Abstracts for 2003 (Also see 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Hale, Margie: Essentials of mathematics: Introduction to theory, proof, and the professional culture (Resource Materials Series)Mathematical Association of America, Washington, DC, 2003
Panza, Marco: Mathematical proofsSynthese, 134, 2003, 119-158This novel philosophical paper discusses so-called mathematical proofs (e.g., for primality of large numbers) which the author believes are more primitive and clearer than logical proofs, which hinge on finite sequences of well-formed formulas and rules of inference. The author's philosophy and line of reasoning has a distinctive taste of intuitionism at several crucial points. Numerous specific illustrative examples of the contexts in which mathematical proofs take place are given and discussed, e.g., the proof of commutativity of addition in Peano's arithmetic and the use of Hilbert's contentual elementary arithmetic. Although the author states (p. 146) that a mathematical proof is a succession of human acts, it is never discussed when a human act is misbegotten. Still, one must ask: Is the proof of Gödel's first incompleteness theorem a mathematical proof or a formal logical proof? Finally, I believe the author's notion of a mathematical proof would be clarified by discussion of the difference between mathematical truth and mathematical proof. (A. A. Mullin)
Peressini, Anthony: Proof, reliability, and mathematical knowledgeTheoria (Stockholm), 69, 2003, 211-230With respect to the confirmation of mathematical propositions, proof possesses an epistemological authority unmatched by other means of confirmation. This paper is an investigation into why this is the case. I make use of an analysis drawn from an early reliability perspective on knowledge to help make sense of mathematical proof's singular epistemological status.
Alvarez, Carlos: Two ways of reasoning and two ways of arguing in geometry: Some remarks concerning the application of figures in Euclidean geometry.Synthese, 134, 2003, 289-323This paper contains: (1) an analysis of what one may call the weak spots in the Elements' plane geometry: (i) the uses of superposition in I.4, I.8, and III.24, and (ii) reasonings involving (arcs and segments of) circumferences, with particular emphasis on I.2, I.7, II.23, and III.25; (2) an exposition of Hilbert's solution to the problems raised by (i); (3) an axiom system for arcs, segments, and circumferences (treated as point-sets), followed by a proof of III.24 based on it.
Corfield, David: Towards a Philosophy of Real MathematicsCambridge University Press, Cambridge, 2003In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. (Publisher - edited)
Fallis, Don: Intentional Gaps in Mathematical ProofsSynthese, 134, 2003, 45-69How do mathematicians know that mathematical propositions are true? This paper argues that the standard answer (viz., that they have worked through the details of a proof) does not capture all of the justificatory practices that are accepted by the mathematical community. Mathematicians frequently do not verify all of the steps in their proofs. In other words, they often intentionally leave "universally untraversed" gaps in their proofs. Several examples of such gaps from the history of mathematics are given.
Abstracts for 2002 (Also see 2003, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Murawski, Roman: Truth vs. provability - philosophical and historical remarksLogic and Logical Philosophy, 10, 2002, 93-117
Mormann, Thomas: Towards an evolutionary account of conceptual change in mathematics: Proofs and refutations and the axiomatic variation of conceptsin George Kampis, Ladislav Kvasz and Michael Stöltzner (eds.), Appraising Lakatos. Mathematics, methodology, and the man, Kluwer Academic Publishers, Dordrecht, 2002, 139-156The author discusses the relation between truth and proof by surveying the development of these notions in the different stages of the disputes on foundations in the early 20th century. In the early period of his axiomatic programme Hilbert sought to justify mathematical theories (truths) with the help of an axiomatization. Gödel was able to show the completeness of first-order logic, but shortly afterwards also showed that the arithmetic of natural numbers and all richer theories are essentially incomplete, given their consistency. The incompleteness theorems and in particular Gödel's recognition of the undefinability of the concept of truth, the author says, "indicated a certain gap in Hilbert's programme and showed in particular ... that (full) truth cannot be established ... by provability and, generally, by syntactic means. The former can only be approximated by the latter" (p. 108). Hilbert's search for new rules of inference in his proof theory, Gödel's search for new axioms and Zermelo's infinitary logic are presented as attempts to overcome this gap. (Volker Peckhaus - edited)
Thomas, R. S. D.: Mathematics and narrativeMath. Intelligencer, 24, 2002, 43-46The use of simile or analogy is frequently adopted by mathematicians in their attempt to explain features of mathematical knowledge to the general public. For example, mathematics looks like a tree or mathematics can be tour-guided by a scenic map. These, among others, are favorite descriptions appearing in popular mathematics writing. However, a comparison of mathematics with narrative, as suggested in the paper under review, is more to the point if one wants to explore in depth the status of mathematical knowledge. Indeed, Thomas's argument reminds us that the story of mathematics can be told in an axiomatic manner. If "proof and narrative are different ways of working out the consequences of relational hypotheses", then various mathematical structures as "stories engage the attention and fire the imagination of a reader". In short, what Thomas says as a storyteller about mathematics and narrative is attractive and illuminating, especially to those teachers who like to explain to their students, math majors or not, what mathematics is. (Wann-Sheng Horng - edited)
Revuz, André: Y a-t-il une méthode mathématique?in Michel Serfati (ed.), De la méthode: Recherches en histoire et philosophie des mathématiques, Presses Universitaires Franc-Comtoises, Besançon, 2002, 155-176The first obvious answer may be `deduction' or `proof'. That is an essential feature of mathematical activity, but it is not the only one: the challenge of the mathematician is not so much how to prove as what to prove and how to choose the axioms that are the basis of deductive reasoning. Surprisingly for some people, mathematics has shown that it can be wonderfully efficient in describing natural phenomena. The process is the following: 1. Delimitation of a `situation', i.e. the portion of the universe which we want to study. 2. Building of a model that is both mathematically coherent and suitable to the situation. 3. Building of theories using the essential ideas that make models successful. The mathematico-scientific process moves back and forth through these three steps. As for teaching, it is not clear that it has been deeply influenced by the true nature of mathematics.
Michel, Alain: Thèses d'existence et travail mathématiquein Michel Serfati (ed.), De la méthode: Recherches en histoire et philosophie des mathématiques, Presses Universitaires Franc-Comtoises, Besançon, 2002, 247-269The author tries to substantiate the thesis that what philosophers of mathematics tend to view as merely abstract problems are in fact very precise problems faced by real mathematicians in their daily practice. The question of mathematical existence is the example chosen to prove this thesis. The first section of the article ("Remarques sur la constitution en probleme de l'existence mathematique") describes the main stages in the evolution of the problem, from Cauchy and Dedekind to Frege and Brouwer. The second section ("Theses d'existence et travail mathematique: l'exemple de Kronecker") shows the connection between the views taken by an author and his mathematical practice, which is always situated in a precise historical context. (Paloma Pérez-Ilzarbe)
Weber, Erik and Verhoeven, Liza: Explanatory proofs in mathematicsLogique et Anal., N.S. 45, 2002, 299-307
Fallis, Don: What do mathematicians want? Probabilistic proofs and the epistemic goals of mathematiciansLogique et Anal., N.S. 45, 2002, 373-388
Lercher, Aaron: What is the goal of proof?Logique et Anal., N.S. 45, 2002, 389-395
Fallis, Don: Response to: "What is the goal of proof?" by A. LercherLogique et Anal., N.S. 45, 2002, 397-398
Rood, Ron: Proof, cognition, and rationalityLogique et Anal., N.S. 45, 2002, 399-419
Mancosu, Paolo: On the constructivity of proofs: A debate among Behmann, Bernays, Gödel, and Kaufmannin Wilfried Sieg, Richard Sommer and Carolyn Talcott (eds.), Reflections on the foundations of mathematics: Essays in honor of Solomon Feferman, A K Peters, Ltd., Natick, MA, 2002, 349-371The question of whether proofs by contradiction can be eliminated in favor of direct proofs has occupied the minds of many philosophers and mathematicians for centuries. Aristotle, Arnauld, Kant, Bolzano, and many others up to our day have tackled this topic. In the present paper I present the details of a debate centered around the relationship between constructivity and indirect proofs which took place in 1930 and involved some of the most able logicians and philosophers of mathematics of the century, namely Heinrich Behmann, Paul Bernays, Rudolf Carnap, Kurt Gödel, and Felix Kaufmann. (From the introduction - edited)
Sheard, Michael: Truth, Provability, and Naive Criteriain Halbach, Volker (ed.), Principles of Truth, Hansel-Hohenhausen, Frankfurt, 2002, 169-181
Abstracts for 2001 (Also see 2003, 2002, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Kadvany, John: Imre Lakatos and the Guises of ReasonDuke University Press, Durham, NC, 2001This is an account of the work (and life) of Imre Lakatos (1922--1974), the brilliant but enigmatic Hungarian refugee who became an important philosopher of mathematics and science. Kadvany's book is the first to fully explore the import of Hegel's dialectics, and central European thinking in general, in Lakatos' work. This is the deliberate aim of the book, and it succeeds admirably in this goal with considerable thoroughness, depth and insight, culminating in Table 1 (pages 294-5), which identifies 18 Hegelian (or Marxist) themes in Lakatos. This book, however, it is not yet the definitive work on Lakatos' philosophy or even his philosophy of mathematics, for two reasons. First of all, like too many commentators, Kadvany draws exclusively on Lakatos' posthumous publications. Secondly, in the book Kadvany is primarily advancing his thesis about the Hegelian underpinnings of Lakatos' work. Nevertheless, this is the most important book that has appeared on Lakatos' work to date, and it contains much that is novel and of real interest and importance to philosophers and mathematicians.(Paul Ernest - edited)
Cupillari, Antonella: The Nuts and Bolts of ProofsAcademic Press, Inc., San Diego, CA, 2001
Gardner, Martin: Is mathematics "out there"?Math. Intelligencer, 23, 2001, 7-8In this short piece the celebrated mathematical puzzler addresses the problem of the universality of mathematics. He takes issue with the expressed view of R. Hersh (1997) that mathematics is a social and cultural construction. In particular, Gardner reaffirms several "myths" about mathematics challenged by Hersh: that mathematical truth is universal, timeless, certain and absolutely true. Since these "myths" were until recently held to be part of "common sense" and above challenge such criticism involves complex and subtle arguments, careful definitions and the clarification of claims. Unfortunately Gardner fails to engage with any of the subtlety and complexity involved in the issues and simply reasserts the challenged claims, exemplifying his assertions with a pile of pebbles, four dinosaurs and a little green man. Doubtless some mathematicians will be sympathetic to this approach, but it isn't philosophy and it doesn't clarify this deep and contested issue. (Paul Ernest)
Senechal, Marjorie: Between discovery and justificationMath. Intelligencer, 23, 2001, 16-17Marjorie Senechal, editor of The Mathematical Intelligencer, reflects on the relevance of the "science wars" to mathematics and the contributions of Graham, Harris and Ruskai on this issue in the journal she edits. In a balanced overview Senechal argues that the traditional hard and fast distinctions between the contexts of discovery and justification and between whether mathematics is invented (socially constructed) or discovered (Platonism) are not fruitful in elucidating the nature of mathematics. (Paul Ernest)
Harris, Michael: Contexts of justificationMath. Intelligencer, 23, 2001, 18-22This paper is a further response to Graham's claim that mathematics can be deeply influenced by its social context. It discusses the "science wars" and their relevance to mathematics. Although no single central point is made, Harris argues convincingly that the issues of intuition, truth, the acceptance of results, and the contexts of discovery and justification are very complex in mathematics. For example, Harris questions the claim that "A mathematical constant like pi doesn't change, even if the idea one has about it may change" (A. Sokal and J. Bricmont, 1998). He asks what kind of object pi was before the definition of real numbers. To assume the real numbers were there all along, waiting to be defined, is to adhere to a form of Platonism, itself a form of idealism. Although there is nothing wrong with Platonism, indeed it may well be an intuitively fruitful position for many working mathematicians, Harris argues that requiring adherence to its doctrine is a strange touchstone for rationality. (Paul Ernest - edited)
Legris, Javier: Deduction and knowledge in the origins of proof theory (Spanish)Theoria (San Sebastián), 16, 2001, 521-538This paper is divided into six sections (in English): (1) The Hilbertian theory of demonstration, (2) The "combinatorial moment" in demonstrations, (3) Intuitive knowledge, (4) Deduction and intuition, (5) Finitary deduction, (6) Intuitionistic deduction and finitary deduction. The author refers mainly to Paul Bernays (Hilbert's student and associate), who argued in 1930 that Hilbert's program was the correct response to the problems raised in the foundations of mathematics as a consequence of the set-theoretic paradoxes. The two leading principles of the program were, on the one hand, the requirement of consistency proofs, and, on the other hand, the finitary nature of the methods employed in those proofs. The author shows that, for Bernays, the whole program is based upon an intuitive knowledge. (Ignacio Angelelli - edited)
Gurevich, Yuri: Platonism, constructivism, and computer proofs vs. proofs by handin G. Päun, G. Rozenberg and A. Salomaa, (eds.), Current trends in theoretical computer science, River Edge, NJ, 2001, 281-302
Tappenden, Jamie: Recent work in philosophy of mathematics (Review Article)The Journal of Philosophy, 98, 2001, 488-97A review article discusses three books on the philosophy of mathematics. The books discussed are Naturalism in Mathematics, by Penelope Maddy; Philosophy of Mathematics: Structure and Ontology, by Stewart Shapiro; and Mathematics as a Science of Patterns, by Michael Resnik.
