"Creation and Completeness"
(with Douglas K. Erlandson, J. Clark Heston, and Charles M. Young.) "Computation with Roman Numerals." Archive for History of Exact Sciences (1976), 15(2):141-148.
"The Importance of Gödel's Second Incompleteness
Theorem for the Foundations of Mathematics." PhD Dissertation, John
Hopkins University, 1976.
Abstract in Dissertation Abstracts
International (July 1976),
37(1):374-A.
"Current Work on Mathematical Truth." In A Christian Perspective on the Foundations of Mathematics. Christian Perspective on the Foundations of Mathematics, Wheaton College, 1977. S.l.: s.n., 1977.
"On Interpreting Gödel's Second Theorem."
Journal of Philosophical Logic (August 1979), 8(3):297-313.
Reprinted with a postscript in S. G. Shanker, ed., Gödel's
Theorem in Focus,
pp. 131-154. Croom Helm Philosophers in Focus Series. London & New York:
Croom Helm, 1988.
"Interpretazione del secondo teorema di Godel." In S.G. Shanker, ed.,
Il teorema di Godel: una messa a fuoco,
pp. 163-188. Translated by Paolo Pagli. Muzzio Scienze, 6. Padova: F.
Muzzio, 1991.
"In this paper I critically evaluate the
most widespread
philosophical
interpretations of Gödel's second incompleteness theorem. My
approach is to say what I think is wrong with these interpretations as
they presently stand, and, where possible, to try to indicate what
would have to be achieved were those interpretations to be revived,
though revival is not, in my opinion, a reasonable hope. Sections 2-7
discuss that cluster of interpretations that I choose to call the
skeptical interpretations of Gödel's second theorem (hereafter G2). in
section 8 i consider that interpretation of G2 which attributes its
significance to some alleged ill effects it has on Hilbert's program.
I
shall argue there that G2 does not imply the failure of Hilbert's
program."
"The Arithmetization of Metamathematics in a Philosophical Setting."
Revue Internationale de Philosophie (1980), 34(1-2)
[131-132]:268-292.
"This paper raises questions
concerning the arithmetizability of
various bodies of constructivistic thought. One type of
constructivistic thought (well-motivated by a consideration of
certain elements of the motion of constructive provability) is shown,
by an argument from Tarski's theorem, to be non-arithmetizable. Here,
the central point is that this notion of constructive provability
"cannot" be treated as a predicate. If one treats it as some sort of
sentential operator, the prohibition posed by Tarski's theorem
disappears. Subtleties (stemming mainly from Lob's theorm) involved in
attempts to arithmetize both finitism and intuitionism are discussed."
(with Mark Luker.) "The Four-color Theorem and Mathematical Proof."
Journal of Philosophy (December 1980), 77(2):803-820.
"We criticize a recent paper by Thomas
Tymoczko in which he
attributes fundamental philosophical significance and novelty to the
lately-published computer-assisted proof of the four-color theorem
(4ct). Using reasoning precisely analogous to that employed by
Tymoczko, we show that much of traditional mathematical proof must
be seen as resting on what Tymoczko must take as being "empirical"
evidence. Hence, the new proof of the 4ct with its use of what
Tymoczko
calls "empirical" evidence is "not" as novel as he would have us
believe.
Finally, without attempting to give a full account of the notion of
empirical mathematical evidence, we sketch a view showing how the use
of calculation injects an empirical ingredient into proof."
"On a Theorem of Feferman."
Philosophical Studies (August 1980), 38(2):129-140.
"What conditions must a formula satisfy
in order to "express" the
consistency of a formal system T? In this paper, I outline an answer
to
this question. Several writers (e.g., Rosser, Mastowski, Feferman)
have
given formulas for which Gödel's second theorem fails. My argument
focuses on Feferman's. I argue that his formula does "not" "express"
the
consistency of T if what one wants to do with it is to show that
epistemologically gainful proofs of T's consistency can be given.
Others (e.g., Feferman and M. D. Resnik) have also held this view, but
their
arguments for it are mistaken."
(with Loren E. Lomasky.) "Medical Paternalism Reconsidered." Pacific Philosophical Quarterly (January 1981), 62(1):95-98.
Review of Judson Webb's Mechanism, Mentalism, and Metamathematics: An Essay on Finitism. Nous (November 1981), 15(4):559-566.
Review of W.H. Newton-Smith's The Rationality of Science. Revue Internationale de Philosophie (1983), 37(3) [146]:364-371.
Hilbert's Program: An Essay on Mathematical Instrumentalism. Synthese Library, 182. Dordrecht & Boston: Reidel/Kluwer Academic, 1986.
"A Philosophical Analysis of Formalism." In Robert L. Brabenec, ed., A Sixth Conference on Mathematics from a Christian Perspective. Proceedings of the conference sponsored by the Association of Christians in the Mathematical Sciences and held at Calvin College, May 27-30, 1987. Wheaton, Ill.: Wheaton College Mathematics Department, 1987.
