Claude P. Bruter (editor), Mathematics in Art: Mathematical Visualization in Art and Education, Springer-Verlag, New York, 2002, pp. X + 337, ISBN 3-540-43422-4.

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)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 37.1 (2003) 1-12 [Access article in PDF] Art and Mathematics in Education Richard Hickman and Peter Huckstep We begin by asking a simple question: To what extent can art education be related to mathematics education? One reason for asking this is that there is, on the one hand, a significant body of claims that assert that mathematics is an art, and, (...) on the other, work in art that has a mathematical basis. Observations of these kinds are not trivial. They have significant implications for the teaching of these areas of the curriculum in at least two ways. First, there is the methodological issue of the extent to which we should teach mathematics and art separately, and second, the teleological question of why they appear in the curriculum at all. So the relationship between the nature of mathematics and art, perceived or real, bears down on questions of the individuation and the justification of these disciplines, or, in other words, upon pedagogy and purpose. Although in principle both the pedagogy and the purpose of any discipline are distinct, there are important connections between them, as we shall draw out.As far as mathematics goes, it has been stressed that the purposes of even rudimentary aspects of mathematics such as counting do need to be made explicit to pupils. 1 What have been seen as errors in understanding are internally linked with pupils' ideas of the purposes of counting. Anna Sierpinska discusses a similar situation within the broader aspect of what she calls "theory" in higher mathematics. 2 More generally, John Passmore, in his analysis of pupils' understanding includes the case where a pupil sees no need to understand since he or she cannot understand the point of learning the subject. 3 If the purposes of art and mathematics are strikingly similar then there is reason to suppose that certain pedagogical approaches that [End Page 1] follow from this will improve understanding of both mathematics and art, and therefore understanding of reasons for engaging in math-based and art-based activities.In schools we see what appears to be some evidence of connections between art and mathematics in cross-curricular work, such as that described in Jacqueline Cossentino and David Schaffer. 4 Apart from providing much-needed motivation, an important value of this approach toward mathematics is that it can illuminate pupils' understanding of some of its purpose by, at the very least, allowing them to recognize instances of mathematics at work, often in unsuspecting contexts. In particular, by facilitating some transference of knowledge pupils can appreciate the so-called "power" of mathematics in its widespread application to the world of artefacts produced by themselves and others.Can we go further than displaying connections between these disciplines? Can we treat mathematics itself as art? In particular, can we suppose that pupils can make mathematics just as they can make artefacts? Some writers seem to imply this by invoking the notion of creativity as a common property between the two disciplines. 5 Leaving aside the ironic sense of creativity in, for example, the case of "creative accountancy" pupils must indeed "make" sense of mathematics and they must originate mathematical assertions of their own that have not been directly taught. In both respects mathematics is similar to the learning of a language. Further than this, we value pupils' inventiveness in their approaches to areas of mathematics, calculations for example.The linking of mathematics and art via creativity has its most persuasive force in the change of attitude toward axiomatic foundations. 6 The story is familiar: for centuries Euclidean geometry was (wrongly) supposed to offer truths of the actual space in which we lived, since its axioms were thought to be confirmed by our intuitions of space. However, when it was found that consistent and useful axiomatic systems could be constructed by making certain alterations to these axioms, it became clear that mathematics was not simply discovered. It involved a significant measure of creativity. Yet one thing is clear: the sense that children make of mathematics, the original utterances they make... (shrink)

In modern mathematics the value of applied research increases, for this reason, modern mathematics is initially focused on resolving the situation actually arose in this respect on a par with other disciplines. Using a new tool - computer systems, applied mathematics appealed to the new object: not to nature, not to society or the practical activity of man. In fact, the subject of modern applied mathematics is a problem situation for the actor-person, and the study is (...) aimed at solving the material and practical problems. Developing the laws of its internal logic, mathematical science today covers various practices and makes them its own requirements, subjugating and organizing in their own way a subject of study. This math activity is influenced by such external factors as the condition of the actor-person; capacity of computers together with service professionals; characteristics of the object, and the external circumstances act as constituting the essence of the case and not as minor issues. The modern applied mathematics is becoming more and more similar to the engineering sciences, and the importance of mathematical modeling problem is rising. In the studies the author bases on the broader context of modern science, including works of philosophy and methodology, as well as mathematicians and specialists in the field of natural sciences. (shrink)

