The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter.

This book intends to show that, in philosophy of mathematics, radical naturalism (or physicalism), nominalism and strict finitism (which does not assume the reality of infinity in any format, not even potential infinity) can account for the applications of classical mathematics in current scientific theories about the finite physical world above the Planck scale. For that purpose, the book develops some significant applied mathematics in strict finitism, which is essentially quantifier-free elementary recursive arithmetic (with real numbers encoded as (...) elementary recursive Cauchy sequences of rational numbers). Applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the basic applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitisitc system is perhaps surprising in itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. The first chapter of the book contains an informal introduction to its philosophical motivation and technical strategy. The book is intended for students and researchers in philosophy of mathematics. -/- Contents -/- Preface 1 Introduction 2 Strict Finitism 3 Calculus 4 Metric Space 5 Complex Analysis 6 Integration 7 Hilbert Space 8 Semi-Riemann Geometry References Index. (shrink)

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, (...) such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism. (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)

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)

Excerpt from Vector Analysis and the Theory of Relativity One of the most striking effects of the publication of Einstein's papers on generalized relativity and of the discussions which arose in connection with the subsequent astronomical observations was to make students of physics renew their study of mathematics. At first they attempted to learn simply the technique, but soon there was a demand to understand more; real mathematical insight was sought. Unfortunately there were no books available, not even papers. (...) Dr. Murnaghan's little book is a most successful attempt to supply what is a definite need. Every physicist can read it with profit. He will learn the meaning of a vector for the first time. He will learn methods which are available for every field of mathematical physics. He will see which of the processes used by Einstein and others are strictly mathematical and which are physical. Every chapter is illuminating, and the treatment of the subject is that of a student of mathematics and is not developed ad hoc. The extension of surface and line integrals is most interesting for physicists and the discussion of the space relations in a four-dimensional geometry is one most needed. This is specially true concerning the case of point-symmetry which forms the basis of Einstein's formulae for gravitation as applied to the solar system. I feel personally that I owe to this book a great debt. I have read it with care and shall read it again. It has given me a definiteness of understanding which I never had before, and a vision of a field of knowledge which before was remote. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

The problem of how mathematics and physics are related at a foundational level is of interest. The approach taken here is to work towards a coherent theory of physics and mathematics together by examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, (...) the complex numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by C n which is the set of pairs, R n,I n, of n figure rational numbers in some basis. The properties of these numbers are based on those of numerical measurement outcomes for continuous variables. A model of space and time based on R n is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n→∞ to obtain locally flat continuum-based space and time is outlined. Also R n based space is invariant under scale transformations. These correspond to expansion and contraction of space relative to a flat background. The location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics, based on C n and on R n space are briefly investigated. (shrink)

