The article is devoted to identifying the value of the phenomenon of aesthetic value and beauty of mathematical knowledge and the beauty of mathematical theory of teaching mathematics. The aesthetic potential of mathematical knowledge allows the use of theater technology in the educational process with the active dialogic interaction between teacher and students. The criteria of beauty in mathematical theories are distinguished: the realization of beauty as the unity of the whole, and in the disclosure of the complex through (...) the elementary; methodological interpretation of the beauty in the community of mathematical structures and optimal information content of the meta-language of mathematics; the practical embodiment of beauty in the formalization of the infinite through the finite. The beauty of mathematics is the force that permeates all the ‘layers of knowledge‘ not along, and across, although the effectiveness of mathematical activity due to aesthetic laws, which do not always lend themselves to unambiguous interpretation. In the article it is stated that, depending on the educational goals of communicative impact on the audience, in fact, ‘mathematical lectures theatricality‘ can have different characteristics, the most important of which are teachers artistry and artistic director’s work of a teacher. This cultural phenomenon that includes the theatrical talent, helps create an atmosphere of cooperation needed in varying degrees of activity for pedagogical interaction. The author believes that such approach, developed on the basis of the Stanislavsky system, allows university professors of mathematics significantly improve mathematical lectures. (shrink)

This volume builds on two recent developments in philosophy on the relationship between art and science: the notion of representation and the role of values in theory choice and the development of scientific theories. Its aim is to address questions regarding scientific creativity and imagination, the status of scientific performances--such as thought experiments and visual aids--and the role of aesthetic considerations in the context of discovery and justification of scientific theories. Several contributions focus on the concept of beauty as employed (...) by practising scientists, the aesthetic factors at play in science and their role in decision making. Other essays address the question of scientific creativity and how aesthetic judgment resolves the problem of theory choice by employing aesthetic criteria and incorporating insights from both objectivism and subjectivism. The volume also features original perspectives on the role of the sublime in science and sheds light on the empirical work studying the experience of the sublime in science and its relation to the experience of understanding. The Aesthetics of Science tackles these topics from a variety of novel and thought-provoking angles. It will be of interest to researchers and advanced students in philosophy of science and aesthetics, as well as other subdisciplines such as epistemology and philosophy of mathematics. (shrink)

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are (...) complementary but both aesthetic. (shrink)

The paper gives an impression of the multi-dimensionality of mathematics as a human activity. This 'phenomenological' exercise is performed within an analytic framework that is both an expansion and a refinement of the one proposed by Kitcher. Such a particular tool enables one to retain an integrated picture while nevertheless welcoming an ample diversity of perspectives on mathematical practices, that is, from different disciplines, with different scopes, and at different levels. Its functioning is clarified by fitting in illustrations based (...) on actual research. (shrink)

The article is devoted to aesthetic nature of the philosophy of dance as a rapidly developing area of studying. The aesthetic issues of choreographies in the cognitive context have not been properly studied. The mathematical component of the classical ballet, which is shown through the internal patterns of the expressiveness of the different types of dance movements in the system of artistic thinking, is analyzed in a wide range of the philosophical problems of art of dancing. The substantial triad of (...) structure of dance is considered at the level of philosophical research. In it, along with static and dynamic forms of dance, the third component, which reflects physicality, related to the ‘sense of movement‘ or the perception of information through body movements is added to the binary structure of dance. (shrink)

It has been observed that whereas painters and musicians are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work. Mathematicians, instead, are fond of passing judgment on the beauty of their favored pieces of mathematics. Even a cursory observation shows that the characteristics of mathematical beauty are (...) at variance with those of artistic beauty. For example, courses in art appreciation are fairly common; it is however unthinkable to find any mathematical beauty appreciation courses taught anywhere. The purpose of the present paper is to try to uncover the sense of the term beauty as it is currently used by mathematicians. (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)

