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Meetings 1998-99

Wednesday, April 28, 1999, 6:00-7:00 Dennes Room (234 Moses)

George Lakoff (Berkeley, Linguistics) 

Where Mathematics Comes From

An Overview of a Book in Progress. Rafael Núñez and I are finishing up a draft of a book (now at 600 pages) called Where Mathematics Comes From: How the Embodied Mind Creates Mathematics . The talk is an overview of the book.

The book is an application of cognitive science to mathematics. Given what we know about the nature of conceptual systems, the book asks,

This includes such questions as, What is the cognitive source of the laws of arithmetic? What is the conceptual structure of set theory and logic? How is e¼i conceptualized in terms of basic cognitive mechanisms? How do we conceptualize infinity in all its forms? What are real numbers?

The book argues that the fundamental mechanisms extending our miniscule innate mathematics are conceptual metaphor and conceptual blending. We analyze the metaphorical mappings used to conceptualize arithmetic, set theory, logic, analytic geometry, trigonometry, exponentials, and imaginary numbers. e¼i is shown to be conceptualized via a blend of many metaphors. We also argue that there is a single basic metaphor for most (if not all) forms of infinity in mathematics, with special cases covering points at infinity, infinite sets, infinite unions, mathematical induction, infinite decimals, infinite sums, limits, least upper bounds, real numbers, transfinite numbers, infinitesimals, and infinite objects. We argue on the basis of these results that the real numbers do not exhaust the continuum, that space-filling curves do not fill space, that the sum of an infinite series is not always equal to its limit, and that Weierstrass continuity does not characterize conceptual continuity.

The main philosophical thrust of the book is that mathematics is a creation of the embodied human mind; that it is not an arbitrary creation but rather is structured by aspects of the mind, brain, and bodily experience; that it is largely metaphorical; and that it has no objective existence. At the same time, the cognitive properties of mathematics explain why arithmetic works where it does, why mathematics is effective in scientific descriptions of the world, why mathematics is stable, and why theorems that are proved stay proved.

We further argue that the principal philosophies of mathematics - Platonism, formalism, and constructivism - are all inadequate and that a new philosophy of mathematics, one responsive to scientific knowledge about the mind, is required.

The talk will try to give the flavor of these results and arguments.

Our current draft of all or most of the book will be made available to seminar participants well in advance.

Incidentally, the philosophical context in which the book was written is my recent book with Mark Johnson, Philosophy in the Flesh.

Wednesday, March 31, 1999, 6:00-7:00 Dennes Room (234 Moses)

Aldo Antonelli (UC Irvine, Logic and Philosophy of Science) 

Motivating the Axioms

This talk is concerned with the way different axiom systems for set theory can be justified by appeal to such intuitions as limitation of size, predicativity, stratification, etc.  While none of the different conceptions historically resulting from the impetus to provide a solution to the paradoxes turns out to rests on an intuition providing an unshakable foundation, each supplies a picture of the set-theoretic universe that is both useful and internally well motivated. The same is true of more recently proposed axiom systems for non-well-founded universes, and an attempt is made to motivate such axiom systems on the basis of an old and respected "algebraic" intuition.

Wednesday, March 3, 1999, 6:00 (sharp)-7:00 Dennes Room (234 Moses)

Martin Davis (NYU, Courant Institute, and Berkeley, Mathematics) 

Gödel's Absolutely Undecidable Proposition

When writing the introduction (for vol. III of Gödel's Collected Works) to some unpublished notes by Gödel for a lecture on Diophantine undecidability, I was struck by Gödel's conjecture in those notes concerning a possible "absolutely undecidable" proposition of second order number theory. Trying to track this down, I had a conjecture that I published in a review in Philosophia Mathematica [vol.6 (1998), 116-128].  Recently I found additional evidence supporting my conjecture.

Wednesday, February 3, 1999, 6:00-7:00 Dennes Room (234 Moses)

Ladislav Kvasz (Comenius University, Bratislava, Mathematics, visiting Berkeley, Philosophy) 

History of geometry and the development of the form of its language 

The aim of this paper is to introduce Wittgenstein's concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning from the Tractatus. Its basic idea is to use the Picture Theory to understand the pictures of geometry. I will try to show that the historical evolution of geometry can be interpreted as the development of the form of its language.  This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein.

Wednesday, December 2, 1998, 6:00-7:00 Townsend Center Seminar Room (220 Stephens)

Paolo Mancosu (Berkeley, Philosophy) 

On the constructivity of proofs.  A debate between Behmann, Bernays, Kaufmann, and Gödel

The talk presents an analysis of a debate on the constructivity of proofs which took place in 1930 and concerned the effects of the constructivist (intuitionist) point of view on mathematical practice. Felix Kaufmann had proposed that proofs of existence claims which do not depend on the axiom of choice implicitly rely on the exhibition of an instance satisfying the existential claim. As proofs by contradiction of existential claims seem to be an obvious counterexample to Kaufmann's claim, attention focused on whether proofs by contradiction could in fact be shown also to be based implicitly on the exhibition of an instance. A theorem purporting to prove the latter claim was contained in a paper by Heinrich Behmann entitled "On the constructivity of proofs". The paper was submitted for publication to the Mathematische Annalen in 1930 but various objections by Bernays and Gödel convinced Behmann not to publish the paper. Despite the problems with the main claim, the paper contains much of interest. It contains, among other things, the first proposal to consider proofs as graph-theoretical objects and a strategy for transforming indirect proofs into direct ones. Moreover, the objections by Bernays and Gödel provide interesting information about the activity of these two logicians in the period.

Wednesday, November 4, 1998, 6:00-7:00Townsend Center Seminar Room (220 Stephens)

Richard Zach (Berkeley, Logic) 

Completeness before Post: Hilbert and Bernays on Propositional Logic, 1917-18

The year 1921 is considered a milestone in the history of logic: Wittgenstein introduced the truth-table method Emil Post proved the completeness of propositional logic, and he and Jan Lukasiewicz introduce logics with more than two truth values. Three years earlier, David Hilbert and Paul Bernays achieved all these results and more in an (unpublished) lecture course in in Bernays's Habilitationsschrift. The talk will describe these achievements, and trace some of the related advances of mathematical logic in the 1920s.

Wednesday, September 30, 1998, 6:00-7:00 Townsend Center Seminar Room (220 Stephens)

Paolo Mancosu (Berkeley, Philosophy) 

The Early Reception of Gödel's Incompleteness Theorems 

Readings: John W. Dawson, "The reception of Gödel's Incompleteness Theorem" in: S. G. Shanker (ed.), Gödel's Theorem in Focus, London: Routledge, 1988. The material is available in the Tarski Room (729 Evans) and on reserve in Howison Philosophy Library (305 Moses).

Wednesday, September 9, 1998, 6:10-7:00 Dennes Room (234 Moses)

Organizational meeting

Symbolic logic has played a very important role in the development of mathematics and of analytic philosophy in this century.  Even though it is now a well established discipline, many philosophical issues connected with it remain contentious; the questionable status of second-order logic being a familiar example.  These issues presuppose a long and varied development of the subject, and are closely tied up with issues in other areas of philosophy, in particular in the philosophy of mathematics.  The Working Group in History and Philosophy of Logic and Mathematics provides a forum for the exploration and discussion of questions related to the history and philosophy of logic, and to philosophy of mathematics, especially as it relates to the logical tradition.  Organizers: Johannes Hafner (Logic), Paolo Mancosu (Philosophy), Richard Zach (Logic)