Emmy Noether’s many articles around the time that Felix Klein and David Hilbert were arranging her invitation to Göttingen include a short but brilliant note on invariants of finite groups highlighting her creativity and perspicacity in algebra. Contrary to the idea that Noether abandoned Paul Gordan’s style of mathematics for Hilbert’s, this note shows her combining them in a way she continued throughout her mature abstract algebra.

Here is a fine semipopular book about the ideas which have motivated the much-talked-about revolution in the theories of information, control and communication. Jagjit Singh is one of those rare science writers who knows how to present intricate technical concepts to the less-than-expert reader without compromising the original sense or significance. The book begins, appropriately enough, with a discussion of the concept of information, culminating in the technical definition which enables us to assign numerical values to its quantity. The following (...) chapters present the conceptual heart of the Shannon-Weaver theory, the theories of automatic computing machines and the intriguing neurological theories of Warren McCulloch and John Von Neumann. The chapter on Turing Machines does not pay enough attention to the purely mathematical applications of these devices for the present reviewer's taste. The closing chapters deal with mathematical theories of intelligence and the brain. The text is accompanied by many good drawings and appropriate mathematical demonstrations. A book of this quality could have been produced only by an author with absolute command of his topic.—H. P. K. (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)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...) mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

Mathematical correspondence offers a rich heritage for the history of mathematics and science, as well as cultural history and other areas. It naturally covers a vast range of topics, and not only of a scientific nature; it includes letters between mathematicians, but also between mathematicians and politicians, publishers, and men or women of culture. Wallis, Leibniz, the Bernoullis, D'Alembert, Condorcet, Lagrange, Gauss, Hermite, Betti, Cremona, Poincaré and van der Waerden are undoubtedly authors of great interest and their letters (...) are valuable documents, but the correspondence of less well-known authors, too, can often make an equally important contribution to our understanding of developments in the history of science. Mathematical correspondences also play an important role in the editions of collected works, contributing to the reconstruction of scientific biographies, as well as the genesis of scientific ideas, and in the correct dating and interpretation of scientific writings. This volume is based on the symposium “Mathematical Correspondences and Critical Editions,” held at the 6th International Conference of the ESHS in Lisbon, Portugal in 2014. In the context of the more than fifteen major and minor editions of mathematical correspondences and collected works presented in detail, the volume discusses issues such as • History and prospects of past and ongoing edition projects, • Critical aspects of past editions, • The complementary role of printed and digital editions, • Integral and partial editions of correspondence, • Reproduction techniques for manuscripts, images and formulae, and the editorial challenges and opportunities presented by digital technology. (shrink)

This collection presents the first sustained examination of the nature and status of the idea of principles in early modern thought. Principles are almost ubiquitous in the seventeenth and eighteenth centuries: the term appears in famous book titles, such as Newton’s _Principia_; the notion plays a central role in the thought of many leading philosophers, such as Leibniz’s Principle of Sufficient Reason; and many of the great discoveries of the period, such as the Law of Gravitational Attraction, were (...) described as principles. Ranging from mathematics and law to chemistry, from natural and moral philosophy to natural theology, and covering some of the leading thinkers of the period, this volume presents ten compelling new essays that illustrate the centrality and importance of the idea of principles in early modern thought. It contains chapters by leading scholars in the field, including the Leibniz scholar Daniel Garber and the historian of chemistry William R. Newman, as well as exciting, emerging scholars, such as the Newton scholar Kirsten Walsh and a leading expert on experimental philosophy, Alberto Vanzo. _The Idea of Principles in Early Modern Thought_ charts the terrain of one of the period’s central concepts for the first time, and opens up new lines for further research. (shrink)

It has been almost eighty years since Paul Dirac delivered a lecture on the relationship between mathematics and physics and since 1960 that Eugene Wigner wrote about the unreasonable effectiveness of mathematics in the natural sciences. The field of cosmology and efforts to define a more comprehensive theory have changed significantly since the 1960s; thus, it is time to refocus on the issue. This paper expands on ideas addressed by these two great physicists, specifically, the ultimate effectiveness of mathematics (...) to describe nature. After illustrating how theoretical physic predicted discoveries, the concepts of established theories are summarized. Then, the “world equation” which combines all key physics theories is conceptually described. As the possible last piece of the puzzle, String Theory is briefly defined. And last, a cursory overview of an extreme proposal is presented — the world as a mathematical object. However, there is no fundamental reason to believe that math will allow mankind to completely comprehend nature. If math does not provide a comprehensive theory, nature will retain her secrets. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.3 (2003) 426-427 [Access article in PDF] Timothy Smiley, editor. Mathematics and Necessity: Essays in the History of Philosophy. New York: Oxford University Press, 2000. Pp. ix + 166. Cloth, $35.00.Mathematics and Necessity contains essays by M. F. Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given to the British Academy in 1998. All concern the history of the philosophical treatment of (...) mathematics, necessity, or both, although there is no single thread running through all three essays.Burnyeat focuses on Plato's understanding of mathematics in the Republic, making use of the Timeaus in particular to defend his interpretation. Plato claimed that in the ideal society mathematics should be studied for a full ten years, which many have thought excessive. To explain it, Burnyeat argues that the role of mathematics is pervasive: ethics and politics are mathematical, because in Plato's view mathematics is a constitutive part of ethical understanding. On Burnyeat's interpretation, goodness is part of the objective world, and mathematics is a science of values. The link between mathematics and goodness in the objective world is forged by the concord, harmony, and unity of proportions, which express the goodness of the Divine Craftsman's beneficent design. Proportions are both intrinsically good and the proper study of mathematics.Burnyeat explains why Plato thinks abstract kinematics (the pure non-sensible counterpart to astronomy) and abstract harmonics lead to knowledge of the good. Abstract kinematics concerns the "harmony of the spheres" described in the Republic. Drawing heavily on the Timeaus, Burnyeat argues that the circular motions of the spheres are motions in the intelligence of the world soul and that there are similar motions in human souls. The world soul and human souls are spatial and have circular motions (invisibility and intangibility make them incorporeal). Souls are the most important subject matter of abstract harmonics—the study of good proportions. A soul that understands the various mathematical disciplines and their relations takes on the harmony and hence goodness of the world soul. Such a soul serves as a model for organizing the social world, a model which properly trained rulers can use to shape a community. Burnyeat's well-argued essay is compelling and exciting. [End Page 427]Ian Hacking's views provide a great contrast to Burnyeat's Plato. Hacking conducts a phenomenological inquiry into why some philosophers have become obsessed by mathematics as a source of philosophical inspiration. He holds that mathematics concerns a relatively minor part of human culture and that the influence of mathematics on philosophy is pernicious; he even describes it as an infection.Hacking examines the views of six philosophers: Plato, Leibniz, Mill, Descartes, Lakatos, and Wittgenstein. Hacking's assessment of the six philosophers is at times disputable, but it is also interesting and stimulating, and serves his phenomenological inquiry.Hacking traces philosophy's infection by mathematics to two factors. A mathematical proof can be perspicuous; that is, a proof can be "grasped," which requires seeing directly why something must be true and being able to recapitulate it. A mathematical proof can also anticipate facts about the world. He thinks that these two factors affect us in two ways: they give us the idea of a priori knowledge, and they dazzle us into unreasoned conviction in proofs. One might respond that, setting aside the feeling proofs give us, we are still justified in being impressed by them. Hacking argues that much of what impresses is based on a misunderstanding of their nature, which he attempts to correct.According to Hacking, a mathematical demonstration involves shifts of meaning that analytically bind the proof and the theorem it establishes. They thereby mutually support each other and provide what he calls self-authentication. The correctness of the proof is supported by the "analytified" truth of the theorem and the theorem is considered true because proven.This bootstrapping feature of mathematical proofs supports Hacking's deflationary account of necessity. Hacking suggests that the "must" of logical necessity rests upon our unwillingness to accept empirical counter-examples to the... (shrink)