PLEASE NOTE: This is the corrected 2nd eBook edition, 2021. ●●●●● _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly, technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost. ●●●●● The printed edition was released on September 1, 2021 and is now available through (...) all booksellers, including Barnes & Noble, Amazon, and brick-and-mortar bookstores under ISBN 978-0-578-88646-6. ●●●●● In light of the length of this book, readers who would like to have a more detailed description of the book's objectives and method may find it helpful to read the detailed and clearly written Wikipedia entry about this work: From the Wikipedia search page, use the search phrase "Critique of Impure Reason". At least at the time of this writing (11/29/2021), the Wikipedia entry is well-researched and accurate. ●●●●● In addition, a "Primer on Bartlett's CRITIQUE OF IMPURE REASON" has been made available by the author. It is available under its title through PhilPapers and other philosophy online archives. ●●●●● COMMENDATIONS OF THIS WORK, from the back cover of the published edition: ●●●●● “I admire its range of philosophical vision.” – Nicholas Rescher, Distinguished University Professor of Philosophy, University of Pittsburgh, author of more than 100 books. ●●●●● “Bartlett’s _Critique of Impure Reason_ is an impressive, bold, and ambitious work. Careful scholarship is balanced by original analyses that lead the reader to recognize the limits of meaning, knowledge, and conceptual possibility. The work addresses a host of traditional philosophical problems, among them the nature of space, time, causality, consciousness, the self, other minds, ontology, free will and determinism, and others. The book culminates in a fascinating and profound new understanding of relativity physics and quantum theory.” – Gerhard Preyer, Professor of Philosophy, Goethe-University, Frankfurt am Main, Germany, author of many books including _Concepts of Meaning_, _Beyond Semantics and Pragmatics_, _Intention and Practical Thought_, and _Contextualism in Philosophy_. ●●●●● “[This work’s] goal is of a unique and difficult species: Dr. Bartlett seeks to develop a formal logical calculus on the basis of transcendental philosophical arguments; in fact, he hopes that this calculus will be the formal expression of the transcendental foundation of knowledge.... I consider Dr. Bartlett’s work soundly conceived and executed with great skill.” – C. F. von Weizsäcker, philosopher and physicist, former Director, Max-Planck-Institute, Starnberg, Germany. ●●●●● “Bartlett has written an American “Prolegomena to All Future Metaphysics.” He aims rigorously to eliminate meaningless assertions, reach bedrock, and place philosophy on a firm foundation that will enable it, like science and mathematics, to produce lasting results that generations to come can build on. This is a great book, the fruit of a lifetime of research and reflection, and it deserves serious attention.” — Martin X. Moleski, former Professor, Canisius College, Buffalo, NY, studies of scientific method, the presuppositions of thought, and the self-referential nature of epistemology. ●●●●● “Bartlett has written a book on what might be called the underpinnings of philosophy. It has fascinating depth and breadth, and is all the more striking due to its unifying perspective based on the concepts of reference and self-reference.” – Don Perlis, Professor of Computer Science, University of Maryland, author of numerous publications on self-adjusting autonomous systems and philosophical issues concerning self-reference, mind, and consciousness. ●●●●● ●●●●● The _Critique of Impure Reason: Horizons of Possibility and Meaning_ comprises a major and important contribution to philosophy. Thanks to the generosity of its publisher, this massive 885-page volume has been published as a free open access eBook (3.75MB) as well as an open access printed edition. It inaugurates a revolutionary paradigm shift in philosophical thought by providing compelling and long-sought-for solutions to a wide range of philosophical problems. In the process, the work fundamentally transforms the way in which the concepts of reference, meaning, and possibility are understood. The book includes a Foreword by the celebrated German philosopher and physicist Carl Friedrich von Weizsäcker. ●●●●● In Kant’s _Critique of Pure Reason_ we find an analysis of the preconditions of experience and of knowledge. In contrast, but yet in parallel, the new _Critique_ focuses upon the ways—unfortunately very widespread and often unselfconsciously habitual—in which many of the concepts that we employ _conflict_ with the very preconditions of meaning and of knowledge. ●●●●● This is a book about the boundaries of frameworks and about the unrecognized conceptual confusions in which we become entangled when we attempt to transgress beyond the limits of the possible and meaningful. We tend either not to recognize or not to accept that we all-too-often attempt to trespass beyond the boundaries of the frameworks that make knowledge possible and the world meaningful. ●●●●● The _Critique of Impure Reason_ proposes a bold, ground-breaking, and startling thesis: that a great many of the major philosophical problems of the past can be solved through the recognition of a viciously deceptive form of thinking to which philosophers as well as non-philosophers commonly fall victim. For the first time, the book advances and justifies the criticism that a substantial number of the questions that have occupied philosophers fall into the category of “impure reason,” violating the very conditions of their possible meaningfulness. ●●●●● The purpose of the study is twofold: first, to enable us to recognize the boundaries of what is referentially forbidden—the limits beyond which reference becomes meaningless—and second, to avoid falling victims to a certain broad class of conceptual confusions that lie at the heart of many major philosophical problems. As a consequence, the boundaries of _possible meaning_ are determined. ●●●●● Bartlett, the author or editor of more than 20 books, is responsible for identifying this widespread and delusion-inducing variety of error, _metalogical projection_. It is a previously unrecognized and insidious form of erroneous thinking that undermines its own possibility of meaning. It comes about as a result of the pervasive human compulsion to seek to transcend the limits of possible reference and meaning. ●●●●● Based on original research and rigorous analysis combined with extensive scholarship, the _Critique of Impure Reason_ develops a self-validating method that makes it possible to recognize, correct, and eliminate this major and pervasive form of fallacious thinking. In so doing, the book provides at last provable and constructive solutions to a wide range of major philosophical problems. ●●●●● CONTENTS AT A GLANCE ▪▪▪▪▪ Preface ▪▪▪▪▪ Foreword by Carl Friedrich von Weizsäcker ▪▪▪▪▪ Acknowledgments ▪▪▪▪▪ Avant-propos: A philosopher’s rallying call ▪▪▪▪▪ Introduction ▪▪▪▪▪ A note to the reader ▪▪▪▪▪ A note on conventions ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART I ▪▪▪▪▪ ▪▪▪▪▪ WHY PHILOSOPHY HAS MADE NO PROGRESS AND HOW IT CAN ▪▪▪▪▪ 1 Philosophical-psychological prelude ▪▪▪▪▪ 2 Putting belief in its place: Its psychology and a needed polemic ▪▪▪▪▪ 3 Turning away from the linguistic turn: From theory of reference to metalogic of reference ▪▪▪▪▪ 4 The stepladder to maximum theoretical generality ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART II ▪▪▪▪▪ THE METALOGIC OF REFERENCE ▪▪▪▪▪ A New Approach to Deductive, Transcendental Philosophy ▪▪▪▪▪ 5 Reference, identity, and identification ▪▪▪▪▪ 6 Self-referential argument and the metalogic of reference ▪▪▪▪▪ 7 Possibility theory ▪▪▪▪▪ 8 Presupposition logic, reference, and identification ▪▪▪▪▪ 9 Transcendental argumentation and the metalogic of reference ▪▪▪▪▪ 10 Framework relativity ▪▪▪▪▪ 11 The metalogic of meaning ▪▪▪▪▪ 12 The problem of putative meaning and the logic of meaninglessness ▪▪▪▪▪ 13 Projection ▪▪▪▪▪ 14 Horizons ▪▪▪▪▪ 15 De-projection ▪▪▪▪▪ 16 Self-validation ▪▪▪▪▪ 17 Rationality: Rules of admissibility ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART III ▪▪▪▪▪ PHILOSOPHICAL APPLICATIONS OF THE METALOGIC OF REFERENCE ▪▪▪▪▪ Major Problems and Questions of Philosophy and the Philosophy of Science ▪▪▪▪▪ 18 Ontology and the metalogic of reference ▪▪▪▪▪ 19 Discovery or invention in general problem-solving, mathematics, and physics ▪▪▪▪▪ 20 The conceptually unreachable: “The far side” ▪▪▪▪▪ 21 The projections of the external world, things-in-themselves, other minds, realism, and idealism ▪▪▪▪▪ 22 The projections of time, space, and space-time ▪▪▪▪▪ 23 The projections of causality, determinism, and free will ▪▪▪▪▪ 24 Projections of the self and of solipsism ▪▪▪▪▪ 25 Non-relational, agentless reference and referential fields ▪▪▪▪▪ 26 Relativity physics as seen through the lens of the metalogic of reference ▪▪▪▪▪ 27 Quantum theory as seen through the lens of the metalogic of reference ▪▪▪▪▪ 28 Epistemological lessons learned from and applicable to relativity physics and quantum theory ▪▪▪▪▪ ▪▪▪▪▪ PART IV ▪▪▪▪▪ HORIZONS ▪▪▪▪▪ 29 Beyond belief ▪▪▪▪▪ 30 _Critique of Impure Reason_: Its results in retrospect ▪▪▪▪▪ ▪▪▪▪▪ SUPPLEMENT ▪▪▪▪▪ The Formal Structure of the Metalogic of Reference ▪▪▪▪▪ APPENDIX I ▪▪▪▪▪ The Concept of Horizon in the Work of Other Philosophers ▪▪▪▪▪ APPENDIX II ▪▪▪▪▪ Epistemological Intelligence ▪▪▪▪▪ References ▪▪▪▪▪ Index ▪▪▪▪▪ About the author. 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This book offers an innovative introduction to the psychological basis of mathematics and the nature of mathematical thinking and learning, using an approach that empowers students by fostering their own construction of mathematical structures. Through accessible and engaging writing, award-winning mathematician and educator Anderson Norton reframes mathematics as something that exists first in the minds of students, rather than something that exists first in a textbook. By exploring the psychological basis for mathematics at every level - (...) including geometry, algebra, calculus, complex analysis and more - Norton unlocks students' personal power to construct mathematical objects based on their own mental activity, and illustrates the power of mathematics in organizing the world as we know it. Including reflections and activities designed to inspire awareness of the mental actions and processes coordinated in practicing mathematics, the book is geared towards current and future secondary and elementary mathematics teachers who will empower the next generation of mathematicians and STEM majors. Those interested in the history and philosophy that underpins mathematics will also benefit from this book, as well as those informed and curious minds attentive to the human experience more generally. Anderson Norton is Professor in the Department of Mathematics at Virginia Tech, USA, where he has been teaching mathematics courses for future teachers for 15 years. He is the editor of Constructing Number, and the co-author of Developing Fractions Knowledge and Numeracy for All Learners. (shrink)

