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Adler J. E. 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 J. 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. Agassi J. 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. Agassi J. 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. Ambrose A. 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". Anapolitanos D. A.
Andersson G. 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. Avgelis
N. 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 P. 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. Crossley J. N.; Lun A. W.
C. 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. Dawson Jr J. W. 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. Derr P. G. 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. De Souza R. L. 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 M.
(ed.) Detlefsen M. Poincare was a persistent critic of logicism. Unlike most critics oflogicism, 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 Leibnez, 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, aKantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical epistimology which underlies it. Detlefsen
M. It is argued that Brouwer's critique of classical logic was not so much focused on particular principles (e.g., the law of excluded middle) as on the use of any kind of logical inference in mathematical proof. He believed that genuine mathematical reasoning requires genuine mathematical insight (or intuition), and thus cannot accommodate the use of topic-neutral forms of inference. Alternative views of knowledge and language which might underlie such a view are discussed, as are certain connections between the thought of Brouwer and Poincaré. Dominicy M. Popper's criterias 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. Drozdek A., Keagy
T. 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 J., Dubucs
M. Ernest P. 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. Feferman S. 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. Feist R. Wittgenstein's remarks on Godel'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. Ferrari P. L., Dapueto
C. 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. Fetzer J. 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. Gavroglu K. (ed) 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. Gonzalez W. J. 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 favour, and so it is recognized by the supporter of the opposite point of view. Thus, the notion of truth cannot be disregarded. Gonzalez W. J. 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 IproofsD; 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. Grolmusz V. 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. Hacking I. (ed.) 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. Hugly P., Sayward
C. 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. Hull K. 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. Jaeger G. 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. Jesseph D. 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. Kadvany
J. 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's 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's account for contemporary mathematics. Koetsier T. 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's philosophy of mathematics. In the remaining chapters several episodes in the history of mathematica are discussedsuch 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. Lehman H. 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. Marcus R. B. (ed) 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. Mancosu P. 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 analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme 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. Mayberry J. 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. Nickles Th. Lakatos's 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. O'Leary D. J. The paper describes and contrasts two approaches to automated theoremproving 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 TheoryMachine approach is more satisfying in its attempt to understand the human endeavor that is the basis for PM. Orton R. This paper explains Lakatos's 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. Pagin P. 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
C. 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. Pera
M. Lakatos's 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's view, this does not imply that science is irrational. Perminov Y. V. Ramachandran S. 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 proofsn 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. Resnik M. D. This paper presents an account of how proving mathematical theorems can induce us to acquire justified true beliefs about abstract mathematical objects. Robert S. Sarkar H. 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. Shanker S. G. Shapiro S. 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 M. Steiner M. 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? Steinvorth U. I try to apply Lakatos's metacriterion of the rationality of normative philosophies of science to normative political theories, stressing that Lakatos's 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. Stekeler-Weithofer
P. 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 ofform 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 mustcontrol 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 S. Sundholm G. 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. Sundholm G. 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. Tait W. 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 exemplifiedin 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. Thagard P. This paper discusses the tractability thesis, which identifies the intuitive class ofcomputationally 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. Tymoczko T. (ed.) 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 Van Bendegem J.
P. 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. Wagner S. J. 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. Inparticular, 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. Wilson J.
L. 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. Weintraub E. R. 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? Wright C.
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