It has been almost eighty years since Paul Dirac delivered a lecture on the relationship between mathematics and physics and since 1960 that Eugene Wigner wrote about the unreasonable effectiveness of mathematics in the natural sciences. The field of cosmology and efforts to define a more comprehensive theory have changed significantly since the 1960s; thus, it is time to refocus on the issue. This paper expands on ideas addressed by these two great physicists, specifically, the ultimate effectiveness of mathematics (...) to describe nature. After illustrating how theoretical physic predicted discoveries, the concepts of established theories are summarized. Then, the “world equation” which combines all key physics theories is conceptually described. As the possible last piece of the puzzle, String Theory is briefly defined. And last, a cursory overview of an extreme proposal is presented — the world as a mathematical object. However, there is no fundamental reason to believe that math will allow mankind to completely comprehend nature. If math does not provide a comprehensive theory, nature will retain her secrets. (shrink)

The main item in the present volume was published in 1930 under the title Das Unendliche in der Mathematik und seine Ausschaltung. It was at that time the fullest systematic account from the standpoint of Husserl's phenomenology of what is known as 'finitism' (also as 'intuitionism' and 'constructivism') in mathematics. Since then, important changes have been required in philosophies of mathematics, in part because of Kurt Godel's epoch-making paper of 1931 which established the essential in completeness of arithmetic. In the (...) light of that finding, a number of the claims made in the book (and in the accompanying articles) are demon strably mistaken. Nevertheless, as a whole it retains much of its original interest and value. It presents the issues in the foundations of mathematics that were under debate when it was written (and in some cases still are);, and it offers one alternative to the currently dominant set-theoretical definitions of the cardinal numbers and other arithmetical concepts. While still a student at the University of Vienna, Felix Kaufmann was greatly impressed by the early philosophical writings (especially by the Logische Untersuchungen) of Edmund Husser!' He was never an uncritical disciple of Husserl, and he integrated into his mature philosophy ideas from a wide assortment of intellectual sources. But he thought of himself as a phenomenologist, and made frequent use in all his major publications of many of Husserl's logical and epistemological theses. (shrink)

Human language has the characteristic of being open and in some cases polysemic. The word “infinite” is used often in common speech and more frequently in literary language, but rarely with its precise meaning. In this way the concepts can be used in a vague way but an argument can still be structured so that the central idea is understood and is shared with to the partners. At the same time no precise definition is given to the concepts used (...) and each partner makes his own reading of the text based on previous experience and cultural background. In a language dictionary the first meaning of “infinite” agrees with the etymology: what has no end. We apply the word infinite most often and incorrectly as a synonym for “very large” or something that we do not perceive its completion. In this context, the infinite mentioned in dictionaries refers to the idea or notion of the “immeasurably large” although this is open to what the individual’s means by “immeasurably great.” Based on this linguistic imprecision, the authors present a non Cantorian theory of the potential and actual infinite. For this we have introduced a new concept: the homogon that is the whole set that does not fall within the definition of sets established by Cantor. (shrink)

Over Plato's Academy in ancient Athens, it is said, hung a sign: "Let no one ignorant of geometry enter here." Plato thought no one could do philosophy without also doing mathematics. In The Waltz of Reason, mathematician and philosopher Karl Sigmund shows us why. Charting an epic story spanning millennia and continents, Sigmund shows that philosophy and mathematics are inextricably intertwined, mutual partners in a reeling search for truth. Beginning with-appropriately enough-geometry, Sigmund explores the power and beauty of numbers and (...) logic, and then shows how those ideas laid the ground work for everything from the theory of a fair election, to modern conceptions of governance, cooperation, morality, and even of reason itself. Did you know, for example, that John Locke, author of some of the most important texts in the Western theory of government, was motivated in his work by his study of geometry? Or, that Locke was actually terrible at geometry, a seventeenth-century mathematical laughingstock? He was, a fact that might want us to think again about the logic of his life's work. And Locke wasn't the only one! Sigmund reveals how many of our modern ideas about what is true and what is reasonable are based on similarly shaky grounds. The economists and other thinkers who promulgated game theory, classical economics, and behavioral economics-the basis of so much in modern life-were not necessarily good at either math or philosophy, or both. The result is a remarkable book: accessible, funny, and wise, it tells an engrossing history of ideas that spins as dizzyingly and beautifully as a ballroom full of expert dancers. But it doesn't just celebrate the past. Instead, by making all these great ideas accessible to all, The Waltz of Reason empowers as it entertains, giving each of us the tools to ask, what do we know, how do we know it, and what do we want to do, with all these ideas? (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Advances in Peircean Mathematics: The Colombian School ed. by Fernando ZalameaGianluca CaterinaFernando Zalamea (Ed.) Advances in Peircean Mathematics: The Colombian School Berlin, Boston: De Gruyter, 2022. 212 pp. (incl. index).The volume Advances in Peircean Mathematics is an important, very much needed contribution towards a deeper understanding of the impact of Peirce's work especially in the fields of mathematics, logic, and semiotic. It fills a gap in the current (...) literature by recasting some of [End Page 373] Peirce's profound mathematical insights into a formal framework. This is a finely edited editorial project, which summarizes the work of more than two decades done by a group of scholars—the Colombian School—lead by Fernando Zalamea, a pioneer of Peirce studies, and pragmatism in general, in Latin America.The structure of the volume unfolds in three dense chapters, each corresponding to a specific mathematical context connected to various aspects of Peirce's work. Broadly speaking, in the first chapter, Gustavo Arengas proposes the theory of higher category theory as a natural framework to represent the main traits of Peirce's semiotic; in the second chapter, Francisco Vargas develops a thorough set-theoretical model of Peirce's continuum; in the third chapter, Arnold Oostra, in a truly Peircean spirit, formalizes the notion of Intuitionistic Existential Graphs, emphasizing the idea that intuitionistic logic is the most natural context to represent and understand Peirce's diagrammatic logic. The fourth and last chapter, penned by Zalamea, provides the reader with a solid summary of the main results presented in the volume along with a robust road map to navigate through the technical details of the text.Mathematics, as implied by the title of the volume, is at the core of this project, and the authors do not shy away from the complexity, details, and depth of it in order to shed light on some of the most fundamental and groundbreaking nature of Peirce's ideas. That level of rigor and formal soundness is certainly one of the strengths of the volume, although readers who may not be acquainted with at least some basic notions of category theory or with the mathematics of Peirce's work, may find some of the text a bit terse—this is not a casual reading and it does require a certain level of commitment in order to appreciate the depth of the provided insights. On the other hand, all the background material is clearly laid out, making the volume a self-contained work which can be thoroughly enjoyed by a general audience.The real upshot of recasting some of Peirce's intuitions within the fabric of formal mathematics is not only that of providing the instruments to clarify those ideas, but also that of highlighting the role that Peirce's work has been playing in the development of modern mathematics in many relevant aspects, from the very genesis of category theory—which today is widely regarded as a necessary and unifying language in several fields of the mathematical and physical sciences—to the development of non-standard analysis, just to mention a few of them.An elegant instance of such perspective is offered by Gustavo Arengas in the first chapter, where it is argued that ∞-categories can be used to model the natural order of hierarchies implicit in Peirce's conception of the basic triadic relation between sign, object, and interpreter. What, at first glance, may seem just a simple analogy, unfolds onto a methodical and enlightening analysis of Peirce's semiotic, as the author builds [End Page 374] the connections with the relevant categorical constructions in a precise manner. Most of the rest of the chapter focuses on Peirce's notions of the pragmatic and pragmaticist maxim, and the use of basic structures such as monads, limits, presheaves and sheaves as the natural environment to better convey their meaning, both from a mathematical and a philosophical perspective. The reader will find it interesting to realize that, by delving deeper into the understanding of the relational nature of category theory, a great chunk of Peirce's work can be seen as fundamentally intertwined with the apparatus and the language of modern mathematics... (shrink)

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was (...) also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated (...) a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics. (shrink)