This book intends to show that radical naturalism, nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry. The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of (...) those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. Both professional researchers and students of philosophy of mathematics will benefit greatly from reading this book. (shrink)

A relative tensor calculus is formulated for expressing equations of mathematical physics. A tensor time derivative operator ▽ b a is defined which operates on tensors λia...ib. Equations are written in a rigid, flat, inertial or other coordinate system a, altered to relative tensor notation, and are thereby expressed in general flowing coordinate systems or materials b, c, d, .... Mirror tensor expressions for ▽ b a λic...id and ▽ b a λic...id exist in a relative geometry G (...) if and only if a rigid coordinate system a exists in G, where ▽ b a λic = λ ,0c ic + λkev ckc aic + λ kc ic v b ckc , ▽jcλic = λ ,jc ic + λkcΓ jc kc ie , and v b aic is the velocity of b relative to a with components in c. These operators are convenient in theoretical analyses and can be incorporated into machine programs for the numerical solution of physical problems. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation (...) can be also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Philosophy of PhysicsMartin CurdRoberto Torretti. The Philosophy of Physics. Cambridge: Cambridge University Press, 1999. Pp. xvi + 512. Cloth, $64.95. Paper, $23.95.This is the first volume in a new Cambridge series, "The Evolution of Modern Philosophy." It is a historical work, tracing the interaction between physics and philosophy from the scientific revolution of the seventeenth century through general relativity and quantum mechanics in the twentieth century. The (...) emphasis is on the philosophy in physics (rather than philosophizing about physics) and, with the exceptions noted below, the focus is on scientists (scientist-philosophers in many cases) rather than philosophers outside of science. Most of the book is about the conceptual and mathematical development of the core branches of physics (mainly mechanics, but also geometry, thermodynamics, and field theory). There are seven chapters plus three supplements giving the mathematical essentials of vector analysis, lattice theory, and topology.The hero of chapter one is Galileo who, more than anyone else prior to Newton, set physics on its modern path towards mathematization and away from Aristotle. Also discussed in chapter one are Descartes, Huyghens, and Leibniz. Chapter two is devoted to Newton (both the foundations of Newtonian mechanics and the derivation of the law of universal gravitation) and follows the development of analytical mechanics up to Lagrange and Hamilton. Chapter three, appropriately entitled "Kant," is the only chapter devoted to a philosopher. After a brief discussion of Leibniz, Berkeley, and Kant's pre-critical writings, it covers the important scientific topics in the Critique of Pure Reason (space, time, the three Analogies) but avoids the Metaphysical Foundations of Natural Science (referring the reader to Michael Friedman's Kant and the Exact Sciences). Chapter four looks at scientific developments in three areas that proved to be important for twentieth-century physics: non-Euclidean geometry from Lobachevsky to Riemann, electromagnetic field theory (Maxwell), and thermal physics (classical thermodynamics, the kinetic theory, and statistical mechanics) from Carnot to Boltzmann. The chapter concludes, somewhat untidily, with a section on four philosophers of science (Whewell, Peirce, Mach, and Duhem), selected, we are told, because of their influence on philosophy of science in the second half of the twentieth century. (For the "two great philosopher-scientists" Helmholtz and Poincaré, the reader is referred to Torretti's earlier books, Philosophy of Geometry from Riemann to Poincaré, and Relativity and Geometry.) Chapter five covers both the special and general theory of relativity. After a masterful exposition of the special theory and its Minkowski spacetime representation, the usual topics of philosophical interest are discussed; these include the conventionality of simultaneity thesis, the twin paradox, the relativistic concept of mass, and determinism. The rather brief treatment of the general theory is, unavoidably, mathematically demanding and carries the subject through to recent inflationary cosmological models. Chapter six (also mathematically demanding at times) examines the development and structure of quantum mechanics, aptly praised as "one of the most elegant creations of the human spirit" (340). Torretti wisely ignores the complexities of quantum field theory since the philosophical problems discussed—measurement, the EPR paradox, the interpretation of the y-function—remain when quantum mechanics is made Lorentz invariant; but he urges philosophers of science to follow the lead of Michael Redhead and [End Page 602] Paul Teller in exploring its metaphysical ramifications. Of special note are the author's trenchant criticisms of attempts to solve the quantum mysteries by Bohr (complementarity), Bohm (hidden variables), Putnam (quantum logic), and Everett (the many worlds interpretation). The book concludes with the author's reflections on issues in the philosophy of science. Torretti defends the structuralist explication of physical theories (more familiarly known as the semantic conception or the model-theoretic view) that informs much of his book, arguing that it succeeds where the "received view" of logical empiricism failed while avoiding Kuhnian incommensurability—"a pseudoproblem that made philosophy of science the laughing stock of practicing physicists" (404). The chapter ends with an attack on inductive logic: "one of the blindest alleys in twentieth-century philosophy" (437).The writing is admirably clear and the author has done a commendable job of explaining the arguments of historical scientists while presenting their theories in a clear... (shrink)

