October 04, 2000, 6-7:30 PM in 234 Moses Hall

Katherine Brading (Wolfson College, Oxford)

A Theme from Hermann Weyl’s 1918 ‘Grand Symphony’, and Emmy Noether’s Variations

October 11, 2000, 6-7:30 PM in 305 Moses

Jerrold J. Katz (CUNY)

Mathematics and Metaphilosophy

November 01, 2000, 6-7:30 PM in 234 Moses Hall

Joel Friedman (Philosophy, UC Davis)

A Modalist Approach to Constructive Empiricism

Consider the following internal problem for Bas van Fraassen’s philosophy of Constructive Empiricism (CE): Scientists are not required to believe the literal truth of scientific theories, only that these theories are empirically adequate. Yet, theory T is empirically adequate (EA) if and only if, by van Fraassen’s definition, there exists a model of T in which every observable phenomenon is embeddable. But such models are themselves unobservable abstract entities. Thus, believing that T is EA entails believing in unobservables. This is just what CE was designed to avoid, especially when coupled with van Fraassen’s anti-Platonism. Hence the internal problem!

My solution to this problem is modalism, a philosophical view providing a simple modal method for avoiding ontological commitment to abstract entities, without the need to translate talk about such entities into talk involving only concrete entities, e.g., modal nominalist talk (a la Geoffrey Hellman) or modal constructivist talk (a la Charles Chihara).

Think of Modalism as promoting systematic use of Occam’s Razor, shaving our beliefs about unobservable abstract entities in mathematics, language, and metaphysics. Modalism’s best application so far is to CE, wherein the notion of empirical adequacy is modalized. Possibility and actuality operators are introduced (diamond symbol and “@”, resp.), so that the statement form, “T is modalistically empirically adequate (MEA)” is not ontologically committed to abstract entities. As a simplified example, consider that “(EM)(M is a model of T)” is ontologically committed to models, whereas “possible(EM)(M is a model of T)” is not so committed. Modalist Constructive Empiricism (MCE) is then formulated on the basis of MEA, completely analogous to van Fraassen’s formulation of CE on the basis of EA.

A key intuitive idea behind Modalism is that one may relate actually existing observable objects to possibly existing abstract entities, through judicious uses of the possibility and actuality operators. Another key intuition is that an actually existing abstract entity is empirically indistinguishable from a possibly existing abstract entity (consider van Fraassen’s Parable of Oz and Id).

The best technical result in my paper is the Maximalist Platonist Theorem (MPT) (proved in the Appendix). Given certain excessive ontological assumptions of Maximalist Platonism, including a new Axiom of Modal Maximality of Cardinals (AMMC), the theorem MPT affirms the equivalence between van Fraassen’s notion of empirical adequacy (EA) and my notion of modalist empirical adequacy (MEA). Thus, MPT gives significant protection against counterexamples (especially from van Fraassen) to the modalist notion, MEA, since any such counterexample would refute Maximalist Platonism. But what are the chances a minimalist such as van Fraassen would refute a maximalist? Pretty low, I would say. At least I have slept well ever since proving MPT!

November 29, 2000, 6-7:30 PM in 234 Moses Hall

Patrick Suppes (Philosophy, Stanford University)

Inequalities for GHZ-type experiments look almost like the Bell inequalities

December 06, 2000, 6-7:30 PM in 234 Moses Hall

David Stump (University of San Francisco)

The Independence of the Parallel Postulate and Development of Rigorous Consistency Proofs

Standard histories of mathematics all agree that Bolyai-Lobachevskii geometry (BL) was discovered around 1830, but generally ignored at that time, and that it was reintroduced and widely accepted around 1870. The claim that BL was established by 1870 is somewhat problematic since the formal axiomatic view of geometry needed to formulate a logically rigorous proof of consistency did not exist until the turn of the century. What were the arguments for the consistency of BL in the 1870s, when a formal axiomatic view was not yet available? What about the geometers who accepted the possibility of BL even before the work of Riemann, Beltrami and Klein, that which most historians recognize as the work that put BL and other non-Euclidean geometries on a firm foundation? In order to contribute to an understanding of the development of rigorous consistency proofs, this paper will analyze the kinds of arguments offered by Jules Höuel in 1870 for the independence of the parallel postulate and for the existence of non-Euclidean geometries, especially his interpretation of Beltrami?s model. Some of the steps that were taken towards developing a formal axiomatic conception of geometry will then be brought into strong relief.

February 07, 2001, 6-7:30 PM in 234 Moses Hall

Thomas Ryckman (University of California, Berkeley)

Weyl’s Debt to Husserl: The Transcendental Phenomenological Roots of the Gauge Principle

The “Gauge Principle” – the requirement of invariance of field laws under local symmetry transformations – was first formulated in 1918 by Hermann Weyl as the result of his epistemological reconstruction of the theory of general relativity in accordance with central precepts of Husserlian transcendental phenomenology. I show that the usual stories told by physicists regarding the origin of the gauge principle are anachronistic and conclude with a reflection upon the equally empirically unmotivated ‘geometrical method’ of present-day theoreticians, who seek to attain a deeper and more consistent understanding of basic physical laws by generalizing their geometrical basis.

February 21, 2001, 6-7:30 PM in 234 Moses Hall

Edward M. Zalta (CSLI, Standford University)

Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics

In this paper, we describe “metaphysical reductions”, in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Though I shall not claim that they are logical axioms, some philosophers have argued that we should accept (something like) them as being logical.