The lecture is an argument for a marriage of theory and experience. It contains something old, something new, something borrowed and something true. The argument is that the dichotomy between science and art, between theory and experience is resolved and the components unified when the role of consensus in the acceptance of theory and the conception of experience is made clear. Moreover, the unification achieved brings with it a method for unifying the empiricism of Moritz Schlick1 with the consensualism of (...) Otto Neurath.2 In earlier work, I developed a mathematical model of consensus with Carl Wagner. That work concerned the aggregation of probabilities by weights of respect assigned by members of a community to each other that converged toward a consensual probability. The convergence depended on connectedness among the members of the group in terms of positive weights. We thought of the probabilities as probabilities of scientific theories or hypotheses. The process converging toward consensus was mathematically equivalent to finding consensual weights to be assigned to members of the group that could be used as a measure of competence and expertise. The consensual weights can be applied to find consensus concerning factors other than probabilities. The use of probabilities was in fact an incidental, though formally well articulated, application of the underlying aggregation of weights to obtain convergence toward consensual weights. (shrink)

This paper concerns the two arts of measurement discussed at Statesman 283-287b. In particular, it argues against the standard interpretation of the first art of measurement, according to which the various branches of mathematics are instances of the first art. Having argued against this standard view, this paper then supplies a more accurate interpretation in its place. Furthermore, it discusses the consequences of this interpretive disagreement for how we understand the relationship between the Statesman's art of measurement and Aristotle's (...) doctrine of the mean. (shrink)

Georg Agricola’s De Re Metallica (1556) is an impressive work about mining arts and sciences, still used as a reference work to understand the mining world of his time and to analyze the relationships between scholars and practitioners. This essay begins by studying how Agricola presents the underground surveying, also known as geometria subterranea. How does the author’s mathematics relate to contemporary geometry or to actual surveying practices? Second, the essay contrasts his scholarly approach with sixteenth-century administrative and technical (...) documents, as well as mining sermons. Comparing the practical geometry contained in the De Re Metallica with different kinds of sources helps to question the relationships between the arts and natural and exact sciences in the practical settings of early modern Europe. Finally, it argues for a distinction between the inquiry on the historical progress of mathematics in civil life and philosophical questions about the rise of new sciences. (shrink)

In this paper, we introduce a suppositional view of linguistic practice that ranges over fiction, science, and mathematics. While having similar con- sequences to some other views, in particular Linsky and Zalta’s plenitudinous platonism, the view advocated here both differs fundamentally in approach and accounts for a wider range of phenomena and scientific discourse.

It is well known that the largest philosophers differently explain the origin of mathematics. This question was investigated in antiquity, a substantial and decisive role in this respect was played by the Platonic doctrine. Therefore, discussing this issue the problem of interaction of philosophy and mathematics in the teachings of Plato should be taken into consideration. Many mathematicians believe that abstract mathematical objects belong in a certain sense to the world of ideas and that consistency of objects and (...) theories really describes mathematical reality, as Plato quite clearly expressed his views on math, according to which mathematical concepts objectively exist as distinct entities between the world of ideas and the world of material things. In the context of foundations of mathematics, so called ‘Gödel’s Platonism‘ is of particular interest. It is shown in the article how Platonic objectification of mathematical concepts contributes to the development of modern mathematics by revealing philosophical understanding of the nature of abstraction. To substantiate his point of view, the author draws the works of contemporary experts in the field of philosophy of mathematics. (shrink)

