"Creation and Completeness"

Compiled by

Eddie Yeghiayan

(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.

Cortois, P.

Irvine, A.D.

Largeault, J.

Steiner, Mark.

Engel, P.

Folina, Janet.

Larvor, Brendan.

Engel, P.

Heinzmann, Gerhard.

Larvor, Brendan.

Steiner, Mark.

**Discussions of Detlefsen**

Ignjatovic, Aleksandar. "Hilbert's Program and the Omega-Rule."
*Journal of
Symbolic Logic* (March 1994), 59(1):322-343.

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