This article reclaims mathematics from the measures of profit and control by first presenting an anarchist analysis of mathematics? status quo societal uses and pedagogic activities. From this analysis, a vision for an anarchist math education is developed, as well as suggestions for how government school practitioners sympathetic to anarchism can insert this vision into their current work. Aspects to this vision include teacher autonomy, freedom from hierarchical curriculum structure and math class as a non-coercive, happy place. Finally, (...) mathematics is argued to be essential knowledge for anarchistic society for three potentialities: in solving social and technological problems through application, as an analytic technology and for increasing individual happiness via the aesthetic dimension. (shrink)

Mathematics and aesthetics are closely intertwined. Not only mathematical concepts, relationships and theorems can be aesthetically pleasing, but we also often find harmony between their results and the patterns of the world around us, and we like that. Yet, apart from rare exceptions, the beauty of mathematics, particularly in education, is mostly unrecognized: this science rarely meets the favour of students. Vedic mathematics is an approach which encapsulates the enjoyment and power of this knowledge, not only (...) in the sphere of thought process, but also in its practical utility. It highlights and develops the aesthetic dimension of learning in a very immediate sense. The aim of this article is to introduce the method – what it is and how it works – to give comparative examples of techniques and their efficacy, and to emphasize the aesthetic value it conveys. (shrink)

This paper explores the nature of mathematical beauty from a Kantian perspective. According to Kant’s Critique of the Power of Judgment, satisfaction in beauty is subjective and non-conceptual, yet a proof can be beautiful even though it relies on concepts. I propose that, much like art creation, the formulation and study of a complex demonstration involves multiple and progressive interactions between the freely original imagination and taste. Such a proof is artistic insofar as it is guided by beauty, namely, the (...) mere feeling about the imagination’s free lawfulness. The beauty in a proof’s process and the perfection in its completion together facilitate a transition from subjective to objective purposiveness, a transition that Kant himself does not address in the third Critique. (shrink)

Mathematicians often say that their theorems, proofs, and theories can be beautiful. They say mathematics can be like art. They know how to move creatively and freely in their domains. But ordinary people usually cannot do this and do not share this view. They often have unpleasant memories from school and do not have this experience of freedom and creativity in doing mathematics. I myself have been a mathematician, and I wish to highlight some of the creative aspects (...) in doing mathematics. I always had the feeling that there is much freedom in mathematics and that one can do as one pleases as long as one avoids contradictions (and one can even live with contradictions for a while). In mathematics one only needs to define something and there it is! Just imagine it, and it immediately exists. Where else, besides fiction, does one have such power? --- This essay has three parts. In the first, I lay out some historical facts and views that I will need later. Second, I insert an interlude with Kant, pointing out some of his claims and insights in aesthetics, and some aspects in mathematics that I think he overlooked or underestimated. Third, I will bring out some aspects of higher mathematics that I will show are similar to art. These aspects are in the doing and creating of mathematics, not in the finished theories. They are usually not found in textbooks. I want to show that a researcher in mathematics is like a painter or composer, exploring and creating. To see this, one has to adopt a first-person perspective. In this way I will show that aesthetics plays a role and leaves its traces also in mathematics. (shrink)

This introduction to the special issue on "Aesthetics of Science" reviews recent philosophical research on aesthetic aspects of science. Topics represented in this research include the aesthetic properties of scientific images, theories, and experiments; the relation of science and art; the role of aesthetic criteria in scientific practice and their effect on the development of science; aesthetic aspects of mathematics; the contrast between a classic and a Romantic aesthetic; and the relation between emotion, cognition, and rationality.

Comparisons made between art and mathematics so often centre on the beauty of mathematics and how its forms might be seen as aesthetically pleasing. Yet the prominence of beauty as an attribute is less prevalent in contemporary art. Rather, art has a much broader scope of concern, perhaps with a greater emphasis on providing apparatus through which we might better understand who we are. This paper considers some performative aspects of contemporary art and draws parallels with some examples (...) of mathematical activity within educative contexts. It argues that the performative dimension of mathematics is underemphasised in school activity and that the social and linguistic conditioning of mathematics within performance is a crucial aspect of the discipline being addressed in school and vocational courses. In particular, proficiency with concretisations of mathematics and the social dynamics that attend these is integral to the broader proficiency of moving between concrete and abstract domains. (shrink)