This book offers an engaging and comprehensive introduction to scientific theories, and the evolution of science and mathematics through the centuries. It discusses the history of scientific thought and ideas and the intricate dynamic between new scientific discoveries, scientists, culture, and societies. Through stories and historical accounts, the volume illustrates the human engagement and preoccupation with science and the interpretation of natural phenomena. It highlights key scientific breakthroughs from the ancient to later ages, giving us accounts of the work of (...) ancient Greek and Indian mathematicians and astronomers as well as of the work of modern scientists like Descartes, Newton, Planck, Mendel and many more. The author also discusses the vast advancements which have been made in the exploration of space, matter, and genetics and their relevance in the advancement of the scientific tradition. He provides great insights into the process of scientific experimentation and the relationship between science and mathematics. He also shares amusing anecdotes of scientists and their interactions with the world around them. Detailed and accessible, this book will be of great interest to students and researchers of science, mathematics, the philosophy of science, science and technology studies, and history. It will also be useful for general readers who are interested in the history of scientific discoveries and ideas. (shrink)

The lives of few mathematicians offer the drama that is presented by the life of L. E. J. Brouwer, correctly identified on the cover of this book as a topologist, intuitionist, and philosopher, and before we go any further, it will be worth indicating why.It is not just that Brouwer would rank high among mathematicians for his work in topology alone: he set standards for rigour and created a theory of dimension for topological spaces, and his fixed-point theorem is of (...) great importance. Nor is it just that his philosophy of intuitionism created a new and vibrant branch of logic, that is, arguably, the only viable alternative to naïve, classical logic outside the enclave of professional logicians. Rather, it is that there is a tension, popularly taken to amount to a contradiction, in the fundamental ideas behind his topology and his intuitionism. Added to that, he took a highly principled stand on mathematical issues that led him into confrontations with major figures and a certain degre .. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.4 (2002) 555-575 [Access article in PDF] Measurer of All Things:John Greaves (1602-1652), the Great Pyramid, and Early Modern Metrology Zur Shalev [Figures]Writing from Istanbul to Peter Turner, one of his colleagues at Merton College, Oxford, John Greaves was deeply worried: Onley I wonder that in so long time since I left England I should neither have received my brasse quadrant which (...) I left to be finished for my journey thither, nor any notice of it [...]. I agreed with mr. Allen upon price and the time that he should finish it, if he hath failed me he hath done me the greatest injury that can be. 1A great injury indeed, because Greaves's journey to Italy and the Levant was all about measuring—luckily the instrument did reach him at some later stage. The thirty-six-year-old Professor of Geometry at Gresham College was taking the measurements of countless monuments and objects in the locations he visited. In Rome he measured, among many other ancient structures, Cestius' Pyramid and St. Peter's basilica. In Lucca, deeply impressed, he counted his paces around the beautiful city walls. In Siena he observed together with a "Mathematical Professor" one of the Sidera Medicea using "a glass." In Egypt he even hurt his eyes gazing at the sun, looking for sunspots and measuring its diameter. 2 His measuring mission, however, culminated in the fixing of the [End Page 555] latitudes of Istanbul, Rhodes, and Alexandria. As Bishop Juxton wrote before the trip to Greaves's employers at Gresham College (apparently using Greaves's own promotional language): This worke I find by the best astronomers, especially by Ticho Brache [ sic ] and Kepler, hath beene much desired as tending to the advancement of that science, and I hope it wil be an honour to that nation and prove ours if we first observe it. 3A mathematician-Orientalist, commanding the ancient and modern astronomical and geographical literature of Latin, Greek, Hebrew, Arabic, and Persian authors, Greaves was arguably the best qualified European at the period to perform the task. That he miscalculated the latitude of Rhodes is of less consequence for his present-day readers. 4 However, one of the most obscure and therefore telling of his measurement activities was the survey of the Pyramids of Giza, which resulted in the Pyramidographia (1646). 5 This remarkable learned treatise and travel account hybrid, which is at the focus of the present study, gives us a glimpse into the rich and complex world of scientific antiquarians.Greaves is most conveniently remembered today as an Orientalist. While we must be thankful to Edward Said for broadening the meaning of Orient-alism—from an academic discipline, accumulating objective knowledge of the East, into a much wider cultural discourse, his emphasis on the nineteenth and twentieth centuries and on the European colonial mindset is less useful for making historical sense of what early modern Orientalists were doing—physically and culturally. We can understand Greaves's "Oriental" enterprise in its [End Page 556] depth and variety only within long established European traditions of scholarship and keeping the intellectual concerns of his time in mind. 6But beyond articulating for us the nature of early Orientalism, the Pyramidographia, as well as Greaves's entire scientific career, situated in their wider European context, provide a fine entry point into a foreign world of learned practices and methods. It teaches us how well into the seventeenth century astronomy and philology, observation and bookishness, could coalesce in one figure, in one enterprise. 7 We still lack a modern biography of Greaves, a complex protagonist of a complex period, and even a full evaluation of his intellectual work. While this paper surely cannot compensate for that, I do attempt here a brief exposition of Greaves's Pyramidographia. 8Modern historians of archaeology and Egyptology, preoccupied mostly with the disciplines' progress and development of scientific standards, have noted the Pyramidographia in passing and praised its precise language and rigorous research... (shrink)

Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics--algebra, geometry, number theory, analysis, logic and set theory--with narratives (...) to show how they are linked. Classic works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare are reproduced in reliable translations and many selections from writers such as Gauss, Cantor, Kronecker and Zermelo are here translated for the first time. The collection is an invaluable source for anyone wishing to gain an understanding of the foundation of modern mathematics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Method and Mathematics:Peter Ramus's Histories of the SciencesRobert GouldingPeter Ramus (1515–72) was, at first sight, the least likely person to write an influential history of mathematics. For one thing, he was clearly no great mathematician himself. His sympathetic biographer Nicholas Nancel related that Ramus would spend the mornings being coached in mathematics by a team of experts he had assembled, and in the afternoon would lecture on the (...) very same subjects.1 But there can be no doubt that Ramus held mathematics in particular esteem and even played a crucial role in linking philosophical discourse to mathematics and in promoting the use of quadrivial reasoning in the study of the natural world.2The origins for his enthusiasm for mathematics are to be found in his account of the nature of the arts and critique of the curriculum of the universities. From his earliest career, Ramus was a logician and remained one in all his works, whether writing on mathematics or Virgil, and he conceived [End Page 63] of all the arts, especially mathematics, as unchanging structures of necessarily true propositions—not prima facie a position which lends itself to notions of historical change and development.3 His career as a historian of mathematics, I will argue, was directed by problems that arose in his theoretical understanding of mathematics as he began to immerse himself in the sciences of the ancient world.Ramus and the Reform of DialecticRamus set out his fundamental logical positions in his very first printed work, the Dialecticae institutiones (Education in Dialectic) of 1543, a contribution to the ongoing humanist attack on scholastic logic, in which Ramus rehearsed many of the commonplace criticisms of the university dialectic. Humanists complained that logic no longer concerned itself with real human reasoning; instead, it had become a discipline studied for its own sake, using its own incomprehensible jargon, and was of no practical interest. The new humanist dialectics of Lorenzo Valla and Rudolph Agricola attempted to teach the kind of practical reasoning useful for composing a speech or letter, and they borrowed extensively from rhetorical works of Cicero and Quintilian to develop a highly rhetoricized logic. Questions about the formal validity of arguments were of little interest; what mattered was whether the arguments were persuasive.4In his Dialecticae institutiones, Ramus took this humanist reformulation of dialectic a step further. Dialectic—or, in fact, any art or science—consisted of nature, doctrine, and exercise. The natural workings of the mind formed the most significant element. Doctrine was nothing more than a record of natural reasoning; in importance it paled next to nature and practice.5 The logic of the universities bore no resemblance to the true, natural dialectic, as he argued at length in the companion volume to the Institutiones, the innocuously titled but exuberantly offensive Aristotelicae animadversiones [End Page 64] (Observations on Aristotle).6 The question, then, remained: how can we gain access to this "natural reasoning"?Ramus's answer was surprising. He asks us to find a group of men—not scholars, but completely uneducated vineyard workers. Question them about the coming year: the fertility of the soil, the quality and quantity of the crop. "And then," he writes "from their minds as from a mirror an image of nature will be reflected."7 In the reasoned replies of these uneducated men, says Ramus, we discover every part of logic needed for any purpose, whether everyday discourse or composition of poetry: the invention of arguments, the assessment of their truth and their proper and orderly presentation. Other humanists had praised man's natural logical faculties, elevating them over artificial technique, but Ramus was the first to look beyond the walls of the university and the writings of the ancients to find natural dialectic at work.If even the uneducated have some possession of the arts, then the arts taught at the university should do no more than clear away the misleading junk in the mind, and allow its natural clarity to shine through.8 Logic should be easy to learn—not because this is a pedagogic ideal, but from the very nature of the art. And if logic as it is... (shrink)

Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics--algebra, geometry, number theory, analysis, logic and set theory--with narratives (...) to show how they are linked. Classic works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare are reproduced in reliable translations and many selections from writers such as Gauss, Cantor, Kronecker and Zermelo are here translated for the first time. The collection is an invaluable source for anyone wishing to gain an understanding of the foundation of modern mathematics. (shrink)

This paper is in two parts. The first presents an analysis of the epistemology underlying the practice of classical Indian mathematical astronomy, as presented in three works of Nīlakaṇṭha Somayāji (1444–1545 CE). It is argued that the underlying concepts put great value on careful observation and skill in development of algorithms and use of computation. This is reflected in the technical terminology used to describe scientific method. The keywords in this enterprise include parīkṣā, anumāna, gaṇita, yukti, nyāya, siddhānta, (...) tarka and anveṣaṇa. The concepts that underlie these terms are analysed and compared with such ideas as theory, model, computation, positivism, empiricism etc. In a short second part, it is proposed that the primacy awarded to number and computation in classical Indian science led to an artificial language that did include equations but emphasized displays that facilitated calculation, as in the Bakshali manuscript (800 CE?). It is further argued that echoes of these concepts can be recognized in current science, where computation is once again playing a greater role triggered by spectacular developments in computer technology. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns by Arkady PlotnitskyNoam CohenPLOTNITSKY, Arkady. Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns. Cham: Springer, 2023. xvi + 294 pp. Cloth, $109.99The limits of thought in its relations to reality have defined Western philosophical inquiry from its very beginnings. The shocking discovery of the incommensurables in Greek mathematics (...) not only reintroduced these limits with greater force but also instigated two different forms of coping with the unthinkable in Western philosophy, mathematics, and science. As Arkady Plotnitsky sets forth, in response to the incommensurability crisis in the ancient Greek intellectual world, there appeared two different attitudes toward the alogon, that is, the unthinkable or irrational. Plato and his followers sought to overcome the unthinkable, of which the incommensurables in mathematics are only an example, through philosophy. In contrast, the Pythagoreans continued to hold on to mathematics as the primary means of knowing the world, but put a larger emphasis on geometry. Plotnitsky [End Page 359] claims that by not putting aside the primacy of mathematics, the Pythagorean shift from arithmetic to geometry established in Western thought a certain tolerance toward the unthinkable within the realm of the thinkable, “a possibility of the alogon in the logos.” Until modern times this remained no more than a possibility. However, in modern mathematics and mathematical physics Plotnitsky identifies a certain fulfillment of this potential, a “radical Pythagorean mathematics.” In his comprehensive and thought-provoking study, Plotnitsky recounts the development of such radical modern Pythagoreanism, not only to tell its history but also to suggest that the unthinkable is a condition of possibility for thought, and that a proper recognition of this role it plays is crucial for the advancement of mathematical thinking and thinking in general.Plotnitsky begins by portraying in broad strokes the two defining characteristics of radical Pythagorean mathematics: the presence of the unthinkable within mathematical thought itself, and the interplay of algebra and geometry. Relying on an extensive knowledge of the history of modern mathematics and physics, in the first two chapters of the book Plotnitsky gradually demonstrates the key role that the alogon plays in some theories of modern mathematical thought, such as Gödel’s incompleteness theorems and quantum mechanics. He also presents and clarifies his own conception of the history of Western thought as a creative process of concept-forming. By creating rather than discovering concepts, great mathematicians break new ground for thinking. The book as a whole argues in length for these two main theses.In chapter 3, the author addresses the thought of Bernhard Riemann to make the case that Riemann’s mathematical thought puts an emphasis on creative conceptual thinking and is not defined merely by calculations or logic. Such thinking is problem-posing rather than axiomatic. Riemann’s central innovation in this respect is the concept of manifold (Mannigfaltigkeit), which enabled him to think anew space and geometry in a radical non-Euclidean fashion, implying an infinite number of possible geometries. Plotnitsky argues—borrowing Deleuze and Guattari’s terminology—that this opened up a “plane of immanence,” which is to say that Riemann created a concept-generating movement of thought. However, we cannot grasp the generation of concepts in modern mathematics without giving proper attention to the interplay of algebra and geometry that defines it. Chapter 4 demonstrates this interweaving of the two fields by recounting the history of the idea of the curve in modernity, from the analytic geometry and calculus of Fermat and Descartes to its modification in the concept of a Riemann surface, up to its more recent developments in the algebraic geometry of Weil and Grothendieck. It gives a historical account of the algebraization of the curve that at the same time brings into relief the irreducible geometrical aspect, continuity, of all modern mathematical conceptualizations of the curve.The next two chapters provide further case studies. Chapter 5 expands and solidifies the argument of chapter 2. It discusses three examples of mathematical practice, each displayed by a pair of important [End Page 360] mathematicians: Abel and Galois, Lobachevsky and Riemann, Weil and Grothendieck. Each case... (shrink)

A fascinating and hugely original book that explains how a vexing technical puzzle was solved, making possible some of the most exquisite music ever written. From the days of the ancient Greeks, the creation of music was thought to be governed by divine and immutable mathematical certainties. But over time skeptics came to understand that those rules limited harmonic possibilities. In Temperament , we see the traditionalists and the innovators battling across the centuries, engaging great thinkers like Newton, (...) Kepler, and Descartes as well as musicians, craftsmen, church leaders, and heads of state. At the heart of their dispute is the question of how the tones of a musical scale should be selected. The breakthrough came in the eighteenth century, when the modern keyboard was given perfect musical symmetry through a tuning of equal temperament, each pitch reliably equidistant from the ones that precede and follow it. This tuning allows a musical pattern begun on one note to be duplicated when starting on any other; it creates a musical universe in which the relationships between tones are reliably, uniformly consistent--a universe of greatly expanded possibility, one that allowed Liszt, Chopin, Brahms, Debussy, and all those who followed to compose the piano music we listen to today. Stuart Isacoff relates the story of the reinvention of the piano--a story that encompasses social history, religion, philosophy, and science as well as musicology--in a concise and sparkling narrative. Temperament is a jewel of a book. (shrink)

Although conceived as a textbook, this extraordinary work contains a great deal of material which is either completely new or which has not appeared before in book form. It is intended as an upperlevel text for those with some familiarity with the subject already. After the introduction, there is a long chapter on formal systems which contains new material on algorithms and the theory of definition; epitheory of formal systems is then discussed, followed by an elegant algebraic treatment of (...) logic. Curry then formulates systems for negation and implication in the next two chapters, follows them by quantification theory, and ends with a sketch of modal logic. What distinguishes this from other logic texts which try to cover about the same ground is this: Curry exercises virtually exquisite care in his analysis of some of the more difficult points, variables and substitution, for example, that others often tend to gloss over. Each chapter has a section dealing with supplementary but related topics so as to give the reader some idea where the subject goes. There is an enormous bibliography and hundreds of references, including historical ones; these also increase its scholarly value. The author takes an informal semantical viewpoint about logic—trying to treat meaning as well as form as essential to logic. This view and a pellucid style make things move freely in the most difficult spots; only Curry's occasionally peculiar terminology might be confusing.—P. J. M. (shrink)