In this revolutionary work, the author sets the stage for the science of the 21st Century, pursuing an unprecedented synthesis of fields previously considered unrelated. Beginning with simple classical concepts, he ends with a complex multidisciplinary theory requiring a high level of abstraction. The work progresses across the sciences in several multidisciplinary directions: Mathematical logic, fundamental physics, computer science and the theory of intelligence. Extraordinarily enough, the author breaks new ground in all these fields. In the field of (...) fundamental physics the author reaches the revolutionary conclusion that physics can be viewed and studied as logic in a fundamental sense, as compared with Einstein's view of physics as space-time geometry. This opens new, exciting prospects for the study of fundamental interactions. A formulation of logic in terms of matrix operators and logic vector spaces allows the author to tackle for the first time the intractable problem of cognition in a scientific manner. In the same way as the findings of Heisenberg and Dirac in the 1930s provided a conceptual and mathematical foundation for quantum physics, matrix operator logic supports an important breakthrough in the study of the physics of the mind, which is interpreted as a fractal of quantum mechanics. Introducing a concept of logic quantum numbers, the author concludes that the problem of logic and the intelligence code in general can be effectively formulated as eigenvalue problems similar to those of theoretical physics. With this important leap forward in the study of the mechanism of mind, the author concludes that the latter cannot be fully understood either within classical or quantum notions. A higher-order covariant theory is required to accommodate the fundamental effect of high-level intelligence. The landmark results obtained by the author will have implications and repercussions for the very foundations of science as a whole. Moreover, Stern's Matrix Logic is suitable for a broad spectrum of practical applications in contemporary technologies. (shrink)

The narrative about the nineteenth century favored by many philosophers of mathematics strongly influenced by either logic or algebra, is that geometric intuition led real and complex analysis astray until Cauchy and Kronecker in one sense and Dedekind in another guided mathematicians out of the labyrinth through the arithmetization of analysis. Yet the use of geometry in most cases in nineteenth century mathematics was not misleading and was often key to important developments. Thus the geometrization of complex numbers (...) was essential to their acceptance and to the development of complex analysis; geometry provided the canonical examples that led to the formulation of group theory; and geometry, transformed by Riemann, lay at the heart of topology, which in turn transformed much of modern mathematics. Using complex numbers as my case study, I argue that the best way to teach students mathematics is through a repertoire of modes of representation, which is also the best way to make mathematical discoveries. (shrink)

The nature of time is yet to be fully grasped and finally agreed upon among physicists, philosophers, psychologists and scholars from various disciplines. Present paper takes clue from the known assumptions of time as - movement, change, becoming - and the nature of time will be thoroughly discussed. -/- The real and unreal existences of time will be pointed out and presented. The complex number notation of nature of time will be put forward. Natural scientific systems and (...) various cosmic processes will be identified as constructing physical form of time and the physical existence of time will be designed. -/- The finite and infinite forms of physical time and classical, quantum and cosmic times will be delineated and their mathematical constructions and loci will be narrated. -/- Thus the physics behind time-construction, time creation and time-measurement will be given. -/- Based on these developments the physics of Timelessness will be developed and presented. -/- . (shrink)

The concept of inertial frame of reference in classical physics and special theory of relativity is analysed. It has been shown that this fundamental concept of physics is not clear enough. A definition of inertial frame of reference is proposed which expresses its key inherent property. The definition is operational and powerful. Many other properties of inertial frames follow from the definition, or it makes them plausible. In particular, the definition shows why physical laws obey space and time symmetries (...) and the principle of relativity, it resolves the problem of clock synchronization and the role of light in it, as well as the problem of the geometry of inertial frames. (shrink)

The physicist's conception of space-time underwent two major upheavals thanks to the general theory of relativity and quantum mechanics. Both theories play a fundamental role in describing the same natural world, although at different scales. However, the inconsistency between them emerged clearly as the limitation of twentieth-century physics, so a more complete description of nature must encompass general relativity and quantum mechanics as well. The problem is a theorists' problem par excellence. Experiment provide little guide, and the inconsistency mentioned (...) above is an important problem which clearly illustrates the intermingling of philosophical, mathematical, and physical thought. In fact, in order to unify general relativity with quantum field theory, it seems necessary to invent a new mathematical framework which will generalise Riemannian geometry and therefore our present conception of space and space-time. Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new kind of geometry may enter physics, and space-time itself can be reinterpreted as an approximate, derived concept. The main purpose of this article is to show the great significance of space-time geometry in predetermining the laws which are supposed to govern the behaviour of matter, and further to support the thesis that matter itself can be built from geometry, in the sense that particles of matter as well as the other forces of nature emerges in the same way that gravity emerges from geometry. Scientific research is not a process of steady accumulation of absolute truths, which has culminated in present theories, but rather a much more dynamic kind of process in which there are no final theoretical concepts valid in unlimited domains. (David Bohm). (shrink)

