Mathematics stand in a privileged relationship with aesthetics: a relationship that follows two main directions. The first concerns the introduction of mathematical considerations into aesthetic discourse. For instance, it is common to mention the mathematical architecture of certain artistic productions. The second leads from aesthetics to mathematics. In this case, the question is that of the role and meaning that aesthetic considerations may assume in mathematics. It is indeed a widely held view among mathematicians, of whatever socio-historical context, not only (...) to see their discipline as presenting a strong aesthetic dimension, but also to consider that this dimension plays a fundamental role in the process of developing and understanding mathematics. The main ambition of this paper is to show how Nelson Goodman’s aesthetics can be used to justify this point of view and to propose a thesis concerning the aesthetic functioning of mathematics. This first result allows to resituate Goodman’s aesthetics within a very classical tradition that will be described. Finally, the underlying ambition is to show the keys provided by Goodman’s theory for the philosophy of mathematics. (shrink)

I am going to compare the strategies and communication bees use in order to locate and retrieve nectar to the world of science and the scientist. The analogy is intentionally anthropomorphic but I wish to argue that if successful bees made assumptions they would be similar to those of the scientist: flowers can be regarded as facts, nectar as knowledge, honey as technology and their ‘waggle-dance’ as communication of ideas. I would like to say that this is to be used (...) as an analogy and should not be taken to be a statement of the scientific method as an emergent property of nature, as evolution ultimately does not care about what is true or false, whereas science does. However, what i do wish to convey is that in the same way that the life of bees can be limited by the process of their enquiries; science can also limit itself by the assumptions that are taken to be true or worthwhile in the quest for new knowledge. (shrink)

Although “theory” has been the prevalent unit of analysis in the meta-study of science throughout most of the twentieth century, the concept remains elusive. I further explore the leitmotiv of several authors in this issue: that we should deal with theorizing (rather than theory) in biology as a cognitive activity that is to be investigated naturalistically. I first contrast how philosophers and biologists have tended to think about theory in the last century or so, and consider recent calls to upgrade (...) the role of theory in the life sciences against the background of the recent “data deluge” in molecular biology, systems biology, etc. I then review thinking about theory in biology in relation to physical theory as a positive or negative exemplar. I conclude by discussing various aspects of a positive program for “naturalizing theorizing.”. (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)

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)

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)

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)

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)

This paper discusses the viability of claims of mathematical beauty, asking whether mathematical beauty, if indeed there is such a thing, should be conceived of as a sub-variety of the more commonplace kinds of beauty: natural, artistic and human beauty; or, rather, as a substantive variety in its own right. If the latter, then, per the argument, it does not show itself in perceptual awareness – because perceptual presence is what characterises the commonplace kinds of beauty, and mathematical beauty is (...) not among these. I conclude that the reference to mathematical beauty merely expresses the awe in the mathematician about the intricate complexities and simplicity of certain proofs, theorems or mathematical “objects.”. (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)

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.

Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘ mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence (...) may possess a common explainer, they have no common explanation ; that two mathematical facts have a unified explanation makes their truth non-coincidental. I suggest that any motivation we may have for thinking that there are mathematical coincidences should also motivate us to think that there are mathematical explanations, since the notion of a mathematical coincidence can be understood only in terms of the notion of a mathematical explanation. I also argue that the notion of a mathematical coincidence plays an important role in scientific explanation. When two phenomenological laws of nature are similar, despite concerning physically distinct processes, it may be that any correct scientific explanation of their similarity proceeds by revealing their similarity to be no mathematical coincidence. (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)

During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are ‘internally’ connected to their canonical applications, i.e. , that mathematical theorems are ‘hardened’ empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications for all (...) of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay. (shrink)

Abstract: Particularism is usually understood as a position in moral philosophy. In fact, it is a view about all reasons, not only moral reasons. Here, I show that particularism is a familiar and controversial position in the philosophy of science and mathematics. I then argue for particularism with respect to scientific and mathematical reasoning. This has a bearing on moral particularism, because if particularism about moral reasons is true, then particularism must be true with respect to reasons of any sort, (...) including mathematical and scientific reasons. (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)

