The article examines the current state of the mathematical education of the students-philosophers that depends on language of the humanitarian mathematics, evidence of its statements and methodological problem of the cognition of the mathematical facts. One of important tasks of philosophy of mathematical education consists in motivation of the need for training mathematics of students-philosophers. The main criterion of the usefulness of mathematics for philosophers is revealed in the ways of justification of its truth and completeness (...) of reasoning of mathematical statements. This reflects the attractiveness of mathematical knowledge for philosophers that is characterized by the fact that the truth is revealed along with a proof in mathematics. One of the main so-called ‘acupuncture points‘ of mathematical education of philosophers, as the author believes, is the language of modern mathematics. In General, the linguistic aspect is very important in modern science. The second ‘acupuncture point‘ of mathematical education of philosophers is the level of evidence claims. Third, according to the author, is responsible for knowledge of mathematical truths, is involved in not only our existence, but also transcendental knowledge. In the context of philosophy of the future of modern civilization, mathematics is indispensable as a necessary element of scientific and philosophical picture of the world. To support the conclusions the author draws a broad philosophical and scientific context. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What (...) could mathematics be knowledge about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

In the article, the phenomenon of understanding in mathematics is studied. This topic is relevant in contemporary philosophy of science, in which the classic dichotomy of ‘understanding-explanation‘, characteristic of classical science, undergoes serious transformations. One reason for this transformation is a quantitative increase in the flow of information in modern science. It is clear that in such a situation you want to hold a serious understanding of this information in the context of modern scientific picture of the world. As (...) the modern science is inextricably linked to mathematical science, it is required to rely on the results of studies of the phenomenon of understanding in mathematics. Such studies are already under way in the modern philosophical and scientific literature, although this issue is still relatively new. The author identifies the most important, in his view, types of understanding in math, taking into account the situation of the teaching of mathematics, the situation of mathematical discovery, and ‘mathematical symbolism‘. A study of the main types of understanding in math and the identification of its specificity would have been impossible without the support of the concept of tacit knowledge taken as a methodological tool and author’s previously published works taken as a basis. In this article, the author relies on the classic work on the philosophy of mathematics, as well as the works of famous contemporary Russian and foreign scientists and philosophers. The results of the study are summarized in the form of conclusions. (shrink)

The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or (...) “complimentary” to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices. (shrink)

The article examines the history of the development of the ideas of semiotics, from the works of St. Augustine to the present. The author shares the semiotic approach, which, judging by the literature, was formulated by Augustine, and semiotics as a scientific discipline, and in two versions, as an analogue of mathematics and natural science. The characteristic of the semiotic approach presented by Augustine in the scheme is given, which, the author shows, can be extended to various humanitarian (...) objects. Based on the semiotic approach and classifications of signs, various variants of semiotics as a science were created in the XIX and XX centuries. The difference of scientific semiotics is explained: semiotics solved different problems and tasks, semiotically comprehended different subject areas, proceeded from a different understanding of science. Nevertheless, in all variants of semiotics, relations between the components of the sign were established. The semiotics reform project proposed by G.P. Shchedrovitsky is considered, and what came of it. Based on the analysis of two cases, the author outlines another version of semiotics, which he calls "expressionism". Although the methodology proposed by him allows analyzing and comprehending a fairly wide range of expressions and works of art, the author suggests not to consider it universal. (shrink)

En el presente artículo me ocupo de la discusión acerca de cuán exigentes son nuestras obligaciones de contribuir con dinero y tiempo a las agencias humanitarias que asisten a personas en situación de pobreza extrema en el mundo. Defiendo una posición intermedia, moderada, frente a la posición extrema formulada por Peter Singer y frente a la posición según la cual nuestras obligaciones son mínimas. La objeción principal contra esas dos posiciones es que, cuando analizan la situación en que los potenciales (...) donantes se encuentran frente a las personas en situación de pobreza extrema, omiten el carácter iterativo que es propio de esa situación. La posición moderada, en cambio, tiene en cuenta ese carácter, gracias a que entiende a nuestras obligaciones hacia los pobres globales como obligaciones imperfectas. In this article I engage in the debate about the demandingness of our duties to contribute with money and time to humanitarian agencies that assist people who live in extreme poverty around the world. I defend an intermediate, moderate, view against Peter Singer's extreme view, and also against the view according to which our duties are minimal. The main objection regarding those two views is that, when they analyze the situation in which potential donors are vis-à-vis people who live in extreme poverty, they miss its distinctive iterative character. The moderate view, on the contrary, pays heed to that character, thanks to its account of our duties towards the global poor as imperfect duties. (shrink)

