Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

There have been a wide range of any human activities concerning the term of “Art and Mathematics”. Regarding directly to the historical root, there are a great deal of discussions on art and mathematics and their connections. The paper elaborates the connection between the two discourses of art and mathematics and how they influence each other.

It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the common view (...) that Kant's aesthetics leaves no room for beauty in mathematics. More specifically, I show that on the Kantian account beauty in mathematics is a non-conceptual response felt in light of our own creative activities involved in the process of mathematical reasoning. The Kantian proposal I thus develop provides a promising alternative to Platonist accounts of beauty widespread among mathematicians. While on the Platonist conception the experience of mathematical beauty consists in an intellectual insight into the fundamental structures of the universe, according to the Kantian proposal the experience of beauty in mathematics is grounded in our felt awareness of the imaginative processes that lead to mathematical knowledge. The Kantian account I develop thus offers to elucidate the connection between aesthetic reflection, creative imagination and mathematical cognition. (shrink)

In Beauty and Revolution in Science, James McAllister advances a rationalistic picture of science in which scientific progress is explained in terms of aesthetic evaluations of scientific theories. Here I present a new model of aesthetic evaluations by revising McAllister’s core idea of the aesthetic induction. I point out that the aesthetic induction suffers from anomalies and theoretical inconsistencies and propose a model free from such problems. The new model is based, on the one hand, on McAllister’s original model and (...) on further developments by Theo Kuipers in his “Beauty, a Road to the Truth?”. On the other hand, it is based on empirical findings about affection and emotion, and a naturalistic aesthetic theory. The new model is thus a naturalistic model with a wider explanatory range and much more internal consistency that McAllister’s. (shrink)

I propose to sketch my views on several aspects of the philosophy of mathematics that I take to be especially relevant to philosophy as a whole. The relevance of my discussion would, I think, become more evident, if the reader keeps in mind the function of (the philosophy of) mathematics in philosophy in providing us with more transparent aspects of general issues. I shall consider: (1) three familiar examples; (2) logic and our conceptual frame; (3) communal agreement and objective certainty; (...) (4) the transcommunal universality of mathematics; (5) the big jump to the potential infinite; (6) the reconciliation of local creation with the hypothesis of discovery; (7) Platonism as realism plus conceptualism; (8) foundational studies and mathematical practice; and (9) the decomposition of philosophical disagreements. The views of Gödel and Wittgenstein are emphasized in order to add specificity to the discussions. (shrink)

Abstract: Philosophy is often conceived in the Anglophone world today as a subject that focuses on questions in particular “core areas,” pre-eminently epistemology and metaphysics. This article argues that the contemporary conception is a new version of the scholastic “self-indulgence for the few” of which Dewey complained nearly a century ago. Philosophical questions evolve, and a first task for philosophers is to address issues that arise for their own times. The article suggests that a renewal of philosophy today should turn (...) the contemporary conception inside out, attending to and developing further the valuable work being done on the supposed “periphery” and attending to the “core areas” only insofar as is necessary to address genuinely significant questions. (shrink)

A rationalist and realist model of scientific revolutions will be constructed by reference to two categories of criteria of theory-evaluation, denominated indicators of truth and of beauty. Whereas indicators of truth are formulateda priori and thus unite science in the pursuit of verisimilitude, aesthetic criteria are inductive constructs which lag behind the progression of theories in truthlikeness. Revolutions occur when the evaluative divergence between the two categories of criteria proves too wide to be recomposed or overlooked. This model of revolutions (...) depends upon a substantial new treatment of aesthetic criteria in science with which much of the paper will therefore be occupied. (shrink)

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. (...) The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures. (shrink)