March 14, 2001, 6-7:30 PM in 234 Moses Hall

Andrew Janiak (Indiana University, Bloomington)

Relationism and Absolutism in Kantian Perspective

It is typically claimed that Kant’s conception of space should be understood as a philosophical competitor to Newtonian absolutist and Leibnizian relationist conceptions of space. Despite its prima facie plausibility, I contend that this common view is mistaken. I argue that Kant’s concern with space in the Critique of Pure Reason, if properly construed, should be seen as orthogonal to the concerns that typically divide relationists from absolutists. Broadly speaking, Kant is concerned not with whether space is absolute or relative, but with whether it is real or ideal. The details of this claim will be explored.

I hope to accomplish two things in this talk. First, to indicate how to understand what Kant’s theory of space is a theory of. Seeing that Kant’s theory does not answer questions about space’s ontological status in the way that relationist and absolutist conceptions answer such questions, I think, represents a first step in this direction. Second, and more generally, I hope to show how the contemporary philosophy of science can be used to illuminate the history of philosophy while avoiding the danger of anachronism.

April 11, 2001, 6-7:30 PM in 234 Moses Hall

Henry R. Mendell (California State University, Los Angeles)

Phenomena and Models in Early Greek Astronomy

How central were planetary models to 4th century BCE Greek astronomers? Did they attempt to model retrograde motions of the planets? These questions have become central in our reconstructions of their work. Assuming that Plato reflects 4th century BCE practice, what does Plato actually think astronomy is about? Which phenomena contributory to kinematic models do extant sources consider important? I shall argue that the primary concern of early Greek astronomers in the construction of mathematical (as opposed to descriptive) astronomy was the establishment of periods. This involves many sorts of phenomena such as the length of day and night, but also and fundamentally synodic phenomena including, eclipses and occultations and horizon phenomena (risings, settings), all of which are crucial to the measurement of time. What Plato adds in his description are more general synodic phenomena. Hence, we should expect that 4th century astronomers made measurements of intervals, which may or may not have required some dated observations, especially between long intervals. I shall also show from an analogy with historical texts why the lack of literary evidence is inadequate to determine whether dated measurements were made.

If this is correct, then we should expect that this would have been reflected in the earliest theoretical models and in the ample traces of evidence of physical models. The primary interest was in modeling solar/lunar phenomena, while the planetary models would have been of secondary interest.

April 18, 2001, 6-7:30 PM in 234 Moses Hall

Antony Valentini (Imperial College, London)

The Early History of Louis de Broglie’s Piolet-Wave Dynamics

In 1927 Louis de Broglie proposed what we would now call a hidden-variables interpretation of quantum theory (revived by Bohm in 1952). In this talk I examine de Broglie’s early thinking from 1923 onwards, and the reception of his pilot-wave theory at the widely-misunderstood Fifth Solvay Congress of 1927. The standard history of pilot-wave theory is shown to be quite false in a number of key respects.

May 02, 2001, 6-7:30 PM in 234 Moses Hall

Otávio Bueno (California State University, Fresno)

Can Set Theory Be Nominalized?

In recent years a number of nominalist approaches to mathematics have been developed: Field’s mathematical fictionalism (Field [1980] and [1989]), Hellman’s modal-structuralism (Hellman [1989] and [1996]), and–under a certain interpretation–Lewis’s megethology (basically, the use of mereology and second-order quantification to re-interpret mathematics; Lewis [1990] and [1993]). In this paper, I argue that all these approaches face a serious problem: they fail to provide an adequate nominalization strategy for set theory. Field’s approach ultimately presupposes such a nominalization, but I argue that it is unable to provide one. It is part of Hellman’s approach to tentatively attempt to provide a nominalistic reconstrual of set theory. However, the approach faces difficulties with the interpretation of the second-order quantifier and the modal operators introduced. Lewis’s proposal–under a certain interpretation–also attempts to provide a nominalization of set theory. However, if the nominalization were to be carried out, it would make set theory contingent upon the size of reality, which conflicts with the way in which the theory is typically conceived. I conclude that set theory becomes the crucial area in which nominalists should concentrate their efforts if nominalism is to become a plausible account.

May 09, 2001, 6-7:30 PM in 234 Moses Hall

R. Lanier Anderson (Stanford University)

The Traditional Logic and Kant’s Problem of Synthetic Judgment

The basic thesis of Kant’s philosophy of mathematics is that mathematical truths are synthetic, not analytic. I will argue that the thesis grows out of Kant’s insight into the expressive limitations of the traditional logic of concepts. That logic provides a perfectly defensible and clear sense of Kant’s notion of analyticity. Once that sense is specified, however, it follows that even elementary mathematical propositions turn essentially on features that cannot be explicitly represented in the resulting system of analyticities. Kant’s view thereby offers a decisive refutation of the program of Wolffian metaphysics to reconstruct all genuine scientific knowledge (and paradigmatically mathematics) in an austere form provided by the traditional logic.

We are pleased to welcome to Berkeley two new faculty members in the Department of Philosophy with interestes falling in the area of the HPLM Working Group, Professors Guido Bacciagaluppi and John MacFarlane On this occasion we have decided to acknowledge the fact that much of our work deals with history and philosophy of science in general by including “Science” in the title of the group. We will continue to focus on historical and philosophical issues in symbolic logic, mathematics, and science in general, and with the help of Prof. Bacciagaluppi also include philosophy of physics specifically.