John Burgess - Academia.edu (original) (raw)

Papers by John Burgess

Interpreting Gödel

Gödel judges certain consequences of the continuum hypothesis to be implausible, and suggests tha... more Gödel judges certain consequences of the continuum hypothesis to be implausible, and suggests that mathematical intuition may be able to lead us to axioms from which that hypothesis could be refuted. It is argued that Gödel must take the faculty that leads him to his judgments of implausibility to be a different one from the faculty of mathematical intuition that is supposed to lead us to new axioms. It is then argued that the two faculties are very hard to tell apart, and that as a result the very existence of mathematical intuition in Gödel's sense becomes doubtful.

Fictionalism in Philosophy

In this chapter, it will be argued that a fictionalist account, in one central sense of that ambi... more In this chapter, it will be argued that a fictionalist account, in one central sense of that ambiguous phrase, of an area of discourse is not something that can be called correct or incorrect once and for all. Rather, areas of discourse often evolve from phases for which an error theory is most appropriate to phases for which a fictionalist account is appropriate. They can then evolve further into phases for which a straightforward account is appropriate. The latter can occur when the meaning of a key term changes. The phenomenon will be illustrated by examples from several areas of discourse.

Notices of the American Mathematical Society, 2014

Book Review desk work again and again proves useful in dealing with the world outside the mathema... more Book Review desk work again and again proves useful in dealing with the world outside the mathematician's office. How by just sitting and thinking we (or some of us) can arrive at results applicable to the world around us has puzzled thinkers from Kant to Wigner [4]. The two features are separate. That results should prove applicable to the physical universe even though they were obtained by pure desk work, without controlled experimentation on or systematic observation of the material world, can be surprising even if what the desk work produces is not compelling deductive proofs but "only" suggestive heuristic arguments. And with the two factors being separate, the material in the book is divided into two more or less separate parts, though with a lot of back and forth between them: one devoted to proof, the other to applications. Neither the part about proof nor the part about applications is concerned only with their role in perennially drawing the attention of philosophers to mathematics. And, beyond the general division into these two broad topics, the book is rather loosely organized and digressive, not to say rambling, in a way that makes it quite impossible for the reviewer to summarize its contents in an even halfway adequate fashion. The analytical table of contents goes on for six pages, and there is nothing I would leave out, but this means that even to list the topics addressed would take up more space than is reasonable for a review. One thing just leads to another: If a philosophical view is stated, some mathematical example will be wanted to illustrate it, but then at least an informal explanation of the key concepts in the example will be wanted also, and perhaps a capsule bio of the author or authors of the relevant result or results, and even perhaps in cases where they December 2014 Notices of the Ams 1345

Annals of Mathematical Logic, 1978

These theorems ot Jet,sen cte:~rly ~ ~ ~, o v /" " ~ ~4 estaHisa t~,~. ~ap-two prmcq?~e:for any a... more These theorems ot Jet,sen cte:~rly ~ ~ ~, o v /" " ~ ~4 estaHisa t~,~. ~ap-two prmcq?~e:for any and any reg~flar h.: For Singular ~.i:the proof is tmpublished ~nd requires sometlhi,~ worse than a moraSS (cfi [t8]), Jensen derives the e~stence of morasses from V= L using his fine Stnmture theoEv (see [6]; Chapter 13)~ tle also has a proof 'unpt~blished, but see [18])of the consistency of morasses by a Shnpler; bti'[ still c ifficutt, torcing constraction. We wilt snow'-how t? ~rove, by. an almost purely~ modebtheoretic argument, entirely a, oiding more, sses, the :following result, which can be substituted for Jensen's 0.2 and 0.3 in provfi~g the consistency of gap-two princip.~es:

A Companion to W.V.O. Quine, 2013

Philosophy of Mathematics in the Twentieth Century, 2013

procedure in building a glossary of Fregean concepts is exemplary too (cf. pp. ix-x and xxxiii-xx... more procedure in building a glossary of Fregean concepts is exemplary too (cf. pp. ix-x and xxxiii-xxxvi). Beginning more than 10 years ago with small translation bulletins, the three core members of the project team discussed existing approaches and alternative interpretations, objections and revisions stage-by-stage on a weekly basis. This was their modus operandi for a long time. It was only when the first draft of the translation of Volume I was finished that a workshop with international Frege scholars was organized in 2006 to work through the full manuscript. A second followed in 2008 for Volume II, and in 2010 a collaborative revision of the entire translation began. In the introductory passage 'Translating Frege's Grundgesetze' (pp. xvi-xxix) the editors not only present the resulting glossary of technical terms (pp. xxvi-xxviii) 3 but also a detailed explanation of their translations (pp. xvi-xxvi) for more than 20 (in some cases of course difficult) concepts. The editors even convincingly explain why they chose not to transform Frege's concept-script into modern notation (pp. xxix-xxx). Should readers need assistance with their understanding of Frege's calculus, the edition contains an appendix 'How to read Grundgesetze' (pp. A-1-A-42) by Roy T. Cook. The edition ends with an 'Index' for both volumes (11 p.). One point of criticism: in their 'Introduction' Philip A. Ebert and Marcus Rossberg speak (like many other proponents of a type-free higher-order logic) of 'the inconsistency of Basic Law V' (p. xv). For Frege scholars this is a very careless remark. It falsely suggests that Basic Law V is a contradiction. In fact the inconsistency lies in the formal system of Grundgesetze, a system that contains alongside Basic Law V a type-free higher-order logic. Why should we consider Basic Law V inconsistent if the underlying logic is likewise necessary to deduce the antinomy? It is only fair to remember that Basic Law V is absolutely harmless if treated within simple Type Theory. But that is my only criticism. There is no doubt that this English edition will become a basic work in international Frege scholarship in general and the standard reference for neologicist programs in particular. The translation and editing were undertaken on the highest level and in veneration of Gottlob Frege. PhilipA. Ebert, Marcus Rossberg and Crispin Wright (as well as all the other members of this project) did a great job. Congratulations and kudos! References