Pinto, Silvio: The Justification of DeductionSorites, 13, 2001, 33-47According to Michael Dummett, deductive inference stands in need of justification which must be provided by the theory of meaning for natural language. Such a theory, he insists, should deliver an explanation for the two essential features of deduction: validity and fruitfulness. Dummett claims that only a molecularist theory of meaning could offer the desired justification. In this paper, I will consider and criticize his solution to the problem of the justification of deduction: the so-called molecular verificationist explanation. My aim here will be to show that Dummett's solution does not succeed in reconciling the conflicting demands of the respective explanations of validity and fruitfulness.
Ganeri, Jonardon: Objectivity and Proof in a Classical Indian Theory of NumberSynthese, 129, 2001, 413-437In India, seemingly as elsewhere, things related with the foundations of mathematics more attracted philosophers than mathematicians. After a short introductory section (Mathematics and the philosophical theory of number), the author investigates how the Nyaya-Vaisesika school of Indian philosophy treated, or rather struggled with, the concept of number, in seven sections: Number as a property of objects, Number as a relation of non-identity, Number and collection, Prasastapada's "eight moments" theory, Objectivity in the theory of number, The epistemology of number, and The meaning of `many'. In the last, ninth section (The theory of number and Indian concepts of rationality and proof), the author points out the importance of empirical grounding in India, both in philosophical theorising, including that of number theory, and in mathematical proof. This article will be welcomed by historians not only of Indian philosophy but also of Indian mathematics. (Takao Hayashi)
Fitelson, Branden and Wos, Larry: Finding Missing Proofs with Automated ReasoningStudia Logica, 68, 2001; 329-356This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are of added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. (edited)
Bronkhorst, Johannes: Panini and Euclid: Reflections on Indian GeometryJournal of Indian Philosophy, 29, 2001 43-80This article deals with the question whether and to what extent the grammarian Panini's method has, historically speaking, occupied in Indian thought a place comparable to that held by Euclid's method in Western thought. In order to test this claim, an early classical Indian text on geometry has been singled out for in-depth analysis: the commentary by Bhaskara I on the Aryabhatiya. The absence of proofs in this work is remarkable in view of the fact that contemporary Indian philosophers and logicians were familiar with this notion. The question is raised whether influence from Panini's grammar might be one of the factors responsible for this absence.
Salmon, Nathan: The Limits of Human MathematicsNous Supplement, 15, 2001, 93-117Gödel derived a philosophical conclusion from his second incompleteness theorem: Either the human mind infinitely surpasses any finite machine, or there are mathematical problems that are humanly unsolvable in principle; and therefore, either the human mind surpasses the human brain or it is not responsible for the creation of mathematics. Gödel's derivation is here examined, reformulated, and defended. Remarks of Alonzo Church to the effect that nothing is a mathematical proof unless it is effectively decidably so are critically examined and found to be unconvincing. The investigation makes room for Gödel's optimism that in principle any mathematical problem is solvable.
Mancosu, Paolo: Mathematical Explanation: Problems and ProspectsTopoi, 20, 2001, 97-117In this article I have three major aims. The first is to introduce the topic of mathematical explanation by listing a number of problems followed by a reflection on the status of research and prospects for further development. The general discussion in the first part motivates the specific contributions to be presented in the remaining two parts. In the second part I draw attention to an important tradition in philosophy of mathematics for which explanation is a concern. Here I discuss Mill, Russell, Godel, Lakatos and other philosophers of mathematics on mathematical explanation. The last part of the articles presents a case study of a development in mathematical practice that originates from explanatory concerns, i.e., Alfred Pringsheim's "explanatory" approach to the foundations of complex analysis.
Abstracts for 2000 (Also see 2003, 2002, 2001, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Graham, Loren R.: Do mathematical equations display social attributes?Math. Intelligencer 22, 2000, 31-36The "science wars" fought between defenders of the absolute objectivity of science and mathematics and social constructivists have been raging for some years. Many exchanges are acrimonious, with opponents questioning each other's integrity. In contrast, this paper is a carefully reasoned contribution that steers a more moderate path. In it Graham argues the radical position that even mathematical equations can display social attributes - meaning that the form of equations adopted in some branches of mathematics and physics can be socially influenced. Graham demonstrates this position through a carefully documented case study of the respected Russian relativity theorist V. Fock, who reformulated Einstein's gravitational equation in terms of harmonic coordinates. This paper illuminates this area of controversy by introducing new facts, thus bringing a discussion that can become dangerously detached from reality down to earth. (Paul Ernest - edited)
Knobloch, Eberhard: Archimedes, Kepler, and Guldin: the role of proof and analogyin Rüdiger Thiele (ed.), Mathesis: Festschrift zum siebzigsten Geburtstag von Matthias Schramm, Berlin, 2000, 82-100In this very interesting paper the author discusses mathematical reasoning and proof methods in Archimedes, Kepler and Guldin, by comparing in particular the role of proof and analogy in Archimedes's writings and Kepler's New stereometry. In doing so, the author provides an accurate philological analysis of some relevant passages of Archimedes' On the sphere and cylinder, and supports his arguments with examples from Archimedes' text and Kepler's Supplement to Archimedes. In the last section of the paper Guldin's criticism of Kepler's proof methods is discussed. (Umberto Bottazzini )
McClure, J. E.: Start where they are: geometry as an introduction to proofAmer. Math. Monthly, 107, 2000, 44-52
Casselman, Bill: Pictures and proofsNotices Amer. Math. Soc., 47, 2000, 1257-1266Casselman draws on his experience as editor of the Notices, his teaching and his research, to demonstrate why the Japanese proverb "one look is better than a hundred hearings" offers an important direction for the development of the presentation of mathematics. Casselman explicitly recommends the use of many more diagrams in mathematical texts and research papers. The very richness of the imagistic mode of perception, however, means that multiple interpretations are likely. A lesson to be learned from this is that how you draw a diagram may reveal how you think about the relationships which it encapsulates. The sequence in which you draw elements is likely to be of significance, but obscured by the final picture. Thus structured diagrams make use of, and can be used to expose, a basic theme of mathematics. Those not experienced with diagrams need to learn techniques of locating what is invariant and what is allowed to change, and those who draw diagrams need to develop ways of indicating these to students. (John H. Mason - edited)
Parsons, Charles: Reason and intuitionSynthese, 125, 2000, 299-315In modern mathematics, the axiomatic method makes it possible to give a workable model of proof as deductions from axioms with the help of logic. Then one has identified a factor in mathematical knowledge that would be attributed to reason, namely logic. A wide variety of statements are accepted without carrying the argument further. In mathematical thought, the axioms of arithmetic are embedded in a dense network of ideas. Arithmetic illustrates as a feature of mathematical principles that there is a decrease of clear and evident character as one introduces more abstract and logically powerful conceptual apparatus. (Alvin M. White - edited)
Di Leonardo, M. V.: Logical laws and schemes of reasoning (Italian)Quaderni di Ricerca in Didattica, 9, 2000, 85-104This paper is directed (yet not explicitly) to teachers of mathematics. The author discusses informally the concept of mathematical proof and calls the attention of the reader to its importance in mathematical reasoning. Classical propositional calculus and first-order predicate calculus are recalled with some detail for providing the basic stuff for the discussion. Some examples of the most common methods of proof are given, as for instance direct and indirect proof of a conditional and of reductio ad absurdum. The author concludes by emphasizing how the systems discussed are useful as "models" for mathematical reasoning. (Décio Krause - edited)
Heintz, Bettina: Die Innenwelt der Mathematik: Zur Kultur und Praxis einer beweisenden DisziplinSpringer-Verlag, Vienna, 2000Mathematics is particular in that it furnishes logically compelling proofs of its propositions. Rather than producing contradictory results, research in widely remote branches of mathematics regularly uncovers deep relations between subjects previously thought of as entirely unconnected. Although split up into a great many separate specialties, mathematics shows a coherent unity with a title to universality. Serious alternatives seem not to exist. The object of this book is to explore the possibility of a sociological characterization of mathematics which accounts for the above epistemic particularities. In three chapters the author presents her results concerning the principles that guide individual mathematicians in the process of discovery and validation, the mechanisms in the mathematical community that account for the external validation of individual contributions, and the role of proofs as vehicle for communicative consensus formation. In the concluding chapter she presents her sketch of a sociological explanation of the particular character of mathematics. (Eduard Glas - edited)
Hacking, Ian: What mathematics has done to some and only some philosophersProc. Br. Acad., 103, 2000, 83-138The author starts from the following remark: "A great many of the philosophers whom we still read have been deeply impressed by mathematics, and have gone so far as to tailor much of their philosophy to their vision of mathematical knowledge, mathematical reality, or, what I think crucial, mathematical proof." Nevertheless the "interesting and difficult" question "How is pure mathematics possible?" (Chapter 1) troubled "some and only some" of them. Hacking refers to six different philosophies, and places them in two groups: "inflationary" (Mill, Plato, Leibniz: Chapters 2--4) and "deflationary" (Descartes, Lakatos, Wittgenstein: Chapters 6--8). Before studying the second group, he inserts a Chapter 5, "The terms of art", in order to clarify the terms a priori, necessary, analytic, inconceivable, certainty, apodictic, and sums up his reflections with a paragraph on "theory and phenomena". (Pierre Crépel - edited)
Antonelli, Aldo and May, Robert: Frege's new scienceNotre Dame J. Formal Logic 41, 2000, 242-270After presenting the differences between Hilbert's and Frege's conceptions of independence proofs, the authors propose a characterization of logical notions that is consistent with Frege's conception of language as already interpreted, i.e. "a system of signs, where a sign is made up of a symbol and a sense". The proposed characterization builds upon the Fregean 1906 suggestions for establishing independence proofs in geometry in the context of his metatheory (the New Science), inspired by the Erlangen Programme, which also informs the Tarski-Sher-McGee approach to the characterization of logical notions. (Victor V. Pambuccian)
McLarty, Colin: Voir-dire in the case of mathematical progressin Emily Grosholz and Herbert Breger, (eds.), The growth of mathematical knowledge, Kluwer, Dordrecht, 2000, 269-280This article is concerned with the relation between rigor and progress in mathematics. In particular, the author is sympathetic to claims made by logicists and formalists that mathematics is not about content, but expression, and that progress in expression comes in the form of progress in rigorization. The author claims that rigorization has a unifying effect, both internal to mathematics and external to it, resulting in the encouragement of multidisciplinary research, as opposed to the raising of barriers between experts and laypersons. On the other hand, he admits that there is a difference between content and expression, and cautions against associating mathematics as a discipline solely with the implementation of rigor. Finally, he disagrees with the claim that contemporary mathematics has witnessed a turn from theory to applications; instead, he sees a unification of theory with applications. (Jonathan Bain - edited)
Knobloch, Eberhard: Analogy and the growth of mathematical knowledgein Emily Grosholz and Herbert Breger, (eds.), The growth of mathematical knowledge, Kluwer, Dordrecht, 2000, 295-314The opening sentence of the article already states the author's interest: "High esteem for the mathematical discoveries and rigorous argumentation of the ancient mathematicians like Euclid or Archimedes has always been accompanied by astonished speculation about how after all they had found their results, which were subsequently demonstrated in such an exemplary way they became paradigms of rigorous argumentation". Although it is the result of the research of a historian of mathematics, the paper follows neither a systematic interest nor a chronological path, but is rather structured according to the following areas: analogy and theory; analogy and proof; analogy and problem; analogy and notion. (Reviewer: Michael Otte - edited)
Mancosu, Paolo: On mathematical explanationin Emily Grosholz and Herbert Breger, (eds.), The growth of mathematical knowledge, Kluwer, Dordrecht, 2000, 35-54The author ranges over wide geographical and temporal domains while considering the subject. His remarks and quotations are provocative. "Mathematical truths are constructed or demonstrated, not explained." "It is thus clear that the main problem is whether these are explanations in mathematics." Included is a survey of contemporary and earlier literature on explanation in mathematics. He quotes Aristotle, B. Bolzano, K. Menger, M. Steiner, P. Kitcher and many others. The reader must be impressed with the long and rich history of mathematics. (Alvin White)
Rowe, David: The calm before the storm: Hilbert's early views on foundationsin Vincent F. Hendricks, Stig Andur Pedersen and Klaus Frovin Jørgensen, (eds.), Proof Theory, Kluwer, Dordrecht, 2000, 55-93The paper provides a comprehensive survey of D. Hilbert's views on the foundations of mathematics in the time before the foundational crisis of the 1920s and before the formalist programme and metamathematics were formulated, later leading to modern proof theory. He convincingly shows that Hilbert's early work on invariant theory and his research on algebraic number fields already showed the tendency towards abstract generalization which was advanced further in his Grundlagen der Geometrie of 1899. The author stresses that the formulation of the second problem of Hilbert's Paris lecture on "Mathematische Probleme" (1900) calling for a proof of the consistency of the axiom system for the real numbers clearly arose from Hilbert's research on the axiomatic method as presented in the Grundlagen der Geometrie. The author deals with the dispute between Hilbert and G. Frege about the axiomatic method, Hilbert's reaction on the publication of the Zermelo-Russell paradox, and the reasons for his silence on foundational issues between 1905 and 1917. The paper ends with a description of Hilbert's return to the "foundational front" as marked by his lecture on "Axiomatisches Denken" (1917). (Reviewer: Volker Peckhaus - edited)
Corry, Leo: The empiricist roots of Hilbert's axiomatic approachin Vincent F. Hendricks, Stig Andur Pedersen and Klaus Frovin Jørgensen, (eds.), Proof Theory, Kluwer, Dordrecht, 2000, 35-54Hilbert's work on logic and proof theory appeared almost two decades after the publication of the epoch-making Grundlagen der Geometrie. A close historical examination of these works, however, brings to light important differences between them. In the present paper I analyze an often overlooked aspect of the background to Hilbert's work on the foundations of geometry, namely, the empiricist elements that inform it. In order to do so, I describe works of three scientists, Heinrich Hertz, Carl Neumann and Paul Volkman. The footprints of these works may then be clearly identified in Hilbert's early axiomatic work, and in particular in the Grundlagen. This analysis helps in understanding the proper context of ideas from which the axiomatic approach originally arose, and it thus opens the way to a more comprehensive analysis of the historical background behind the development of Hilbert's proof theory. (From the Introduction - edited)
Sieg, Wilfried: Toward finitist proof theoryin Vincent F. Hendricks, Stig Andur Pedersen and Klaus Frovin Jørgensen, (eds.), Proof Theory, Kluwer, Dordrecht, 2000, 95-114The paper deals with the foundational research of D. Hilbert and his school between 1917 and 1922. The author gives impressive evidence for the dynamics of Göttingen foundational research in this period mainly by analyzing a remarkable series of (so far unpublished) lecture courses given by Hilbert. He stresses the importance of the first course in this series, in the winter term 1917/18, dealing with "Prinzipien der Mathematik": "The notes present more than a system for the development of parts of higher mathematics. Hilbert's finitist proof theory is finally developed in a course in the winter term 1921/22 on "Grundlagen der Mathematik", where for the first time the terms "finite Mathematik", "transfinite Schlussweisen", and "Hilbertsche Beweistheorie" are used. Mathematical and metamathematical considerations are clearly distinguished. (Reviewer: Volker Peckhaus - edited)
Shapiro, Stewart: Thinking about mathematics: The philosophy of mathematicsOxford University Press, Oxford, 2000This is a philospohy book about mathematics. There are, first, matters of metaphysics: What is mathematics all about? Does it have a subject matter? What is this subject matter? What are numbers, sets, points, lines, functions, and so on? Then there are semantic matters: What do mathematical statements mean? What is the nature of mathematical truth? And epistemology: How is mathematics known? What is its methodology? Is observation involved, or is it a purely mental exercise? How are disputes among mathematicians adjudicated? What is a proof? Are proofs absolutely certain, immune from rational doubt? What is the logic of mathematics? Are there unknowable mathematical truths? (From the Preface)
Horsten, Leon: Models for the logic of possible proofsKluwer: Dordrecht, 2000The present paper investigates the logical structure of possible proofs. We present and philosophically motivate a class of possible proof models that describes in some detail the modal-epistemic propositional logical structure of possible proofs. This class of models is then recursively axiomatized.