The philosophy of nature, which encompasses the comprehensive study of the natural world, became intimately linked with the interdisciplinary approach of self-organization theory, or synergetics, as it was revealed in the latter third of the 20th century. This novel understanding of reality and its connection to synergetics becomes evident when comparing the panlogism of G.W.F. Hegel and the dialectical materialism of F. Engels, both based on 19th-century scientific achievements, with contemporary issues in natural science. This comparison is justified as the (...) worldviews formulated by Hegel, Marx, and Engels significantly influenced the development of civilization in the 20th century. Similarly, synergetics appears poised to become the cornerstone of the emerging scientific worldview. The philosophical legacies of these classical thinkers play a crucial role in shaping the theory of self-organization. Hegel examined the question of self-development within complex systems and analyzed culture through this perspective, consequently, he can be considered a precursor to synergetics. His viewpoint on the philosophy and methodology of science as a means of reflection within the domain of knowledge remains influential. The evolutionary approach, which Engels regarded as one of the major accomplishments of 19th-century science, now serves as the foundation for numerous contemporary scientific disciplines, including synergetics itself. The philosophical approach to seeking elementary entities, from which the properties of the whole could be discerned, emerged as the driving force in the development of 20th-century science. Concurrently, the challenges confronting humanity have significantly transformed the realm of scientific knowledge. Emphasis has shifted toward the laws governing the interactions of elementary entities, as well as the associated issues of structure, chaos, and self-organization. The roles of mathematical modeling, intra-scientific reflection, and large-scale projects have proven to be more critical than initially anticipated by the classical philosophers. The ongoing humanitarian and technological revolution necessitates new responses to the profound and significant questions originally posed by Hegel and Engels. (shrink)

Global ignorance about Africa continues to sustain inappropriate global interventions to resolve public health crises, often with disastrous consequences. To explain why this continues to happen, I marshal two theorems that predict basic statistical properties, called ‘the doctrinal paradox’ and ‘the discursive dilemma’, which underlie scientific consensus formation and evidence-based decision making on a global scale. These mathematical results illuminate the epistemic and material injustices committed by the protocols of medical research conducted at the highest level of global knowledge production (...) in the service of international humanitarian aid for Africans. These social choice theorems reveal that a global scientific consensus projecting claims and proposing policies about Africa’s disease burden is likely to yield a low degree of reliability, in that the probability of its accuracy is less than chance. The solution to this anomaly is to remove from the global scientific agenda the statistically unrealisable demand of satisfying too many multinational corporate and foreign governments’ priorities as equally entitled to benefit from the knowledge produced to improve Africa’s public health sector. Instead, foreign funding targeted to support medical science and policy in Africa should be directed by those specialists in situ who are most familiar with their own national health challenges and potential solutions, rather than relying upon foreign decision makers to interpret Africa’s emergency public health care needs. Keywords Doctrinal paradox, discursive dilemma, scientific consensus, epistemic trust, global health ethics, group agency. (shrink)

The first part of my hypothesis, then, is simple enough, and would be accepted in principle by most students of Plato: the dramatic structure of the dialogues is an essential part of their philosophical meaning. With respect to the poetic and mathematical aspects of philosophy, we may distinguish three general kinds of dialogue. For example, consider the Sophist and Statesman, where Socrates is virtually silent: the principal interlocutors are mathematicians and an Eleatic Stranger, a student of Parmenides, although one who (...) is not always loyal to his master's teaching of what might be called monadic homogeneity. In these dialogues, the mathematical character of philosophy is not merely emphasized but exaggerated, and any attempt to interpret them must take this fact into account. Otherwise, we shall not be able to understand why the Stranger seems to classify the general's art in the same species of the genus hunting as the louse catcher. The significance of this step, which does not stem from humanitarian considerations, but rather illustrates how the human becomes invisible from the mathematical viewpoint, contributes its share to the obscurity of these two dialogues. Second, there are dialogues like the Phaedrus and Symposium, in which the style, the interlocutors, and even the subject-matter seem to be largely poetic and rhetorical. Here, the overwhelming impression is of enthusiasm, divine madness, and intoxication in speech and deed. Finally, there are dialogues like the Philebus and Republic, in which it is not so easy to say whether poetry or mathematics predominates, if indeed either may be said to do so. (shrink)