"Fregean Hierarchies and Mathematical Explanation."
International Studies in the Philosophy of Science (1988), 3:
97-116.
"This paper investigates the conception
of explanatory proof
(coming down from Aristotle through Leibniz to Bolzano and Frege)
which sees it as based upon an objective ordering of mathematical
truths (called a 'grounding hierarchy'). Coupled with this idea, in
Frege's
thought, is a global conception of logic (which sees truth and
implication as everywhere the same, both inside and outside
mathematics, and both in the context of the objective grounding of
truth as well as in other contexts). This combination of ideas is
criticized; the suggestion being that in order to accommodate a
grounding hierarchy, one must, at the very least, adopt a local
conception of logic. Detailed versions of this argument are developed
for two different models of grounding hierarchies."
Review of Aleksandar Pavkovic's Contemporary Yugoslav Philosophy: The Analytic Approach. Canadian Philosophical Reviews (1989), 9(12):492-496.
"Brouwerian Intuitionism."
Mind (October 1990), 99 [396]:501-534.
Reprinted in Michael Detlefsen, ed., Proof and Knowledge in
Mathematics (1992), pp. 208-250.
"It is argued that Brouwer's critique of
classical logic was not so
much focused on particular principles
(e.g., the law of excluded middle) as on the use of any kind of
logical inference in mathematical proof. He
believed that genuine mathematical reasoning requires genuine
mathematical insight (or intuition), and thus
cannot accommodate the use of topic-neutral forms of inference.
Alternative views of knowledge and
language which might underlie such a view are discussed, as are
certain connections between the thought
of Brouwer and Poincaré."
"On an Alleged Refutation of Hilbert's Program Using Gödel's
First
Incompleteness Theorem."
Journal of Philosophical Logic (November 1990), 19(4):343-377.
Reprinted in Michael Detlefsen, ed., Proof,
Logic and Formalization (1992), pp. 199-235.
"It is argued that an instrumentalist
notion of proof such as that
represented in Hilbert's viewpoint is not
obligated to satisfy the conservation condition that is generally
regarded as a constraint on Hilbert's
Program. A more reasonable soundness condition is then considered and
shown not to be
counter-exemplified by Godel's First Theorem. Finally, attention is
given to the question of what a theory
is; whether it should be seen as a "list" or corpus of beliefs, or as
a method for selecting beliefs. The
significance of this question for assessing "intensional" results like
Godel's Second Theorem, and their
bearing on Hilbert's Program are discussed."
"Poincaré Against the Logicians."
Synthese (March 1992), 90(3):349-378.
"Poincaré was a persistent critic of
logicism. Unlike most critics
of logicism, however, he did not focus
his attention on the basic laws of the logicists or the question of
their genuinely logical status. Instead, he
directed his remarks against the place accorded to logical "inference"
in the logicist's conception of
mathematical proof. Following Leibnitz, traditional logicist dogma has
held that reasoning or inference is
every-where the same--that there are no principles of inference
specific to a given local topic. Poincaré,
a Kantian, disagreed with this. Indeed, he believed that the use of
non-logical reasoning was essential to
genuinely mathematical epistimology which underlies it."
Edited. Proof and Knowledge in Mathematics. London & New York: Routledge, 1992.
Edited. Proof, Logic and Formalization. London & New York: Routledge, 1992.
"Hilbert's Formalism."
Revue Internationale de Philosophie (1993), 47(4) [186]:285-304.
This issue is entitled "Hilbert."
"Hilbert's Work on the Foundations of Geometry in Relation to his
Work on the Foundations of Arithmetic."
Acta Analytica (1993), 8(11):27-39.
"Hilbert's foundational work has
commonly been divided into two
historically and perhaps
philosophically distinct parts, the one concerning the foundations of
geometry, the other the foundations of
arithmetics. How should one conceive the relation between these two
parts? It is argued that what unifies
Hilbert's geometrical and arithmetical foundational work is neither
the same general abstract conception of
theory (as Bernays and Weyl erroneously thought) nor the concern for
the purity of the proof (the view
held by Kreisel and Cellucci) but the same epistemological theme,
deriving from Kant's general critical
epistemology and based primarily on his distinction between judgments
of the understanding and ideas of
reason (which corresponds roughly to Hilbert's distinction between
real and ideal elements, propositions
and proofs)."
"The Kantian Character of Hilbert's Formalism." In Johannes Czermak, ed., Philosophy of Mathematics: Proceedings of the 15th International Wittgenstein-Symposium: 16th to 23rd August 1992, Kirchberg am Wechsel (Austria)/Philosophie der Mathematik: Akten des 15. Internationalen Wittgestein-Symposiums: 16. bis 23. August 1992, Kirchberg am Wechsel (Österreich), Volume 1, pp. 195-205. Schriftenreihe Wittgenstiengesellschaft, 20. Volume I. Vienna: Hölder-Pichler-Tempsky, 1993.