Andrew Marr has built this masterful study of Mutio Oddi on a set of ironies. He begins with a bitter blow of fortune: Oddi, in the middle of an apparently promising life as mathematician and architect in his native Urbino, had fallen afoul of his lord the Duke, accused of participating in a plot to depose him. After years of apparently unjust imprisonment, he was released in 1610, but into exile. Yet Oddi managed to recast his career in Milan and (...) then in Lucca, building upon a varied set of skills, and even returned eventually to Urbino. That varied set of skills had resulted from yet another, earlier, set of reversals and recoveries: he had turned to mathematics only after first training as an artist in the studio of Federico Barocci, a field he had been forced to abandon due to problems with his eyesight. Oddi was widely respected in his day, not only for his achievements themselves but also for his persistence and ingenuity in overcoming such obstacles. Yet to modern historians of .. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 38.1 (2004) 12-22 [Access article in PDF] Art of a Child with Autism:Drawing Systems and Proto Mathematics Julia Kellman Sung, a five year old girl with autism, was enrolled by her mother in the university Saturday art program with the hope that Sung's favorite church school teacher, a graduate student in art education, would be able to tutor her daughter during the weekly (...) classes. Though hesitant because of her inexperience with autistic children and because of Sung's frequently angry tantrums on Sundays and the little girl's inability to speak, the young teacher, nonetheless, agreed to the plan. Her concerns proved unfounded, for Sung was able to work in the midst of the churning mass of noisy pre-K classmates without any signs of the fury that drove her to screaming, biting fits in church school. Additionally, and to everyone's delight, Sung's ability to develop as an image-maker was itself confirmed within the first few weeks of art class, dispelling any remaining doubts of her art aptitude or the value of art classes for the little girl.Additional questions and considerations arose about Sung and her art during the classes. How might one understand Sung's changing imagery? What might such explanations and images suggest about Sung herself and her activities? What might her art indicate about her similarity to other children and their art making? The possible answers to these and related queriespresented themselves during the semester, suggesting new avenues for inquiry surrounding both the wordless little girl and her Saturday morning creations.What follows is the story of Sung and her graphic development as it unfolded during the art class. Further, it is an attempt to uncover what Sung's drawings suggest about methods that are useful in the examination of the art of an autistic child. Documentation was carried out with a 35 mm. camera and color copier throughout the semester. Ethnographic methods including participant observation and interviews were employed to gather contextual data. Alfred Schutz's phenomenological sociology allowed for [End Page 12] consideration of nonverbal and verbal social interactions and behaviors. 1 A search of the literature of early mathematical development and drawing systems provided the underpinnings for early insights and later hypotheses. Here is Sung's story. Sung Sung, thick black hair swung forward to hide her face, lies prone in the middle of the busy classroom oblivious to the surging, chattering children around her, so intent is she on the sketchbook in which she draws. Next to her left elbow teeters a stack of a half dozen boxes of thick craypas from which she snatches colors, never looking at the hue she grabs. Her drawing behavior has a pattern; it looks like this. Shuffle through the identical boxes. Choose a box. Open it. Snatch out a bright waxy craypa. Hold it close to the eyes. Sniff it. Lick it. If the craypa passes muster, scribble with it in rich heavy lines — banana-like arcs, large solid dots the size of cherries, rectangular patches, and a variety of forms that come and go from one drawing to the next, like heat lightening on a summer evening. Her less frequent forms include: an enclosed yellow irregular shape with a dark dot in the middle suggesting a fried egg; a two-sided zig-zag; X forms; and solid and empty trapezoids made sail-like with superimposed crosses serving as mast and boom.Unlike the chosen craypas, however, those that do not pass Sung's inspection suffer a terrible fate; they are broken in two and their pieces tossed without ceremony to opposite sides of the room to be trodden into flat, bright smears by the other children in the class, pathetic traces of Sung's art. The craypas boxes themselves also play a role. For when they are closed they are used as blocks or toys, objects to be neatly stacked and restacked or placed side by side on the floor. When they are open, however, they are mere receptacles, passively holding... (shrink)

