On a stroll down a neighbourhood street in Togo, one is likely to see: little boys playing with homemade toys that roll and can be pushed with a stick or pulled on a string; girls helping their mother around the house and tending younger siblings; men sitting chatting with friends, smoking and playing dice games or zipping by on a variety of two-wheelers; women waiting at the pump for their turn to fill their basins before 2 p.m. when the pump (...) is locked for the day and girls doing errands or selling small items. (shrink)

This paper considers two models of embodied mathematical cognition, and analyses the image of mathematics that they support.

In this paper I explore possibilities of bringing post-positivist philosophies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical (...) conceptual innovations. (shrink)

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

Economic methodology has been dominated by developments in the philosophy of science. This book's central thesis is that a great deal can be gained by refocusing attention on developments in the philosophy of mathematics, in particular those that took place over the course of the twentieth century. In this book the authors argue that a close examination of the major developments in the philosophy of mathematics both deepens and enriches our understanding of the formalisation of economics, while also (...) offering novel methods of critically evaluating the strengths and limitations of formalism in economics. This book represents an innovative attempt to more fully understand the complexity of the interaction between developments in the philosophy of mathematics and the process of formalisation in economics. (shrink)

A thorough account of the philosophy of mathematics. In a cogent account the author argues against the view that mathematics is solely logic.

A thorough account of the philosophy of mathematics. In a cogent account the author argues against the view that mathematics is solely logic.

A thorough account of the philosophy of mathematics. In a cogent account the author argues against the view that mathematics is solely logic.

_Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.

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)

Time is fascinating and important to everyone. This book does two things: it reviews a great range of images of time, that is, theories of time, from a diversity of perspectives: historical, religious, biological, mathematical, and scientific. In addition to a wide-ranging discussion of many aspects of time, the book shows how a spreadsheet can be used easily to explore various aspects of time such as time travel and light cones in relativity. But it goes further. It introduces the (...) concept of generalized proposition, which can be used by the reader to distinguish those images of time that are metaphysical (which means they cannot be scientifically validated) and those that could in principle be put to empirical test. This is of particular importance in this day and age, when we are flooded by a plethora of competing images of time. Many of these have no scientific basis, empirical support, or content but are presented as if they do. (shrink)

In this introduction we discuss the motivation behind the workshop “Towards a New Epistemology of Mathematics” of which this special issue constitutes the proceedings. We elaborate on historical and empirical aspects of the desired new epistemology, connect it to the public image of mathematics, and give a summary and an introduction to the contributions to this issue.

The following paper, together with its sequel ("The use and non-use of mathematics"), is a study in the mathematization of nature. It looks into the history of one of the most emblematic achievements of this fundamental aspect of the making of modem science - the Inverse Square Law of universal gravitation - before its celebrated application by Newton to celestial mechanics. What did it take, we ask, to tum a particular mathematical ratio into a candidate for a law of (...) nature? (shrink)

Reale points out that the good and the demiurgic intelligence are radically distinct, a conclusion denied by J. Seifert in the last paper of the book. Fourteen characteristics of the idea of the good are listed by T. A. Szlezák. It is obvious, he argues, that the theory of principles of Plato’s unwritten doctrines is not identical with what Republic 6 and 7 say about the good, but there is no real opposition. In the next paper, however, H. W. Ausland, (...) after a sober evaluation of the esoteric teachings, suspects that Reale and Szlezák succumb to a certain romanticism. According to L. Brisson, the good belongs to the sphere of being, even if it is not being ; it is not in every way beyond being. Plotinus was the first to place it above being. M. Erler shows what must be the attitude of the philosopher with regard to the information orally received by him. Although Plato did not want to define the good, he pointed out several of its characteristics. The good, M. Migliori notes, combines in itself beauty, proportion, and unity. The Philebus pursues the doctrine of the Republic and goes into the direction of the unwritten doctrines. The ultimate reality, however, is beyond our understanding. E. Cattanei attempts to explain how for Plato mathematics is a way of access to the knowledge of the good. R. Ferber, on the other hand, stresses that Plato nowhere directly says that the essence of the good is the one. Beauty, symmetry, and truth are not the good itself. According to C. Gill, Gadamer demystified the idea of the good, a view which shows some common ground with the position of analytical philosophy: the good is not a transcendent entity. M. Vegetti, however, upholds the transcendent character of the good as a cause of being. G. A. Press argues that over the past hundred years Plato was considered a dogmatic philosopher, and it was believed that the leading speaker in the dialogues states Plato’s own views, but the situation is far more complex. Plato wants us to acquire a vision rather than propositional knowledge. The idea of the good should not be accorded a special position. C. Rowe tries to demystify some current interpretations, whereas F. Ferrari insists on the causality of the good, and sees Plato’s teaching as the origin of the entire tradition of negative theology. Yet some texts give the impression that the good belongs to the ontological order. C. Porebski reminds us that problems of ethics are the starting point of what Plato says about the good in the Republic. According to F. Trabattoni, Aristotle’s description of Plato’s theory of principles is wrong and his philosophy differs fundamentally from Plato’s. The School of Tübingen is a prisoner of Aristotle’s interpretation. Plato’s reluctance to describe the good in detail may mark the beginning of a skeptic trend in the Academy. The Parmenides tells us, E. Berti says, that one has first to remove all hypotheses before a definition of the Good can be given. E. Moutsopoulos draws attention to some simplistic and wrong qualifications of Plato’s thought as related to idealism and mentions Kant and Hegel’s attitudes with regard to Platonism. J. F. Crosby points to a clear leaving behind of the Socratic “no one knowingly does wrong” in the Alcibiades 216b: as Plato explains in the Protagoras, one may live according to one’s lower appetite or follow the insight of the rational soul. G. Santos makes a strong stand in favor of the identity of goodness and reality. L. P. Gerson examines theories of Platos’s development: there is no evidence of a theory of the form of the good earlier than the Republic. Aristotle, J. Dudley writes, upheld an ethical ideal resembling that of Plato, substituting the contemplation of the unmoved mover for that of the idea of the good. In the last paper J. Seifert argues that Plato identified the demiurge and the good. (shrink)