Translated by Drew S. Burk and Anthony Paul Smith. Excerpted from Struggle and Utopia at the End Times of Philosophy , (Minneapolis: Univocal Publishing, 2012). THE END TIMES OF PHILOSOPHY The phrase “end times of philosophy” is not a new version of the “end of philosophy” or the “end of history,” themes which have become quite vulgar and nourish all hopes of revenge and powerlessness. Moreover, philosophy itself does not stop proclaiming its own death, admitting itself to be half dead (...) and doing nothing but providing ammunition for its adversaries. With our sights set on clearing up this nuance, we differentiate philosophy as an institutional entity, and the philosophizability of the World and History, of “thought-world,” which universalizes the narrow concept of philosophy and that of “capital.” We also give an eschatological and apocalyptical cause to this end, of “times” or “ages” rather than those of philosophical practice. Last but not least, it is the Future itself in the performativity of its ultimatum that determines this end times, reversing these times from the Identity the Future accorded to it, withdrawing the thought-world from the lie of its death. Why this style of axioms and oracles, these more or less subtle distinctions, old and debased, with an appeal to the ultimata , to end times and last words? We fight to give, parallel to the concept of Hell, its new theoretical position, for its philosophical return and its non-philosophical transformation. No more so than any other word, Hell is not a metaphor here, just the Principle of Sufficient World. Every man, no doubt, has his “hell” readily available, connivance, control, conformism, domestication, schooling, alienation, extermination, exploitation, oppression, anxiety, etc. We have our little list that the Contemporaries established in the previous century in the same way one used to construct lists of virtues and vices or honors and wealth. They invented it for us without knowing it, for us-the-Futures who have as our responsibility to invent its use. In the Christian and Gnostic tradition, the struggle of the End Times takes place “on earth.” The most sophisticated of believers have it taking place in Heaven as well, above all in Heaven. The various kinds of gnosis imagine infinite falls and vertiginous highs, the vertigo of salvation. On Earth as in Heaven, a hell is available. The Marxists have the law of profit and the control of production, the class struggle Capital imposes on man. The Nietzscheans, the dull grumble of the struggle in the foundations of World and History, the domestication of man, the society of control. The phenomenologists, the capture of being, the most superficial amongst them, the age of suspicion. But all of these “hells” are taken from World, History, Society, and Religion. What we call Hell is no longer of the order of these specific and total intra-worldly generalities, it is both more singular and more universal, no longer being at all of the same order, it is the determinant Identity of these small hells strung out through history but unified here in the name of the last Humaneity. It is even found within the French idiom for hell [ enfer ], en-fer literally means “in-irons.” We are as much “in-irons” as we are “alive” [ en-Vie ]. We believe in Hell but as non-philosophers and it is even because we are non-philosophers that we can believe in it outside of any sort of religion. Hell is less mythological than ideological, it combines philosophizability with universal capital. Under what form? A single term could work for them without being a metaphor or something they would participate in by analogy, a more innovative and conjectural term than control, more universal than profit. It would denote the growing and permanent extortion of a surplus-value of communication, of speed and of urgency in change, in productivity and in work, in the pressure of images and slogans. It would be worse than solicitation, more tenacious than capture, more active and persecutory than control, softer and more insidious than a frontal attack, just as perverse as questioning and accusation, less brutal and offensive than extermination, less ritualized than inquisition, it would be soft and dispersed, instantaneous and vicious, it would be a crime without declared violence. Collusion and conformism, it would be afraid to show itself. Related to rumor, from which it borrows its infinite and tortuous ways. It is harassment. As a modernized form of Hell, perhaps harassment has a long future in front of it, of innocent torture, slow assassination, in short the fall, but radical with no way of recovering and which tolerates only salvation. THE PHILOSOPHICAL PAST OF NON-PHILOSOPHY Non-philosophy is thus Man as the utopian identity of the philosophical form of the World, a utopia destined to transform it. We still have to understand these equations, in particular that of the being-uni-versed from Man, and this book adheres to this by re-exposing non-philosophy in a different way via one of its new possibilities. It uses this opportunity to once again take up formulations that lead to objections answering certain external critics, as well as revisiting diverging interpretations specific to non-philosophers. A portion of this book is devoted to going through these theories via a strict or “lengthy” presentation of non-philosophy, and its defense against more expeditious solutions. This work of rectification is the occasion, merely the occasion, for refocusing non-philosophy on Man (the “Man-in-person,” “Humaneity”), and in a more innovative manner, on its utopian vocation established since the book Future Christ . As for this “occasion,” it is quite obvious. A school of posture, not to say a school of thought, supposes a minimum of closure from the most liberating of knowledge, a heritage, its utilization, and its no less certain dispersal. Within its development, a variety of interpretations will no doubt appear, deviations that are as much normalizations, and a struggle against this multiplication of divergent tendencies. These are perhaps not inevitable evils, especially here, merely a normal development according to the twisting paths of history. But the problem is made worse by the fact that this school of non-philosophers is that of utopia. Not the former attempts devoted to commenting on the worst authoritarian and criminal forms of the past and the present, but utopia as the determinant principle for human life, or to put it another way, of the Future as an irreducible presupposition of (for) thinking the World and History. Non-philosophers are engaged in an aleatory navigation between the respect for the most rigorous utopia, whose rules are not that of the reproductive imagination but those from the Future determining imagination itself, as well as the temptations, diversions, and remorse of history. Little by little, we will begin to understand that the Future as we understand it no longer has any temporal consistency or positive content, without being an empty form or a nothingness, but that it is foreclosed to past and present History, just as it is foreclosed to the place of places, the World, and that it is the only method for establishing the practice of thinking in a non-imaginary instance. Because it is the World and History that are imaginary and have a terrible materiality, it is not necessarily utopia. We will overlap two objectives here: the defense of non-philosophy against the (non-) philosophers that we are, only occasionally, and the introduction of philosophy to a rigorous future. Together they set out to definitively render, without any possible return to philosophical conformism or towards the facilities of the past and present, the non-philosophical enterprise understood as utopia or uchronia. Imagination and speculation, left to themselves and thus undistinguished, are quite good for participating in the grand game of History but have little value or worse for the Future which is unimaginable and unintelligible and must be maintained as such. MAN-IN-PERSON AS SUSPENSION OF THE PHILOSOPHICAL CHORA The point where philosophical resistance is concentrated is without a doubt the invention of the Name-of-Man, first name, oracular as much as axiomatic, of the determining cause for the non-philosophical posture. And that which concentrates the differends is the style of non-philosophy as identity that possesses the dual aspect, of discipline and of the oeuvre, of the theorem and of the oracle. But the real difficulty in understanding the simplicity of non-philosophy is profoundly hidden in the depths of philosophy itself. Because philosophy, from Parmenides to Derrida, even Levinas, continues to be a divided gesture, without a veritable immanence, transforming its thematic contents of transcendence in also forgetting to transform the operative transcendence in the element from which the ontology of surface is established which we will call, in memory of Plato, the chorismos. The general effect of the chora literally gives place to philosophy, demanding binding and sutures to which we will once and for all “oppose” the Man-in-person, his power of cloning, and his future being. All philosophy contains a hinter-philosophy in which it deploys its operations and weaves its tradition like an understudy of a topographical nature and in the best of cases being itself topological. Philosophy as well consists of two levels, its pre-ontological operative conditions on the one hand, and its superficial theme on the other, it too has its presupposed, but is not aware of it or erases it within the unity of appearance named Logos. Rightly, the Logos, and its flash or lightning nature, possesses a “dark precursor,” the chora, which is as much a virtual image, and philosophy, dazzled by its own lightning flash, seems to completely forget about it at the same time it sets itself up within it. Non-philosophy risks taking this same path, of confusing what it believes to be the real with its phantom double, contenting itself to working on the thematic level of philosophy, not its surface objects and its idle chatter (we stopped talking about this a long time ago and in any case they are merely simple materials for inducing a work of transformation), but the transcendence-form of its objects. In the end it risks, through precipitation, taking back up the heritage of philosophy, a heritage of a misunderstood presupposed, even more profound than the play of transcendences. This is what the imperative of the radicality of immanence meant, to treat immanence in an immanent manner, not to make a new object out of it. And from here we get non- (philosophy) and its refusal of the Platonic chorismos , symbol of all abstraction, and thus all transcendental appearance. There are no illusions. The message will leave a heritage in tattered pieces and interpretations. But it was difficult not to dispute the differend to its core. There will be complete confusion of the multiple, possible, and necessary effectuations of non-philosophy with its interpretations. The non-philosophical or human freedom of philosophical effectuation and the philosophical freedom of interpretation. Effectuations demand non-philosophy to return to zero from the point of view of its philosophical material and thus also but within these limits the formulation of its axioms , but in no way providing from the outset divergent interpretations of the aforementioned axioms. They are divergent because they do not take into account the material from which these axioms are derived within non-philosophy, and because they do not see themselves as symptoms of another vision of the World. The utterances of non-philosophy are not mathematical theorems and pure axioms, they merely have a mathematical aspect . They are, by their extraction or origin, mathematical and transcendental. And by their determined function in-Real, within non-philosophy, they are identically in-the-last-Humaneity entities which have an aspect of an axiom and an aspect of interpretation (or an oracular aspect as we say) that attempts (sometimes it is ourselves who provide the occasion) to isolate and transform, in complete freedom of interpretation. There will be an opportunity to complain about