The idea of human cruelty to animals so consumes novelist Elizabeth Costello in her later years that she can no longer look another person in the eye: humans, especially meat-eating ones, seem to her to be conspirators in a crime of stupefying magnitude taking place on farms and in slaughterhouses, factories, and laboratories across the world. Costello's son, a physics professor, admires her literary achievements, but dreads his mother’s lecturing on animal rights at the college where he teaches. His colleagues (...) resist her argument that human reason is overrated and that the inability to reason does not diminish the value of life; his wife denounces his mother’s vegetarianism as a form of moral superiority. At the dinner that follows her first lecture, the guests confront Costello with a range of sympathetic and skeptical reactions to issues of animal rights, touching on broad philosophical, anthropological, and religious perspectives. Painfully for her son, Elizabeth Costello seems offensive and flaky, but—dare he admit it?—strangely on target. In this landmark book, Nobel Prize–winning writer J. M. Coetzee uses fiction to present a powerfully moving discussion of animal rights in all their complexity. He draws us into Elizabeth Costello’s own sense of mortality, her compassion for animals, and her alienation from humans, even from her own family. In his fable, presented as a Tanner Lecture sponsored by the University Center for Human Values at Princeton University, Coetzee immerses us in a drama reflecting the real-life situation at hand: a writer delivering a lecture on an emotionally charged issue at a prestigious university. Literature, philosophy, performance, and deep human conviction—Coetzee brings all these elements into play. As in the story of Elizabeth Costello, the Tanner Lecture is followed by responses treating the reader to a variety of perspectives, delivered by leading thinkers in different fields. Coetzee’s text is accompanied by an introduction by political philosopher Amy Gutmann and responsive essays by religion scholar Wendy Doniger, primatologist Barbara Smuts, literary theorist Marjorie Garber, and moral philosopher Peter Singer, author of Animal Liberation. Together the lecture-fable and the essays explore the palpable social consequences of uncompromising moral conflict and confrontation. (shrink)

First English translation of revolutionary paper that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.

We investigate a rst-order predicate logic based on Wittgenstein's suggestion to express identity of object by identity of sign, and difference of objects by difference of signs. Hintikka has shown that predicate logic can indeed be set up in such a way; we show that it can be done nicely. More specically, we provide a perspicuous cut-free sequent calculus, as well as a Hilbert-type calculus, for Wittgensteinian predicate logic and prove soundness and completeness theorems.

There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly (...) precisely, it is to be found in the margin of Brouwer's notes for his course on Pointset Theory of 1915/16. The course was repeated in 1916/17 and he must have inserted his first formulation of the continuity principle in the fall of 1916 as new material right at the beginning of the course.In modern language, the principle readswhere α and β range over choice sequences of natural numbers, m and x over natural numbers, and stands for ⟨α, α, …, α⟩, the initial segment of α of length m.An immediate consequence of WC-N is that all full functions are continuous, and, as a corollary, that the continuum is unsplittable [28]. Note that WC-N is incompatible with Church's thesis, [22], section 4.6.After Brouwer asserted WC-N, Troelstra was the first to ask in print for a conceptual motivation, but he remained an exception; most authors followed Brouwer by simply asserting it, cf. [18].Let us note first that in one particular case the principle is obvious indeed, namely in the case of the lawless sequences. The notion of lawless sequence surfaced fairly late in the history of intuitionism. Kreisel introduced it in [17] for metamathematical purposes. There is a letter from Brouwer to Heyting in which the phenomenon also occurs [7]. This is an important and interesting fact since it is the only time that Brouwer made use of a possibility expressly stipulated in, e.g., [5], see below. (shrink)

Michael Friedman maintains that Carnap did not fully appreciate the impact of Gödel's first incompleteness theorem on the prospect for a purely syntactic definition of analyticity that would render arithmetic analytically true. This paper argues against this claim. It also challenges a common presumption on the part of defenders of Carnap, in their diagnosis of the force of Gödel's own critique of Carnap in his Gibbs Lecture. The author is grateful to Michael Friedman for valuable comments. Part of the research (...) towards this paper was carried out while the author was a Visiting Fellow at the Center for Philosophy of Science at the University of Pittsburgh. The paper was presented to the Center's Fellowship Reunion Conference in Athens in 1992. It was committed for publication in the Proceedings of that conference, but those Proceedings never appeared. By the time it became evident that they would never appear, both the hard copy and the source file had been mislaid. The hard copy re-surfaced in 2007. The literature on this topic since 1992 appears to leave some space for the ideas and arguments presented here. Although the paper has been updated in light of the more recent literature, its basic thesis, presented in 1992, remains the same. Only 3 is new, questioning a basic presumption made by more recent commentators in their presentation of Gödel's criticism of Carnap in his Gibbs Lecture. For helpful comments on the current version, the author is indebted to Robert Kraut, Stewart Shapiro, and Adam Podlaskowski. CiteULike Connotea Del.icio.us What's this? (shrink)

Mark Balaguer's Platonism and Anti-Platonism in Mathematics presents an intriguing new brand of platonism, which he calls plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that Balaguer's attempts to characterise full-blooded platonism fail. They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it.

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians (...) might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish. (shrink)

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...) measurable cardinals, whether or not those facts are knowable by us. (shrink)

The basic relations and functions that mathematicians use to identify mathematical objects fail to settle whether mathematical objects of one kind are identical to or distinct from objects of an apparently different kind, and what, if any, intrinsic properties mathematical objects possess. According to one influential interpretation of mathematical discourse, this is because the objects under study are themselves incomplete; they are positions or akin to positions in patterns or structures. Two versions of this idea are examined. It is argued (...) that the evidence adduced in favor of the incompleteness of mathematical objects underdetermines whether it is the objects themselves or our knowledge of them that is incomplete. Also, holding that mathematical objects are incomplete conflicts with the practice of mathematics. The objection that structuralism is committed to the identity of indiscernibles is evaluated and it is also argued that the identification of objects with positions is metaphysically suspect. (shrink)

Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a deflationary one, there surprisingly (...) turns out to be logical space for a response to the improved challenge where no such space appeared to exist. (shrink)

We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing.

§1. Mathematics and the physical world. Genuine scientific knowledge cannot be certain, nor can it be justified a priori. Instead, it must be conjectured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics.This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, “the universe is written in the (...) language of mathematics”. Galileo's introduction of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on supposedly necessary a priori principles, to its modern status as a theoretical, conjectural and empirical science. Instead of seeking an infallible universal mathematical design, Galilean science usesmathematics to express quantitative descriptions of an objective physical reality. Thus mathematics became the language in which we express our knowledge of the physical world — a language that is not only extraordinarily powerful and precise, but also effective in practice. Eugene Wigner referred to “the unreasonable effectiveness of mathematics in the physical sciences”. But is this effectiveness really unreasonable or miraculous?Numbers, sets, groups and algebras have an autonomous reality quite independent of what the laws of physics decree, and the properties of these mathematical structures can be just as objective as Plato believed they were. (shrink)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.