Available from UMI in association with The British Library. ;Plato's philosophy of mathematics must be a philosophy of 4th century B.C. Greek mathematics, and cannot be understood if one is not aware that the notions involved in this mathematics differ radically from our own notions; particularly, the notion of arithmos is quite different from our notion of number. The development of the post-Renaissance notion of number brought with it a different conception of what mathematics is, and (...) we must be able to see the differences before we can begin to appreciate Plato's views, which must never be seen as applying to a revolutionary notion of mathematics . For this reason the first part of this dissertation must comprise a detailed analysis of these differences, along with some criticisms of those of Plato's modern interpreters who have not taken this point. Much of what will be found here can be found in Jacob Klein's Greek Mathematics and the Origin of Algebra, which will be extensively quoted and interpreted. This book is often quoted, but very seldom understood; or at least, those lessons which should be taken from it are rarely applied to Platonic exegesis. In the second part I attempt a better explanation of Plato's mathematical ontology, beginning with the most extensive passage in the dialogues, the triple image of Sun, Line and Cave in Republic. This reveals a relation between sensibles and forms which is at odds with the traditional view that mathematical objects are never precisely accurate instances of their forms--but this view can be refuted. Plato's view of the nature of mathematical objects will be seen to be a metaphorical way of saying what Aristotle says logically, and is entirely appropriate to the aims and methods of his contemporary mathematicians. Aristotle's criticism of Plato's views is examined and found to make sense and to be appropriate to views which Plato could consistently have held. (shrink)

The present contribution offers an overview of the main problems concerning the representation of the planets in the pre-Islamic Iranian world, the origin of their denominations, their astral roles and the reasons behind their demonization in the Zoroastrian and Manichaean frameworks. This is a preliminary attempt to resume the planetary iconography and iconology in western and eastern Iranian sources, involving also external visual data, such as those coming from Dunhuang and the Chinese world. The article offers an intellectual journey into (...) a net of mutual cultural and spiritual relations, focusing on the image of the heaven (and of its celestial beings), thereby proposing a new synthesis and highlighting a number of intercultural contaminations. (shrink)

In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ℝ2 and ℝ. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non-measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non-measurable.

The ArgumentRecent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of thebodyandimagesof mathematics.Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics (...) also has reflexive capacities unlike those of any other exact science. In no other exact science can the standard methodological framework usedwithinthe discipline also be used to study the nature of the discipline itself.Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics. (shrink)

Mathematicians typically invoke a wide range of reasons as to why their research is valuable. These reveal considerable differences between their personal images of mathematics. One of the most interesting of these concerns the relative importance accorded to conceptual reformulation and development compared with that accorded to the achievement of concrete results. Here I explore the conceptualists' claim that the scales are tilted too much in favour of the latter. I do so by taking as a case study (...) the debate surrounding the question as to whether groupoids are significantly more powerful than groups at capturing the symmetry of a mathematical situation. The introduction of groupoids provides a suitable case as they score highly according to criteria relating to theory-building rather than problem-solving.Several of the arguments for the adoption of the groupoid concept are outlined, including claims as to its capacity for reformulating existing theory, its ability to measure symmetry more systematically, and its ‘naturalness’. This last notion is given an extensive treatment. (shrink)