"Logicism and the Nature of Mathematical Reasoning." In A. D. Irvine and G. A. Wedeking, eds., Russell and Analytic Philosophy, pp. 265-292. Toronto Studies in Philosophy. Toronto: University of Toronto Press, 1993.
"Poincaré vs Russell on the Role of Logic in Mathematics."
Philosophia Mathematica (1993)
, 1:24-49.
"In the early years of this century,
Poincaré and Russell engaged in a debate concerning the nature of
mathematical reasoning. Siding with Kant, Poincaré argued that
mathematical reasoning is characteristically
non-logical in character. Russell urged the contrary view, maintaining
that 1) the plausibility originally
enjoyed by Kant's view was due primarily to the underdeveloped state
of logic in his (i.e., Kant's) time,
and that 2) with the aid of recent developments in logic, it is
possible to demonstrate its falsity. This
refutation of Kant's views consists in showing that every known
theorem of mathematics can be proven by
purely logical means from a basic set of axioms."
Review of John Etchemendy's The Concept of Logical Consequence. Philosophical Books (1993), 3491):1-10.
Review of Denia Mieville, ed., Kurt Gödel: actes du colloque, Neuchatel, 13 et 14 juin 1991. History and Philosophy of Logic (1994), 15(1);135-136.
"Wright on the Non-mechanizability of Intuitionist Reasoning."
Philosophia Mathematica (January 1995), 3(1):103-119.
Special issue on "The Mechanization of Reason," edited by
Michael Detlefsen and Stuart G. Shanker.
"In his paper, 'Intuitionists are not
(Turing) Machines', Crispin
Wright joins the ranks of those who have
sought to refute mechanist theories of mind by invoking Godel's
incompleteness theorems. His
predecessors include Gödel himself, J. R. Lucas and, most recently,
Roger Penrose. The aim of this essay is
to show that, like his predecessors, Wright, too, fails to make his
case, and that, indeed, he fails to do so
even when judged by standards of success which he himself lays down."
"Philosophy of Mathematics in the Twentieth Century." In Stuart G. Shanker, ed., Philosophy of Science, Logic, and Mathematics in the 20th Century, pp. 50-123. Routledge History of Philosophy, 9. London & New York: Routledge, 1996.
"Constructive Existence Claims." In Matthias Schirn, ed.,
The Philosophy of Mathematics
Today, pp. 307-335. Oxford & New York: Clarendon Press, 1998.
Papers from a conference held in Munich from June 28 to
July 4, 1993.
"Gödel's Theorems." In Routledge Encyclopedia of Philosophy, Volume 4, pp. 107-119. London & New York: Routledge, 1998.
"Hilbert's Programme and Formalism." In Routledge Encyclopedia of Philosophy, Volume 4, pp. 422-429. London & New York: Routledge, 1998.
"Mathematics, Foundations of." In Routledge Encyclopedia of Philosophy, Volume 6, pp. 181-192. London & New York: Routledge, 1998.
(with David Charles McCarty and John B. Bacon.)
Logic from A to Z. London & New York: Routledge, 1999.
First published in the Routledge
Encyclopedia of Philosophy.
"A Subject with No Object." Philosophical Books (2000),
41(3):153-163.
Review of John P. Burgess and Gideon
Rosen's A Subject with No Object: Strategies for Nominalistic
Interpretation of Mathematics.
"What Does Godel's Second Theorem Say?"
Philosophia Mathematica (January 2001), 9(1):37-71.
"The George Boolos Memorial Symposium,
II, Notre Dame, Indiana, 1998."
"We consider a seemingly popular
justification (we call it the reflexivity defense) for the third
derivability condition of the Hilbert-Bernays-Lob generalization of
Gödel's second incompleteness
theorem (G2). We argue that (i) in certain settings, use of the
reflexivity defense to justify the third
condition induces a fourth condition, and that (ii) the justification
of this fourth condition faces serious
obstacles. We conclude that, in the types of settings mentioned, the
reflexivity defense does not justify the
usual 'reading' of G2--namely, that the consistency of the represented
theory is not provable in the
representing theory."
Reviews of Michael Detlefsen's Books
Michael Detlefsen's Hilbert's Program: An Essay on Mathematical
Instrumentalism (1986)
Auerbach, David D. Journal of Symbolic Logic (June 1989),
54(2):620-622.
Cortois, P. Tijdschrift voor Filosofie (December 1988),
50(4):730-731.
Irvine, A.D. Canadian Philosophical Reviews (April 1989),
9:145-148.
Largeault, J. Archiv für Geschichte der Philosophie
(April-June 1989), 52(2):304.
Steiner, Mark. Journal of Philosophy (June 1991),
88(6):331-336.
Discussions of Detlefsen
Ignjatovic, Aleksandar. "Hilbert's Program and the Omega-Rule."
Journal of
Symbolic Logic (March 1994), 59(1):322-343.