Hendricks, V., Pedersen, S. A. and Jorgensen, K. F. (eds): Proof Theory: History and Philosophical SignificancePacific Philosophical Quarterly, 81, 2000. 49-66
Vermeulen, C F M: Text Structure and Proof StructureJournal of Logic, Language and Information, 9, 2000, 273-311This paper is concerned with the structure of texts in which a proof is presented. Some parts of such a text are assumptions; other parts are conclusions. We show how the structural organization of the text into assumptions and conclusions helps to check the validity of the proof. Then we go on to use the structural information for the formulation of proof rules, i.e., rules for the (re-)construction of proof texts. (edited)
Krishna, Daya: 'Shock-Proof', 'Evidence-Proof', 'Argument-Proof' World of Sampradayika Scholarship of Indian PhilosophyJournal of Indian Council of Philosophical Research, 17, 2000, 143-159
Abstracts for 1999 (Also see 2003, 2002, 2001, 2000, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Harel, Guershon: Students' understanding of proofs: a historical analysis and implications for the teaching of geometry and linear algebraLinear Algebra Appl. 302/303, 1999, 601--613This article contains the text of an oral presentation at the International Linear Algebra Society Conference of 1998. The author has investigated students' understanding of mathematical proofs. The (modern) idea that geometric properties describe structures, and are not supposed to evoke specifically spatial imagery, is virtually absent from students' conceptions. The author refers to this inability to detach from a specific imaginable context as the "contextual" conception of mathematical proof (category 1). In the author's view, these students are not content when a proof merely proves "that" a proposition is true, they also want a clear demonstration of "why" it is true. Inquiring after the "why" of something is inquiring after its "cause" in the Aristotelean sense of "cause". Hence we could speak of a "causal' conception of proof (category). It corresponds with the view of various great mathematicians of the early modern period, among them Descartes, who rejected indirect proofs and demanded direct and ostensive demonstrations of mathematical truths. The author thus draws from his historical analyses various valuable insights into the possible conceptual basis of students' difficulties with modern conceptions of mathematical proof. (E. Glas - edited)
Benson, Donald: The moment of proofOxford University Press, New York, 1999I hope in this book to communicate something of the experience and the joy of mathematical discovery. In the popular literature of science, it is usual to tell what is true rather than why it is true. For example, a mathematician can tell us that there are infinitely many prime numbers. It is interesting to know that this true, but it is much more exciting to learn why. An elegant proof must meet the following requirements. (a) It must have mathematical importance. (b) It must be subtle, but not obscure. (c) It must be short---one or two pages at most. (d) It must have a surprise---if not in its conclusion, then in its execution. This book is eclectic in its choice of subject matter; however, it provides a self-contained introduction to many topics of mathematics as well as an introduction to the intellectual pleasures of mathematics. (From the Introduction - edited)
Sabatier, Xavier: La logique dans la science: place et statut de la logique dans la philosophie de Jean Cavaillès (French)Rev. Histoire Sci. 52, 1999, 81--106Cavaillès' essays deal with epistemological problems concerning the development of "modern" mathematics and set theory, and, in particular, attempt to answer the following questions: "What kind of relation is to be thought between mathematics and logic? Does logic coincide with the foundations for mathematics? To what extent is logic useful or indispensable in building philosophical views on the nature and growth of modern mathematics?" As résistant and as philosopher Cavaillès has a very good reputation. However, some of his writings, and especially his book, Sur la logique et la théorie de la science (1947), are difficult to understand. The latter was written in jail and Cavaillès had no time to polish it up. The present paper summarizes in twenty-five pages a longer work entirely devoted to an explanation of the meaning of the words "logic" and "logical" in Sur la logique et la théorie de la science. The main argument of the paper is the following: It is necessary to distinguish between "logic" and "the logical"; moreover, "the logical" must not be mistaken for logic, a science with a history, which is only a part of "the logical", cut from its origin and defined as an independent science by a specific epistemology, namely, logicism. (Hourya Sinaceur - edited)
Taylor, Paul: Practical foundations of mathematicsCambridge University Press, Cambridge, 1999A moderate form of the logicist thesis is established in the tradition from Weierstrass to Bourbaki, that mathematical treatments consist of the manipulation of assertions built using symbolic logic. Automated deduction is still in its infancy, but such awareness may also be expected to help with computer-assisted construction, verification and dissemination of proofs. This suggests a new understanding of foundations apart from the mere codification of mathematics in logical scripture. A large part of this book is category theory, but that is because for many applications this seems to me to be the most efficient heuristic tool for investigating structure, and comparing it in different examples. I have deliberately not given any special status to any particular formal system, whether ancient or modern, because I regard them as the vehicles of meaning, not its cargo. (From the Introduction - edited)
Bressoud, David M.: Proofs and confirmations: The story of the alternating sign matrix conjectureMAA, Washington, DC; Cambridge University Press, Cambridge, 1999This book recieved the Beckenbach Prize by the MAA in January, 2000. It is a carefully crafted, compelling, chronological cruise through a rivetingly beautiful topic. There is careful attention to history. A number of photos of the principal actors are included to flesh out the story with a human touch. The title of the book, Proofs and Confirmations, was clearly chosen by the author to contrast with the title of Imre Lakatos' book Proofs and Refutations (1976). We may reasonably view the book under review as an optimistic counterpart to the work of Lakatos. The author to some extent reveals his educational object in the following short paragraph.
"My intention in this book is not just to describe this discovery of new mathematics, but to guide you into this land and to lead you up some of the recently scaled peaks. This is not an exhaustive account of all the marvels that have been uncovered, but rather a selected tour that will, I hope, encourage you to return and pursue your own explorations."(G. E. Andrews - edited)
Murawski, Roman: On new trends in the philosophy of mathematicsin Logic at work, Heidelberg: 1999, 15--24,The aim of this paper is to present some new trends in the philosophy of mathematics. The theory of proofs and counterexamples of I. Lakatos, the conception of mathematics as a cultural system of R. L. Wilder, as well as the conception of R. Hersh and the intensional mathematics of N. D. Goodman and S. Shapiro are discussed. All those new trends and tendencies in the philosophy of mathematics are attempts to overcome the limitations of the classical theories (logicism, intuitionism and formalism) by taking into account the actual practices of mathematicians.
Peressini, Anthony: Confirming mathematical theories: an ontologically agnostic stanceSynthese 118, 1999, 257--277This paper provides an interesting twist to the consideration of a common argument, due to W. V. O. Quine and Hilary Putnam, for realism concerning mathematics. According to this argument, one concludes that mathematical objects, like sets and numbers, exist because mathematics is indispensable in science (e.g., Putnam, 1971, Chap. 5). The idea seems to be that a mathematical theory is confirmed the same way that a scientific theory is - by playing a central role in the scientific web of belief. The present author invokes a distinction, borrowed from discussions of "constructive empiricism", between accepting a theory and believing that a theory is literally true (van Fraassen, 1980). According to the provided sketch, a scientific theory is accepted if it is empirically adequate - if it gets the appearances right. According to the present author, an empirically adequate theory is believed to be true if it has explanatory value and evidential relevance. So scientists are sometimes justified in taking accepted theories to be true (against constructive empiricism). In contrast, mathematical theories do not have direct empirical consequences, and so cannot be said to be empirically adequate or empirically inadequate. Thus, we cannot use the scientific criteria. He concludes that one should be agnostic about the truth of accepted mathematical theories, and, in particular, agnostic on the existence of mathematical objects - pending the articulation of criteria for believing a mathematical theory to be literally true. (S. Shapiro - edited)
Aigner, Martin and Ziegler, Günter M.: Proofs from The BookSpringer-Verlag, Berlin, 1999Paul Erdös maintained that God kept a Book with only the most elegant mathematical arguments. This volume, conceived in consultation with Erdös and published in his memory, suggests some of the Book's contents. Thirty sections treat results drawn from number theory, geometry (mainly combinatorial), analysis, combinatorics and graph theory; these can be followed by one versed in undergraduate mathematics including discrete topics. The proofs date mainly from the entire span of the twentieth century; many are due to Erdös himself. The authors have done a fine job of arranging diverse material into a thematic progression. Many readers will find unfamiliar results along with some old favorites: a decade-old proof of Fermat's two-square theorem, Hilbert's equidecomposible polyhedra problem, Sylvester's problem on lines determined by pairs of points, applications of Euler's formula V - E + F = 2, maximizing the number of touching pairs of d-simplices, Hall's marriage theorem, a Dinitz coloring problem for an n X n chessboard, the art gallery theorem, probabilistic combinatorics, and much more. The presentation is clear and attractive with wide margins for portraits, diagrams and sketches.
Krivosheev, D. N.: Mathematical proof and intuitionism (Russian)Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Reg. Estestv. Nauki 1999, 72-79, 97This interesting note is presented under the section on philosophical problems of science. The point of entry is with well-known comments of Eugene Wigner on the unusual effectiveness of mathematics in the physical sciences, with special emphasis on the use of symmetry ideas in physics, mathematics and art. The author presents two procedures from mathematics that are fundamental for physics. It seems one cannot eliminate the role of intuition and intuitionism in mathematics or physics. Indeed, intuition is basic to the discovery/invention of new mathematical knowledge and physical knowledge just as formal logic/proof is fundamental for their preservation. Not discussed is the important role of intuition in the formalization of deep and fruitful problems/conjectures in mathematics and physics. In any case, the author believes that intuition cannot be removed from mathematical knowledge.
