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 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)

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.

Feminist standpoint theory remains highly controversial: it is widely advocated, used to guide research and justify its results, and yet is also vigorously denounced. This essay argues that three such sites of controversy reveal the value of engaging with standpoint theory as a way of reflecting on and debating some of the most anxiety-producing issues in contemporary Western intellectual and political life. Engaging with standpoint theory enables a socially relevant philosophy of science.

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)

Are there arguments in mathematics? Are there explanations in mathematics? Are there any connections between argument, proof and explanation? Highly controversial answers and arguments are reviewed. The main point is that in the case of a mathematical proof, the pragmatic criterion used to make a distinction between argument and explanation is likely to be insufficient for you may grant the conclusion of a proof but keep on thinking that the proof is not explanatory.

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (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)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

Paradoxes have played an important role both in philosophy and in mathematics and paradox resolution is an important topic in both fields. Paradox resolution is deeply important because if such resolution cannot be achieved, we are threatened with the charge of debilitating irrationality. This is supposed to be the case for the following reason. Paradoxes consist of jointly contradictory sets of statements that are individually plausible or believable. These facts about paradoxes then give rise to a deeply troubling epistemic problem. (...) Specifically, if one believes all of the constitutive propositions that make up a paradox, then one is apparently committed to belief in every proposition. This is the result of the principle of classical logical known as ex contradictione (sequitur) quodlibetthat anything and everything follows from a contradiction, and the plausible idea that belief is closed under logical or material implication (i.e. the epistemic closure principle). But, it is manifestly and profoundly irrational to believe every proposition and so the presence of even one contradiction in one’s doxa appears to result in what seems to be total irrationality. This problem is the problem of paradox-induced explosion. In this paper it will be argued that in many cases this problem can plausibly be avoided in a purely epistemic manner, without having either to resort to non-classical logics for belief (e.g. paraconsistent logics) or to the denial of the standard closure principle for beliefs. The manner in which this result can be achieved depends on drawing an important distinction between the propositional attitude of belief and the weaker attitude of acceptance such that paradox constituting propositions are accepted but not believed. Paradox-induced explosion is then avoided by noting that while belief may well be closed under material implication or even under logical implication, these sorts of weaker commitments are not subject to closure principles of those sorts. So, this possibility provides us with a less radical way to deal with the existence of paradoxes and it preserves the idea that intelligent agents can actually entertain paradoxes. (shrink)

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is (...) that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (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)

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.

Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into a single framework of axiomatic systems (...) and that much of mathematics could be expressed in a single language. The new framework was the product of an interdisciplinary coalition whose ideas resemble those later adopted by the Vienna Circle and logical empiricists. (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)

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)

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)

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)

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 author describes the Cartesian way of solving the problem of the universal method in mathematics, in particular, the problem of applying algebra in geometry when it comes to the convergence of a discrete number and a continuous quantity. The article shows that the solution to this problem proposed by F. Viète is imperfect, since it introduces vague pseudo-geometric objects, and the geometric quantity is still far from an algebraic number. The author proves that Descartes' solution to this problem through (...) the use of Eudoxus proportions is based on such Cartesian epistemological principles as: the requirement of clarity and expressiveness of thinking; the idea of the central role of a holistic mathematical science; the idea of the existence of a simple and obvious nature of length as a basis for comparing all extended things; the elevation of the concept of ratio to the rank of a single subject of mathematical disciplines. (shrink)

In the past three decades scholars in virtually every humanistic and social scientific research discipline, and in some natural sciences, have drawn attention to quite striking instances of gender bias in the modes of practice and theorizing typical of traditional fields of research. They generally begin by identifying explicit androcentric biases in definitions of the subject domains appropriate to specific scientific fields. Their primary targets, in this connection, have been research that leaves women out altogether, research that ignores women’s contributions (...) or victimization, and research that conceptualizes its subject, male or female, human or non-human, in explicitly gender biased terms. (shrink)

I explain Kant’s critique of Leibniz’s rejection of real opposition in the Amphiboly in the context of Kant’s pre-Critical writings on vis viva and negative magnitudes and his Metaphysical Foundations of Natural Science. Properly contextualized in terms of the vis viva controversy, I argue, Kant’s arguments against Leibniz succeed, even though they are laden with theoretical inconsistencies and operate under false physical premises.

At the heart of my pragmatic theory of truth and ontology is a view of the relation between language and reality which I term internal justification: a way of explaining how sentences may have truth-values which we cannot discover without invoking the need for the mystery of a correspondence relation. The epistemology upon which the theory depend~ is fallibilist and holistic ; places heavy reliance on modal idioms ; and leads to the conclusion that current versions of realism and anti-realism (...) are deficient. Just as my theory avoids the need for an epistemic 'given', it avoids the need for a metaphysical 'given' or 'joints'. I offer a view of the nature of philosophy and what it can properly achieve with respect to ontological questions ; since those views lead me to believe that philosophical discussion about what exists should be restricted to 'entities' discussed in non-philosophical contexts, my views on how we should understand claims made about the existence of middle-sized physical objects, theoretical entities in science, and abstract entities in mathematics, give the thesis a schematic completeness. My theory leads me to a conception of inquiry which defends the cognitive status of moral statements whilst being critical of Kantian and utilitarian approaches to morality. Chapter 1 explores the views of my closest philosophical allies: William James and Nelson Goodman. (shrink)

It is my intention in this thesis to demonstrate that there exists a clear and explicit formal relationship between the seemingly exclusive descriptions of spatio-temporal and purely temporal continuity, and further, that this relationship manifests itself within our most fundamental understanding of the physical world itself, namely; within our understanding of the identity, diversity and re-identification of material bodies. It may therefore be claimed that behind that cultural understanding which leads us to imagine that the physical world is located in (...) both space and time, whereas our thoughts and feelings are located in time alone, there lies a formal logical framework, or an explicit formal description of how being in space and time relates to being in time alone - leading us to wonder, perhaps, whether these two things are really as distinct as we might at first imagine. That I should then go on to apply to this analysis a philosophical interpretation of Bergson's conception of the relationship between the intuition and the intellect is of lesser importance - indicating as it does little more than my own philosophical inclinations. However, something will be gained, I hope, from this further exercise. Along the way it will allow me to clarify a number of technical points of which the general philosopher may be unaware; for example the unobservable nature of numerical identity and re-identification, the importance of the principle of special relativity to the topic of mind and the technical difficulties of claiming that mental events are 'in time' at all. Notwithstanding these latter points, however, the intentions of this work are predominantly analytical and are adequately described as an attempt to consolidate spatio-temporal and purely temporal description under a unified logical framework. (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)

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)