Professor Rahn takes the approach to the analysis of Western art music developed recently by theorists such as Benjamin Boretz and extends it to address non-Western forms. In the process, he rejects recent ethnomusicological formulations based on mentalism, cultural determinism, and the psychology of perception as potentially fruitful bases for analysing music in general. Instead he stresses the desirability of formulating a theory to deal with all music, rather than merely Western forms, and emphasizes the need to evaluate an analysis (...) and compare it with other interpretations, and demonstrates how this may be done. The theoretical concepts which form the basis of Rahn's approach are discussed and applied: first to individual pieces of non-Western music which have enjoyed a fairly high profile in ethnomusicological literature, and second to repertoires or groups of pieces. The author also discusses the fields of anthropology and psychology, showing how his approach serves as a starting point for studies of perception and the concepts, norms, and values found in specific music cultures. In conclusion, he lists what he considers to be musical universals and takes up the more controversial issues implicit in his discussion. (shrink)

W.D. Hamilton, Narrow Roads of Gene Land: The Collected Papers of W.D. Hamilton.

A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly (...) within the framework of progress Maddy offers, is developed with a special focus on mathematicians’ peer-review practice. (shrink)

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine's "Indispensability Argument", which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge (...) the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field's nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field's fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer's argument for his thesis that fictionalism is "the best version of anti-realistic anti-platonism". (shrink)

We characterize abstraction in computer science by first comparing the fundamental nature of computer science with that of its cousin mathematics. We consider their primary products, use of formalism, and abstraction objectives, and find that the two disciplines are sharply distinguished. Mathematics, being primarily concerned with developing inference structures, has information neglect as its abstraction objective. Computer science, being primarily concerned with developing interaction patterns, has information hiding as its abstraction objective. We show that abstraction through information hiding is a (...) primary factor in computer science progress and success through an examination of the ubiquitous role of information hiding in programming languages, operating systems, network architecture, and design patterns. (shrink)

What is the nature of the concept BEAUTY? Does it differ fundamentally from nearby concepts such as PRETTINESS? It is argued that BEAUTY, but not PRETTINESS, is a dual-character concept. Across a number of contexts, it is proposed that BEAUTY has a descriptive sense that is characterised by, inter alia, having intrinsically pleasing appearances; and a normative sense associated with deeply-held values. This account is supported across two, pre-registered, studies (N=500), and by drawing on analysis of corpus data. It is (...) suggested that this can help to explain why beauty, unlike prettiness, is thought to be deep in both the sense of being important, and in the sense of being less closely tied to sensory surfaces. (shrink)

The way of thinking of mathematicians and chemists in their respective disciplines seems to have very different levels of abstractions. While the firsts are involved in the most abstract of all sciences, the seconds are engaged in a practical, mainly experimental discipline. Therefore, it is surprising that many luminaries of the mathematics universe have studied chemistry as their main subject. Others have started studying chemistry before swapping to mathematics or have declared some admiration and even love for this discipline. Here (...) I reveal some of these mathematicians who were involved in chemistry from a biographical perspective. Then, I analyze what these remarkable mathematicians and statisticians could have learned while studying chemical subjects. I found analogies between code-breaking and molecular structure elucidation, inspiration for statistics in quantitative analytical chemistry, and on the role of topology in the study of some organic molecules. I also analyze some parallelisms between the way of thinking of organic chemists and mathematicians in terms of the use of backward analysis, search for patterns, and use of pictures in their respective researches. (shrink)

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same critique. (...) The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse. (shrink)

En este artículo se presenta una revisión de los diferentes enfoques de enseñanza de la ecuación de segundo grado para Educación Secundaria que se conjetura serían necesarios con el fin de aumentar la calidad del aprendizaje con el rigor y el tratamiento requerido, resaltando el valor de este tópico en sí mismo y de sus aplicaciones. Además, se aportan los resultados de un estudio descriptivo exploratorio realizado sobre los conocimientos de la ecuación de segundo grado de alumnado de Enseñanza Secundaria (...) para finalmente extraer la conclusión de que su conocimiento no quedó consolidado y se precisa hacer cambios. (shrink)