In his influential 1960 paper ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Eugene P. Wigner raises the question of why something that was developed without concern for empirical facts—mathematics—should turn out to be so powerful in explaining facts about the natural world. Recent philosophy of science has developed ‘Wigner’s puzzle’ in two different directions: First, in relation to the supposed indispensability of mathematical facts to particular scientific explanations and, secondly, in connection with the idea that aesthetic (...) criteria track theoretical desiderata such as empirical success. An important aspect of Wigner’s article has, however, been overlooked in these debates: his worries about the underdetermination of physical theories by mathematical frameworks. The present paper argues that, by restoring this aspect of Wigner’s argument to its proper place, Wigner’s puzzle may become an instructive case study for the teaching of core issues in the philosophy of science and its history. (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.

This essay will inform the reader about Kant’s views on mathematics and aesthetics. It will also critically discuss these views and offer further suggestions and personal opinions from the author’s side. Kant (1724-1804) was not a mathematician, nor was he an artist. One must even admit that he had little understanding of higher mathematics and that he did not have much of a theory that could be called a “philosophy of mathematics” either. But he formulated a (...) very influential aesthetic theory that is contained in his “Critique of the Power of Aesthetic Judgment” (1790), and his views on mathematics, especially those that compare it with philosophy, are distinctive and worthy of our attention. Hence, combining mathematics and aesthetics, asking whether mathematics can be beautiful and why and how it can or cannot be so called, according to Kant’s theory, will be particularly interesting. This essay has three parts: it discusses mathematics, aesthetics, and the aesthetics of mathematics, primarily in Kantian perspectives but also in ways that critically evaluate those perspectives. (shrink)

Comparisons made between art and mathematics so often centre on the beauty of mathematics and how its forms might be seen as aesthetically pleasing. Yet the prominence of beauty as an attribute is less prevalent in contemporary art. Rather, art has a much broader scope of concern, perhaps with a greater emphasis on providing apparatus through which we might better understand who we are. This paper considers some performative aspects of contemporary art and draws parallels with some examples (...) of mathematical activity within educative contexts. It argues that the performative dimension of mathematics is underemphasised in school activity and that the social and linguistic conditioning of mathematics within performance is a crucial aspect of the discipline being addressed in school and vocational courses. In particular, proficiency with concretisations of mathematics and the social dynamics that attend these is integral to the broader proficiency of moving between concrete and abstract domains. (shrink)

The paper proposes a reflection on mathematical beauty, considering the possibility of aesthetic qualities for formal language. Through a concise overview of the way this question is understood by some famous scientists and mathematicians, we turn our attention to Gian-Carlo Rota’s theoretical proposal: his reflections as a mathematician and philosopher offer a perspective, of phenomenological matrix, fruitful for looking at the question. Rota’s contribution allows us to focus on the role of competence, acquired through effort, sedimentation and habit of repetition, (...) in cultivating the potential to recognise the aesthetic quality of formal language. Finally, we draw on some contributions from Gestalt theory, closely connected to twentieth-century phenomenology for chronological and conceptual reasons, applying the idea of gestalt wholeness to the mathematical product and proposing some reflections that may help to clarify some of the dynamics underlying the attribution of qualities of beauty to formulas or the detection of its lack. (shrink)

Einstein proclaimed that we could discover true laws of nature by seeking those with the simplest mathematical formulation. He came to this viewpoint later in his life. In his early years and work he was quite hostile to this idea. Einstein did not develop his later Platonism from a priori reasoning or aesthetic considerations. He learned the canon of mathematical simplicity from his own experiences in the discovery of new theories, most importantly, his discovery of general relativity. Through his neglect (...) of the canon, he realised that he delayed the completion of general relativity by three years and nearly lost priority in discovery of its gravitational field equations. (shrink)

We analyze two problems in mathematics – the first (stated in our title) is extracted from Wittgenstein’s “Philosophy for Mathematicians”; the second (“What set of numbers is non-denumerable?”) is taken from Cantor. We then consider, by way of comparison, a problem in musical aesthetics concerning a Brahms variation on a theme by Haydn. Our aim is to bring out and elucidate the essentially riddle-like character of these problems.