Chaotic dynamics has been hailed as the third great scientific revolution in physics this century, comparable to relativity and quantum mechanics. In this book, Peter Smith takes a cool, critical look at such claims. He cuts through the hype and rhetoric by explaining some of the basic mathematical ideas in a clear and accessible way, and by carefully discussing the methodological issues which arise. In particular, he explores the new kinds of explanation of empirical phenomena which modern dynamics (...) can deliver. Explaining Chaos will be compulsory reading for philosophers of science and for anyone who has wondered about the conceptual foundations of chaos theory. (shrink)

Polish philosophy of science has been the beneficiary of three powerful creative streams of scientific and philosophical thought. First and fore­ most was the Lwow-Warsaw school of Polish analytical philosophy founded by Twardowski and continued in their several ways by Les­ niewski, Lukasiewicz, and Tarski, the great mathematical and logical philosophers, by Kotarbinski, probably the most distinguished teacher, public figure, and culturally influential philosopher of the inter-war and post-war period, and by Ajdukiewicz, the linguistic philosopher who was intellectually (...) sympathetic with the anti-irrationalist (as he would say), logistic and meta-theoretical inquiries of the Vienna Circle. Second was independent and lively Polish Marxism, with its fine development of social research under Krzywicki, a social anthropologist and younger contemporary of Engels, and then after the war the economist Lange, the philosophers Schaff, Kolakowski, Baczko, and many others. Finally there has been a wide range of philosophical, scientific and humanistic scholar­ ship which lends its various qualities to the understanding of both the logic of science and the historical situation of the sciences: we mention only that great and humane physicist Infeld, the phenomenologist with deep epistemological interest Ingarden, the historian of scientific ideas Zawirski, the historian of philosophy and aesthetics Tatarkiewicz, and the mathematical logicians such as Mostowski and Szaniawski. (shrink)

Alan Weir’s new book is, like Darwin’s Origin of Species, ‘one long argument’. The author has devised a new kind of have-it-both-ways philosophy of mathematics, supposed to allow him to say out of one side of his mouth that the integer 1,000,000 exists and even that the cardinal ℵω exists, while saying out of the other side of his mouth that no numbers exist at all, and the whole book is devoted to an exposition and defense of this new view. (...) The view is presented in the book in a way that can make it difficult for the reader to trace the main line of argument: with a great deal of apparatus, and with a great many digressions into subordinate issues. In what follows I will try to stick to what I take to be the essentials, even at the risk of oversimplifying some central but complicated issues, and at the cost of neglecting some interesting but peripheral ones.In chapter 1, the author introduces a distinction between what he calls ‘two aspects of meaning’ and dubs informational content and metaphysical content. Informational content is the aspect of meaning of primary interest to linguists, and the one of which speakers themselves are generally aware, at least upon reflection. Metaphysical content is supposed to be another aspect of meaning primarily of interest to philosophers. The basic idea is that if there are standards of correctness for assertions of a certain kind, then such an assertion may be called ‘true’ when those standards are met, even though the kind of correctness involved is not correctness in representing how the world is. What the world must be like in order for the utterance to be true is the metaphysical content of the assertion, but it need not be part of its …. (shrink)

Making Sense of Inner Sense 'Terra cognita' is terra incognita. It is difficult to find someone not taken abackand fascinated by the incomprehensible but indisputable fact: there are material systems which are aware of themselves. Consciousness is self-cognizing code. During homo sapiens's relentness and often frustrated search for self-understanding various theories of consciousness have been and continue to be proposed. However, it remains unclear whether and at what level the problems of consciousness and intelligent thought can be resolved. Science's greatest (...) challenge is to answer the fundamental question: what precisely does a cognitive state amount to in physical terms? Albert Einstein insisted that the fundamental ideas of science are essentially simple and can be expressed in a language comprehensible to everyone. When one thinks about the complexities which present themselves in modern physics and even more so in the physics of life, one may wonder whether Einstein really meant what he said. Are we to consider the fundamental problem of the mind, whose understanding seems to lie outside the limits of the mind, to be essentially simple too? Knowledge is neither automatic nor universally deductive. Great new ideas are typically counterintuitive and outrageous, and connecting them by simple logical steps to existing knowledge is often a hard undertaking. The notion of a tensor was needed to provide the general theory of relativity; the notion of entropy had to be developed before we could get full insight into the laws of thermodynamics; the notice of information bit is crucial for communication theory, just as the concept of a Turing machine is instrumental in the deep understanding of a computer. To understand something, consciousness must reach an adequate intellectual level, even more so in order to understand itself. Reality is full of unending mysteries, the true explanation of which requires very technical knowledge, often involving notions not given directly to intuition. Even though the entire content and the results of this study are contained in the eight pages of the mathematical abstract, it would be unrealistic and impractical to suggest that anyone can gain full insight into the theory that presented here after just reading abstract. In our quest for knowledge we are exploring the remotest areas of the macrocosm and probing the invisible particles of the microcosm, from tiny neutrinos and strange quarks to black holes and the Big Bang. But the greatest mystery is very close to home: the greatest mystery is human consciousness. The question before us is whether the logical brain has evolved to a conceptual level where it is able to understand itself. (shrink)

The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will (...) wish to refer to the complete edition). It contains the whole of the preliminary sections (which present the authors' justification of the philosophical standpoint adopted at the outset of their work); the whole of Part 1 (in which the logical properties of propositions, propositional functions, classes and relations are established); section 6 of Part 2 (dealing with unit classes and couples); and Appendices A and B (which give further developments of the argument on the theory of deduction and truth functions). (shrink)