The main aim of this paper is to examine the practice of describing intellectual pursuits in aesthetic terms, and to investigate whether this practice can be accounted for in the framework of a standard conception of aesthetic experience. Following a discussion of some historical approaches, the paper proposes a way of conceiving of aesthetic experience as both epistemically motivating and epistemically inventive. It is argued that the aesthetics of intellectual pursuits should be considered as central rather than marginal to our (...) philosophical accounts of aesthetic experience, and that our views about the relation between the aesthetic and cognitive domains should be reconfigured accordingly. (shrink)

Discussions of aesthetic reasons and normativity are becoming increasingly popular. This piece outlines six basic questions about aesthetic reasons, normativity, and value and discusses the space of possible answers to these questions. I divide the terrain into two groups of three questions each. First are questions about the shape of aesthetic reasons: what they favour, how strong they are, and where they come from. Second are relational questions about how aesthetic reasons fit into the wider normative landscape: whether they are (...) distinctive, what their normative status is, and how they interact with each other and with non-aesthetic reasons. This piece aims to provide a taxonomy to clarify and organise the burgeoning literature and to make a few concrete suggestions for avenues of future research. (shrink)

I offer the first sustained defence of the claim that ugliness is constituted by the disposition to disgust. I advance three main lines of argument in support of this thesis. First, ugliness and disgustingness tend to lie in the same kinds of things and properties (the argument from ostensions). Second, the thesis is better placed than all existing accounts to accommodate the following facts: ugliness is narrowly and systematically distributed in a heterogenous set of things, ugliness is sometimes enjoyed, and (...) ugliness sits opposed to beauty across a neutral midpoint (the argument from proposed intensions). And third, ugliness and disgustingness function in the same way in both giving rise to representations of contamination (the argument from the law of contagion). In making these arguments, I show why prominent objections to the thesis do not succeed, cast light on some of the artistic functions of ugliness, and, in addition, demonstrate why a dispositional account of disgustingness is correct, and present a novel problem for warrant-based accounts of disgustingness (the ‘too many reasons’ problem). (shrink)

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...) meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)

In the Foundations of Mathematics, Ramsey attempted to amend Principia Mathematica’s logicism to meet serious objections raised against it. While Ramsey’s paper is well known, some questions concerning Ramsey’s motivations to write it and its reception still remain. This paper considers these questions afresh. First, an account is provided for why Ramsey decided to work on his paper instead of simply accepting Wittgenstein’s account of mathematics as presented in the Tractatus. Secondly, evidence is given supporting that Wittgenstein was not moved (...) by Ramsey’s objection against the Tractarian account of arithmetic, and a suggestion is made to explain why Wittgenstein reconsidered Ramsey’s account in the early thirties on several occasions. Finally, a reading is formulated to understand the basis on which Wittgenstein argues against Ramsey’s definition of identity in his 1927 letter to Ramsey. (shrink)

This paper uses a case study of a 1970s controversy in artificial-intelligence (AI) research to explore how scientists understand the relationships between research and practical applications. It is part of a project that seeks to map such relationships in order to enable better policy recommendations to be grounded empirically through historical evidence. In 1972 the mathematician James Lighthill submitted a report, published in 1973, on the state of artificial-intelligence research under way in the United Kingdom. The criticisms made in the (...) report have been held to be a major cause behind the dramatic slowing down (subsequently called an ‘AI winter’) of such research. This paper has two aims, one narrow and one broad. The narrow aim is to inquire into the causes, motivations and content of the Lighthill report. I argue that behind James Lighthill's criticisms of a central part of artificial intelligence was a principle he held throughout his career – that the best research was tightly coupled to practical problem solving. I also show that the Science Research Council provided a preliminary steer to the direction of this apparently independent report. The broader aim of the paper is to map some of the ways that scientists (and in Lighthill's case, a mathematician) have articulated and justified relationships between research and practical, real-world problems, an issue previously identified as central to historical analysis of modern science. The paper therefore offers some deepened historical case studies of the processes identified in Agar's ‘working-worlds’ model. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