This volume was initially conceived as a thematic issue of the Sfera Politicii journal and some of its chapters (written by Gabriela Tănăsescu, Henrieta A. Şerban, Lorena Stuparu and Cristian-Ion Popa) were published as such in the 9 (163), September 2011 issue under the title „Theory and Political Ideology”. To enlarge the discussion on the theme, new papers have been added to the previous ones for inclusion in this book. By choosing to title it „Political theories versus ideologies?” we wanted (...) to suggest from the beginning the difficulty of a consensus on such a disputable topic. As it has multiple facets, there are several possible ways to deal with it. Each contribution in this collection is an attempt to clarify a different aspect of the issue. So, a category of texts seeks to explore the nature of political theory, the value neutrality thesis concerning the social sciences, and the distinctions between theory and ideology. One of them is devoted to a trans-theoretical analysis of the naturalist and the interpretivist models of political theory; another one, by scrutinizing the original meaning of „philosophy”, aims at showing that, when philosophy neglects the ancient harmony between philo and sophia, by focusing only on the last one, it runs the risk of failing into ideology; the first efforts of some modern thinkers, (fascinating by mathesis universalis) to apply mathematics to politics, or Michael Oakeshott ‘s concept of ideology are also dealt with in other texts. By broadening the discussion on the neutrality thesis, one of the articles brings to light the philosophical prejudices underlying both „the constitutional” and „the welfare” models, i.e., a „rule ethics” and, respectively, a utilitarian one. Several papers investigate the status of the theories of international relations. Some of them address the peculiar question whether the so called theories of European integration satisfy the criteria of what political scientists mean by an empirical study of phenomena. To put it differently: Can we refer to such theories as a scientifically approaching to the unification European process, i.e., as being able to explaining and making predictions? A subsidiary question would be: are they ideologically neutral? The problem is explicitly put in another article, whose author, by suspecting such theories’ claim to neutrality, is asking if it is possible not a scientific approach to the European integration, but a normative one, namely, a philosophy of European unification „without ideology”. Some articles, pointing out the weakness and vulnerabilities of the traditional theories of international relations when confronted with new political realities, endorse a constructivist interpretation that is likely to offer a better account for actions such as humanitarian interventions. In a similar line of thinking, constructivism is seen by another contributor as a perspective in terms of which some traditional political concepts should be revised. An example is „the national interest”, a concept that can be explained in all its aspects neither by the realist nor by liberal theories. Subjects such as „the end of ideology”, or the ideological left-wing deviations of today liberalism for electoral reasons, or that of a likely „international solidarity” ideology are discussed in this volume too. If it is to draw a conclusion from most contributions, we would say that, irrespective of its versions, normative, or empirical (scientific), or analytical, political theory conceived of itself as being not only different from, but also opposed to ideology. And it still pretends to be so. Lots of the facets of the topic have remained, obviously, unexplored. Among them, the contemporary efforts to reconcile the normative and the empirical dimensions of political theory would have been worth of addressing – maybe, in a future enterprise. Engaging in the difficult endeavor of assembling and integrating individual texts into a coherent account, we hope to be successful in stimulating further debate on the theme, a debate in which the present volume is only a modest contribution. (shrink)

The purpose of this article is to make apparent Hannah Arendt’s thought on the practical dimension of universality alluded throughout her works. The issue of universality has been one of the most pivotal questions in political philosophy until today. Beneath of her philosophical endeavor there is always her deep concern for it. In this article I will show the practical dimension of universality unintentionally pursued by Arendt and its political implications. By harshly criticizing Plato Arendt successfully shows how violent the (...) truth claim in the political realm can be. What Arendt is critical of is the attitude to dictate philosophical ideals in politicalrealm, the attitude I named the “Philosopher King Complex.” Arendt’s bitter criticism of this attitude makes believe that she is critical of any claim of universality in the political realm, but her praise of Socrates clearly shows that she does not altogether blame the claim itself. We also need to note the debate with Gershom Scholem. Scholem’s charge that Arendt dealt with the Eichmann case from a humanitarian, universalistic viewpoint while neglecting the Jewish perspective could be serious considering Arendt’s emphasis on the pariah’s perspective. Arendt’s standpoint, however, includes both a particularistic view as a Jew and auniversalistic view as a human at the same time. This is possible because Arendt correctly understood the role of speech. Arendt’s use of Pastor Grueber’s example in Eichmann in Jerusalem implies that she believes in the power of speech to relate our consciousness to reality and to make communication possible. To retain human plurality along with universality, it is necessary but not sufficient to focus on speech alone. In this regard we learn a lesson from the Habermas-Henrich debate: that self-relation of consciousness plays the role of establishing self-identity. A successful example of combining these two insights is Arendt’s position since she delivered her hermeneutic insights in the language of philosophy of consciousness. Albrecht Wellmer’s unjustifiable charge of Arendt’s concept of judgment to be “mystic” is an example of misunderstanding of her peculiar position. If we admit this, we can say that Arendt establishes aposition to give an answer to the question “is democracy a universal truth?”: democracy, as far as it is a political truth, is neither equivalent to a universal truth as a mathematical one, nor just a particular cultural opinion of westerners. This interpretation of Arendt puts her position near to liberal-communitarian views of Michael Sandel or Charles Taylor. Sandel criticizes Arendt to be universalistic, but in fact his position is quite near to hers. For example, his method of analogy is almost the same as her concept of exemplary validity. Rather, it seems to me that Arendt provides more solid theoretical ground than Taylor or Sandel does. For Arendt’s position stands on a firm understanding of the power of speech and on an insight of self-relation of consciousness. (shrink)