Albert Lautman (1908-1944) was a French philosopher of mathematics whose work played a crucial role in the history of contemporary French philosophy. His ideas have had an enormous influence on key contemporary thinkers including Gilles Deleuze and Alain Badiou, for whom he is a major touchstone in the development of their own engagements with mathematics. Mathematics, Ideas and the Physical Real presents the first English translation of Lautman's published works between 1933 and his death in 1944. Rather than being preoccupied (...) with the relation of mathematics to logic or with the problems of foundation, which have dominated philosophical reflection on mathematics, Lautman undertakes to develop an understanding of the broader structure of mathematics and its evolution. The two powerful ideas that are constants throughout his work, and which have dominated subsequent developments in mathematics, are the concept of mathematical structure and the idea of the essential unity underlying the apparent multiplicity of mathematical disciplines. This collection of his major writings offers readers a much-needed insight into his influence on the development of mathematics and philosophy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 26.2 (2002) 465-468 [Access article in PDF] Cogito Ergo Sum: The Life of René Descartes, by Richard Watson; vii & 375 pp. Boston: David R. Godine, 2002, $35.00. Scholarly in what it delivers, but delightful in how it delivers what it delivers, Cogito Ergo Sum is highly informative and fun to read. Touching on all the key places, players and events in the philosopher's life, Watson (...) tells us (at least) everything we wanted to know about Descartes as he cuts through the myths that have been passed down to us about him. Cutting through the myths meant dissociating his biography ("the first biography of Descartes since 1920 that is based on substantial new research, and the only one ever written for general readers"; p. 23) from the two still-operative hagiographic traditions: the French Catholic apologetic tradition and the scientific apologetic tradition. The result is a skeptical biography, full of distrust toward tradition and authority, written very much in the spirit of methodical doubt practiced by its subject.Descartes emerged from a powerful family tradition of law and medicine but mentions his childhood only six times. With good reason. He didn't like either his father or his brother and his mother died thirteen months after he was born. He was considered the family failure, not only wasting time in philosophy but squandering his share of the family fortune. He received an excellent eight-year education at the outstanding Jesuit school at La Flèche where he was immediately attracted to the certitude that mathematics offered him. He earned a law degree in Poitiers, spent time in the Army, though not in battle, and refused a judgeship.It was Isaac Beeckman who radically influenced Descartes's future when he awakened him to the possibility of applying mathematics to the problems of natural science. Watson examines sympathetically Descartes's eventual revolt against this father figure. Watson humanizes Descartes greatly when he examines both his relationship with his daughter, Francina, who died of scarlatina at the age of five, and his relationship with the child's mother, Helena Jans. He also rehabilitates the reputation of Helena who has been traditionally maligned by biographers of the philosopher. Here, and elsewhere, he speaks out loudly and clearly against "the denizens of the Saint Descartes Protection Society" who purport that Descartes broke his celibacy just this once. "Actually," writes Watson, "I think Descartes was enough of a scientist that he would have repeated the experiment several times, if only to see if the experience was the same the second and a third time" (pp. 181; 182).In 1628, totally unwilling to get involved in religious squabbles in France and desiring to live in "joyful anonymity," Descartes takes off for Friesland. This constituted a flight from family, social responsibility, and royalist Catholic totalitarian oppression. He moved around often to situate himself at universities where he wanted to do research and to avoid the plague. This great mathematical genius and father of modern philosophy wanted to replace [End Page 465] Aristotelian philosophy and science and thought correctly that it would be easier to get his ideas accepted here. He became friendly with Princess Elizabeth of Bohemia who spoke five languages and astutely critiqued his metaphysics. Despite the myths, neither was in love with the other. She inspired him to write The Passions of the Soul, which, oddly enough, he dedicated to Queen Christina of Sweden. He did, however, dedicate his Principles to Elizabeth. Christina was a bookish intellectual who loved the theater. In 1649, even though he had exhibited a clear dislike of court life, he agreed to be her tutor. He died in Sweden in 1650 during one of the coldest winters ever recorded in Europe.Why did he go to Sweden? He was broke and had little chance for a pension in France. In addition, while he knew the value of his work, he began to think his life might have been a failure. Perhaps he might really enjoy being a courtier after all. Certainly he feared that he would end... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 91 The passage which permitted such an interpretation is the following: This self-command is very different at different times.... Can we give any reason for these variations, except experience? Where then is the power of which we pretend to be conscious? Is there not here, either in a spiritual or a material substance, or both, some secret mechanism or structure of parts, upon which the effect depends...?" (...) (Enquiry, p. 68). This rhetorical question is, on the contrary, intended as the part of the third argument for the thesis "that even this command of the will gives us no real idea of force or energy" (Enquiry, p. 67). There is no evidence in the Enquiries that Hume concluded that self is a substance. On p. 45, Anderson "suggests that Hume employs 'impression'...in a less frequent usage, however.., to be a pressure, or pattern of light, or other means whereby objects are applied to the external organs of sense"t On p. 65 and also on p. 173, it is argued fallaciously that, "Hume considers mathematical ideas, like all other ideas, to be existences. If this be true, then the necessary relations ~mpposed to obtain among mathematical ideas cannot be discerned even though these ideas themselves be fully known." Now it is a matter of debate whether Hume believed that all mathematical ideas are derived from impressions (see my controversy with Flew on this issue, in Ratio, Vol. VI, No. 2, "Hume on Pure and Applied Geometry," and Flew's article, "Did Hume Distinguish Pure and Applied Geometry?," Ratio, Vol. VIII, No. 1). But even if we assume that this is his belief, it does not follow at all that the logical relations among these ideas are contingent. As Pap argued a long time ago, "A sensationalist might maintain that all the concepts that constitute a given proposition are derived from experience and might nevertheless admit that proposition itself is a priori in the sense of requiring no empirical verification in order to be assertable as true." Thus, Hume the sensationalist admitted that a priori knowledge is possible in mathematics, which is conversant exclusively about "relations of ideas" and not about "matters of fact." Finally, in the last chapter, it is argued that, "Hume's own suggestion concerning these three natural relations (resemblance, contiguity, and cause and effect) is that they are fundamentally material--that they consist in a certain characteristic movement of the animal spirit in the brain" (p. 163)I1 That, "The necessity is the relationship between cause and effect, Hume's remarks suggest, is ultimately their identity in essence. One passage which we have noted indicates that cause and effect are inseparable, and by use of Hume's own maxims we may strictly infer from their inseparability their identity" (p. 165)II And that, "Although Hume's doctrine of matter is ultimately of central importance in his system, his beginnings in moral philosophy doubtless led him to turn his attention away from the topic of matter" (p. 175). This book, though replete with quotations from Hume's works and exhibiting a great amount of microscopic observations, represents a mistaken view of Hume's epistemology. ~ARHANG ZABEEIq Roosevelt University, Chicago Hegel's Phenomenology. Dialogues on the Life of Mind. By Jacob Loewenberg. (La Salle, Ill. : The Open Court Publ. Co., 1965.Pp. xv + 377.) Hegel's Phenomenology o] Spirit, intriguing in its richness, has long remained one of the most forbidding of philosophical classics. Unfortunately, commentaries, though altogether indispensable, are sparse. Heretofore, the reader found himself referred mainly to two foreign language studies, G. A. Gabler's Kritik des Be~usst~ein~ of 1835, and J. Hyppolite's Gen~se et structure de la Phdnomdnologie de l'esprit; no full-scale work in English was available. And 92 HISTORY OF PHILOSOPHY even the reader prepared to turn to these works might register disenchantment: Gabler deals with two fifths of the book only, and Hyppolite is at times too close to the text to be much help. What remains is comment in the framework of more general studies of Hegel's philosophy (Royce, Haering, Kroner, Stace, Findlay and others). In this situation Loewenberg's contribution marks a... (shrink)

The first English collection of the work of Albert Lautman, a major figure in philosophy of mathematics and a key influence on Badiou and Deleuze.

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. (...) Her development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

The great vision of logical empiricism brought up a hope for ultimate solving the problem of understanding our knowledge. Starting, for the first time in the history of empiricism, from a point of view typical for mathematicians not empiricists, i.e. from the program of logicism, the philosophers of Vienna Circle aimed at creating a consistent philosophy which makes possible both mathematics and empirical knowledge. Today, after more than a half century, their efforts seem to be completely futile. The topic (...) of the paper is to find out the reasons why the program failed. It seems that the deepest reason for the failure lies in the philosophy of mathematics adopted by logical empiricists - namely logicism. The main problem of the contemporary methodology is to find a remedy for this ilness. the paper suggests some ideas in this direction. One of them is more serious approach to the philosophy of mathematics. Another is the recognition of the crucial role epistemic and other values play in scientific research. (shrink)

This article, written in Bengali ('Gonit Dorshon' means `philosophy of mathematics' ), briefly reviews a few of the major points of view toward mathematics and the world of mathematical entities, and interprets the philosophy of mathematics as an interaction between these. The existence of these different points of view is indicative that mathematics, in spite of being of universal validity, can nevertheless accommodate alternatives. In particular, I review the alternative viewpoints of Platonism and Intuitionism and present the case that (...) in spite of their great differences, they are not mutually exclusive - that both can be accommodated within the infinite edifice of mathematics. This, in turn, is argued to be consistent with the viewpoint of Category Theory that holds the promise of an entirely new interpretation of the world of mathematics and the relation of that world to the world of our concepts and ideas: mathematics is a human enterprise and mathematical logic is a reflection of how our ideas and concepts are formed and combined with one another. I venture that this, perhaps, is the view of mathematics that Ludwig Wittgenstein would espouse. (shrink)

_ Source: _Volume 29, Issue 1, pp 39 - 65 Since Euclid, optics has been considered a geometrical science, which Aristotle defines as a “mixed” mathematical science. Hobbes follows this tradition and clearly places optics among physical sciences. However, modern scholars point to a confusion between geometry and physics and do not seem to agree about the way Hobbes mixes both sciences. In this paper, I return to this alleged confusion and intend to emphasize the peculiarity of Hobbes’s geometrical (...) optics. This paper suggests that Hobbes’s conception of geometrical optics, as a mixed mathematical science, greatly differs from Descartes’s one, mainly because they do not share the same “mechanical conception of nature.” I will argue that Hobbes and Descartes also have in common the quest for a different kind of geometry for their optics, different from that of the Ancients. I will show that this departure is not recent since Hobbes’s approach is already evident in 1636, when he judges the demonstrations of his contemporary friends, Claude Mydorge and Walter Warner. Finally the paper broadly suggests what is noteworthy in Hobbes’s optics, that is, the importance of the idea of force in his mechanics, although he was not able to conceptualize it in other terms than “quickness.”. (shrink)