In this research we will examine the paradox nature of self-reference. This concept appears in the form of pure feedback loops in language and mathematics and naturally extends towards many different domains such as biology, sociology, art and philosophy. The basic elements of human experience show the manifestations of such loops. Their results are noticeable in internal or external, mental or body processes. Our interest with these loops focuses on the domain of brain processes in observing, thinking and interpreting (...) as it is meticulously analysed in the books Gödel Escher Bach and I Am a Strange Loop. Through the interactive artistic experiments ‘Ouroboros VR Self-Reference Apparatus’ by John Bardakos (2015) and ‘Anima’ by Elhem Younes (2016) we explore applied and theoretical concepts and experiences in logic, dream aesthetics and the human creativity. These experiments were conducted using ‘perception’ as a natural model of aesthetics generator via self-reference. Within this model, the loops were enriched with an artistic metamorphose of data from measurements of the human space to the aesthetic objects in the virtual space. Through the quantifiable perception feedback loop and the digital interaction, the spectator changes his role and becomes an active artist expressing himself or herself to a digital output with the support of data. The now post-conscious artist, with the use of visual and audio-visual tool sets, observes the fusion of art technology and aesthetics being transformed into basic elements that form consciousness. (shrink)

Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory, this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians’ sensitivity to aesthetics of the abstract.

In this paper, we explore the possibility of applying traditional and modern aesthetical theories to logical and mathematical proofs, with the goal of better understanding the intuitive concept of mathematical beauty. This informal concept takes a central role in the work of logicians and mathematicians and can be thought of as their main motivation. In the present paper, we try to define concepts connected to mathematical beauty or beauty in mathematical proofs, so that we may lay the foundations for a (...) more precise definition of mathematical beauty which would be obtained through a detailed survey among logicians and mathematicians, presented in a future paper. The present paper brings crucial results to be used for constructing the survey. (shrink)

This article analyses the conditions under which mathematics could enter the field of fourteenth-century music. It distinguishes between descriptive and argumentative uses of mathematics. Jean de Murs’ uses of arithmetic to study musical time is an example of the former, Jean de Boen’s study of the division of the whole tone an example of the latter. It is furthermore explained how the mathematical descriptions appear to bring into agreement two types of constraint, namely the physical characteristics of sound (...) and the aesthetic principles of the medieval discourse about music. Within these constraints, mathematics manages to fulfill different argumentative roles: it has an ontological function when music is seen as a part of the quadrivium; but an explicative function in the framework of the scientia media and, in an more innovative spirit for Jean de Boens, it provides a definition of the possible in the argumentation about the division of the whole tone. (shrink)

Einstein proclaimed that we could discover true laws of nature by seeking those with the simplest mathematical formulation. He came to this viewpoint later in his life. In his early years and work he was quite hostile to this idea. Einstein did not develop his later Platonism from a priori reasoning or aesthetic considerations. He learned the canon of mathematical simplicity from his own experiences in the discovery of new theories, most importantly, his discovery of general relativity. Through his neglect (...) of the canon, he realised that he delayed the completion of general relativity by three years and nearly lost priority in discovery of its gravitational field equations. (shrink)

The idea of ‘gesture’ is present in the philosophical world in various forms. All of them might find an important theoretical grounding in pragmatist philosophy, if we combine pragmatism with some French philosophies of mathematics and read it as a way out of the Kantian philosophy of representation. The paper uses the insights of Jean Cavaillès to set out the problem of the weakness of the epistemic Kantian defense of mathematical and logical thought. Cavaillès rejected the possible amendments to (...) Kant’s explanation provided by both Husserl and Bolzano and their heirs. He used the word ‘gesture’ in order to explain the activity of mathematicians who have to act synthetically, following rules, with some physical representation, and being aware of the possibility of failure. Cavaillès conceived the use of gesture as an alternative to the Heidegerian idea of event defended by Albert Lautman. The paper then follows the idea of gesture in the French philosophy of mathematics of Gilles Châtelet and Giuseppe Longo. Finally, the paper illustrates how Peirce’s study of Existential Graphs and the main insights of pragmatism complete Cavaillès’s idea by giving to gestures a phenomenological and semiotic structure. The pragmatist philosophy of gesture is thus a new way of overthrowing Kant’s philosophy of representation without surrendering to irrationalism. (shrink)