Gasser, James: Informations visuelles et évidence déductive (French)Logique, discours et pensée, Sci. Comm., 1999, 52, 61-84This article is a defence of the role played by visual information in making explicit the link between premises and conclusion in deductive arguments. The starting point is a conception of deduction as intended to bring to light some hidden information that is implicitly contained in the premises. According to the author, deducing is essentially an epistemic activity: it is based on the recognition of some evident inferential steps that contribute to making manifest the validity of a non-evident argument. Thus, the activity of deducing presupposes the ability of "seeing" the "logical way" between premises and conclusion, and different means can help to illuminate this way. On the one hand, words or formulae are not the only modes of expressing a deduction: a diagram is a representation that may well show the way from premises to conclusion. On the other hand, the author even affirms that any proof, in order to actually prove, must be "visualised" in some sense: for example, a vertical disposition of deductive steps helps the reader to easily follow a written proof, and the choice of a particular system of notation can be seen as an aid to intuition. Finally, the author carefully analyses the Aristotelian method of ekthesis: this shows the usefulness of introducing "real" objects into proofs, in order to make their validity perceivable. In summation, the author wants to demonstrate that the traditional opposition between linguistic expression and graphic expression is a false dichotomy. "In a certain 'sense, every deductive evidentness depends on some visual information" (p. 83).
Ferraro, Giovanni: Rigor and proof in the mid-eighteenth century (Italian)Physis Riv. Internaz. Storia Sci. (N.S.) 36 (1999), 137--163Physis Riv. Internaz. Storia Sci. (N.S.) 36 (1999), no. 1, 137--163. An investigation of the standards of rigour in mathematical analysis between 1740 and 1770, drawing heavily on material from Euler's pen, shows that, although analysts paid attention to rigour and eliminated geometrical references from their proofs, the standards of rigour of the day where vastly different from modern ones, as witnessed by the widespread use of proving a certain statement on "typical" examples, the reasoning in question being assumed to work analogously in any other case, often aided by wishful inductive generalizations.
Richman, Fred: Existence proofsAmer. Math. Monthly 106, 1999, 303--308
Mancosu, Paolo: Bolzano and Cournot on mathematical explanationRev. Histoire Sci. 52, 1999, 429--455.Recent discussions on the topic of `mathematical explanation' have focused on the distinction between explanatory and non-explanatory proofs. The former proofs are supposed to differ from the latter in that they not only establish that a result is true but also show why it is true. This opposition is at the core of the philosophies of mathematics of Bolzano and Cournot. In the paper we analyze Bolzano's theory of Grund and Folge, and Cournot's opposition between the logical and the rational order, emphasizing their relevance to the issue of mathematical explanation. In the final part of the paper we investigate the shortcomings of Bolzano's and Cournot's theories as explications of mathematical explanation.
Chateaubriand Filho, Oswaldo: Proof and logical deductionPRATICA, 1999, 79-98
da Silva, Jairo José: On the notion of proofPRATICA, 1999, 213--219
Barwise, Jon and Etchemendy, John: Language, Proof and LogicCSLI Publications: Stanford, 1999
Netz, Reviel: The Shaping of Deduction in Greek MathematicsCambridge University: Cambridge, 1999An innovative study of Greek geometry. The author particularly addresses the issues of the cognitive function of language and lettered diagrams. The book also gives an explanation of how Greek geometry achieved necessity and generality.
Brown, James Robert: Philosophy of Mathematics: An Introduction to the World of Proofs and PicturesRoutledge: New York, 1999This is a valuable and rather untraditional introduction to some major philosophical issues concerning mathematics. The author entertains the view that mathematical objects exist independently of our awareness of them, though the concepts we use to characterize them are man-made and fallible. One remarkable feature of the book, reflected in the subtitle, is the adoption of the "representational" view of pictorial models. The author makes a well-argued case for the claim that diagrammatic evidence is perfectly valid and that pictures can indeed furnish fully legitimate proofs of theorems. In the author's view, the living philosophical issues of present-day mathematics cluster around visualisation and experimentation, and it is from this perspective that the philosophy of mathematics is approached in this book. (Eduard Glas - edited)
Rav, Yehuda: Why Do We Prove Theorems?Philosophia Mathematica. 1999; 7, 5-41Ordinary mathematical proofs--to be distinguished from formal derivations--are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarized in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalized in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, 'systemic cohesiveness', as proposed here, seeks to explicate why mathematical knowledge is coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics.
Sherry, David: Thales's Sure PathStudies in History and Philosophy of Science, Series A. 1999 30, 621-650Kant says that mathematics entered the sure path of science only "only among that wonderful people, the Greeks." This view is still standard among philosophers and historians. I criticize the standard view in light of examples of mathematical reasoning from Egyptian, Indian and Greek tests. First, I consider what makes Euclid's "Elements" deductive, given that his arguments aren't formally valid. In light of this analysis I argue that each of the pre-Euclidean texts employs deductive methodology as well.
Heinzmann, Gerhard: Poincare on Understanding MathematicsPhilosophia Scientiae. 1998-99; 3, 43-60Poincare holds up that the varieties of formal logical theories don't express the essential proof-theoretical structure in order to understand mathematics. Intuition and aesthetic reasoning, the latter depending from the criterion "harmony by a "surprising" order-character", are other decisive proof-aspects. In what sense elements from Peirce's semiotics and Goodman's aesthetics contribute something to Poincare's aim of mathematical reasoning besides logical inference?
Folina, Janet: Pictures, Proofs, and 'Mathematical Practice': Reply to James Robert BrownBritish Journal for the Philosophy of Science. 1999; 50, 425-429This paper argues for two things. First, James Robert Brown has not shown that pictures can be mathematical proofs. Second, pictures and diagrams provide a different form of evidence from a mathematical proof. Brown's argument is here linked to other recent arguments in favor of nontraditional mathematical "proofs". Underlying such arguments is a conflation between what is mathematical and what is mathematically provable, and/or a mis-assimilation between mathematical evidence and mathematical proof.
Otte, M.: Mathematical Creativity and the Character of Mathematical ObjectsLogique et Analyse. 1999; 42, 387-410Developments in informal mathematics can usefully be construed as thought-experiments, even when they are not usually regarded as such. Consideration of different thought-experiments reveals that they share the tentative use of concepts, rules, and methods beyond their paradigmatic range of application and in contexts where their validity is dubious or hypothetical. Their novelty lies in an experimental approach that blends different kinds of procedural and conceptual resources. Thought-experiments play an essential role in the argumentation through which the outcomes of such experimental mathematical developments are warranted "ex post facto." Mathematics develops by "trying and testing," but in a somewhat broader, less theory-orientated sense than that envisioned by Imre Lakatos.
Glas, Eduard: Thought-experimentation and mathematical innovationStudies in History and Philosophy of Science, Part A, 30, 1999, 1-19Developments in informal mathematics can usefully be construed as thought-experiments, even when they are not usually regarded as such. Consideration of different thought-experiments reveals that they share the tentative use of concepts, rules, and methods beyond their paradigmatic range of application and in contexts where their validity is dubious or hypothetical. Their novelty lies in an experimental approach that blends different kinds of procedural and conceptual resources. Thought-experiments play an essential role in the argumentation through which the outcomes of such experimental mathematical developments are warranted "ex post facto." Mathematics develops by "trying and testing," but in a somewhat broader, less theory-orientated sense than that envisioned by Imre Lakatos.
Livingston, Eric: Cultures of ProvingSocial Studies of Science, 29, 1999, 867-888.Although studies of scientific practice are now common, relatively little attention has been given to the technical detail of the social objects of the sciences. That detail, and the ordinariness of that detail, establish the legitimacy of practitioners' work and make that work recognizable as the work of a discovering science. In particular, the close examination of mathematical proofs exhibits the embeddedness of mathematics within a surrounding culture of proving, and leads to an appreciation of what it means to be a member of such a culture and to be engaged in the work of doing mathematics.
MacKenzie, Donald: Slaying the Kraken: The Sociohistory of a Mathematical ProofSocial Studies of Science, 1999; 29, 7-60.This paper outlines the history of the four-colour conjecture (that four colours suffice to colour in any map drawn on a plane in such a way that no countries that share a border are the same colour). It describes the conjecture's origins, the first claimed proof (in 1879), the refutation of that proof (in 1890), and the developments that led to Kenneth Appel and Wolfgang Haken's celebrated, computer-assisted solution of the problem in 1976. There is a brief discussion of the significance of the new computerized proof by Robertson, Sanders, Seymour and Thomas. The paper describes fierce controversy over whether or not the Appel-Haken solution should be regarded as a 'proof', and contrasts the case of the four-colour theorem with Imre Lakatos' history of the proof of Euler's polyhedral formula. While Lakatos showed the negotiation of concepts such as 'polyhedron', 'face' and 'edge', the history of the four-colour theorem reveals the negotiability of 'proof' itself, and therefore of the boundary of what constitutes mathematical knowledge.
Rumfitt, Ian: Logic and Existence: Frege's LogicismAristotelian Society. 1999; Supp. 73. 151-180Frege's logicism in the philosophy of arithmetic consisted, "au fond", in the claim that in justifying basic arithmetical axioms a thinker need appeal only to methods and principles which he already needs to appeal in order to justify paradigmatically logical truths and paradigmatically logical forms of inference. Using ideas of Gentzen to spell out what these methods and principles might include, I sketch a strategy for vindicating this logicist claim for the special case of the arithmetic of the finite cardinals.
Peressini, Anthony: Confirming Mathematical Theories: An Ontologically Agnostic StanceSynthese. 1999; 118, 257-277The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between "accepting the adequacy" of a mathematical theory and "believing in the truth" of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the prospects for justifying this move are qualitatively worse in mathematics than they are in science.
Sherry, David: Construction and "Reductio" ProofKant-Studien. 1999; 90, 23-39This article discusses geometrical construction in light of its role in reductio proofs, especially those in Saccheri's "Euclid Freed of Every Fleck" (1733), which contains results of non-Euclidean geometry. Reductio proof presupposes two types of geometrical construction--rule/compass construction and letting a figure be a figure of a certain type. Because of the latter type of construction, non-Euclidean concepts have as much claim to objective validity as Euclidean concepts.
Van Bendegem, Jean-Paul: The Creative Growth of MathematicsPhilosophica. 1999; 63, 119-152Jean Paul Van Bendegem tries to bring together some elements that must be part of a description of mathematical practices. He begins at the most general level - the mathematical community as a whole - and goes slowly down to the level of the working mathematician who is, among other things, trying to find a specific proof for a particular theorem. (edited)
Czermak, Joannes: Was ist ein mathematischer Beweis?Kriterion. 1999; 13, 16-23
Abstracts for 1998 (Also see 2003, 2002, 2001, 2000, 1999, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Balaguer, Mark: Platonism "and" Anti-Platonism "in" MathematicsOxford Univ. Pr. : New York, 1998.In this book, Mark Balaguer demonstrates that there are no good arguments for or against mathematical Platonism (i.e., the view that abstract, or nonspatio-temporal, mathematical objects exist, and that mathematical theories are descriptions of such objects). Balaguer does this by establishing that both Platonism and anti-Platonism are defensible positions. (edited)
Corfield, David: Beyond the Methodology of Mathematics Research ProgrammesPhilosophia Mathematica. 1998; 6(3), 272-301.In this paper I assess the obstacles to a transfer of Lakatos's methodology of scientific research programmes to mathematics. I argue that, if we are to use something akin to this methodology to discuss modern mathematics with its interweaving theoretical development, we shall require a more intricate construction and we shall have to move still further away from seeing mathematical knowledge as a collection of statements. I also examine the notion of rivalry within mathematics and claim that this appears to be significant only at a high level. In addition, ideas of progress' in mathematics are outlined.
Dusek, Val: Brecht and Lukacs as Teachers of Feyerabend and Lakatos: The Feyerabend-Lakatos Debate as Scientific Recapitulation of the Brecht-Lukacs DebateHistory of the Human Sciences. 1998; 11(2), 25-44.Feyerabend and Lakatos were invited to be assistants of the literary Marxists Brecht and Lukacs, respectively. In the 1930s Expressionism Debate, Lukacs associated artistic expressionism with irrationalism and fascism, while Brecht criticized Lukacs's antimodernism. Lakatos's criticisms of Kuhn echo Lukacs's denunciations of German idealism, and Lukacs influenced the terminology and topics in Lakatos's methodology. Lakatos, concerned with progress, and fearful of irrationalism and degeneration, recapitulates positions of his teacher, Lukacs, in the latter's attack on modern art. Feyerabend's criticisms of Lakatos parallel Brecht's critique of Lukacs.