Philosophical arguments usually are and nearly always should be abductive. Across many areas, philosophers are starting to recognize that often the best we can do in theorizing some phenomena is put forward our best overall account of it, warts and all. This is especially true in esoteric areas like logic, aesthetics, mathematics, and morality where the data to be explained is often based in our stubborn intuitions. -/- While this methodological shift is welcome, it's not without problems. Abductive arguments involve (...) significant theoretical resources which themselves can be part of what's being disputed. This means that we will sometimes find otherwise good arguments which suggest their own grounds are problematic. In particular, sometimes revising our beliefs on the basis of such an argument can undermine the very justification we used in that argument. -/- This feature, which I'll call self-effacingness, occurs most dramatically in arguments against our standing views on the esoteric subject matters mentioned above: logic, mathematics, aesthetics, and morality. This is because these subject matters all play a role in how we reason abductively. This isn't an idle fact; we can resist some challenges to our standing beliefs about these subject matters exactly because the challenges are self-effacing. The self-effacing character of certain arguments is thus both a benefit and limitation of the abductive turn and deserves serious attention. I aim to give it the attention it deserves. (shrink)

There is a tendency in contemporary (analytic) aesthetics to consider- ably restrict the scope of things that can be beautiful or ugly. This peculiarly modern tendency is holding back progress in aesthetics and robbing it of its potential contribution to other domains of inquiry. One view that has suffered neglect as a result of this tendency is the moral beauty view, whereby the moral virtues are beautiful and the moral vices are ugly. This neglect stems from an assumption to the (...) effect that virtues and vices simply cannot be beautiful or ugly. The aim of this paper is twofold. First, it develops an account of form, under which, it argues, possession of form suffices for an object’s candidature for beauty and ugliness. Second, it argues that, under the foregoing proposal, the moral beauty view turns out to be a coherent position, and so should be taken seriously in both aesthetics and ethics. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

One of the most unsettling problems in the history of philosophy examines how mathematics can be used to adequately represent the world. An influential thesis, stated by Eugene Wigner in his paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," claims that "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." Contrary to this view, this thesis delineates and implements (...) a strategy to show that the applicability of mathematics is very reasonable indeed. I distinguish three forms of the problem of the applicability of mathematics, and focus on one I call the problem of uncanny accuracy: Given that the construction and manipulation of mathematical representations is pervaded by uncertainty, error, approximation, and idealization, how can their apparently uncanny accuracy be explained? I argue that this question has found no satisfactory answer because our rational reconstruction of scientific practice has not involved tools rich enough to capture the logic of mathematical modelling. Thus, I characterize a general schema of mathematical analysis of real systems, focusing on the selection of modelling assumptions, on the construction of model equations, and on the extraction of information, in order to address contextually determinate questions on some behaviour of interest. A concept of selective accuracy is developed to explain the way in which qualitative and quantitative solutions should be utilized to understand systems. The qualitative methods rely on asymptotic methods and on sensitivity analysis, whereas the quantitative methods are best understood using backward error analysis. The basic underpinning of this perspective is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. In addition, this perspective is used to discuss the nature of theories, the role of scaling, and the epistemological and semantic aspects of experimentation. In conclusion, we argue for a method of local and global conceptual analysis that goes beyond the reach of the tools standardly used to capture the logic of science; on their basis, the applicability of mathematics finds itself demystified. (shrink)

Pre-publication certification through peer review stands in need of philosophical examination. In this paper, philosopher-psychologist Steven James Bartlett recalls the arguments marshalled four hundred years ago by English poet John Milton against restraint of publication by the "gatekeepers of publication," AKA today's peer reviewers.

ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and (...) assessing the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

An interview with Charles McCarty by Piotr Urbańczyk concerning mathematical explanation.

Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and checking to (...) make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed. (shrink)

It is well known that reductio ad absurdum arguments raise a number of interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? What kind of absurdity do we obtain? Moreover, in the mathematics education literature number of studies have shown that students find it difficult to truly comprehend the idea of reductio proofs, which indicates the cognitive complexity of these constructions. In (...) this paper, I start by discussing four philosophical issues pertaining to reductio arguments. I then briefly present a dialogical conceptualization of deductive arguments, according to which such arguments are best understood as a dialogue between two participants—Prover and Skeptic. Finally, I argue that many of the philosophical and cognitive difficulties surrounding reductio arguments are dispelled or at least further clarified once one adopts a dialogical perspective. (shrink)

In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content that does not imply the existence of mathematical objects. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by (...) mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more about the physical world than we would otherwise be able to. Mathematical objects do not need to exist for mathematics to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences that reveals they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which reveals that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show that the emerging hermeneutic fictionalist position is preferable to any alternative interpretation of mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects. (shrink)

The history of the continuous inclusion of mathematics in liberal education in the West, from ancient times through the modern period, is sketched in the first two sections of this chapter. Next, the heart of this essay (Sects. 3, 4, 5, 6, and 7) delineates the central role mathematics has played throughout the history of Western civilization: not just a tool for science and technology, mathematics continually illuminates, interacts with, and sometimes challenges fields like art, music, literature, and philosophy – (...) subjects now universally considered to be liberal arts. Section 8 adds an international perspective to the contemporary liberal arts story by describing some instructive mathematical achievements from many cultures and societies. Finally, Sect. 9 addresses how contemporary mathematics teaching can use the history of mathematics viewed as a liberal art to enhance the appreciation and understanding of mathematics for all students. (shrink)

This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...) field, it lays down a much-needed marker of progress to date and provides a platform for informed and coherent future analysis and research of the subject. -/- The publication comes at a time of heightened worldwide concern over the standard of science and mathematics education, attended by fierce debate over how best to reform curricula and enliven student engagement in the subjects There is a growing recognition among educators and policy makers that the learning of science must dovetail with learning about science; this handbook is uniquely positioned as a locus for the discussion. -/- The handbook features sections on pedagogical, theoretical, national, and biographical research, setting the literature of each tradition in its historical context. Each chapter engages in an assessment of the strengths and weakness of the research addressed, and suggests potentially fruitful avenues of future research. A key element of the handbook’s broader analytical framework is its identification and examination of unnoticed philosophical assumptions in science and mathematics research. It reminds readers at a crucial juncture that there has been a long and rich tradition of historical and philosophical engagements with science and mathematics teaching, and that lessons can be learnt from these engagements for the resolution of current theoretical, curricular and pedagogical questions that face teachers and administrators. (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)

This thesis constructs a model of the material causes of the capacity of individuals to act at work, by using the ontology of scientific realism to facilitate a synthesis between Marx and Foucault. This synthetic model is submitted as a solution to the long-standing problem of Industrial Relations theory, now manifest in the deconstruction of the organon of 'control'. The problems of 'control' are rooted in the radical concept of power and traditional, base/superstructure, interpretations of Marx. Developing an alternative to (...) the last provides the means of transcending the limitations of the first and the second. A realist, chronological-bibliographic reading of Marx provides an alternative to traditional interpretations, by creating a novel concept of his object, his initial explicandum and his putative explicans. This reading identifies a fresh problem with his model of capital: it cannot explain how labour is organized into a productive power and subsumed to capital. Foucault provides the means of resolving these deficiencies of Marx's explicans. A realist interpretation turns Marx and Foucault around to face each other, renders them compatible and establishes points of contact between their work. Together they constitute a model of the operative logic of production relations capable of explaining the organization of labour, its subsumption to capital and the materialism of civil society and the idealism of the state. On this basis, the contemporary form of the problem of Industrial Relations theory is explained. (shrink)