the complex character of the language of non-philosophy, an idiom saturated with classical references, sophisticated in a contemporary way. Its freedom of decision up against the whole of philosophy demands these effects of “complication” and “privatization,” as the saying goes. But it also demands fighting against the drift [ dérive ] of the pedagogical-all and the mediatic-all that leads philosophy into the shallow depths of opinion, which is the site of its impossible death. The noble idealism of “pop-philosophy” has been consumed into a “philo-reality;” against this we propose philo-fiction. Parricide, which is at the bottom of these interpretations and which we can judge as being quite fertile, although it has informed tradition, only takes place once or within one lone meaning. In regards to Parmenides, it was possible; Plato introduced the Other as non-being and language, bringing into existence the philosophical system of the World, but is it possible to repeat it again with the same fecundity in regards to non-philosophy, this time in introducing (non) religion or (non) art, still mixing them without taking into account this mixture, alternatively as a philosophical or religious resentment? If philosophy begins via a crime, it is no doubt obliged to continue down similar pathways, to the effect that the crimes of philosophy, once the founding crime has been committed, are a reaction of self-defense. It is undoubtedly from this that we get Marx’s declaration that history begins by tragedy and repeats itself or ends in farce. The preservation of rigor and fecundity is, in every respect, a psychologically difficult task within a theory such as non-philosophy. Having posited an essential objective of liberation in regards to philosophy and its services, one has often understood this objective as an authorization of providing particular interpretations of its axioms and ends up obliterating their scope. This ends up confusing, on one hand, two kinds of freedoms in regards to non-philosophy, the freedom of its interpretations and the freedom of its effectuations. On the other hand, any defense of “principles” against precipitated interpretations is immediately taxed with a will to orthodoxy, a prohibitive objection when we are dealing with, as is the case here, a heretical theory of thought. Nevertheless, it is time to stop confusing heresy as the cause of thought with an ideology of heresies, which is certainly not at all our object, but rather a form of normalization. As for the “disciplinary” aspect, which is not the only aspect, it demands something other than philosophical “answers to objections,” a precision in the definition and use of its procedures in the formation of utterances, since non-philosophy is neither a supplementary doctrine interior to philosophy nor a vision of the world but one whose priority is a “vision of Man,” or rather Man as “vision” that implies a theory and a practice of philosophy. In the end, struggle is only one aspect of non-philosophy, not its whole or telos, struggle coming only from its materiality. In particular, if the discipline of non-philosophy is inseparable from struggle, it is not a question of reducing the monomaniacal obsession of its “marching orders.” This would reduce its complexity and kill its indivisibility, deploying it in a “long march” and a form of Maoization whose philosophical presuppositions no longer have any pertinence here, a case of the One and the Two, which are now cloned and no longer tied together. More generally, non-philosophy is a complex thought composed of a multitude of aspects, which is to say, unilateral interpretations, of a philosophical origin but reduced by their determination in-the-last-instance . The “liberalism” of non-philosophy is merely one of the aspects of which it is capable, not an essence. Similarly, it is only capable of having a “Maoist” aspect. Let us generalize. The weakness of non-philosophy is due to a specific cause, the determination-in-the-last-Humaneity of a subject for the World. Everything that has a right to the philosophical city can be said about it in turn and in a retaliatory mode since Man contributes nothing of himself that Man takes from the World. We can consider non-philosophy as being pretentious, absurd, idealistic, empty, materialistic, formalistic, contradictory, modern, post-modern, Zen, Buddhist, Marxist; it endures or tolerates, perhaps “appeals” to, or at least renders possible, sarcasms, ironies, and insults without even talking about the misunderstandings, partly for the same reason as psychoanalysis. All of this goes beyond simple “deviations.” They are its aspects, which is to say, its “unilateral” philosophical interpretations in both senses of the word, being either sufficient coming from the mouth of philosophers, or reduced to their absolute dimension of sufficiency and totality in the mouths of non-philosophers, and both times due to the weakness and strength of Man-in-person as their determination only in-the-last-instance. The non-philosopher is certainly not a Saint Paul fantasizing about a new Church. The non-philosopher is either a (Saint) Sebastian whose flesh is pierced with as many arrows as there are Churches, or a Christ persecuted by a Saint Paul. What is engaged in here is the practice of retaliation. A negative rule of the non-philosophical ethics of outlawed discussion by way of argumentation (the sufficient is you, the orthodox one is always you, you are the fashionable one, and when a master you are someone else) that is founded on the confusion of effectuations of non-philosophy and of its overall interpretations. Retaliation is the law but as with any too-human law, it must acquire a dimension that displaces it, or rather emplaces it and takes away its authority but not all of its effectiveness. If the non-philosopher is only authorized by himself, which is to say by philosophy but limited by the Real-of-the-last-instance, its critique of other non-philosophers can merely be retaliatory under the same conditions, only by the Real limited in-the last-instance. THE TREE OF PHILOSOPHICAL SAINTLINESS The thematic horizon or material of these debates is in the relationships between philosophy, religion as gnosis, and non-philosophy. It is inevitable, regarding non-philosophers in general (whether they are non-philosophers by name or simply its neighbors) that we often end up evoking Marx’s Holy Family and imagining, arranged on the neighboring branches of the tree of philosophy and annexed, sometimes abusively, to non-philosophy, authors who would quite evidently and quite rightly refuse this label. So it is that we find, for example, a Saint Michel, a Saint Alain, a Saint Gilles 1 without even mentioning the youngest who aspire as well to the freedom of “saintliness” and who make their muted voices heard here. If there is a Holy Family of non-philosophers, it extends completely beyond these three, provided that the sectarian spirit can save us. This book is organized in the following manner: To begin, in order to recall the essential part of the problematic, we have organized a Summary of Non-Philosophy , a vade-mecum of notions and basic problems, in a classical style. Secondly, there is Clarifications On the Three Axioms of Non-Philosophy , designed to posit their proper use as much as to elucidate their meaning. Thirdly, an analysis of Philosophizability and Practicity , both being extreme constituents of philosophical material or the contents of the third axiom. Fourthly, the heart of this work: Let us Make a Tabula Rasa of the Future or of Utopia as Method . Fifthly, we have a theoretical outline of a non-institutional utopia, The International Organization of Non-Philosophy, L’Organisation Non-Philosophique Internationale, (ONPHI) already created in practice but under the conditions of possibility and functioning from which here we put into question “de jure,” thus not without a perplexity concerning “facts,” in any case, without the capability of “getting to the bottom of things.” Sixthly, an essay characterizing The Right and the Left of Non-Philosophy , a brief topology of several philosophizing and normalizing positions of proponents or tenants of this problematic. Seventhly, Rebel in the Soul: A Theory of Future Struggle , a systematic discussion starting from a confrontation of non-philosophical gnosis and non-religious gnosis to the extent that they pose, posed or perhaps still will pose themselves as rivals to non-philosophy in a mixture of fidelity and infidelity. Despite the fact that it can also be read as putting non-philosophy into perspective: it pits against a standard Platonism two contemporary appropriations of Gnosticism. On the basis of the paradigm of Man who never ceases to come as the Future-in-person, each one of these moments strives to reestablish not the “true” non-philosophy and its orthodoxy, but the minimal conditions to respect in order to allow for its maximum fecundity. And in order to bring about one of the last possibilities of its development, making explicit Humaneity as a utopia-for-the-World. In introducing these considerations in the form of a “testament” and “ultimatum,” we want to indicate two things: First, that this is the last time we will intervene in order to caution non-philosophers against the temptation of returning and looking backwards towards philosophy. Only a disillusioned nostalgia for the former World and its traditions barely remain permissible to us.… Secondly, that non-philosophy is also a sort of ultimatum for considering one’s life and transforming one’s thought from the perspective of a uni-version rather than a conversion. Man as future is this ultimatum in action, not an impatient self-proclaimed genius, and philosophy is his testament. It is obviously the ultimatum that determines this testament as “old” with a view towards a life that is, itself, non-testamentary. In and of itself, the “old” can never bear a veritable eschaton. Thus, this book intersects according to the logic of this paradigm, under the sign of the ultimate or “last” as future, philosophy as testament and cautionary note for maintaining the non-philosophical oeuvre as “future” or “utopian.” We will see that between these two dimensions it cradles a theory of struggle. In the end, this book envisions non-philosophers in multiple ways. It inevitably sees them as subjects of knowledge, most often academics insofar as life in the world demands, but above all as close relatives of three great human types. The analyst and political militant are quite obvious, for non-philosophy is close to psychoanalysis and Marxism insofar as it transforms the subject in transforming philosophy. Here again, one must have a sense not of certain nuances but of aspects (of the interpretations, albeit unilateralized) and not in order to construct a simple proletarization or militarization of thought as theory. To be rigorous, rather than authoritarian or spiteful, is its task. And lastly, non-philosophy is a close relative of the spiritual but definitely not the spiritualist. Those who are spiritual are not at all spiritualists, for the spiritual oscillate between fury and tranquil rage, they are great destroyers of the forces of Philosophy and the State, which are united under the name of Conformism. They haunt the margins of philosophy, gnosis, mysticism, science fiction and even religions. Spiritual types are not only abstract mystics and quietists; they are heretics for the World. The task is to bring their heresy to the capacity of utopia, and their utopia to the capacity of the paradigm. NOTES We will recognize allusions, and sometimes references, to closely related or distant themes, but which are related, in the work of Michel Henry, Alain Badiou and via the representative of “non-religious” gnosis of a Platonic origin, in Gilles Grelet. It goes without saying that these discussions are current and local, neither concerning the ensemble of doctrines nor prejudging the eventual evolution of certain amongst them. This concerns defining certain proximities with non-philosophy (rather than adversaries which in some sense they are) and typological and emblematic differends (rather than conflicts with a certain author). (shrink)