Ernest, Paul: Social Constructivism as a Philosophy of MathematicsSUNY-Pr. : Albany, 1998.Proposing social constructivism as a novel philosophy of mathematics, this book is inspired by current work on sociology of knowledge and social studies of science. It extends the ideas of social constructivism to the philosophy of mathematics, developing a whole set of new notions. The outcome is a powerful critique of traditional absolutist conceptions of mathematics, as well as of the field of philosophy of mathematics itself. Proposed are a reconceptualization of the philosophy of mathematics and a new set of adequacy criteria. (edited)
Glas, Eduard: Thought Experimentation and Mathematical InnovationStudies in History and Philosophy of Science. 1999; 30A(1), 1-19.The article expands on Lakatos's view of informal mathematical proofs as thought-experiments. It is argued that the role of thought-experiments is by no means confined to delivering the informal proofs and refutations of mathematical propositions that Lakatos's methodology relies upon. On the basis of historical case-studies, thought-experimentation is characterized as a major analytical tool for conceptual development and change. What the various historical cases considered had in common, was the tentative (quasi-experimental) use of concepts and methods beyond their paradigmatic range of application, thereby, enabling progress to more comprehensive, integrated and unified theories, providing more profound understanding.
Gurr, Corin; Lee, John; Stenning, Keith: Theories of Diagrammatic Reasoning: Distinguishing Component ProblemsMinds and Machines. 1998; 8(4), 533-557.Theories of diagrams and diagrammatic reasoning typically seek to account for either the formal semantics of diagrams, or for the advantages which diagrammatic representations hold for the reasoner over other forms of representation. Regrettably, almost no theory exists which accounts for both of these issues together, nor how they affect one another. We do not attempt to provide such an account here. We do, however, seek to lay out larger context than is generally used for examining the processes of using diagrams in reasoning or communication. A context in which detailed studies of subproblems, such as the formal semantics or cognitive impact of specific diagrammatic systems, may be embedded. (edited)
Hintikka, Jaakko: On Godel's Philosophical AssumptionsSynthese. 1998; 114(1), 13-23.Godel was a one-world theorist who did not use the idea of other possible worlds or scenarios. Logical truths were for him not truths in all possible worlds, but truths about certain abstract entities in "this" world. As a consequence, Godel failed to distinguish between different kinds of (in)completeness. He proved the "deductive" incompleteness of elementary arithmetic, but this implies "descriptive" incompleteness only if the underlying theory is "semantically" complete. Because of the same one-world stance, Godel had to postulate a special supersensory access to his abstract entities, viz. mathematical intuition.
Lomas, Dennis: Diagrams in Mathematical Education: A Philosophical AppraisalPhilosophy of Education 1998, ed. Steve Tozer Phil.-Education-Soc.: Urbana, 1998.This paper addresses an epistemological issue that is posed in using diagrams in mathematical reasoning: What is the epistemological status of diagrammatic reasoning? Specifically, is diagrammatic reasoning a full-fledged form of mathematical reasoning or is it merely an intuitive appendage (a pedagogical or psychological helping hand for the student rather than true mathematical reasoning that, it is typically thought, is characterized by purely abstract or logical thought)? This issue especially pertains to the work of education theorists who hold that diagrams are purely intuitive aids, yet seem to implicitly accept them as legitimate components of proofs, for example, of the Pythagorean theorem.
O'Neill, John: Practical Reason and Mathematical ArgumentStudies in History and Philosophy of Science. 1998; 29A(2), 195-205.The separation of theoretical from practical reason form one of a series of contrasts employed to divide human inquiry into different spheres. Theoretical reason addresses us as knowers concerned to answer questions of what is true; practical reason addresses us as agents concerned to answer problems of what to do. The two forms of reason are taken to involve different forms of argument: the forms employed in practical reasoning are taken to differ in kind from those in theoretical reasoning. In this paper I examine the role of practical reason in a theoretical disciple – mathematics. I suggest that this gives us reason to question the way these contrasts have been drawn. (edited)
Paulos, John Allen: Once Upon a Number: The Hidden Mathematical Logic of StoriesBasic Books: New York, 1998."Once Upon a Number" shows that stories and numbers aren't as different as you might imagine, and in fact, they have surprising and fascinating connections. The concepts of logic and probability both grew out of intuitive ideas about how certain stories would play out. Now, logicians are inventing ways to deal with real world situations by mathematical means--by acknowledging, for instance, that items that are mathematically interchangeable may not be interchangeable in a story. And complexity theory looks at both number strings and narrative strings in remarkably similar terms. (publisher, edited)
Potter, Michael D: Classical Arithmetic Is Part of Intuitionistic ArithmeticGrazer Philosophische Studien. 1998; 55, 127-141.One of Michael Dummett's most striking contributions to the philosophy of mathematics is an argument to show that the correct logic to apply in mathematical reasoning is not classical but intuitionistic. In this article I wish to cast doubt on Dummett's conclusion by outlining an alternative, motivated by consideration of a well-known result of Kurt Godel, to the standard view of the relationship between classical and intuitionistic arithmetic. I shall suggest that it is hard to find a perspective from which to arbitrate between the competing views.
Sandborg,-Davi: Mathematical Explanation and the Theory of Why QuestionsBritish Journal for the Philosophy of Science. 1998; 49(4), 603-624.Van Fraassen and others have urged that judgments of explanations are relative to why-questions; explanations should be considered good in so far as they effectively answer why-questions. In this paper, I evaluate van Fraassen's theory with respect to "mathematical" explanation. I show that his theory cannot recognize any proofs as explanatory. I also present an example that contradicts the main thesis of the why-question approach--an explanation that appears explanatory despite its inability to answer the why-question that motivated it. This example shows how explanatory judgments can be context-dependent without being why-question-relative.
Shanks, Niall (ed): Idealization IX: Idealization in Contemporary PhysicsRodopi : Amsterdam, 1998This volume consists of a comprehensive review and analysis of the several roles played by idealization procedures in the logic, mathematics and models that lie at the heart of twentieth century physics. It is only through idealization of one form or another that the objects and processes of modern physics become tractable. The essays in this volume will be of interest to all those who are concerned with the uses of models in physics and the relationships between models and the real world. The essays cover the role of idealization in all the main areas of modern physics, ranging from quantum theory, relativity theory and cosmology to chaos theory.
Sklar, Lawrence: The Language of Nature Is Mathematics--But Which Mathematics? and What Nature?Proceedings of the Aristotelian Society. 1998; 98, 241-261.In theoretical physics the physical states of systems are represented by components of mathematical structures. This paper explores three ways in which the representation of states by mathematics can give rise to foundational problems, sometimes on the side of the mathematics and sometimes on the side of understanding what the physical states are that the mathematics represents, that is on the side of interpreting the theory. Examples are given from classical mechanics, quantum mechanics and statistical mechanics.
Thompson, Paul: The Nature and Role of Intuition in Mathematical EpistemologyPhilosophia. 1998; 26(3-4), 279-319.Great intuitions are fundamental to conjecture and discovery in mathematics. In this paper, we investigate the role that intuition plays in mathematical thinking. We review key events in the history of mathematics where paradoxes have emerged from mathematicians' most intuitive concepts and convictions, and where the resulting difficulties led to heated controversies and debates. Examples are drawn from Riemannian geometry, set theory and the analytic theory of the continuum, and include the Continuum Hypothesis, the Tarski-Banach Paradox, and several works by Godel, Cantor, Wittgenstein and Weierstrass. We examine several fallacies of intuition and determine how far our intuitive conjectures are limited by the nature of our sense-experience, and by our capacities for conceptualization. Finally, I suggest how we can use visual and formal heuristics to cultivate our mathematical intuitions and how the breadth of this new epistemic perspective can be useful in cases where intuition has traditionally been regarded as out of its depth.
Titiev, Robert J : Finiteness, Perception, and Two Contrasting Cases of Mathematical IdealizationJournal of Philosophical Research. 1998; 23, 81-84.Idealization in mathematics, by its very nature, generates a gap between the theoretical and the practical. This article constitutes an examination of two individual, yet similarly created, cases of mathematical idealization. Each involves using a theoretical extension beyond the finite limits which exist in practice regarding human activities, experiences and perceptions. Scrutiny of details, however, brings out substantial differences between the two cases, not only in regard to the roles played by the idealized entities, but also in regard to appropriate criteria for justifying the use of such entities. The background information supplied and the examples chosen for analysis in this paper were selected from the areas of measurement theory, probability theory, mathematical logic and philosophy of mathematics.
Troelstra, A S: Concepts and AxiomsPhilosophia Mathematica. 1998; 6(2), 195-208.The paper discusses the transition from informal concepts to mathematically precise notions; examples are given and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning".
Tymoczko, Thomas (ed): New Directions in the Philosophy of Mathematics: An AnthologyPrinceton Univ. Pr. : Princeton, 1998.The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics and what processes a proof undergoes before it reaches publishable form.
Abstracts for 1997 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Brown, James Robert: Proofs And PicturesBrit. J. Phil. Sci., 48(2), 161-180, 1997.Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy ; Shin ; and Giaquinto ), I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.
Corfield, David: Assaying Lakatos' Philosophy of MathematicsStud. Hist. Phil. Sci., 28(1), 99-121, Mar 1997.While various of his critics are shown to have been off the mark, two principal theses of Lakatos' philosophy of mathematics are here put to the test in the field of algebraic topology. First, that the "method of proofs and refutations" predominates in the passage from native conjecture to research programme. Second, that axiomatization of a theory marks the end its most dynamic and creative phase. I conclude that axiomatization may act as a spur in the production of a more refined informal thought and that uncertainty in mathematics resides largely in the question of a theory's importance.
Devlin, Keith: The Logical Structure of Computer-Aided Mathematical ReasoningThe American Mathematical Monthly, 104, 632-646, 1997.Computer-aided mathematical reasoning is discussed. Much of the activities of mathematicians now involve the use of computers, and the rapid change to a computer-based mode of working has fundamentally altered the nature of how mathematics is carried out. Computer-aided mathematics is qualitatively different from that carried out in the traditional manner. In particular, the logical structure of computer-aided mathematical reasoning is different from the more traditional form of mathematical reasoning. The use of computer-aided reasoning in proofs and refutations, real deduction, information and representation, situation theory, a stacked blocks problem, and the structure of computer-aided proofs are examined.
Fallis, Don: The Epistemic Status of Probabilistic ProofThe Journal of Philosophy, 94(4), 165-86, 1997.A deductive argument is considered by mathematicians (and many philosophers) to be the only legitimate way to establish that a mathematical claim is true. In this paper, I argue that mathematicians do not have good grounds for their rejection of probabilistic methods as a means of establishing mathematical truths. Specifically, I argue that one particular probabilistic method (which utilizes recent DNA technology) has no epistemic drawbacks that are not shared by standard methods of establishing mathematical truths.
Gower, Barry S.: Scientific Method: An Historical and Philosophical IntroductionRoutledge, New York 1997.The results, conclusions and claims of natural science are often taken to be reliable because they arise from the use of a distinctive method. Yet today, there is widespread scepticism as to whether we can validly talk of method in modern science. This outstanding new survey explains how this controversy has developed since the seventeenth century and explores its philosophical basis. Questions of scientific method are discussed through key figures such as Galileo, Bacon, Newton, Bayes, Mill, Poincare, Duhem, Popper, and Carnap. The concluding chapters contain stimulating discussions of attacks on the idea of scientific method by key figures such as Kuhn, Lakatos, and Feyerabend.(publisher,edited)
Hersh, Reuben: Prove-Once More And AgainPhil. Math. 5, 153-165, 1997.There are two distinct meanings to ‘mathematical proof’. The connection between them is an unsolved problem. The first step in attacking it is noticing that it is an unsolved problem.
Jaffe, Arthur: Proof And The Evolution of MathematicsSynthese, 111(2), 133-146, 1997.
Kleiner, Israel: Proof: A Many-Splendored ThingThe Mathematical Intelligencer, 19, 16-26, 1997.The debate over the nature and function of proof in mathematics and over the nature of the mathematical enterprise itself is a fascinating one. This debate, which has been going on for 2,000 years, is complicated by the fact that it can be difficult to separate the process of proving from the general fabric of doing mathematics. Throughout the history of mathematics, a general lack of established criteria as to what constituted an acceptable proof and considerable variations in perceptions of the role of proof have meant that mathematicians have disagreed on the best means of doing mathematics and on the techniques to employ in attacking problems and establishing results. Historical examples that illustrate the pluralistic nature of mathematical practice are presented.
Larvor, Brendan P. : Lakatos as Historian of MathematicsPhil. Math. 5, 42-64, 1997.This paper discusses the connection between the actual history of mathematics and Lakatos' philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos' own view of mathematics fails that test.