Batterman raises a number of concerns for the inferential conception of the applicability of mathematics advocated by Bueno and Colyvan. Here, we distinguish the various concerns, and indicate how they can be assuaged by paying attention to the nature of the mappings involved and emphasizing the significance of interpretation in this context. We also indicate how this conception can accommodate the examples that Batterman draws upon in his critique. Our conclusion is that ‘asymptotic reasoning’ can be straightforwardly accommodated within (...) the inferential conception. 1 Introduction2 Immersion, Inference and Partial Structures3 Idealization and Surplus Structure4 Renormalization and the Stability of Mathematical Representations5 Explanation and Eliminability6 Requirements for Explanation7 Interpretation and Idealization8 Explanation, Empirical Regularities and the Inferential Conception9 Conclusion. (shrink)

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual (...) thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

Standard approaches to counterfactuals in the philosophy of explanation are geared toward causal explanation. We show how to extend the counterfactual theory of explanation to non-causal cases, involving extra-mathematical explanation: the explanation of physical facts by mathematical facts. Using a structural equation framework, we model impossible perturbations to mathematics and the resulting differences made to physical explananda in two important cases of extra-mathematical explanation. We address some objections to our approach.

This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which (...) he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field. (shrink)

The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.

In these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: Brouwer, Hilbert, Bernays, Weyl, Gödel, Russell, Quine, Putnam, Wang, and Tait.

A beneficial effect of gesture on learning has been demonstrated in multiple domains, including mathematics, science, and foreign language vocabulary. However, because gesture is known to co-vary with other non-verbal behaviors, including eye gaze and prosody along with face, lip, and body movements, it is possible the beneficial effect of gesture is instead attributable to these other behaviors. We used a computer-generated animated pedagogical agent to control both verbal and non-verbal behavior. Children viewed lessons on mathematical equivalence in which (...) an avatar either gestured or did not gesture, while eye gaze, head position, and lip movements remained identical across gesture conditions. Children who observed the gesturing avatar learned more, and they solved problems more quickly. Moreover, those children who learned were more likely to transfer and generalize their knowledge. These findings provide converging evidence that gesture facilitates math learning, and they reveal the potential for using technology to study non-verbal behavior in controlled experiments. (shrink)

This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. (...) The famous impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities. (shrink)

Rarely has the history or philosophy of mathematics been written about by mathematicians, and the analysis of mathematical texts themselves has been an area almost entirely unexplored. _Figures of Thought_ looks at ways in which mathematical works can be read as texts, examines their textual strategies and demonstrates that such readings provide a rich source of philosophical issues regarding mathematics: issues which traditional approaches to the history and philosophy of mathematics have neglected. David Reed, a professional mathematician (...) himself, offers the first sustained and critical attempt to find a consistent argument or narrative thread in mathematical texts. In doing so he develops new and fascinating interpretations of mathematicians' work throughout history, from an in-depth analysis of Euclid's Elements, to the mathematics of Descartes and right up to the work of contemporary mathematicians such as Grothendeick. He also traces the implications of this approach to the understanding of the history and development of mathematics. (shrink)

Suppose we are asked to draw up a list of things we take to exist. Certain items seem unproblematic choices, while others (such as God) are likely to spark controversy. The book sets the grand theological theme aside and asks a less dramatic question: should mathematical objects (numbers, sets, functions, etc.) be on this list? In philosophical jargon this is the ‘ontological’ question for mathematics; it asks whether we ought to include mathematicalia in our ontology. The goal of this (...) work is to answer this question in the affirmative, by drawing on considerations on the the applicability of mathematics to natural science. (shrink)

