventually get us there. In the context of the current slow read I suspect that Hilbert would have agreed with the position that Joe Ransdell outline (though surely with insufficient rigor to satisfy Hilbert), and then to the degree that this view reflects that of Peirce it reflects that of Hilbert. Hilbert certainly knew of Peirce and gave him the highest praise in the introduction to his work on Mathematical Logic with Ackermann. Hilbert's reference to "Bar Stools and Beer Mugs" appears in his "Foundations of Geometry" as I recall (I do not immediately have access to my copy of the work). I agree that Hilbert's remark reflects his formal view, echoed in his and Ackermann's work on mathematical logic, but I am unclear as to whether this reflects his view of Logic in general as a subject of study (esp. given his appreciation of Peirce). Perhaps you can clarify for me the Kantian phrase that Google translates as: "Thus all human cognition begins with intuitions, proceeds from thence to concepts and ends with ideas." What are the distinctions that Kant seeks though the notions of "cognition," "intuition," "concept" and "idea." I'll accept both that Google's translation is imperfect and that my appreciation of Kant is lacking a good understanding of German. From the point of view of the argument given in Joe Ransdell's paper (and consistent with my own view) these notions are ways of speaking about one and the same thing and Kant's statement on the face of it would appear to be empty (or, at least, redundant). I agree when you say: I think that what is wanted is a deep clarification of what Peirce may > or may not have meant in asserting that logic is "an experiential, or > positive science." I can't say that I am in a position to perform this "deep clarification" but I suspect a simplistic analysis is not far from the truth. For Peirce, Logic relies upon Semeiotic Theory and not merely the syntax of "Symbolic Logic" and it's semantic rules. While Hilbert was no doubt the great formalist, I have never believed from my reading of him and his biography that Hilbert ignored semeiotic considerations. Indeed, I either read or I dreamed that I read that Hilbert rather wished that Tarski had used the notion of "valid" rather than "truth" - which reflects a concern with matters of apprehension. This is one of two references - the other being a reference to something that Benjamin Peirce said about "Will" in an astronomy lecture at Harvard - that I can no longer find and that cause me to be more disciplined in future scholarship. I also recall that Hilbert wanted at various times to return to these matters but that the war and the more tractable formal exercise always got in the way. Although my recall of dreaming analysis of Hilbert and the immediate study of Hilbert need to be confirmed by hard references*. *I make this point in the context of Joe Ransdell's paper. With respect, Steven On Nov 6, 2011, at 1:44 PM, Irving wrote: > > Steven, > > You quote Peirce as saying in CP 7.526 that "Logic is a branch of > philosophy. That is to say it is an experiential, or positive science, > but a science which rests on no special observations, made by special > observational means, but on phenomena which lie open to the observation > of every man, every day and hour. There are two main branches of > philosophy, Logic, or the philosophy of thought, and Metaphysics, or > the philosophy of being. Still more general than these is High > Philosophy which brings to light certain truths applicable alike to > logic and to metaphysics. It is with this high philosophy that we have > at first to deal." > > A few paragraphs later, you then say: > > 'To echo Hilbert, "We can know, we will know." Only it is not > mathematics alone that will inform us (and a revolution in the > Foundations of Logic is required).' > > > I do not think that Hilbert would have accepted the interpretation that > seems to be implied in placing his remark in juxtaposition with the > quote from Peirce calling logic an experiential or positive science. At > the very least, this juxtaposition of Peirce and Hilbert runs counter > to Hilbert’s conception of logic and mathematics as purely formal. > > When Hilbert quotes his Königsberg brethren Kant as the motto for his > _Grundlagen der Geometrie_, he does so to make what I would call a > Kantian-Piagetan point; true, we learn numbers by counting objects, and > in counting different collections of objects, begin to extrapolate the > concept of number; but there is a further abstraction of the > abstraction before we reach the concept of number as something > fundamentally UN-experiential. Or, in the passage that Hilbert quotes > from Kant‘s K.d.r.V., "So fängt denn alle menschliche Erkenntnis mit > Anschauungen an, geht von da zu Begriffen und endigt mit Ideen." I > suggest that the import of Hilbert's remark, as recorded in his > biography by Otto Blumenthal, that we should be able to replace points, > lines, and planes with tables, chairs, and beer mugs as the primitives > with which our axioms deal and which we manipulate when deriving > theorems from our axioms, means that our concern with logic and > mathematics is entirely formal and abstract. Hilbert as we know, was a > formalist, and whether. > > When Hilbert made the remark that we can know, we will know, he did so > within the context of his Problems list at the ICM in 1900, listing and > sketching what he considered to be the most interesting and important > open problems in mathematics remaining at the start of the twentieth > century, and which he hoped mathematicians would work on and solve. > What he was saying is that he had the expectation that new and sharper > mathematical tools would be devised which would give mathematicians the > analytical means to solve those open problems. What he was NOT saying, > I would argue, is that there are physical experiments or observations > that would be undertaken that would allow mathematicians to point to > some so-to-speak new or hitherto undiscovered mathematical animal as a > result of experiment or observation. > > I think that what is wanted is a deep clarification of what Peirce may > or may not have meant in asserting that logic is "an experiential, or > positive science." > > Therefore, I guess my point is that I initially feel uncomfortable if > the suggestion, in quoting Hilbert, is that Hilbert would endorse an > empiricist reading of logic or mathematics. And I guess my question is > whether Hilbert and Peirce would or would not agree with this > "Kantian-Piagetan" position and with each other regarding the > "Kantian-Piagetan" point as I have outlined it. > > > Being an historian of logic and mathematics, rather than a philosopher > of logic or mathematics, and probably a bit dense in general, I will > not myself attempt to unpack this any further. Rather, I would require > a tutorial to elucidate in what sense Peirce was calling logic "an > experiential, or positive science" and what connection, if any, this > has with (a) Kant's views, and (2) with Peirce's. > > > > 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