The author of the work proposes a philosophical and methodological interpretation of the mathematical objects, using the system triad of the main directions of substantiation of mathematics: the formalism of Hilbert, Brouwer’s intuitionism and Godel’s Platonism. The need for these directions in the concept of substantiation of mathematics from the point of view of the current state of the philosophy of mathematics is shown on the mathematical examples. The philosophical and methodological analysis of objects of mathematics (...) has never been unambiguous, therefore in this paper the results of studies of philosophers, logicians and mathematicians, in which the problem of substantiation is explicated in the context of trends in the development of mathematics, are used. Their professional view on philosophical characteristics of the objects of mathematics contributes to the identification of the unity of all mathematical knowledge maintaining the initial mathematical base of knowledge and revealing new ways of integrating the directions of substantiation in the philosophy of mathematics. The practical problem of substantiation of mathematics is realized through the elaboration of metatheoretical knowledge under the paradigm shift in the philosophy of mathematics to the productive direction from analysis to synthesis. (shrink)

In the present paper the roots of the Art of the Catalan philosopher Ramon Llull are examined. Moreover the impact of the Art on seventeenth mathematics is briefly discussed.

Mathematical images occur in lectures, books, notes and posters, and on the internet. We extend Kennedy's proposal for classifying these images. In doing so we distinguish three uses of images in mathematics: iconic images; incidental images; and integral images. An iconic image is one that so captures the essence of a concept or proof that it serves for a community of mathematicians as a motto or a meme for an area or a result. A system such as Euclid's can (...) combine such apprehensions with other forms of logical inference and an image that is built into a system of exposition is called an integral image. An incidental image is an image used by a mathematician to reason with a particular concept. In addition to this thematic characterization, we also explore one concept, infinity, in some depth by comparing representations of the infinite by mathematicians and artists. (shrink)

Greek philosophical tradition, not only the Aristotelian one, is strongly associated with proportion (Eco, 1993: 90). This principle of symmetry is generalisable; forasmuch as it is used as a normative rule in figurative arts. Nonetheless, the proportion for Ancient Greeks does not only describe a mathematical relation, but also represents a metaphysical principle. Thus, beauty is the measurement of the elements of the external form (in the case of tragedy, the meter, the symmetry of the parts, the number of syllables, (...) the number of verses) as well as the aesthetic harmony which the play transmits as total to the spectators (in the same case, the content must be neither superior nor inferior to the chosen form; otherwise the form would give the impression of grotesque or inadequate). (shrink)

The Liberal Arts deal with the human being as a whole and hence with what lies at the essence of being human. As a result, the Liberal Arts have a far greater capacity to do good than other fields of study, for their foundation in philosophy enables them to bring students into contact with the ultimate questions which they are free to accept. Even if these questions have little or no ‘market value’, it should be obvious that the way they (...) are taught and learned is going to have a powerful impact upon the future of the students and society. It is suggested here that mathematics has an integral role in the study of the liberal arts in a first degree at a university where the ‘meal ticket’ is subsequently studied in the graduate or professional school. (shrink)

Wenn die fuhrende "paradigmatische" Philosophie Voraussetzungen wie die der Spaltung der Wirklichkeit in sich schliesst und wenn diese Philosophie empirisch die absolut leere -- ohne jede Moglichkeit der wissenschaftlichen Untersuchung --ist, so ist auch die Behauptung der Unmoglichkeit des Irrefuhrens und der Uberflussigkeit solcher Philosophie sinnvoll. Wenn aber auch die bedeutendste Kritik dieses paradigmatischen Rahmens eine Schwache zeigt, ist auf das Bedurfnis einer "Verweigerung" zu schliessen. Darum bedarf es einer Redefinition des Begriffs des aprioristischen Wissens. "A priori" bezeichnet in diesem (...) Sinne die Beschaftigung mit Wissen, das von den Objekten der empfindlich vernehmbaren Welt absolut unabhangig ist. In Ubereinstimmung mit dieser Definition ist das Verstandnis der Mathematik als der Art des aprioren Wissens moglich, oder epistemologisch gesagt: unter der Aktualitat der erreichten mathematischen Moglichkeit. (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)

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)