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)

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)

A green philosopher's peripeteia.--Physics and metaphysics of music.--The roots of arithmetic.--Critique of new geometrical abstractions.--The philosophical value of science.

This paper offers an analysis of Kant’s account of the mathematical sublime with reference to his claim that ‘Nature is thus sublime in those of its appearances the intuition of which brings with them the idea of its infinity’. In undertaking this analysis I challenge Paul Crowther’s interpretation of this species of aesthetic experience, and I reject his interpretation as not being reflective of Kant’s actual position. I go on to show that the experience of the mathematical sublime is necessarily (...) connected with the progression of the imagination in its move towards the infinite. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation. (shrink)

Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’. And he rests these upon consistently opposite grounds. In the Antinomy we are told that we can have (...) no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive. (shrink)

This book offers an archeology of the undeveloped potential of mathematics for critical theory. As Max Horkheimer and Theodor W. Adorno first conceived of the critical project in the 1930s, critical theory steadfastly opposed the mathematization of thought. Mathematics flattened thought into a dangerous positivism that led reason to the barbarism of World War II. The Mathematical Imagination challenges this narrative, showing how for other German-Jewish thinkers, such as Gershom Scholem, Franz Rosenzweig, and Siegfried Kracauer, mathematics offered (...) metaphors to negotiate the crises of modernity during the Weimar Republic. Influential theories of poetry, messianism, and cultural critique, Handelman shows, borrowed from the philosophy of mathematics, infinitesimal calculus, and geometry in order to refashion cultural and aesthetic discourse. Drawn to the austerity and muteness of mathematics, these friends and forerunners of the Frankfurt School found in mathematical approaches to negativity strategies to capture the marginalized experiences and perspectives of Jews in Germany. Their vocabulary, in which theory could be both mathematical and critical, is missing from the intellectual history of critical theory, whether in the work of second generation critical theorists such as Jürgen Habermas or in contemporary critiques of technology. The Mathematical Imagination shows how Scholem, Rosenzweig, and Kracauer’s engagement with mathematics uncovers a more capacious vision of the critical project, one with tools that can help us intervene in our digital and increasingly mathematical present. (shrink)

Many mathematicians, physicists, and philosophers have suggested that the fact that mathematics—an a priori discipline informed substantially by aesthetic considerations—can be applied to natural science is mysterious. This paper sharpens and responds to a challenge to this effect. I argue that the aesthetic considerations used to evaluate and motivate mathematics are much more closely connected with the physical world than one might presume, and (with reference to case-studies within Galois theory and probabilistic number theory) show that they are (...) correlated with generally recognized theoretical virtues, such as explanatory depth, unifying power, fruitfulness, and importance. (shrink)

Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’ (A25/B39). And he rests these upon consistently opposite grounds. In the Antinomy we are told that we can (...) have no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive. (shrink)

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, (...) and strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)

In this paper I reflect on the nature of mathematical beauty, and examine the connections between mathematics and the arts. I employ Plutarch’s distinction between the intelligible and the sensible, to compare the beauty of mathematics with the beauties of music, poetry and painting. While the beauty of mathematics is almost exclusively intelligible, and the beauties of these arts primarily sensible, it is pointed out that the latter share with mathematics a certain kind of intelligible beauty. (...) The paper also contains reflections on the formal beauty and timelessness of mathematics, beauty as richness flowing from simplicity, form and content in mathematics, and mathematics and fiction. It concludes with some remarks on the question of why mathematical beauty is so little appreciated by non-mathematicians. (shrink)

The cultures here in question are those of mathematics and of physics that I shall interpret with the goal of exploring different modes of creativity. As case studies I will consider two scientists who were exemplars of these cultures, the mathematician Henri Poincaré (1854-1912) and the physicist Albert Einstein (1879-1955). The modes of creativity that I will compare and contrast are their notions of aesthetics and intuition. In order to accomplish this we begin by studying their introspections.