I offer a critical review of several different conceptions of the activity of foundational research, from the time of Gauss to the present. These are (1) the traditional image, guiding Gauss, Dedekind, Frege and others, that sees in the search for more adequate basic systems a logical excavation of a priori structures, (2) the program to find sound formal systems for so-called classical mathematics that can be proved consistent, usually associated with the name of Hilbert, and (3) the historicist (...) alternative, guiding Riemann, Poincaré, Weyl and others, that seeks to perfect available conceptual systems with the aim to avoid conceptual limitations and expand the range of theoretical options. I shall contend that, at times, assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images. To round the discussion, I mention some traits of an alternative program that investigates the epistemology of mathematical knowledge. (shrink)

The sociocultural landscape of contemporaneity can be represented as a porous inhomogeneous space consisting of border zones. These zones, understood by P. Galison as “local coordination of beliefs and actions,” can be deployed due to already existing conditions, for example, a project may arise at a university. However, such zones do not create sustainable social configurations and infrastructures associated with new areas of knowledge. In this article, we expand the concept of trading zones and call them zones requiring local ontologies (...) that arise on the border between different knowledge and imagination systems and, in turn, require their own infrastructure. This approach supports the view on humanities not as “pre-packaged” knowledge, but as special “technique” forming local ontologies. For a humanities-as-technique to emerge, it is necessary that two or more systems of knowledge coincide in time and form an intermediate zone, on the basis of which some local ontology can unfold. Thus, for example, when combining contemporary philosophy and contemporary mathematics, new images of knowledge emerge. Humanities-as-technique are considered as the basis for the establishment of trading zones, which are understood as local ontologies. We also propose a system of concepts that can be used to describe and construct these trading zones on the territory of the humanities in a techno-oriented cultural environment. (shrink)

Cover -- Title -- Copyright -- Dedication -- Contents -- List of Figures -- List of Tables -- Acknowledgements -- 1 Physics, Knowledge and Semiosis -- 2 Language, Knowledge and Description -- 3 Mathematical Statements and Expressions -- 4 Mathematical Symbols and the Architecture of the Grammar of Mathematics -- 5 Genres of Language and Mathematics -- 6 Images and the Knowledge of Physics -- 7 Physics and Semiotics -- Appendix A System Network Conventions -- Appendix B (...) Full System Networks for Mathematics -- Appendix C Details of Corpus -- Index. (shrink)

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the (...) training of preservice as well as inservice mathematics teachers. (shrink)

This is not a mathematics book, but a book about mathematics, which addresses both student and teacher, with a goal as practical as possible, namely to initiate and smooth the way toward the student’s full understanding of the mathematics taught in school. The customary procedural-formal approach to teaching mathematics has resulted in students’ distorted vision of mathematics as a merely formal, instrumental, and computational discipline. Without the conceptual base of mathematics, students develop over time (...) a “mathematical anxiety” and abandon any effort to understand mathematics, which becomes their “traditional enemy” in school. This work materializes the results of the inter- and trans-disciplinary research aimed toward the understanding of mathematics, which concluded that the fields with the potential to contribute to mathematics education in this respect, by unifying the procedural and conceptual approaches, are epistemology and philosophy of mathematics and science, as well as fundamentals and history of mathematics. These results argue that students’ fear of mathematics can be annulled through a conceptual approach, and a student with a good conceptual understanding will be a better problem solver. The author has identified those zones and concepts from the above disciplines that can be adapted and processed for familiarizing the student with this type of knowledge, which should accompany the traditional content of school mathematics. The work was organized so as to create for the reader a unificatory image of the complex nature of mathematics, as well as a conceptual perspective ultimately necessary to the holistic understanding of school mathematics. The author talks about mathematics to convince readers that to understand mathematics means first to understand it as a whole, but also as part of a whole. The nature of mathematics, its primary concepts (like numbers and sets), its structures, language, methods, roles, and applicability, are all presented in their essential content, and the explanation of non-mathematical concepts is done in an accessible language and with many relevant examples. (shrink)

This paper deals with Hobbes's theory of optical images, developed in his optical magnum opus, ‘A Minute or First Draught of the Optiques’, and published in abridged version in De homine. The paper suggests that Hobbes's theory of vision and images serves him to ground his philosophy of man on his philosophy of body. Furthermore, since this part of Hobbes's work on optics is the most thoroughly geometrical, it reveals a good deal about the role of mathematics (...) in Hobbes's philosophy. The paper points to some difficulties in the thesis of Shapin and Schaffer, who presented geometry as a ‘paradigm’ for Hobbes's natural philosophy. It will be argued here that Hobbes's application of geometry to optics was dictated by his metaphysical and epistemological principles, not by a blind belief in the power of geometry. Geometry supported causal explanation, and assisted reason in making sense of appearances by helping the philosopher understand the relationships between the world outside us and the images it produces in us. Finally the paper broadly suggests how Hobbes's theory of images may have triggered, by negative example, the flourishing of geometrical optics in Restoration England. (shrink)