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)

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)

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)

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)

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)

This essay revisits the topic of how we should measure the things that matter, at a time when we continue to mismeasure our lives, as we hold fast to outworn myths of usefulness, popularity, and the desire to influence others. /// Three central, unquestioned presumptions have come to govern much of contemporary society, education, and the professions. They are: the high value placed on usefulness, on the passion to achieve popularity, and on the desire to influence others. In this essay, (...) the psychologist-philosopher author makes the case against these presumptions, presumptions which lead to exclusionary commitments that stand in the way of human cultural development. (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)

ABSTRACT Mathematicians frequently use aesthetic vocabulary and sometimes even describe themselves as engaged in producing art. Yet aestheticians, in so far as they have discussed this at all, have often downplayed the ascriptions of aesthetic properties as metaphorical. In this paper I argue firstly that the aesthetic talk should be taken literally, and secondly that it is at least reasonable to classify some mathematics as art.

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)

Prior Analytics by the Greek philosopher Aristotle and Laws of Thought by the English mathematician George Boole 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.… using mathematical methods … has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to … when Boole's masterpiece was published.Paul Rosenbloom 1950. (shrink)

ArgumentIn the decades following World War II, the Cowles Commission for Research in Economics came to represent new technical standards that informed most advances in economic theory. The public emergence of this community was manifest at a conference held in June 1949 titledActivity Analysis of Production and Allocation. New ideas in optimization theory, linked to linear programming, developed from the conference's papers. The authors’ history of this event situates the Cowles Commission among the institutions of postwar science in-between National Laboratories (...) and the supreme discipline of Cold War academia, mathematics. Although the conference created the conditions under which economics, as a discipline, would transform itself, the participants themselves had little concern for the intellectual battles that had defined prewar university economics departments. The authors argue that the conference signaled the birth of a new intellectual culture in economic science based on shared scientific norms and techniques un-interrogated by conflicting notions of the meaning of either science or economics. (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)

This paper examines the view of ethical language that Wittgenstein took in later years. It argues that according to this view, ethics falls into place as a part of our natural history, while every sense of the mystical or supernatural that once surrounded it is irrevocably lost. Moreover, Wittgenstein argues that ethical language does not correspond to reality “in the way” in which a physical theory does. I propose an interpretation of this claim that shows how it sets his view (...) apart from a “realist” theory of ethics. The reality of which he speaks is the reality of human life. (shrink)

The concept of the "linear model of innovation" was introduced by authors belonging to the field of innovation studies in the middle of the 1980s. According to the model, there is a simple sequence of steps going from basic science to innovations - an innovation being defined as an invention that is profitable. In innovation studies, the LMI is held to be assumed in Science the endless frontier , the influential report prepared by Vannevar Bush in 1945. In this paper, (...) it is argued that: the LMI was introduced with critical purposes, as part of the questioning of the conception of science and the proposals for science policies put forward in Sef; at a first level of analysis, the LMI appears as a straw man, defended neither in Sef, nor anywhere else; the LMI is a weapon against the importance attributed to basic science in Sef, and its defense of the financing of basic research by the state; the LMI is a component of the process of commodification of science promoted by neoliberalism. The last section of the paper presents a qualified defense of basic science and basic research. (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)

This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem — that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena (...) cannot correctly answer. The connections between explanation, depth, unification, power, and coincidence in mathematics and science are compared. (shrink)

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

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)

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

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)

Mathematicians often conduct aesthetic judgements to evaluate mathematical objects such as equations or proofs. But is there a consensus about which mathematical objects are beautiful? We used a comparative judgement technique to measure aesthetic intuitions among British mathematicians, Chinese mathematicians, and British mathematics undergraduates, with the aim of assessing whether judgements of mathematical beauty are influenced by cultural differences or levels of expertise. We found aesthetic agreement both within and across these demographic groups. We conclude that judgements of mathematical beauty (...) are not strongly influenced by cultural difference, levels of expertise, and types of mathematical objects. Our findings contrast with recent studies that found mathematicians often disagree with each other about mathematical beauty. (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)

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)

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)