This paper attempts to elucidate Wittgenstein’s remark about the “strange resemblance between a philosophical investigation (especially in mathematics) and an aesthetic one” from 1937 by looking at its textual and philosophical context. The conclusion is that the remark can be seen both as a description of a particular conception of philosophy, a prescription or declaration of intent (to proceed in a particular way), and a reminder (to Wittgenstein himself) about the form of a philosophical investigation. Furthermore, it is concluded (...) that the Darstellungsform he has in mind is the one that finds expression especially in the first part of the PI. (shrink)

Both in Force of Imagination: The Sense of the Elemental and in his very recent Logic of Imagination: The Expanse of the Elemental, John Sallis enacts a reconfiguration of the relationship of geometry to elementology, which might be regarded more generally as a rethinking of the relation of mathematics to philosophy. The paper will trace this reconfiguration in two ways: as it lies present but concealed in the history of philosophy, for example, in Descartes’ so-called “dualism” and in Kant’s (...) pure productive imagination, and in its present creative evolution in fractal geometry, as Sallis interprets it. Sallis draws together the mathematical affinity with a fundamental aesthetic drive, likening mathematical patterns to choreographic ones. I conclude by following this strain as it points to specific dance companies, and to my own sense of aesthetic homecoming as presented in my Imagination in Kant’s Critique of Practical Reason. (shrink)

In philosophical discourse, vagueness is commonly regarded as an undesirable and problematic aspect of human experience. Such standpoints are not unfounded. However, in this article, I argue that vagueness may in certain instances also possess an instrumental role that supports specific modes of human aspiration, including the artistic and the mathematical. In particular, I investigate the ways in which vagueness not only hinders but also fosters the emergence of an aesthetic quality of experience during the imaginative endeavours of fine art (...) and mathematics. The manner in which the benefit from felt vagueness ties into the formation of artistic and mathematical scenarios helps to illuminate deep and often unnoticed relations between these two domains of human ingenuity. In this article, the overall concept of experience and the particular features of human experience, such as the aesthetic and the vague, are examined in the context of John Dewey’s philosophical pragmatism. In Dewey’s philosophical framework, the human mind and culture are understood as natural phenomena. As such, the Deweyan approach fits the paradigm of a twenty-first century scientific understanding of the inherent ontological unity of all modes of human existence and activity. (shrink)

Christine Buci-Glucksmann’s__ _The Madness of Vision_ is one of the most influential studies in phenomenological aesthetics of the baroque. Integrating the work of Merleau-Ponty with Lacanian psychoanalysis, Renaissance studies in optics, and twentieth-century mathematics, the author asserts the materiality of the body and world in her aesthetic theory. All vision is embodied vision, with the body and the emotions continually at play on the visual field. Thus vision, once considered a clear, uniform, and totalizing way of understanding the (...) material world, actually dazzles and distorts the perception of reality. In each of the nine essays that form _The Madness of Vision_ Buci-Glucksmann develops her theoretical argument via a study of a major painting, sculpture, or influential visual image—Arabic script, Bettini’s “The Eye of Cardinal Colonna,” Bernini’s _Saint Teresa_ and his 1661 fireworks display to celebrate the birth of the French dauphin, Caravaggio’s _Judith Beheading Holofernes,_ the Paris arcades, and Arnulf Rainer’s selfportrait, among others—and deftly crosses historical, national, and artistic boundaries to address Gracián’s_ El Criticón_; Monteverdi’s opera _Orfeo; _the poetry of Hafiz, John Donne, and Baudelaire; as well as baroque architecture and Anselm Kiefer’s Holocaust paintings. In doing so, Buci-Glucksmann makes the case for the pervasive influence of the baroque throughout history and the continuing importance of the baroque in contemporary arts. (shrink)