On the 10th of April, 2019 the Event Horizon Telescope Collaboration presented the first image of the black hole. The image was obtained with a planet-scale array of eight ground-based radio telescopes. The observation relied on a technique called very long base interferometry which synchronises telescope facilities around the world. The image of a black hole together with the recent detections of gravitational waves confirms one of the most intriguing predictions of Einstein’s gravity theory, namely, the existence of black holes. (...) I will provide more details on this remarkable observation and explore its consequences for our understanding of nature. The physical reality of black holes is strongly supported by recent advances of astronomy. I claim that this fact is the key to understanding the relation between our world and the world of mathematics. (shrink)

The following is the second part of our Archaeology of the Inverse Square Law. Together these papers examine the transformation of the inverse square ratio from its origins in a metaphysical image of medieval thought in Grosseteste and the perspectivist tradition, through a playful magical practice in the Renaissance with Cusanus and Dee, and into a mathematical tool, applicable to the physical world. This last transformation allowed Newton to condense the geometrical image into a celebrated algebraic equation for universal gravity, (...) but it proved particularly confusing for the people who wrought it, Kepler and Hooke. (shrink)

The article presents philosophical views on music in the context of the transformations of the worldview from Antiquity to the Modern Time. In this research author also mentions the contemporary issues, and uses her own philosophical concept of the music, which can be described as following: the value of music as a substance and the way of the valuable interaction of a person with the world affirm the essence of musical being, in which the invariable principle of Harmony, the principle (...) of Chaos-Form movement, is preserved. Music expresses the fluid essence of the world and changes of being in space and time. Philosophy of music as a field of philosophical knowledge considers music from ontological-epistemological and phenomenological-axiological prospectives, as something more than just a form of art. It explores the deep, ultimate foundations of the existence of music as such and the philosophical and aesthetic foundations of musical art. Since ancient times music has been a representation of the world in the human conscience and served as the harmonic equivalent of cosmological philosophy, mathematics, astronomy, and astrophysics. The scientific view on music was enriched in the Modern Time by the expanded view on the cosmo-sound space reflected in musical art, which at the same time transforms the mathematical ideas of geometricity, squareness etc. The tendency to create integral world music in the musical practice of the XX-XXI centuries explains the attempt of mankind to present music by modern methods of composing musical art as an expression of fear towards the secret Harmony of universal existence, and, on the other hand - as a form of search for salvation and mental balance, intuitively reflecting what is happening in natural science, which more and more points to the abyss of unknowable universe, and the unstable place of man in the world. (shrink)

Classical results of additive number theory lead to the undecidability of the existence of solutions for diophantine equations in given special sets of integers. Those sets which are images of polynomials are covered by a more general result in the second section. In contrast, restricting diophantine equations to images of exponential functions with natural bases leads to decidable problems, as proved in the third section.

This book gives an analysis of Hertz's posthumously published Principles of Mechanics in its philosophical, physical and mathematical context. In a period of heated debates about the true foundation of physical sciences, Hertz's book was conceived and highly regarded as an original and rigorous foundation for a mechanistic research program. Insisting that a law-like account of nature would require hypothetical unobservables, Hertz viewed physical theories as images of the world rather than the true design behind the phenomena. This paved (...) the way for the modern conception of a model. Rejecting the concept of force as a coherent basic notion of physics he built his mechanics on hidden masses and rigid connections, and formulated it as a new differential geometric language.Recently many philosophers have studied Hertz's image theory and historians of physics have discussed his forceless mechanics. The present book shows how these aspects, as well as the hitherto overlooked mathematical aspects, form an integrated whole which is closely connected to the mechanistic world view of the time and which is a natural continuation of Hertz's earlier research on electromagnetism. Therefore it is also a case study of the strong interactions between philosophy, physics and mathematics. Moreover, the book presents an analysis of the genesis of many of the central elements of Hertz's mechanics based on his manuscripts and drafts. Hertz's research program was cut short by the advent of relativity theory but its image theory influenced many philosophers as well as some physicists and mathematicians and its geometric form had a lasting influence on advanced expositions of mechanics. (shrink)

A special kind of maximum cuts in sequent derivations, actual maximum cuts, is defined. It is shown that (1) each actual maximum cut of a sequent derivation makes maximum segments in its natural deduction image, and (2) each maximum segment of a natural deduction derivation makes an actual maximum cut in its sequent image.

This study examines the theology of the fourth-century bishop, Hilary of Poitiers, concentrating particularly on two commentaries written at different times in his life. The main focus of the study is on Hilary's anthropological theology.