This “fun, brain-twisting book... will make you think” as it explores more than 75 paradoxes in mathematics, philosophy, physics, and the social sciences (Sean Carroll, New York Times–bestselling author of Something Deeply Hidden) Paradox is a sophisticated kind of magic trick. A magician’s purpose is to create the appearance of impossibility, to pull a rabbit from an empty hat. Yet paradox doesn’t require tangibles, like rabbits or hats. Paradox works in the abstract, with words and concepts and symbols, to create (...) the illusion of contradiction. There are no contradictions in reality, but there can appear to be. In Sleight of Mind, Matt Cook and a few collaborators dive deeply into more than 75 paradoxes in mathematics, physics, philosophy, and the social sciences. As each paradox is discussed and resolved, Cook helps readers discover the meaning of knowledge and the proper formation of concepts—and how reason can dispel the illusion of contradiction. The journey begins with “a most ingenious paradox” from Gilbert and Sullivan’s Pirates of Penzance. Readers will then travel from Ancient Greece to cutting-edge laboratories, encounter infinity and its different sizes, and discover mathematical impossibilities inherent in elections. They will tackle conundrums in probability, induction, geometry, and game theory; perform “supertasks”; build apparent perpetual motion machines; meet twins living in different millennia; explore the strange quantum world—and much more. (shrink)