I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

In this article I examine the status of putative aesthetic judgements in science and mathematics. I argue that if the judgements at issue are taken to be genuinely aesthetic they can be divided into two types, positing either a disjunction or connection between aesthetic and epistemic criteria in theory/proof assessment. I show that both types of claim face serious difficulties in explaining the purported role of aesthetic judgements in these areas. I claim that the best current explanation of this role, (...) McAllister's 'aesthetic induction' model, fails to demonstrate that the judgements at issue are genuinely aesthetic. I argue that, in light of these considerations, there are strong reasons for suspecting that many, and perhaps all, of the supposedly aesthetic claims are not genuinely aesthetic but are in fact 'masked' epistemic assessments. (shrink)

We have all heard people say ‘Beauty is only skin-deep’, or ‘Beauty is in the eye of the beholder’: our culture promulgates a conception of beauty as subjective, superficial, and independent of other values like moral goodness or knowledge and understanding. Yet our taste in beauty affects many aspects of our lives, sometimes playing a decisive – and often detrimental – role in areas as wide-ranging as our identity and self-esteem, our morally salient decisions, and our relationship to the environment. (...) This presents us with a choice: we can either ignore the facts – leaving our conception of beauty unchanged and allowing our taste to influence much in our lives while either not acknowledging such influence, or perhaps seeking to reprimand it; or we can take the power of beauty seriously and seek to harmonise our taste with our values. I argue for the latter option and propose a way of bringing beauty and taste in line with what matters to us using the notion of functional beauty. Adopting this strategy, I suggest, can have a powerful – and positive – impact on our self-esteem and wellbeing, our relationship to others, as well as our attitudes towards the environment. (shrink)

Our meta-argumentative vocabulary supplies the conceptual tools used to reflectively analyse, regulate, and evaluate our argumentative performances. Yet, this vocabulary is susceptible to misunderstanding and abuse in ways that make possible new discursive mistakes and pathologies. Thus, our efforts to self-regulate our reason-transacting practices by articulating their norms makes possible new ways to violate and flout those very norms. Scott Aikin identifies the structural possibility of this vicious feedback loop as the Owl of Minerva Problem. In the spirit of a (...) shared concern for the flourishing or our rational, argumentative practices, this paper approaches the Owl of Minerva Problem from a vantage point that, by comparison with Aikin’s, affords perspectives that are more pessimistic in some aspects and more optimistic in others. Pessimistically, the problem at the root of the weaponization of our meta-argumentative vocabulary is motivational, not structural. Its motivational nature explains its resistance to the normal repertoire of reparative argumentative maneuvers, as well as revealing a profound and deeply entrenched misunderstanding of the connection between our reasons-transacting practices and the goods achievable within them. Optimistically, in the absence of this motivational problem, the misunderstandings and errors made possible by our meta-argumentative vocabularies are amenable to remedy by familiar techniques of discursive instruction and repair. More optimistically, even though our meta-argumentative vocabularies are generated only retrospectively, they can be used prospectively, thereby making possible an aspirational motivation resulting in a virtuous cycle of increasingly autonomous normative self-regulation. Properly harnessed, the Owl of Minerva releases the Lark of Arete. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

Many of the methods commonly used to research mathematical practice, such as analyses of historical episodes or individual cases, are particularly well-suited to generating causal hypotheses, but less well-suited to testing causal hypotheses. In this paper we reflect on the contribution that the so-called hypothetico-deductive method, with a particular focus on experimental studies, can make to our understanding of mathematical practice. By way of illustration, we report an experiment that investigated how mathematicians attribute aesthetic properties to mathematical proofs. We demonstrate (...) that perceptions of the aesthetic properties of mathematical proofs are, in some cases at least, subject to social influence. Specifically, we show that mathematicians’ aesthetic judgements tend to conform to the judgements made by others. Pedagogical implications are discussed. (shrink)