Notre Dame Journal of Formal Logic, 1983

Proceedings of the American Mathematical Society, 1978

The equivalence relation on the reals generated by a family of Ha Borel sets has either < Na or e... more The equivalence relation on the reals generated by a family of Ha Borel sets has either < Na or else exactly 2"° equivalence classes.

Pacific Journal of Mathematics, 1979

Notre Dame Journal of Formal Logic, 1981

Notre Dame Journal of Formal Logic, 1998

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpret... more After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility. 1 Historical introduction After discovering the inconsistency in Frege's Grundgesetze der Arithmetik, Russell proposed two changes: first, dropping the assumption that to every higher-order entity there corresponds a first-order entity; and second, restricting the assumptions on the existence of higher-order entities, so that instead of a simple hierarchy of first-order, second-order, third-order, and so on, one has a ramified hierarchy in which each order is subdivided into various types in such a way that a condition involving quantification over all entities of one type is never assumed to determine another entity of the same type, but only of a higher type. But Russell found that with these two changes he could not derive classical mathematics, so in Principia Mathematica he partially compensated for the first change by assuming the axiom of infinity and, for all mathematical purposes, wholly undid the second change by assuming his axiom of reducibility. The predicativist tradition from Weyl [21] to Feferman [2] and beyond accepts infinity but rejects reducibility and is willing to give up parts of classical mathematics. However, predicativists have been unable to derive classical arithmetic and unwilling to give it up and so have simply assumed it as axiomatic. This assumption has its defenders, as with Feferman and Hellman [3], and also its detractors, as with C. Parsons [15]. It is, therefore, of some philosophical as well as historical interest to ask how large a fragment of classical arithmetic can be developed within the Russellian system of Principia Mathematica with infinity but without reducibility. Now many subsystems of classical or Peano arithmetic have been recognized since the work of Skolem [18], Kalmar [9], Grzegorczyk [4], and other pioneers. Among these the most studied have been the subprimitive or Grzegorczyk arithmetics n. These agree in allowing definitions by primitive recursion, but only when the function F being defined recursively is bounded by some function already given; or

History and Philosophy of Logic, 1984

Chairman M a o I. For more on the conrrasl between the two approaches to philosophq of mathematic... more Chairman M a o I. For more on the conrrasl between the two approaches to philosophq of mathematics, see [he opening section of the editorial introduction to Benacerraf and Putnarn 1964.

$Research paper thumbnail of Review: Robert Vaught, A. R. D. Mathias, H. Rogers, Descriptive Set Theory in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><msub><mi>ω</mi><mn>1</mn></msub><mi>ω</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{\omega_1\omega}</annotation></semantics></math>Lω1ω; Robert Vaught, Invariant Sets in Topology and Logic$

The New Theory of Reference, 1998

Did Saul Kripke in his 1970 lecture series on naming and necessity merely repeat, with elaboratio... more Did Saul Kripke in his 1970 lecture series on naming and necessity merely repeat, with elaborations but without acknowledgments, a theory that had been propounded to him by Ruth Barcan Marcus in a 1962 colloquium talk on modalities and intensional languages? That is what seems to be insinuated in footnotes in recent publications by Marcus, and what was explicitly alleged

Logic, Meaning and Computation, 2001

Several nominalist paraphrases, in the sense of methods by which assertions of conventional mathe... more Several nominalist paraphrases, in the sense of methods by which assertions of conventional mathematics that give the appearance of implying that there are such things as numbers can be systematically replaced by other assertions not giving this appearance, have been proposed in the literature. Can the success of such methods help show that the original assertions are not really ontologically committed in the sense of not really being incompatible with the denial that there are such things as numbers? I argue against the claim that it can.

Sankhya A, 2010

It is shown that for invariance under the action of special groups the statements "Every invarian... more It is shown that for invariance under the action of special groups the statements "Every invariant PCA is decomposable into ℵ1 invariant Borel sets" and "Every pair of invariant PCA is reducible by a pair of invariant PCA sets" are independent of the axioms of set theory.

Theoria, 2008

' As Aristotle puts in the 'sea fight' passage of On interpwrurion, ix.