The first part of this paper consists of an exposition of the views expressed by Pierre Duhem in his Aim and Structure of Physical Theory concerning the philosophy and historiography of mathematics. The second part provides a critique of these views, pointing to the conclusion that they are in need of reformulation. In the concluding third part, it is suggested that a number of the most important claims made by Duhem concerning physical theory, e.g., those relating to the Newtonian method, (...) the limited falsifiability of theories, and the restricted role of logic, can be meaningfully applied to mathematics. (shrink)

The ArgumentThe belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of (...) necessary stages inexorably leading to its current state — meaning by this, the specific perspective that Bourbaki had adopted and were promoting. But unlike anyone else, Bourbaki actively put forward the view that their conception of mathematics was not only illuminating and useful for dealing with the current concerns of mathematics, but that this was in fact the ultimate stage in the evolution of mathematics, bound to remain unchanged by any future development of this science. In this way, they were extending in an unprecedented way the domain of validity of the belief in the eternal character of mathematical truths, from the body to the images of mathematical knowledge.Bourbaki were fond of presenting their insistence on the centrality of the modern axiomatic method as a way to ensure the eternal character of mathematical truth as an offshoot of Hilbert's mathematical heritage. A detailed examination of Hilbert's actual conception of the axiomatic method, however, brings to the fore interesting differences between it and Bourbaki's conception, thus underscoring the historically conditioned character of certain, fundamental mathematical beliefs. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

The author finds no support for the claim that J. M. Keynes had severe reservations, in general, as opposed to particular, concerning the application of mathematical, logical and statistical methods in economics. These misinterpretations rest on the omission of important source material as well as a severe misconstrual ofThe Treatise on Probability (1921).

Game theory has proved a useful tool in the study of simple economic models. However, numerous foundational issues remain unresolved. The situation is particularly confusing in respect of the non-cooperative analysis of games with some dynamic structure in which the choice of one move or another during the play of the game may convey valuable information to the other players. Without pausing for breath, it is easy to name at least 10 rival equilibrium notions for which a serious case can (...) be made that here is the “right” solution concept for such games. (shrink)

Zeno’s paradoxes of motion, allegedly denying motion, have been conceived to reinforce the Parmenidean vision of an immutable world. The aim of this article is to demonstrate that these famous logical paradoxes should be seen instead as paradoxes of immobility. From this new point of view, motion is therefore no longer logically problematic, while immobility is. This is convenient since it is easy to conceive that immobility can actually conceal motion, and thus the proposition “immobility is mere illusion of the (...) senses” is much more credible than the reverse thesis supported by Parmenides. Moreover, this proposition is also supported by modern depiction of material bodies: the existence of a ceaseless random motion of atoms—the ‘thermal agitation’—in the scope of contemporary atomic theory, can offer a rational explanation of this ‘illusion of immobility’. Our new approach to Zeno’s paradoxes therefore leads to presenting the novel concept of ‘impermobility’, which we think is a more adequate description of physical reality. (shrink)

At their extremes, the modernization of ancient mathematical texts (absolute presentism) leaves nothing of the source and the refusal to modernize (absolute antiquarism) changes nothing. The extremes exist only as tendencies. This paper attempts to justify the admissibility of broad modernization of mathematical sources (presentism) in the context of a socio-cultural (non-fundamentalist) philosophy of mathematics.

The traditional conception of knowledge is justified, true belief. If one looks at a modern textbook on epistemology, the great bulk of questions with which it deals are to do with personal knowledge, as embodied in beliefs and the proper experiences that someone ought to have had in order to have the right (or justification) to know. I intend to argue that due to the explosive growth of knowledge whose domain is “outside the head”, this conception has outlived its relevance. (...) -/- On the other hand, Karl Popper’s (1934) Falsificationism, with its emphasis on the objective character of knowledge, is not only a sounder, but also a more appropriate theory of knowledge for understanding the nature and growth of civilization. Later, Popper (1945) generalized this approach to obtain critical rationalism, in which all claims to knowledge, whether scientific or otherwise, are understood as objective solutions to objective problems and can be evaluated by other non-observational - types of criticism. I will first argue that Popper’s methodology is quite suited to the view that knowledge is an objective autonomous product and then adduce his theory of world 3, an ontology that neatly wraps up various considerations. World 3 is the domain of abstract products of the human mind that now have a life of their own, outside of human heads: theories, arguments, problems, plans, etc. I then adduce some other arguments for the autonomous quality of knowledge. (shrink)