Maddy, Penelope: Naturalism in MathematicsClarendon-Oxford-Pr : New York, 1997.Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. "Naturalism in Mathematics" investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. (publisher, edited)
Mazur, B.: ConjectureSynthese, 111(2), 197-210, 1997.Much has been written about the formal logical structure of mathematics and the formulation of axioms, definitions, theorems, and proofs, in mathematical literature. But aside from foundational and logical issues there are elements of style and mode of presentation that have seldom been specifically commented upon. This article discusses one such element: the appearance of what could be called “architectural conjectures” in mathematical writing. Architectural conjectures are formally articulated constellations of conjectures which constitute theories in their own right and which can be thought of as detailed research projects.
Peterson, Ivars: Computers And ProofScience News, 151, 176-177, 1997.Last fall, the so-called Robbins conjecture was proven by a computer program. Originally proposed in the 1930s by Herbert Robbins, now at Rutgers University in New Brunswick, New Jersey, the problem, which concerns an aspect of the basic rules of logic, has stymied everyone who tackled it over the years. The automated reasoning software, called EQP for "equational prover," that solved the problem was developed by computer scientist William McCune of the Argonne National Laboratory in Illinois. No one is certain whether the computer proof generated by EQP represents the first of many triumphs or a fluke success, however. McCune's computer-generated proof will appear in a future issue of the Journal of Automated Reasoning.
Rips, Lance J.: Goals For A Theory of Deduction: Reply to Johnson-LairdMind Match, 7(3), 409-424, 1997.This paper outlines the progress that has been achieved in using natural-deduction systems as psychological theories of deductive reasoning. It reviews the motivations behind one such theory (the PSYCOP system described by Rips, The Psychology of Proof, MIT Press) and evaluates the outcome with respect to recent criticism. PSYCOP attempts to provide a theory of first-order reasoning--one that is well-specified formally and computationally, that accounts for human inference patterns and that can serve as the basis for other cognitive tasks.
Sherry, David: On Mathematical ErrorStudies in History and Philosophy of Science, 28(3), 393-416, 1997.Fallibilists point to the historical record as evidence of the fallible, corrigible and tentative nature of mathematics. I argue that errors implicating an entire community of mathematicians do not exist in any but a philosophically problematic sense. This is not a historical accident but a reflection of mathematics’ nature.
Weiss, Bernhard: Proof And Canonical ProofSynthese, 113(2), 265-284, 1997.Certain antirealisms about mathematics are distinguished by their taking proof rather than truth as the central concept in the account of the meaning of mathematical statements. This notion of proof which is meaning determining or canonical must be distinguished from a notion of demonstration as more generally conceived. This paper raises a set of objections to Dummett’s characterization of the notion via the notion of a normalized natural deduction proof. The main complaint is that Dummett’s use of normalized natural deduction proofs relies on formalization playing a role for which it is unfit. Instead, I offer an alternative account which does not rely on formalization and go on to examine the relation of proof to canonical proof, arguing that rather than requiring an explicit characterization of canonical proofs we need to be more aware of the complexities of that relation.
Abstracts for 1996 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Beltran Orenes, Pilar: Siguen las Pruebas Realizadas por Ordenador Algun Patron de Verdad Matematica?Verdad: logica, representacion y mundo, Villegas Forero, L Univ Santiago Comp, Santiago de Compostela 1996.At the end of the 50s, a new problem arises in the field of mathematics. The use of computers as productive tools makes it possible to prove some new theorems. This fact has set forth the question of the accuracy of the results achieved in such a way. However, the problem about tests compiled by computers can be clarified, from our point of view, by including these kinds of tests into the so-called, by Lakatos, "informal proofs".
Hammer, Eric: Symmetry As A Method of ProofJ. Phil. Log., 25(2), 523-543, 1996.This paper is a logical study of valid uses of symmetry in deductive reasoning, of what underlying principles make some appeals to symmetry legitimate but others illegitimate. The issue is first motivated informally. A framework is then given covering a fairly broad range of symmetry arguments and the formulation of symmetry provided is shown to be a valid principle of reasoning, as is a slightly stronger principle of reasoning, one that is shown to be in some sense as strong as possible. The relationship between symmetry and isomorphism is discussed and finally the framework is extended to a more general model-theoretic setting.
Kolata, Gina: With Major Math Proof, Brute Computers Show Flash Of Reasoning PowerNew York Times, 10 Dec 1996, p. C1.A computer program written by researchers at Argonne National Laboratory, Illinois, has developed a pure mathematical proof that may represent a breakthrough in creative thinking by computers. Dr. William McCune asked the EQP (equational prover) program to prove that a set of 3 equations is equivalent to a Boolean algebra, an unproven conjecture first posed by Dr. Herbert Robbins in the 1930s. EQP produced a proof 8 days later and then refined the proof over 10 days. According to Dr. Larry Wos, supervisor of the computer reasoning project, this result is a great leap forward in reasoning power.
Kuipers, T. A. F.: Truth Approximation by the Hypothetico-Deductive MethodStructuralist Theory of Science: Focal Issues, New Results; Moulines, C Ulises (Ed.), de Gruyter, New York 1996.The popular devaluation of the HD-method of testing has been premature, due to the dominating one-sided truth-value perspective. On closer analysis it is essentially a method for the evaluation of a theory in terms of its instantial failures (counterexamples) and its explanatory successes. The resulting, essentially instrumentalist, methodology is argued to be functional for truth approximation conceived in structuralist terms. It is compared with the corresponding realist and empiricist methodologies. Finally, some points of Popper, Lakatos and Nowak are reinterpreted in the new perspective.
Abstracts for 1995 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Kadvany, John: The Mathematical Present as HistoryPhil Forum, Sum 1995.Imre Lakatos' historical philosophy of mathematics, as developed in his "Proofs and Refutations: The Logic of Mathematical Discovery", has, in its historiographic style, strong structural affinities with Hegel's "Phenomenology of Spirit": both written as philosophical-historical "Bildungsromans". This formal similarity is the starting point for a detailed exposition of what Lakatos calls the method of proofs and refutation.
The aim is to identify Lakatos' historical claims relevant to his philosophical account; to show that the method of proofs and refutations is similar in fundamental respects to the organizing method of the "Phenomenology", Hegel's so-called phenomenology criticism; and to identify central implications of Lakatos' account for contemporary mathematics.
MacKenzie, Donald: The Automation of Proof: A Historical and Sociological ExplorationAnnals of the History of Computing, 1995, 17(3), pp. 7-29.Concerns the history of the use of computers to automate mathematical proofs.
Zambrana Castañeda, Guillermo: Wittgenstein On Mathematical ProofIn Santiago Ramirez, Robert S. Cohen (Eds.) Mexican Studies in the History and Philosophy of Science. Dordrecht: Kluwer Academic, 1995, pp. 235-248.
Abstracts for 1994 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1995, 1996, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Crossley, J N; Lun, A W C: The Logic of Liu Hui and Euclid as Exemplified in Their Proofs of the Volume of a PyramidPhil. Hist. Sci., Ap. 1994.We present a comparison of the logic employed in Euclid's "Elements" in the West and the "Jiu Zhang Suan Shu" in China. Previously it has been said that Chinese mathematics was algorithmic and practical, as opposed to the logical and theoretical Euclidean mathematics. We point out that Euclid uses logic which either is, or could be, as constructive as that of Liu Hui and has a number of points of contact even though the traditions are very different.
Drozdek, Adam; Keagy, Tom: A Case for Realism in MathematicsMonist, Jl 1994.The paper takes issue with anti-realist position in philosophy of mathematics, which denies an existence of mathematical entities independent of the cognitive subject. Anti- realism has appeared in various forms, and some of the arguments are discussed (Brouwer, Kitcher, Machover). The paper opts for realism and gives both philosophical arguments substantiating this position and an example of mathematical proof which was possible only due to the realist view (the proof concerns a transform of Levin to be used in approximating the limit of a sequence).
Dubucs, Jacques; Dubucs, Monique: Mathematiques: la Couleur des PreuvesRhetoriques de la Science, De Coorebyter, Vincent (Ed.), Pr. Univ. France, Paris 1994.
Hull, Kathleen: Why Hanker After Logic? Mathematical Imagination, Creativity and Perception in Peirce's Systematic PhilosophyTrans Peirce Soc, Spr 94.According to Peirce, mathematical reasoning is the model for all reasoning. As for Wittgenstein, it is a non-foundational, pre-logical, thinking practice. As for Kant, it involves the construction of diagrams. We do not know mathematical truths by their "proofs", but have "perceptions" of our constructed objects in the mathematical imagination. Mathematical hypotheses (axioms) "exert a force upon us." Peirce provides a unique account of how hypotheses force themselves upon us. The causal links between our cognitive faculties and mathematical objects are his metaphysical categories of reality, especially the secondness of diagrams. The author applies these insights to creativity and religious perception.
Mayberry, John: What is Required of a Foundation for Mathematics?Phil Math, 1994.The business of mathematics is definition and proof, and its foundations comprise the principles that govern them. Modern mathematics is founded upon set theory. In particular, both mathematical logic and the axiomatic method belong to the theory of sets. Accordingly, foundational set theory is not, and cannot logically be an axiomatic theory, in the modern sense. Failure to grasp this leads to confusion. The idea of a set is that of an extensional plurality, limited and definite in size, composed of well-defined objects. It is obtained by extending the Greek notion of number ("arithmos") into Cantor's transfinite.
Pagin, Peter: Knowledge of ProofsTopoi, S 1994.If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic. The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical constants.
Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch's paradox. The third section considers one attempt to save intuitionism while partly giving up verification. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. (edited)
Penco, Carlo: Dummett and Wittgenstein's Philosophy of MathematicsThe Philosophy of Michael Dummett, McGuinness, Brian (Ed.); Kluwer, Dordrecht 1994.In the paper it is argued that: 1) Dummett's analysis of Wittgenstein's philosophy of mathematics is an excellent reconstruction of an early stage of Wittgenstein's research (necessity as free decision). 2) Dummett's position (patterns of proofs come out of our experience of deductive argument and give a rationale for accepting the proof) has been somehow envisaged by Wittgenstein in his later remarks on the phenomenology of proof formation. 3) Wittgenstein's last remarks on rule- following give a solution to the problems arising out of this kind of considerations, without falling into scepticism.
Sundholm, Goran: Existence, Proof and Truth-Making: A Perspective on the Intuitionistic Conception of TruthTopoi, S 1994.Truth- maker analyses construe truth as existence of proof, a well- known example being that offered by Wittgenstein in the "Tractatus". The paper subsumes the intuitionistic view of truth as existence of proof under the general truth-maker scheme. Two generic constraints on truth- maker analysis are noted and positioned with respect to the writings of Michael Dummett and the "Tractatus". Examination of the writings of Brouwer, Heyting and Weyl indicates the specific notions of truth- maker and existence that are at issue in the intuitionistic truth- maker analysis, namely that of proof in the sense of proof- object (Brouwer, Heyting) and existence in the non-propositional sense of a judgment abstract (Weyl). Furthermore, possible anticipations in the writings of Schlick and Pfander are noted.
Abstracts for 1993 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1995, 1996, 1994, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Feferman, Solomon: Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable MathematicsProc. Phil. Sci. Ass., 1993.Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments.
Fetzer, James: Foundations of Philosophy of Science: Recent DevelopmentsParagon House, New York 1993.This collection of twenty-five articles focuses on the crucial theoretical problems fundamental to the philosophy of science today. Included are classic studies by Hempel, Salmon, Popper, Carnap, Quine, Goodman, Kuhn, Lakatos, and many others, as well as contemporary selections reflecting recent developments within the field and discussing their importance. Review questions and suggestions for further reading are provided. The anthology is systematically correlated with a companion text, "Philosophy of Science", authored by the editor, where the two books may be used independently or in combination as appropriate.
Ramachandran, S.: Computers and the Philosophy of MathematicsJ. Indian. Counc. Phil. Res., Ja-Ap 1993.This paper illustrates two easily stated long standing popular problems in mathematics -- the four color problem and the problem of orthogonal latin squares that are recently settled using extensive use of computers. The implication of such proofs on the nature of mathematics is discussed.
The paper concludes by suggesting division of mathematics into two parts -- practical or quasi-empirical mathematics and rigorous mathematics and classification of mathematical research into three types: i) creating rigorous mathematics, ii) creating practical mathematics and iii) providing analytical proofs for what is there in practical mathematics so that they become rigorous mathematics.
Robert, Serge: Les Mecanismes de la Decouverte ScientifiqueUniv. of Ottawa Pr., Ottawa 1993.
Sundholm, Goran: Questions of ProofManuscrito, O 1993.It is shown that the concept of proof is over- determined, in the sense that not all the claims commonly made about proofs are compatible. It is shown how these diverse claims can be reconciled by making a series of distinctions, in particular that between proof- act, proof- object and proof- trace.
Thagard, Paul: Computational Tractability and Conceptual Coherence: Why Do Computer Scientists Believe That P is Not Equal to NP?Can. J. Phil., S 1993.This paper discusses the tractability thesis, which identifies the intuitive class of computationally tractable problems with a precise class of problems whose solutions can be computed in polynomial time. Intimately connected with the tractability thesis is the mathematical conjecture that P is not equal to NP. This conjecture is precise enough to be capable of mathematical proof, but most computer scientists believe it even though no proof has been found. Understanding the grounds for acceptance of the conjecture that P is not equal to NP has implications for general questions in the philosophy of mathematics and science, especially concerning the epistemological importance of explanatory and conceptual coherence.