Dutch Mathematician Luitzen Egbertus Jan Brouwer (1881-1966) was a rebel. His doctoral thesis... was the manifesto of an angry young man taking on the mathematical establishment on all fronts. In a short time he established a world-wide reputation for himself; his genius and originality were acknowledged by the great mathematicians of his time... The Intuitionist-Formalist debate became a personal feud between the mathematical giants Brouwer and Hilbert, and ended in 1928 with the expulsion of Brouwer from the (...) editorial board of the Mathematische Annalen by dictat of Hilbert. Forsaken, humiliated and disillusioned Brouwer abandoned his Intuitionist Programme and withdrew into silence just about the time when the Formalist Programme appeared to be fundamentally flawed and major opposition collapsed... This book attempts to follow the `genetic' development of Brouwer's ideas, linking the man Brouwer, his Weltanschauung, his philosophy of mathematics and his reconstruction of mathematics. Brouwer's own writings, his publications as well as his unpublished papers, are its immediate and main source of reference. It is the second volume in the new series Studies in the History and Philosophy of Mathematics, and is written for the specialist as well as for the general reader interested in mathematics and the interpretation of its status and function. (shrink)

Albert Lautman. Mathematics, Ideas and the Physical Real. Simon B. Duffy, trans. London and New York: Continuum, 2011. 978-1-4411-2344-2 (pbk); 978-1-44114656-4 (hbk); 978-1-44114433-1 (pdf e-bk); 978-1-44114654-0 (epub e-bk). Pp. xlii + 310.

First published in 1978, this reissue presents a seminal philosophical work by professor Putnam, in which he puts forward a conception of knowledge which makes ethics, practical knowledge and non-mathematic parts of the social sciences just as much parts of 'knowledge' as the sciences themselves. He also rejects the idea that knowledge can be demarcated from non-knowledge by the fact that the former alone adheres to 'the scientific method'. The first part of the book consists of Professor Putnam's John (...) Locke lectures, delivered at the University of Oxford in 1976, offering a detailed examination of a 'physicalist' theory of reference against a background of the works of Tarski, Carnap, Popper, Hempel and Kant. The analysis then extends to notions of truth, the character of linguistic enquiry and social scientific enquiry in general, interconnecting with the great metaphysical problem of realism, the nature of language and reference, and the character of ourselves. (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)

Jakob Friedrich Fries (1773-1843): A Philosophy of the Exact Sciences -/- Shortened version of the article of the same name in: Tabula Rasa. Jenenser magazine for critical thinking. 6th of November 1994 edition -/- 1. Biography -/- Jakob Friedrich Fries was born on the 23rd of August, 1773 in Barby on the Elbe. Because Fries' father had little time, on account of his journeying, he gave up both his sons, of whom Jakob Friedrich was the elder, to the Herrnhut Teaching (...) Institution in Niesky in 1778. Fries attended the theological seminar in Niesky in autumn 1792, which lasted for three years. There he (secretly) began to study Kant. The reading of Kant's works led Fries, for the first time, to a deep philosophical satisfaction. His enthusiasm for Kant is to be understood against the background that a considerable measure of Kant's philosophy is based on a firm foundation of what happens in an analogous and similar manner in mathematics. -/- During this period he also read Heinrich Jacobi's novels, as well as works of the awakening classic German literature; in particular Friedrich Schiller's works. In 1795, Fries arrived at Leipzig University to study law. During his time in Leipzig he became acquainted with Fichte's philosophy. In autumn of the same year he moved to Jena to hear Fichte at first hand, but was soon disappointed. -/- During his first sojourn in Jenaer (1796), Fries got to know the chemist A. N. Scherer who was very influenced by the work of the chemist A. L. Lavoisier. Fries discovered, at Scherer's suggestion, the law of stoichiometric composition. Because he felt that his work still need some time before completion, he withdrew as a private tutor to Zofingen (in Switzerland). There Fries worked on his main critical work, and studied Newton's "Philosophiae naturalis principia mathematica". He remained a lifelong admirer of Newton, whom he praised as a perfectionist of astronomy. Fries saw the final aim of his mathematical natural philosophy in the union of Newton's Principia with Kant's philosophy. -/- With the aim of qualifying as a lecturer, he returned to Jena in 1800. Now Fries was known from his independent writings, such as "Reinhold, Fichte and Schelling" (1st edition in 1803), and "Systems of Philosophy as an Evident Science" (1804). The relationship between G. W. F. Hegel and Fries did not develop favourably. Hegel speaks of "the leader of the superficial army", and at other places he expresses: "he is an extremely narrow-minded bragger". On the other hand, Fries also has an unfavourable take on Hegel. He writes of the "Redundancy of the Hegelistic dialectic" (1828). In his History of Philosophy (1837/40) he writes of Hegel, amongst other things: "Your way of philosophising seems just to give expression to nonsense in the shortest possible way". In this work, Fries appears to argue with Hegel in an objective manner, and expresses a positive attitude to his work. -/- In 1805, Fries was appointed professor for philosophy in Heidelberg. In his time spent in Heidelberg, he married Caroline Erdmann. He also sealed his friendships with W. M. L. de Wette and F. H. Jacobi. Jacobi was amongst the contemporaries who most impressed Fries during this period. In Heidelberg, Fries wrote, amongst other things, his three-volume main work New Critique of Reason (1807). -/- In 1816 Fries returned to Jena. When in 1817 the Wartburg festival took place, Fries was among the guests, and made a small speech. 1819 was the so-called "Great Year" for Fries: His wife Caroline died, and Karl Sand, a member of a student fraternity, and one of Fries' former students stabbed the author August von Kotzebue to death. Fries was punished with a philosophy teaching ban but still received a professorship for physics and mathematics. Only after a period of years, and under restrictions, he was again allowed to read philosophy. From now on, Fries was excluded from political influence. The rest of his life he devoted himself once again to philosophical and natural studies. During this period, he wrote "Mathematical Natural Philosophy" (1822) and the "History of Philosophy" (1837/40). -/- Fries suffered from a stroke on New Year's Day 1843, and a second stroke, on the 10th of August 1843 ended his life. -/- 2. Fries' Work Fries left an extensive body of work. A look at the subject areas he worked on makes us aware of the universality of his thinking. Amongst these subjects are: Psychic anthropology, psychology, pure philosophy, logic, metaphysics, ethics, politics, religious philosophy, aesthetics, natural philosophy, mathematics, physics and medical subjects, to which, e.g., the text "Regarding the optical centre in the eye together with general remarks about the theory of seeing" (1839) bear witness. With popular philosophical writings like the novel "Julius and Evagoras" (1822), or the arabesque "Longing, and a Trip to the Middle of Nowhere" (1820), he tried to make his philosophy accessible to a broader public. Anthropological considerations are shown in the methodical basis of his philosophy, and to this end, he provides the following didactic instruction for the study of his work: "If somebody wishes to study philosophy on the basis of this guide, I would recommend that after studying natural philosophy, a strict study of logic should follow in order to peruse metaphysics and its applied teachings more rapidly, followed by a strict study of criticism, followed once again by a return to an even closer study of metaphysics and its applied teachings." -/- 3. Continuation of Fries' work through the Friesian School -/- Fries' ideas found general acceptance amongst scientists and mathematicians. A large part of the followers of the "Fries School of Thought" had a scientific or mathematical background. Amongst them were biologist Matthias Jakob Schleiden, mathematics and science specialist philosopher Ernst Friedrich Apelt, the zoologist Oscar Schmidt, and the mathematician Oscar Xavier Schlömilch. Between the years 1847 and 1849, the treatises of the "Fries School of Thought", with which the publishers aimed to pursue philosophy according to the model of the natural sciences appeared. In the Kant-Fries philosophy, they saw the realisation of this ideal. The history of the "New Fries School of Thought" began in 1903. It was in this year that the philosopher Leonard Nelson gathered together a small discussion circle in Goettingen. Amongst the founding members of this circle were: A. Rüstow, C. Brinkmann and H. Goesch. In 1904 L. Nelson, A. Rüstow, H. Goesch and the student W. Mecklenburg travelled to Thuringia to find the missing Fries writings. In the same year, G. Hessenberg, K. Kaiser and Nelson published the first pamphlet from their first volume of the "Treatises of the Fries School of Thought, New Edition". -/- The school set out with the aim of searching for the missing Fries' texts, and re-publishing them with a view to re-opening discussion of Fries' brand of philosophy. The members of the circle met regularly for discussions. Additionally, larger conferences took place, mostly during the holidays. Featuring as speakers were: Otto Apelt, Otto Berg, Paul Bernays, G. Fraenkel, K. Grelling, G. Hessenberg, A. Kronfeld, O. Meyerhof, L. Nelson and R. Otto. On the 1st of March 1913, the Jakob-Friedrich-Fries society was founded. Whilst the Fries' school of thought dealt in continuum with the advancement of the Kant-Fries philosophy, the members of the Jakob-Friedrich-Fries society's main task was the dissemination of the Fries' school publications. In May/June, 1914, the organisations took part in their last common conference before the gulf created by the outbreak of the First World War. Several members died during the war. Others returned disabled. The next conference took place in 1919. A second conference followed in 1921. Nevertheless, such intensive work as had been undertaken between 1903 and 1914 was no longer possible. -/- Leonard Nelson died in October 1927. In the 1930's, the 6th and final volume of "Treatises of the Fries School of Thought, New Edition" was published. Franz Oppenheimer, Otto Meyerhof, Minna Specht and Grete Hermann were involved in their publication. -/- 4. About Mathematical Natural Philosophy -/- In 1822, Fries' "Mathematical Natural Philosophy" appeared. Fries rejects the speculative natural philosophy of his time - above all Schelling's natural philosophy. A natural study, founded on speculative philosophy, ceases with its collection, arrangement and order of well-known facts. Only a mathematical natural philosophy can deliver the necessary explanatory reasoning. The basic dictum of his mathematical natural philosophy is: "All natural theories must be definable using purely mathematically determinable reasons of explanation." Fries is of the opinion that science can attain completeness only by the subordination of the empirical facts to the metaphysical categories and mathematical laws. -/- The crux of Fries' natural philosophy is the thought that mathematics must be made fertile for use by the natural sciences. However, pure mathematics displays solely empty abstraction. To be able to apply them to the sensory world, an intermediatory connection is required. Mathematics must be connected to metaphysics. The pure mechanics, consisting of three parts are these: a) A study of geometrical movement, which considers solely the direction of the movement, b) A study of kinematics, which considers velocity in Addition, c) A study of dynamic movement, which also incorporates mass and power, as well as direction and velocity. -/- Of great interest is Fries' natural philosophy in view of its methodology, particularly with regard to the doctrine "leading maxims". Fries calls these "leading maxims" "heuristic", "because they are principal rules for scientific invention". -/- Fries' philosophy found great recognition with Carl Friedrich Gauss, amongst others. Fries asked for Gauss's opinion on his work "An Attempt at a Criticism based on the Principles of the Probability Calculus" (1842). Gauss also provided his opinions on "Mathematical Natural Philosophy" (1822) and on Fries' "History of Philosophy". Gauss acknowledged Fries' philosophy and wrote in a letter to Fries: "I have always had a great predilection for philosophical speculation, and now I am all the more happy to have a reliable teacher in you in the study of the destinies of science, from the most ancient up to the latest times, as I have not always found the desired satisfaction in my own reading of the writings of some of the philosophers. In particular, the writings of several famous (maybe better, so-called famous) philosophers who have appeared since Kant have reminded me of the sieve of a goat-milker, or to use a modern image instead of an old-fashioned one, of Münchhausen's plait, with which he pulled himself from out of the water. These amateurs would not dare make such a confession before their Masters; it would not happen were they were to consider the case upon its merits. I have often regretted not living in your locality, so as to be able to glean much pleasurable entertainment from philosophical verbal discourse." -/- The starting point of the new adoption of Fries was Nelson's article "The critical method and the relation of psychology to philosophy" (1904). Nelson dedicates special attention to Fries' re-interpretation of Kant's deduction concept. Fries awards Kant's criticism the rationale of anthropological idiom, in that he is guided by the idea that one can examine in a psychological way which knowledge we have "a priori", and how this is created, so that we can therefore recognise our own knowledge "a priori" in an empirical way. Fries understands deduction to mean an "awareness residing darkly in us is, and only open to basic metaphysical principles through conscious reflection.". -/- Nelson has pointed to an analogy between Fries' deduction and modern metamathematics. In the same manner, as with the anthropological deduction of the content of the critical investigation into the metaphysical object show, the content of mathematics become, in David Hilbert's view, the object of metamathematics. -/-. (shrink)