The conceptual framework of behaviorism is reconstructed in a logical scheme rather than along chronological lines. The resulting reconstruction is faithful to the history of behaviorism and yet meets the contemporary challenges arising from cognitive science, psycholinguistics, and philosophy. In this reconstruction, the fundamental premise is that psychology is to be a natural science, and the major corollaries are that psychology is to be objective and empirical. To a great extent, the reconstruction of behaviorism is an elaboration of behaviorist views (...) of what it is for a science to be objective and empirical. The reconstruction examines and evaluates behaviorist positions on observation and the rejection of introspection, the behavioral data language, theory construction, stimulus-response psychology, the organization of behavior, complex processes, agency, and the interpretation of mentalistic language. The resulting reconstruction shows behaviorism to be a pragmatic psychological version of positivism based on a behavioral epistemology. (shrink)

Systematics based solely on structuralist principles is non-science because it is derived from first principles that are inconsistent in dealing with both synchronic and diachronic aspects of evolution, and its evolutionary models involve hidden causes, and unnameable and unobservable entities. Structuralist phylogenetics emulates axiomatic mathematics through emphasis on deduction, and “hypotheses” and “mapped trait changes” that are actually lemmas and theorems. Sister-group-only evolutionary trees have no caulistic element of scientific realism. This results in a degenerate systematics based on patterns of (...) fact or evidence being treated as so fundamental that all other data may be mapped to the cladogram, resulting in an apparently well-supported classification that is devoid of evolutionary theory. Structuralism in systematics is based on a non-ultrametric analysis of sister-group informative data that cannot detect or model a named taxon giving rise to a named taxon, resulting in classifications that do not reflect macroevolutionary changes unless they are sister lineages. Conservation efforts are negatively affected through epistemological extinction of scientific names. Evolutionary systematics is a viable alternative, involving both deduction and induction, hypothesis and theory, developing trees with both synchronic and diachronic dimensions often inferring nameable ancestral taxa, and resulting in classifications that advance evolutionary theory and explanations for particular groups. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice (...) mathematics teachers. (shrink)

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 (...) Gödel, 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. (shrink)

This article develops a conception of linguistic meaning that treats it as an attitude on the part of language users towards pairs of expressions. As with propositional attitudes, these meaning attitudes are subject to being deliberately altered over time by language users, with the aim of maximizing the efficiency of their language use. Therefore, meaning attitudes can be justified or refuted in practical terms. Our instrumentalist-coherentist approach, which allows for meanings to be advocated for alongside beliefs, provides a viable theory (...) of justification of that kind. This view fits better with the evolutionary nature of linguistic phenomena, and resolves the problem of substitutability in opaque contexts. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)

Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the so-called Convection Theorem. Both approaches are thought to be necessary for cross-verification of the standard (...) differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general four-dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary cross-verification for a reconsidered differential form of continuity equation. Although the approach is developed for one-component (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored. (shrink)

In a recent paper, “The Concept of Logical Consequence,” W. H. Hanson criticizes a formal-structural characterization of logical consequence in Tarski and Sher. Hanson accepts many principles of the formal-structural view. Relating to Sher 1991 and 1996a, he says.

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.

The importance of mathematics in the context of the scientific and technological development of humanity is determined by the possibility of creating mathematical models of the objects studied under the different branches of Science and Technology. The arithmetisation process that took place during the nineteenth century consisted of the quest to discover a new mathematical reality in which the validity of logic would stand as something essential and central. Nevertheless, in contrast to this process, the development of mathematical analysis within (...) a framework that largely involves intuition and geometry is a fact that cannot go unnoticed amongst the mathematics community, as we shall show in this paper through the research made by Bernhard Riemann on complex variables. (shrink)

The roles of classical realism, logical positivism, and pragmatic instrumentalism in the shaping of fundamental ideas in quantum physics are examined in the light of some recent historical and sociological studies of the factors that influenced their development. It is shown that those studies indicate that the conventionalistic form of instrumentalism that has dominated all the major post-World War II developments in quantum physics is not an outgrowth of the Copenhagen school, and that despite the “schism” in twentieth century physics (...) created by the Bohr-Einstein “disagreements” on foundational issues in quantum theory, both their philosophical stands were very much opposed to those of conventionalistic instrumentalism. Quotations from the writings of Dirac, Heisenberg, Popper, Russell, and other influential thinkers, are provided, illustrating the fact that, despite the various divergencies in their opinions, they all either opposed the instrumentalist concept of “truth” in general, or its conventionalistic version in post-World War II quantum physics in particular. The basic epistemic ideas of a quantum geometry approach to quantum physics are reviewed and discussed from the point of view of a quantum realism that seeks to reconcile Bohr's “positivism” with Einstein's “realism” by emphasizing the existence of an underlying quantum reality, in which they both believed. This quantum geometry framework seeks to introduce geometro-stochastic concepts that are specifically designed for the systematic description of that underlying quantum reality, by developing the conceptual and mathematical tools that are most appropriate for such a use. (shrink)