Then involvement is essential. You cannot stand aside as a member of a community. The real involvement presupposes first a connection at the level of understanding the phenomenon, an empathic relationship with the studied phenomena and processes, relationship that generate intuitions, leaps in understanding and knowledge, discoveries, revelations. The paper is based on the idea that information technology and communications today generate a global interaction based on a complex computer network, which forms a sphere that covers the planet, assimilated with (...) a global intelligence, to which we can consciously adhere becoming a synapse. a global neural network. (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)

This work, in continuity with the article published by Ferdinando Casolaro and Giovanna Della Vecchia in Vol 5, 2017 of this series, in which we noted that in the centuries since the eight century B.C. at the 13th century A.D. the evolution of Astronomy and historical events have influenced the development of Mathematics, intends to demonstrate how the Architecture and Literature of the following centuries have further conditioned the development of the sciences in Italy and, in particular, of (...) class='Hi'>Mathematics, identifying the affinities and, in some cases, the coincidences between the different forms of thought that invite us to make a serious reflection on the uniqueness of culture. (shrink)

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.

ACMES is a multidisciplinary conference series that focuses on epistemological and mathematical issues relating to computation in modern science. This volume includes a selection of papers presented at the 2015 and 2016 conferences held at Western University that provide an interdisciplinary outlook on modern applied mathematics that draws from theory and practice, and situates it in proper context. These papers come from leading mathematicians, computational scientists, and philosophers of science, and cover a broad collection of mathematical and philosophical topics, (...) including numerical analysis and its underlying philosophy, computer algebra, reliability and uncertainty quantification, computation and complexity theory, combinatorics, error analysis, perturbation theory, experimental mathematics, scientific epistemology, and foundations of mathematics. By bringing together contributions from researchers who approach the mathematical sciences from different perspectives, the volume will further readers' understanding of the multifaceted role of mathematics in modern science, informed by the state of the art in mathematics, scientific computing, and current modeling techniques. (shrink)

This text is organized around the distinction between finite and infinite. It includes a brief overview of what different philosophers have said about infinity, and looks at some of the arguments to the effect that one should adopt a pro-infinity attitude. Other chapters contain an exposition of the ontological schools; interactions among these schools and various theories of truth; the relationship between mathematics and values; a history of mathematics; an analysis of mathematical knowledge; the role of mathematics (...) in eduction; the implications of religion for the philosophy of mathematics; and referenc eto mathematical objects. (shrink)

Despite progress, gender gaps persist in mathematical and language-related fields, and gender stereotypes likely play a role. The current study examines the relations between parents’ gender-related beliefs and their adolescent child’s motivation and career aspirations through a survey of 172 parent-child dyads. Parents reported their gendered beliefs about ability in mathematics and language arts, as well as their prescriptive gender role beliefs. Students reported their expectancies and values in these two domains, as well as their career aspirations The results (...) of path models suggested that parents’ ability stereotypes about language boosted girls’ motivation for language arts, thereby nudging them away from STEM pathways. Girls’ career aspirations stemmed not only from their valuation of the corresponding domain, but also from their valuation of competing domains. Such findings highlight the need to consider multiple domains simultaneously in order to better capture the complexity of girls’ career decisions. For boys, parents’ language ability stereotypes were directly related to mathematical career aspirations. These results suggest that stereotypes that language arts is not for boys push them instead toward mathematics. Our study also highlighted the unique role of parental beliefs in traditional gender roles for boys’ motivation and career aspirations. Specifically, parents’ gender role stereotypes directly related to less interest in language arts only among boys. This highlights that research into gender gaps in female-dominated fields should consider stereotypes related to appropriate behavior and social roles for boys. (shrink)

Why are mathematicians drawn to art? How do they perceive it? What motivates them to pursue excellence in music or painting? Do they view their art as a conveyance for their mathematics or an escape from it? What are the similarities between mathematical talent and creativity and their artistic equivalents? What are the differences? Can a theatrical play or a visual image capture the beauty and excitement of mathematics? Some of the world's top mathematicians are also accomplished artists: (...) musicians, photographers, painters, dancers, writers, filmmakers. In this volume, they share some of their work and reflect on the roles that mathematics and art have played in their lives. They write about creativity, communication, making connections, negotiating successes and failures, and navigating the vastly different professional worlds of art and mathematics. (shrink)