According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, (...) I argue that what is picked out by the mathematics are structural facts that go beyond any specific physical facts. (shrink)

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, (...) and there are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant (...) compares the Formula of Universal Law, central to his theory of moral judgement, to a mathematical postulate; in the Critique of Judgement , where he considers aesthetic judgment, Kant distinguishes the mathematical sublime from the dynamical sublime. This last point rests on the distinction that shapes the Transcendental Analytic of Concepts at the heart of Kant’s Critical philosophy, that between the mathematical and the dynamical categories. These examples make it clear that Kant's transcendental philosophy is strongly influenced by the importance and special status of mathematics. The contributions to this book explore this theme of the centrality of mathematics to Kant’s philosophy as a whole. This book was originally published as a special issue of the Canadian Journal of Philosophy. (shrink)

Originally published in 1949. This meticulously researched book presents a comprehensive outline and discussion of Aristotle's mathematics with the author's translations of the greek. To Aristotle, mathematics was one of the three theoretical sciences, the others being theology and the philosophy of nature. Arranged thematically, this book considers his thinking in relation to the other sciences and looks into such specifics as squaring of the circle, syllogism, parallels, incommensurability of the diagonal, angles, universal proof, gnomons, infinity, agelessness of (...) the universe, surface of water, meteorology, metaphysics and mechanics such as levers, rudders, wedges, wheels and inertia. The last few short chapters address 'problems' that Aristotle posed but couldn't answer, related ethics issues and a summary of some short treatises that only briefly touch on mathematics. (shrink)

This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox, a challenge to 'classical' mathematics from a world-famous mathematician, a new foundational school, and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. (...) The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My (...) argument shows that Hobbesian natural philosophy relies upon suppositions that bodies plausibly behave according to these borrowed causal principles from geometry, acknowledging that bodies in the world may not actually behave this way. First, I consider Hobbes's relation to Aristotelian mixed mathematics and to Isaac Barrow's broadening of mixed mathematics in Mathematical Lectures (1683). I show that for Hobbes maker's knowledge from geometry provides the ‘why’ in mixed-mathematical explanations. Next, I examine two explanations from De corpore Part IV: (1) the explanation of sense in De corpore 25.1-2; and (2) the explanation of the swelling of parts of the body when they become warm in De corpore 27.3. In both explanations, I show Hobbes borrowing and citing geometrical principles and mixing these principles with appeals to experience. (shrink)

This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then, Wittgenstein's mature philosophy of mathematics is carefully presented and examined. Schroeder explains that it is based on two key ideas: (...) the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity -- calculation -- rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems. (shrink)

In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. (...) We show that the left to right directions of these theorems are equivalent to ACA0 and ATR0, respectively. On the other hand, the opposite directions are both provable in WKL0, but not in RCA0. We also prove the equivalence with ACA0 of the following result of Erdös and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains. (shrink)

How does mathematics apply to something non-mathematical? We distinguish between a general application problem and a special application problem. A critical examination of the answer that structural mapping accounts offer to the former problem leads us to identify a lacuna in these accounts: they have to presuppose that target systems are structured and yet leave this presupposition unexplained. We propose to fill this gap with an account that attributes structures to targets through structure generating descriptions. These descriptions are physical (...) descriptions and so there is no such thing as a solely mathematical account of a target system. (shrink)

This book is intended to fill a gap between popular 'wonders of mathematics' books and the technical writings of the philosophers of mathematics.

Modal logic is the study of modalities - expressions that qualify assertions about the truth of statements - like the ordinary language phrases necessarily, possibly, it is known/believed/ought to be, etc., and computationally or mathematically motivated expressions like provably, at the next state, or after the computation terminates. The study of modalities dates from antiquity, but has been most actively pursued in the last three decades, since the introduction of the methods of Kripke semantics, and now impacts on a wide (...) range of disciplines, including the philosophy of language and linguistics ('possible words' semantics for natural language), constructive mathematics (intuitionistic logic), theoretical computer science (dynamic logic, temporal and other logics for concurrency), and category theory (sheaf semantics). This volume collects together a number of the author's papers on modal logic, beginning with his work on the duality between algebraic and set-theoretic modals, and including two new articles, one on infinitary rules of inference, and the other about recent results on the relationship between modal logic and first-order logic. Another paper on the 'Henkin method' in completeness proofs has been substantially extended to give new applications. Additional articles are concerned with quantum logic, provability logic, the temporal logic of relativistic spacetime, modalities in topos theory, and the logic of programs. (shrink)

In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.