This book presents a moderately revisionist history of the great books idea anchored in the following movements and struggles: fighting anti-intellectualism, advocating for the liberal arts, distributing cultural capital, and promoting a public philosophy, anchored in mid-century liberalism, that fostered a shared civic culture.

This volume tells the story of the legacy and impact of the great German polymath Gottfried Wilhelm Leibniz (1646-1716). Leibniz made significant contributions to many areas, including philosophy, mathematics, political and social theory, theology, and various sciences. The essays in this volume explores the effects of Leibniz’s profound insights on subsequent generations of thinkers by tracing the ways in which his ideas have been defended and developed in the three centuries since his death. Each of the 11 essays is (...) concerned with Leibniz’s legacy and impact in a particular area, and between them they show not just the depth of Leibniz’s talents but also the extent to which he shaped the various domains to which he contributed, and in some cases continues to shape them today. With essays written by experts such as Nicholas Jolley, Pauline Phemister, and Philip Beeley, this volume is essential reading not just for students of Leibniz but also for those who wish to understand the game-changing impact made by one of history’s true universal geniuses. (shrink)

Ideograms have great communicative value. They refer to concepts in a simple manner, easing the understanding of related ideas. Moreover, ideograms can simplify the often cumbersome notation used in the fields of Physics and physical Chemistry. Nonetheless only a few ideograms—like \ and Å—have been defined to date. In this work we propose that the scientific community follows the example of Mathematics—as well as that of oriental languages—and bestows a more important role upon ideograms. To support this thesis we (...) propose ideograms for essential concepts in Physics and Chemistry. They are designed to be intuitive, and their goal is to make equations easier to read and understand. Our symbols are included in a publicly available package. (shrink)

Bertrand Russell, the great philosopher, was extremely prolific in various fields of philosophy, such as metaphysics, mathematical logic and mathematical philosophy, linguistic philosophy, ethics, epistemology, and social and political philosophy, but left little legacy in aesthetics. Some scholars regretted that “If the 20th-century had seen any polymath, Russell is the one. The only branch of philosophy he did not write on is aesthetics.” Although Russell did not write a book or article specifically on aesthetics, discussions on the (...) subject can be found in his various writings, for example, the most prominent proposal of “the beauty of mathematics.” In addition, he also talked about how agnostics explain the beauty and harmony of nature from the perspective of epistemology. Views on aesthetics are scattered in Russell's writings, and no consistent views have been formed; some fragmented ideas about aesthetics are scattered in different philosophical writings. Compared with his profound contribution to other fields of philosophy, Russell did not deal with aesthetic issues systematically, nor did he attempt to answer the basic questions of aesthetics. In addition to personal qualities, there were certain constraints of the greater social environment. One of the main reasons is that in the English world of the 20th century, the analytical philosophy of language occupied the central stage, so that the space reserved for ethics, aesthetics, and political philosophy was quite limited. Furthermore, aesthetics was indeed not a popular philosophical topic culturally acceptable at the time. (shrink)