Mathematical practitioners in seventeenth-century London formed a cohesive knowledge community that intersected closely with instrument-makers, printers and booksellers. Many wrote books for an increasingly numerate metropolitan market on topics covering a wide range of mathematical disciplines, ranging from algebra to arithmetic, from merchants’ accounts to the art of surveying. They were also teachers of mathematics like John Kersey or Euclid Speidell who would use their own rooms or the premises of instrument-makers for instruction. There was a high degree of (...) interdependency even beyond their immediate milieu. Authors would cite not only each other, but also practitioners of other professions, especially those artisans with whom they collaborated closely. Practical mathematical books effectively served as an advertising medium for the increasingly self-conscious members of a new emerging professional class. Contemporaries would talk explicitly of ‘the London mathematicians’ in distinction to their academic counterparts at Oxford or Cambridge. The article takes a closer look at this metropolitan knowledge culture during the second half of the century, considering its locations, its meeting places and the mathematical clubs which helped forge the identity of its practitioners. It discusses their backgrounds, teaching practices and relations to the London book trade, which supplied inexpensive practical mathematical books to a seemingly insatiable public. (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)

Maurits Cornelis Escher’s 1938 lithograph, Cycle, illustrates what mathematical physicist Roger Penrose calls ‘impossible objects’. The illusion of three-dimensionality, the innovative use of tessellation, and the incorporation of traditionally figurative elements induce the viewer to perceive the lithographic print as depicting a visually plausible reality built on the deconstructive metamorphosis of man into cube. It is Escher’s ability to paradoxically combine the radical oppositions of man and cube, landscape and geometric abstraction into an apparently harmonious composition where shapes repeat with (...) subtle variation and almost imperceptibly transition into radically different shapes that I want to use as a visual paradigm for conceptualizing the shifts in human consciousness in the digital age. Roger Penrose, path-breaking theorist of consciousness as a physical but non-computational process, was inspired by and inspired Escher. Penrose’s provisional concepts of consciousness, time and space can be visually compared to Escher’s earlier man-cube renderings, where each of these individual design elements allows the transition to another shape. In other words, Penrose’s theories, like Escher’s design concepts, illustrate a moment of transition in consciousness theory. As in Escher’s Cycle, the individual elements are being continually redefined, reshaped and revealed. Each may be a unique distortion of the previous, but this does not mean that it is disconnected or independent of the final outcome. These distortions are transitional elements that lead us visually to the other side of the composition. Radical transformations and developing connectedness in the boundaries between art, technology and science function like the individual elements in Escher’s Cycle. As we try to define the shifts in human consciousness occurring in the digital age, visually speaking, we are focusing on only one element right in the middle of an Escher metamorphosis, unable to see exactly how this element will be part of the completed design. This thought can be both frightening and liberating: frightening because we humans are used to relying on boundaries and predictable outcomes to feel a sense of control, liberating because anything and everything is possible and the changes will occur naturally as part of the transition. If Escher teaches us anything, it is that each piece in the meta-morphosis is needed to complete the big picture. (shrink)

This article asserts the significance of the portrayal of the political art of statesmanship as weaving, and aims to show how this image emphasizes two main aspects of the political art of statesmanship. Firstly, the image implies a three-dimensionality, both through the process of weaving and through the thickness of the protective fabric this produces, that in turn indicates the vital aspect of corporeality in politics. Secondly, weaving as a paradigmatic example of the art of statesmanship presents a way of (...) incorporating different entities and joining diverse threads into a cohesive unity, without reducing them to a form of sameness as mere mathematical counting would. These two aspects are in turn connected to an emphasis on the importance in good statesmanship of recognizing the specificity of the occasion and the variety and difference of the innumerable qualities to be interlaced in weaving the fabric of a good society. (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)