Van Bendegem, Jean Paul: Real-Life Mathematics Versus Ideal MathematicsThe Ugly Truth in Empirical Logic and Public Debate, Krabbe, Erik C. W. (Ed.), Rodopi, Amsterdam 1993.The paper starts with a description of the ideal community of mathematicians, roughly a set of interchangeable individuals that generate proofs of such nature that control is possible by every member of the set. This ideal picture is compared with a description of how mathematics is really done. The distance between ideal and reality is "the ugly truth" referred to in the title.
Abstracts for 1992 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1991, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Detlefsen, Michael (Ed.) Proof and Knowledge in MathematicsRoutledge, New York 1992.
Detlefsen, Michael: Poincare Against the LogiciansSynthese, Mr 1992.Poincare 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 Leibniz, 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. Poincare, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical epistimology which underlies it.
Feist, Richard: Wittgenstein: On Not Getting Excited About Gödel's ProofDe Phil, 1992.Wittgenstein's remarks on Gödel's incompleteness theorem are notoriously obscure. They are often regarded as irrelevant or indicative of a lack of understanding. I offer a therapeutic interpretation of his remarks. Such an interpretation helps to make sense of Wittgenstein's cavalier talk of contradiction as well as his attitude towards the foundations and philosophy of mathematics.
Grolmusz, Vince: On Mathematical Rigorousness (in Hungarian)Magyar Filozof Szemle, 1992.It was a common view among nineteenth century mathematicians that a theorem is absolutely true and should have been regarded as proved if the proof had been "logically rigorous". But the criteria of this rigorousness depend on the intuitions of the mathematicians. Paradoxa such as those of Russell and Cantor undermined this view as well as Frege's formalized system. As to the fact that there are "proofs" which prove but don't convince us and there are ones which are convincing but don't prove anything, the author argues it is an illusion to regard mathematics as a science of rigorous proofs.
Resnik, Michael D: Proof as a Source of TruthProof and Knowledge in Mathematics, Detlefsen, Michael (Ed.) Routledge, New York 1992.This paper presents an account of how proving mathematical theorems can induce us to acquire justified true beliefs about abstract mathematical objects.
Shapiro, Stewart: Foundationalism and Foundations of MathematicsProof and Knowledge in Mathematics, Detlefsen, Michael (Ed.), Routledge, New York 1992.This essay is a study of foundationalism in mathematics, and the relationship between one's views on foundations and the appropriate, or best, logic. I suggest that some sort of foundationalism dominated work in logic and foundations of mathematics until recently, most notably logicism and the Hilbert program. But foundationalism has now fallen into disrepute. The issues concern how much of the perspective is still plausible, and how logic and foundational studies are to be understood in the prevailing anti-foundationalist spirit. One common orientation seems to be to regard logic and, perhaps, mathematics in general, as an exception to the prevailing anti- foundationalism, or anti-rationalism--a sort of last outpost.
Against this, I argue that we have learned to live with uncertainty in virtually every special subject, and we can live with uncertainty in logic and mathematics. In like manner, we can live without completeness in logic, and live well.
Steiner, Mark: Mathematical Rigor in PhysicsProof and Knowledge in Mathematics, Detlefsen, Michael (Ed.), Routledge, New York 1992.Physicists for centuries have been making correct numerical predictions on the basis of nonrigorous mathematics. Their arguments, though deductively invalid, were bolstered by extra-mathematical considerations and approximation techniques. Recently, however, physicists have been flouting mathematical rigor with no pretense of before-the-fact justification. Mathematical rules for symbol manipulation are employed beyond the range of their validity. Mathematically inconsistent theories are employed.
The result? The greatest accuracy in the history of physics (in the case of quantum electrodynamics, 1 part in 10 billion). How do physicists do it?
Stekeler-Weithofer, Pirmin: On the Concept of Proof in Elementary GeometryProof and Knowledge in Mathematics, Detlefsen, Michael (Ed.), Routledge, New York 1992.The concept of proof used in elementary synthetic geometry relies heavily on the fact that we really can fulfill certain intentions of forming solid bodies. Although we say that a body is solid if it does not change its form by changing position, there is no concept of form independent from that of solidity. The conditions of satisfaction of the intentions of forming solid bodies must be formulated in a holistic way using a "descriptive" language. We must control a plan to construct bodies or pictures of certain forms by "real" observations. An analysis of the basic concepts of idealization and abstraction shows, then, that any justification of Euclidean geometry or general kinetics rests on experience. It cannot be "a priori" in an absolute sense, as Kant, Dingler, and the German constructivists seem to claim.
Stillwell, Shelley: Empirical Inquiry and ProofProof and Knowledge in Mathematics, Detlefsen, Michael (Ed.), Routledge, New York 1992.
Tait, William: Reflections on the Concept of "a Priori" Truth and Its Corruption by KantProof and Knowledge in Mathematics, Detlefsen, Michael (Ed.), Routledge, New York 1992.The distinction is drawn and discussed between a conception of "a priori" truth, which is first found in Plato and is found in Leibniz under the heading of "a priori" truth, "according to which it is truth about a species of structure and can be understood and studied independently of whether or not this kind of structure is exemplified in the natural world, and the conception of "a priori" in Kant and later writers, according to which propositions may be "a priori" true of the empirical world.
Wagner, Steven J: LogicismProof and Knowledge in Mathematics, Detlefsen, Michael (Ed.), Routledge, New York 1992.The "first-generation" logicism of Frege and Dedekind differs from subsequent versions by claiming a purely rational status for mathematics. An updated version of this claim is defensible. Mathematics is "a priori" in the sense of needing empirical data only to compensate for human limits of memory and attention. It is analytic in the sense that any rational beings satisfying minimal conditions would have reason to develop forms of arithmetic and set theory. This version of logicism is immune to standard objections. In particular, it avoids Philip Kitcher's critique. My notion of analyticity, however, is more general than its traditional counterparts. One point is that analytic knowledge may be conjectural. Logicism is, ultimately, a thesis about human cognitive capacities and mechanisms.
Abstracts for 1991 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1990, 1989, 1988, 1987, 1986, 1985, Pre-1985)
Ambrose, Alice: On CertaintyLogical Foundations: Essays in honour of D. J. O'Connor, Mahalingam, Indira (Ed.), St Martin's Pr, New York: 1991.This paper is directed against the thesis that mathematical proofs are "only provisional" and mathematics a "quasi-empirical science", and its concomitant, that no distinction can be made between analytic and synthetic propositions, despite general agreement in some cases on their application. The main support of the thesis is that the similarity between the practice of mathematicians preceding rigorous proof and of natural scientists makes the result plausible but not certain. Computer proofs of theorems, in lacking criteria for the reliability of their programming, are held to yield uncertain results because error is possible. However calculation, by human or machine, does not show a result sometimes holds, but as Wittgenstein pointed out, shows what the result "must" be. This paper argues that dissimilarities between experimental and proof steps undercut the view that these are "quasi-empirical".
Dawson Jr, John W: The Reception of Godel's Incompleteness TheoremsPerspectives on the History of Mathematical Logic, Drucker, Thomas (Ed.), Basle: 1991.This paper examines the extent to which Godel's incompleteness theorems were understood and accepted at the time of their enunciation. It is concluded that Godel's Proofs were most persuasive to formalists; others raised objections on technical or philosophical grounds. Reactions of Skolem, von Neumann, Finsler, Zermelo, Russell and Wittgenstein are discussed in some detail.
Gonzalez, Wenceslao J: Intuitionistic Mathematics and WittgensteinHist Phil Log, 1991.The relation between Wittgenstein's philosophy of mathematics and mathematical intuitionism has raised a considerable debate. My attempt is to analyze if there is a commitment in Wittgenstein to themes characteristic of the intuitionist movement in mathematics and if that commitment is one important strain that runs through his "Remarks on the Foundations of Mathematics". The intuitionistic themes to analyze in his philosophy of mathematics are: firstly, his attacks on the unrestricted use of the Law of Excluded Middle; secondly, his distrust of non-constructive "proofs"; and thirdly, his impatience with the idea that mathematics stands in need of a "foundation". These elements are Fogelin's starting point for the systematic reconstruction of Wittgenstein's conception of mathematics.
Jaeger, Gerhard: Some Proof-Theoretic Contributions to Theories of SetsLogic Colloquium, Paris, J B (Ed.), Amsterdam: 1991, pp. 171-191.Several theories of sets are presented and discussed, mainly from a proof-theoretic point of view. We isolate some proof-theoretically relevant set existence axioms, analyze their strength and consider the question of set existence versus induction principles. The emphasis is put on various theories for iterated admissible sets and their relationship to subsystems of second order arithmetic.
Koetsier, T: Lakatos' Philosophy of Mathematics: A Historical ApproachElsevier Science, New York 1991.In this book, which is both a philosophical and a historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed such as the appearance of deduction in Greek mathematics and the transition from eighteenth century to nineteenth century analysis. The author aims at developing a notion of mathematical rationality that agrees with the historical facts. He proposes a modified version of Lakatos' methodology. The resulting reconstructions show that mathematical knowledge is fallible, but its fallibility is remarkably weak.
Mancosu, Paolo: On the Status of Proofs by Contradiction in the XVIIth CenturySynthese, Jl 1991.In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyze how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programs of providing a development of parts of geometry free of proofs by contradiction. The main protagonist of this part is Wallis.
Finally, I analyze some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's program of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism.
O'Leary, Daniel J: Principia Mathematica' and the Development of Automated Theorem ProvingPerspectives on the History of Mathematical Logic, Drucker, Thomas (Ed.), Birkhauser, Basle 1991.The paper describes and contrasts two approaches to automated theorem proving applied to portions of Russell and Whitehead's "Principia Mathematica" (PM). The Logic Theory Machine by Newell, Shaw, and Simon tried to duplicate the reasoning behind the proofs as a human mathematician might do. Wang's approach uses sequent logic and the computer to prove the theorems. The paper describes both methods in detail. It also resolves an error in PM and in the correspondence between Simon and Russell. The paper concludes that the Logic Theory Machine approach is more satisfying in its attempt to understand the human endeavor that is the basis for PM.
Wright, Crispin: Wittgenstein on Mathematical ProofWittgenstein Centenary Essays, Griffiths, A. Phillips (Ed.), Cambridge Univ Pr, New York 1991.
Abstracts for 1990 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1989, 1988, 1987, 1986, 1985, Pre-1985)
De Souza, Roberta Lima: O Metodo de Analise da Geometria GregaCad. Hist. Filosof. Cie., Ja-Je 1990.This paper presents the method of analysis as a method of discovery used by ancient Greek geometers in looking for proofs of theorems and of constructions to solve problems. The controversy concerning the interpretation of this method came from different approaches given to the Pappusian description of analysis. In dealing with this problem, we try to show that there is a justificationist methodology inherent in the view of historians of mathematics. So according to this view we can remark that analysis might have a certain degree of certainty, though the meaning and field of application of the method be narrowed. Also on the basis of Pappus's account, new possibilities of interpretation of geometrical analysis as a heuristic method come to light by using other historical evidence. Among the main reasons for the loss of the primitive sense of this method, when we follow the lead of historians, there is a subjacent previous conception of rationality.
Detlefsen, Michael: Brouwerian IntuitionismMind, O 1990.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 Poincare.
Ernest, Paul: The Meaning of Mathematical Expressions: Does Philosophy Shed Any Light on Psychology?Brit. J. Phil. Sci., D 1990.The paper reviews a number of approaches to meaning in the philosophical literature, including post-Fregean, syntactical, proof-theoretic, model-theoretic, and holistic approaches. They are evaluated with respect to their applicability to the psychology of learning mathematics. A theoretical model of the meaning of mathematical expressions is proposed, based on this synthetic review. It is elaborated elsewhere.
Gonzalez, Wenceslao J.: Semantica anti-realista: Intuicionismo matematico y concepto de verdad Theoria (Spain)N 1990.Among the philosophical problems recently discussed, the question on the anti- realist semantic is outstanding. Its origin arose when M Dummett tries a Wittgenstenian interpretation of the Intuitionistic Mathematics. He uses the concept of justification as the key concept--understood as proof or verification--, and it faces up to a realistic view centered in the notion of truth. But, carefully analyzed, it shows a clear vulnerability, while the realistic position has got serious elements on its favor, and so it is recognized by the supporter of the opposite point of view. Thus, the notion of truth cannot be disregarded.
Jesseph, Douglas: Rigorous Proof and the History of Mathematics: Comments on CroweSynthese, Je 1990.Duhem's portrayal of the history of mathematics as manifesting calm and regular development is traced to his conception of mathematical rigor as an essentially static concept. This account is undermined by citing controversies over rigorous demonstration from the eighteenth and twentieth centuries.