Christopher Ormell; Is the Uncertainty of Mathematics the Real Source of Its Intellectual Charm?, Journal of Philosophy of Education, Volume 27, Issue 1, 30 May.

This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical modelling, the formation (...) of mathematics on the basis of internal mathematical principles and the mathematical theory of experiment. (shrink)

Is rhetoric just a new and trendy way toépater les bourgeois?Unfortunately, I think that the newfound interest of some economists in rhetoric, and particularly Donald McCloskey in his new book and subsequent responses to critics, gives that impression. After economists have worked so hard for the past five decades to learn their sums, differential calculus, real analysis, and topology, it is a fair bet that one could easily hector them about their woeful ignorance of the conjugation of Latin verbs or (...) Aristotle's Six Elements of Tragedy. Moreover, it has certainly become an academic cliché that economists write as gracefully and felicitously as a hundred monkeys chained to broken typewriters. The fact that economists still trot out Keynes's prose in their defense is itself an index of the inarticulate desperation of an inarticulate profession. (shrink)

The modern philosophy of science has not succeeded in defining conclusively what the scientific method consists in. On the contrary, scientific practice seems to consist in a methodological pluralism, a definition that connects with essential fragments of John Dewey's Logic, the Theory of Inquiry. For Dewey, even the forms of logic emerge from the problems defined in indeterminate situations. A historical example was the introduction of the notion of complementarity in physics, which allowed the interpretation of two confusingly paradoxical experiments (...) in a coherent way. Dewey's thought demonstrates its relevance by helping us to define the pattern of inquiry. Methodological pluralism and the dependence of logic on research problems is not something that will happen, it is something that has happened and does happen in scientific practices. (shrink)

У даній роботі обговорюється процес зародження науки як суспільного інституту. Наука пройшла шлях від фрагментарних успіхів у справі усвідомленні світу до формування комплексу уявлень про природу речей і явищ. Мейнстрімом становлення планетарної цивілізації все ж слід визнати еволюцію наукових уявлень саме в Євразії, поблизу теплих морів і в умовах сприятливого клімату. І тут, за загальною думкою, потрібно віддати належне внеску еллінів, які створили соціальної уклад, який сприяв розвитку філософії, що дала змогу побудувати фундамент для науки. Наприклад, елліни Мілетської школи поставили (...) низку важливих питавинь, що виходили за межі локальних пізнавальних завдань і стосувались загальних проблем філософського характеру. Тільки на цьому ґрунті можна було починати будувати величну споруду Науки. Саме філософія еллінів дозволила їм досягти вражаючих успіхів у розвитку наук, які не мали прецедентів у світі на той час. Свою математику, більше схожу на геометрію, греки будували на аксіомах, які вони вважали інтуїтивно очевидними. Ці аксіоми були фундаментом і гарантували стійкість всіх наступних висновків. Греки розвинули дедукцію і логіку, що було найбільш яскраво представлено в «Началах» Евкліда, метою якої був опис природи. У відомій книзі «Альмагест» К. Птолемей не тільки узагальнив існуючі уявлення про Всесвіт, але й ясно сформулював принцип вибору саме такої теорії, яка є найпростішою. Саме ці дві наукові праці аж до середньовіччя справляли помітний вплив на наукову думку. Вчені тієї епохи не відокремлювали прикладну науку від філософії та етики. Тому будь-які наукові досягнення вони намагалися використовувати для вирішення багатьох проблем і при виникненні суперечностей втрачали інтерес до таких неузгоджених теорій. У сучасних умовах, коли наукові напрями розійшлися разом зі своїми прихильниками далеко один від одного, і останні не завжди схильні пов’язувати й впроваджувати свої досягнення в інші області знання, бажання стародавніх мислителів узгоджувати природничо-наукові дослідження з філософією, етикою і релігією здається дивним. Але саме це спільне існування та взаємне збагачення таких різних областей людської думки і привертає увагу окремих сучасних вчених до спадщини давньогрецьких мислителів. (shrink)