In lieu of an abstract, here is a brief excerpt of the content:Abraham, Planter of Mathematics":Histories of Mathematics and Astrology in Early Modern EuropeNicholas PopperFrancis Bacon's 1605 Advancement of Learning proposed to dedicatee James I a massive reorganization of the institutions, goals, and methods of generating and transmitting knowledge. The numerous defects crippling the contemporary educational regime, Bacon claimed, should be addressed by strengthening emphasis on philosophy and natural knowledge. To that end, university positions were to be created (...) devoted to "Artes and Sciences at large," rather than to the professions. High salaries would render lecturers "able and sufficient," undistracted from their task. Most famously, he argued that teaching of the "operatiue studie of many Scyences" should involve sophisticated technical education. The study of natural philosophy demanded not only books, but globes, astrolabes, and other "instrumentals." Most significantly, yielding reliable and meaningful knowledge from experiential gleanings required a rigorous system of deductive reasoning.The legacy of this colossal proposal has earned Bacon honored status as devisor of the scientific method.1 But Bacon's educational reform extended [End Page 87] beyond the methods of producing and transmitting knowledge. To facilitate more efficient "vse and administration" of the knowledge produced by his system, he also demanded a searching examination of the history of learning. This history would provide a mirror enabling his contemporaries to deploy the fruits of his method, by considering how learning in the past had been used successfully or ill-advisedly. Producing this history was the ambition of the Advancement, for he explained: "no man hath propounded to himselfe the generall state of learning to bee described and represented from age to age, as many have done the works of Nature, & the State civile and Ecclesiastical." He continued:And yet I am not ignorant that in divers particular sciences, as of the Iurisconsults, the Mathematicians, the Rhetoricians, the Philosophers, there are set down some smal memorials of the Schooles, Authors, and Bookes: and so likewise some barren relations touching the Invention of Arts, or usages. But a iust story of learning, containing the Antiquities and Originalls of Knowledges, & their Sects; their Inventions, their Traditions; their diverse Administrations and Managings; their Flourishings, their Oppositions, Decayes, Depressions, Oblivions, Removes, with the causes, and occasions of them, and all other events concerning learning, throughout the ages of the world; I may truly affirme to be wanting.2Bacon thus positioned himself not only as a Father of Modern Science, but also a Father of the History of Science.Following Bacon's suggestion, I will examine the "small memorials" of the history of mathematics—and particularly the mathematical art of astrology—in the sixteenth and seventeenth centuries. My conclusion, however, will not bear out Bacon's claims. Despite his frustration, Bacon was only one of many early modern scholars appraising the role of mathematics within history. And he devoted less energy than others to mapping its origins and tracing its transmissions between communities. In fact, Bacon's 1605 proposal for a history of mathematics was already out-of-date. Discussions [End Page 88] of the history of mathematics had been rife on the continent and in England throughout the previous century.3The evidence that Renaissance scholars inherited was sprawling and inconclusive. Several genealogies for mathematics could be found within classical Greek, Latin, and patristic references. One lineage claimed mathematics began in ancient Assyria, where the priestly caste, the Chaldeans, practiced a form of mathematics that seemed a corrupt admixture of philosophy, medicine, and religion, and relied heavily on observation of the heavens. Other scholars traced the origins to Egypt, claiming that the field developed to survey lands frequently flooded by the Nile. The real problem for ancient, late antique, and medieval scholars had not been the origins of mathematics, but what exactly mathematics were. For some, the term strictly referred to the quadrivium: geometry, arithmetic, astronomy, and music. But for others, mathematics' origins amongst the Egyptians or Chaldeans inextricably linked it to forms of astrological divination, augury, and necromancy that were unsavory to both Latin and Christian traditions. Mathematici were included alongside ghastly lists of Magi, Brahmins, Aruspices, Genethliaci, and other diabolic practitioners of idolatrous magic. These might or might not be distinguished from mathematici such as Pythagoras... (shrink)