Wilson, Jack L.: Sobre la no paradoja de un cretenseRev. Filosof. (Costa Rica), D 1990.A lack of mathematical rigor in formalizing logical terms such as proposition' and sentence' leads to the mistaken postulation of a paradox where none really exists in the case of Crete's Epimenides. Saint Paul himself adds to the confusion. Herein is an attempt to show that rigorous formulation of functional relations gives proof that no paradox exists.
Wright, Crispin: Wittgenstein on Mathematical ProofPhilosophy, 1990 Supp.
Abstracts for 1989 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1988, 1987, 1986, 1985, Pre-1985)
Anapolitanos, D. A.: Proofs and Refutations: A ReassessmentImre Lakatos and Theories of Scientific Change, Norwell Kluwer, 1989.
Avgelis, Nikolaos: Lakatos on the Evaluation of Scientific TheoriesImre Lakatos and Theories of Scientific Change, Norwell Kluwer, 1989.In this paper I consider the problem of evaluation of scientific theories focusing on the claims concerning theory acceptance posed by logical empiricism and critical rationalism (Popper and above all Lakatos). Also I try to point out to some difficulties arising from the criteria of acceptability of scientific theories formulated by Lakatos and trace some implications for the methodology of scientific research.
Clayton, Philip: Disciplining Relativism and TruthZygon, S 1989.Imre Lakatos's philosophy of science can provide helpful leads for theological methodology, but only when mediated by the disciplines that lie between the natural sciences and theology. The questions of relativism and truth are used as indices for comparing disciplines, and Lakatos's theory of natural science is taken as the starting point. Major modifications of Lakatos's work are demanded as one moves from the natural sciences, through economics, the interpretive social sciences, literary theory, and into theology. Although theology may consist of Lakatosian research programs, it also includes programs of interpretation and programs for living. This conclusion must influence our definition of theological truth and our assessment of theological relativism.
Gavroglu, Kostas (Ed.): Imre Lakatos and Theories of Scientific ChangeNorwell Kluwer, 1989.This volume includes texts of the talks given during a conference in 1986 titled "Criticism and the Growth of Knowledge: Twenty Years Later." The articles assess the developments in philosophy of science during the twenty years from the 1965 London Conference.
Hugly, Philip; Sayward, Charles: Can There Be a Proof That Some Unprovable Arithmetic Sentence Is True?Dialectica, 1989.A common theme of logic texts is that the Godel incompleteness result shows that some unprovable statement of arithmetic is true, or, at least, that determining whether arithmetic truth is axiomatizable is a logical or mathematical issue. Against this common theme we argue that the issue is a philosophical issue that has not been settled.
Pera, Marcello: Methodological Sophisticationism: A Degenerating ProjectImre Lakatos and Theories of Scientific Change, Norwell Kluwer, 1989.Lakatos' project is taken as an attempt at looking for a universal methodology which fits scientific practice. Two main presuppositions of this project are examined, namely, that methodology offers a theory of scientific rationality, and that the aim of a theory of rationality is that of eliminating the personal factors which may enter into scientific decisions. It is argued that Lakatos was affected by a "Cartesian syndrome", according to which if there were no universal, sharp and impersonal criteria of demarcation and validation, or a precise logic of discovery, then science would degenerate into "mob psychology." Not differently from Feyerabend, Lakatos could not conceive of any middle ground between these two extremes. It is shown that, in spite of many subtleties and refinements, Lakatos's methodology does not require fewer conventional elements than Popper's, and, contrary to Lakatos' view, this does not imply that science is irrational.
Abstracts for 1988 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1987, 1986, 1985, Pre-1985)
Orton, Robert: Lakatos' Model for Assessing a Research ProgramJ. Thought, Fall-Winter 1988This paper explains Lakatos' model of scientific change. The model posits a relationship between theory and evidence that avoids three problems: the logical problem of attempting to confirm a general statement from a finite number of particular statements, the psychological problem of separating "evidence" from "theory," and the historical problem of accounting for research practice. The decision to accept one theory over another is "rational" if the new theory has excess empirical content over its predecessor, some of which has been corroborated, and if it accounts for all of the facts that its predecessor could.
Perminov, Y. V.: On the Reliability of Mathematical ProofsRev. Int. Phil., 1988.
Abstracts for 1987 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1986, 1985, Pre-1985)
Nickles, Thomas: Lakatosian Heuristics and Epistemic SupportBrit. J. Phil. Sci. 1987.Lakatos' methodology of scientific research programs (MSRP) is an attempt to combine Popper's purely consequentialist epistemology with the old view that theories are rationally derived from a suitable basis. I argue against Lakatos, Zahar, et al. That this basis cannot be purely heuristic and nonepistemic. Besides, MSRP's heuristic constructivism is incompatible with Popper's epistemology; thus MSRP is incoherent. A non-Popperian, "generative" methodology of research programs is more defensible.
Shanker, S G: Wittgenstein and the Turning-point in the Philosophy of MathematicsAlbany Suny Pr., 1987.
Weintraub, E. Roy: Rosenberg's "Lakatosian Consolations for Economists": CommentEcon. Phil., AP 1987.Rosenberg argued that economists have embraced the methodology of scientific research programs (MSRP) while philosophers have been abandoning the approach. Rosenberg claims a) that there is no agreement today on the progressivity of neoclassical economics; and b) MSRP does not "demarcate" science from non-science, and thus cannot be used to identity economics as science.
My response to this is, with respect to a), Rosenberg might not know what would constitute empirical progress, but many others do. And with respect to b), so what?
Abstracts for 1986 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1985, Pre-1985)
Andersson, Gunnar: Lakatos and Progress and Rationality in Science: A Reply to AgassiPhilosophia (Israel), AG 1986.Imre Lakatos' methodology of scientific research programmes leads to a dilemma, to a choice between second level inductivism and epistemological anarchism. For this reason Joseph Agassi is right in maintaining that in the philosophy of science the Lakatos’ era is over.
Marcus, Ruth Barcan (Ed.): Logic, Methodology and Philosophy of Science, VIINetherlands North-Holland, 1986.Proceedings of the invited papers of the Seventh International Congress of Logic, Methodology and Philosophy of Science, held in Salzburg, Austria in 1983. Three papers in each of the following categories were delivered: Proof Theory and Foundations of Mathematics, Model Theory and its Applications, Recursion Theory and Theory of Computation, Axiomatic Set Theory, Philosophical Logic, Methodology of Science, Foundations of Probability and Induction, Foundations and Philosophy of the Physical Sciences, Foundations and Philosophy of Biology, Foundations and Philosophy of Psychology, Foundations and Philosophy of the Social Sciences, Foundations and Philosophy of Linguistics.
Tymoczko, Thomas (Ed.): New Directions in the Philosophy of MathematicsBoston Birkhauser, 1986.The aim of this anthology is to contribute to a revitalization of the philosophy of mathematics. The essays in part (i) criticize the programs to provide mathematics with foundations. The essays in part (ii) explore nonfoundational approaches to the philosophy of mathematics. They focus on mathematical practice and also on informal proof, evolution in mathematics and computer use. The anthology is interdisciplinary and authors include philosophers, mathematicians, logicians, and computer scientists.
Abstracts for 1985 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, Pre-1985)
Abstracts prior to 1985 (Also see 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986, 1985)
Dominicy, Marc: Falsification and Falsifiabilization from Lakatos to GoodmanRev. Int. Phil., 1983.Popper's criteria for verifiability and falsifiability cannot deal with restrictive statements, which express "Ceteris paribus" clauses (e.g., propositions which limit the number of planets in the solar system). Restrictive statements cannot be laws (as is shown by the interpretation of related counterfactuals) nor initial conditions (since they are not verifiable). A methodological principle is put forth, which constraints the use of restrictive statements and provides a new solution to Goodman's "grue and bleen" paradox.
Sarkar, Husain: A Theory of MethodBerkeley Univ. of Calif Pr., 1983.Historians, philosophers, and sociologists of science have long argued for using the history of science as an arbitrator between competing methodologies. "A theory of method" argues otherwise. It also offers: a theory of group rationality, a theory of explaining rational decisions, framework for analyzing methods, a different perspective on the relations between social sciences and methodologies, and explains the importance of heuristic advice which it considers as normative rather than empirical or conventional.
Steiner, Mark: The Philosophy of Mathematics of Imre LakatosJ. Phil., S 1983.
Ferrari, P L; Dapueto, C: Nota Intorno al Nuovo Dibattito Sullo Status DellaMatematica Annali, 1982-83.The aim of this paper is to give a survey of some "Philosophies" of mathematics developed in the last years. In particular, it discusses Bishop's philosophy of mathematics and the related debate about "constructivist" and "conceptual" proofs. Other aspects which are discussed are the opinions of Davis, Hersh and others about the "experimental" nature of mathematics and Morris Kline's historical analysis about the claimed "loss of certainty" in contemporary mathematics.
Agassi, Joseph: Lakatos on Proof and on MathematicsLog. Anal., S-d 1981.Peggy Marchi has interpreted the contribution of Imre Lakatos, his "proofs and refutations", as a non-justificationist theory of the role of proof: proofs should explain mathematical facts and be tested by thought experiments. Lakatos had no comprehensive theory of mathematics. His trailblazing researches thus constitute a challenge and a (non- justificationist) (progressive) research program.
Derr, Patrick G: Reflexivity and Methodology of Scientific Research ProgrammesNew Scholas, Autumn 1981.Two central theses in Imre Lakatos' theory of science are: (1) the unit of appraisal in science is not an isolated theory by a research program, a developing "series of theories"; and (2) the methodology of research programs may be applied to "any" norm-impregnated knowledge--including even ethics, aesthetics, history, mathematics, inductive logic, and scientific methodology. This paper argues that (1) and (2) are not cotenable, and offers a revision of Lakatos' MSRP which progressively resolves the problem.
Hacking, Ian (Ed): Scientific RevolutionsOxford Oxford Univ. Pr., 1981.This anthology contains: editor's introduction; Kuhn, "A function for thought experiments"; Shapere, "meaning and scientific change"; Putnam, "The "Corroboration of scientific theories"; Popper, "The rationality of scientific revolutions"; Lakatos, "History of science and its rational reconstructions"; Hacking, "Lakatos's philosophy of science"; Laudan, "A problem-solving approach to scientific progress"; Feyerabend, "How to defend society against science"; and an annotated bibliography of 95 items useful to students.
Adler, Jonathan E.: Criteria for a Good Inductive LogicApplications of Inductive Logic: Proc. 1978, Cohen, Jonathan (Ed.), 379-405. Oxford, 1980.We critically examine Imre Lakatos' "Changes in the problem of inductive logic." We are particularly concerned with evaluating Lakatos' arguments as they apply to more recent work in inductive logic. Although many of Lakatos' challenges to the programme of inductive logic are worth meeting, we are doubtful that his overall critique succeeds. We try to show how complex and difficult any such general critique would be.
Agassi, Joseph: Was Lakatos an Elitist?Ratio, JE 1980.Applying a criterion of scientific progress may lead to assessments conflicting with the scientific elite. Elitism is readiness to give in. Applying a criterion to historical cases may have the same effect; alternatively, endorsing today's scientific elite's history, elitists may redefine the historical elite (the elite of the mid-nineteenth century rejected field theory, yet today's elite considers the original field theoreticians the true elite). Hence, proving oneself non-elitist is showing willingness to clash with today's elite. Since Lakatos refused to apply his criterion except in retrospect, the current debate as to whether he was an elitist is undecidable.
Lehman, Hugh: An Examination of Imre Lakatos' Philosophy of MathematicsPhil. Forum, Fall 1980.In this paper, I explain Imre Lakatos views concerning the nature and function of proof in mathematics. Lakatos maintained that no mathematical statements are known indubitably. But this claim leads to questions concerning the nature and possibility of proofs and mathematical reasoning.
Steinvorth, Ulric: Lakatos Und Politische TheorieZ. Allg. Wiss., 1980.I try to apply Lakatos' metacriterion of the rationality of normative philosophies of science to normative political theories, stressing that Lakatos' metacriterion is not only an extension of Popper's idea of tests by potentially falsifying "descriptive" basic judgments to tests by potentially falsifying "normative" judgments. Rather, its application is a test by demonstrating the tested theory's capability of reconstructing its own history as rational. Finally I argue that the tradition of utilitarian political theories is fittest to be confirmed by a Lakatosian test.
Agassi, Joseph: The Legacy of LakatosPhil. Soc. Sci., S 1979.Lakatos pretended he had a new revolutionary methodology of science. He had old reactionary fragments-- Appraisal must be retrospective (Hegel); criticism must be constructive (Lenin) since minor modifications may meet it (Duhem)--and a new bizarre notion that scientific theories should be appraised in time-series. This is based on the observation that some modifications are progressive, some regressive. This observation comes from his superb proofs and refutations. Lakatos' disciples can hardly do good work while following his silly methodology of science instead of his wonderful heuristic of mathematics.