This book features more than 20 papers that celebrate the work of Hajnal Andréka and István Németi. It illustrates an interaction between developing and applying mathematical logic. The papers offer new results as well as surveys in areas influenced by these two outstanding researchers. They also provide details on the after-life of some of their initiatives. Computer science connects the papers in the first part of the book. The second part concentrates on algebraic logic. It features a range of (...) papers that hint at the intricate many-way connections between logic, algebra, and geometry. The third part explores novel applications of logic in relativity theory, philosophy of logic, philosophy of physics and spacetime, and methodology of science. They include such exciting subjects as time travelling in emergent spacetime. The short autobiographies of Hajnal Andréka and István Németi at the end of the book describe an adventurous journey from electric engineering and Maxwell’s equations to a complex system of computer programs for designing Hungary’s electric power system, to exploring and contributing deep results to Tarskian algebraic logic as the deepest core theory of such questions, then on to applications of the results in such exciting new areas as relativity theory in order to rejuvenate logic itself. (shrink)

In this double-volume work, a great modern philosopher propounds a system of thought in which Einstein's theory of relativity represents only the latest (albeit the most radical) fulfillment of the motives inherent to mathematics and the physical sciences. In the course of its exposition, it touches upon such topics as the concept of number, space and time, geometry, and energy; Euclidean and non-Euclidean geometry; traditional logic and scientific method; mechanism and motion; Mayer's methodology of natural science; Richter's definite (...) proportions; relational concepts and the activity of the ego; Einstein's relativity and "reality"; and the philosophical concept of truth and its role in relativity theory. 1923 edition. (shrink)

The history of mathematics can be read in two ways. On the one hand, unlike the history of physics, it does not proceed by conjectures and refutations. New theories rarely refute old theories, but give them new foundations, generalize them, and reinterpret them through new concepts. This reading is unifying, highlighting the unity of the history of mathematics from its origins, through the permanence of its truths. On the other hand, many contemporary historians of mathematics have insisted on (...) the diversity of methods, objects, and practices throughout history. They have shown that the objects of the past mathematics are not the same as those of today. That second reading shatters the illusion of unity. It disjoins what the first reading unifies.Rather than deciding between these two readings, this chapter examines what makes them both possible and legitimate, based on an analysis of the acts of reading that we have to perform when confronted with the pages of a mathematical text, its signs, diagrams and images, and the different language modalities it employs. Starting with a discussion of Husserl's theses in The Origin of Geometry on the role of writing in the history of mathematics, and using a number of historical examples, the chapter shows that any reading of a mathematical text involves at least three divisions, three delimitations made by the reader’s gaze: divisions between the true and the false, between language and image, and between what is inside mathematics and what is outside. For the same text, these divisions are traced by the gaze in different ways throughout history.The chapter develops an example in greater detail, concerning the status of the equation in Descartes’ Geometry. It discusses the contrasting recent readings of the equation by two historians of science, Henk Bos and Enrico Giusti. It shows the concepts of The Geometry and the ideas the philosopher develops on reading and the nature of the sign. It shows how these ideas can be used to reconstruct how Descartes might have read a mathematical text, and it explains how successive readings of The Geometry can at the same time insert it into the mathematics of later centuries. (shrink)

Francesca Biagioli’s Space, Number, and Geometry from Helmholtz to Cassirer is a substantial and pathbreaking contribution to the energetic and growing field of researchers delving into the physics, physiology, psychology, and mathematics of the nineteenth and twentieth centuries. The book provides a bracing and painstakingly researched re-appreciation of the work of Hermann von Helmholtz and Ernst Cassirer, and of their place in the tradition, and is worth study for that alone. The contributions of the book go far beyond that, (...) however. It is a clear, accurate, and deep account of fascinating and philosophically momentous implications of the move to relativity theory. (shrink)

Theoretical physics is a deductive discipline which presupposes the validity and applicability of certain other disciplines. Among these are logic, algebra, analysis, and geometry. Before relativity, Euclidean geometry was the only one thought to be important for physical space. These disciplines correlate well with experience, and, in the course of time, a priori validity came to be ascribed to them. To Kant, for example, the universe could not possibly be based on any geometry other than Euclid's. The discovery of (...) non-Euclidean geometries by Bolyai and Lobatchevsky, its systematic extension by Riemann, and the critical scrutiny of foundations by Mach and Poincaré shook the “self-evident truth” of Euclidean geometry as applied to the world of physics. The death knell of the a priori truth of any geometry was supplied by Einstein. He showed that a wider body of experience was coordinated by a Riemannian geometry than by Euclidean. He traced the success of Euclidian geometry back to the existence of so-called rigid bodies, in terms of which operations can be associated with numbers which correlate well with numbers associated in a purely mathematical way with mathematical constructs put in correspondence with these bodies. For the physicist, then, the geometry of the real world is now a physical theory whose “truth” like that of all physical theory, is measured by its degree of success in coordinating experience. (shrink)

Newton's use of mathematics in mechanics was justified by him from his neo-platonician conception of the physical world that was going along with his «absolute, true and mathematical concepts» such as space, time, motion, force, etc. But physics, afterwards, although it was based on newtonian dynamics, meant differently the legitimacy of being mathematized, and this difference can be seen already in the works of eighteenth century «Geometers» such as Euler, Clairaut and d'Alembert (and later on Lagrange, Laplace and (...) others). Despite their inheritance of Newton's achievements, they understood differently the meaning and use of mathematical quantities for physics, in a way that was more neutral to metaphysics. The continental reception and assimilation of Newton's Principia had indeed occured as its budding onto Leibniz' calculus and a cartesian conception of rationality (spread in particular by the malebranchist disciples of Leibniz). This new thought of the legitimacy of mathematization is clearly at variance with Descartes' identification of physics with geometry, but it nevertheless can be traced back to Descartes' conception of magnitudes, as it was developed and analyzed from the notion of dimension in his Regulæ ad directionem ingenii (in particular, rule 14). This idea can be followed afterwards with further philosophical or mathematical specifications through authors such as Kant, Riemann and others. This inquiry into the original thought of magnitudes, and of physical magnitudes conceived through mathematization, leads us to suggest an extension of meaning for the concept of physical magnitude that puts emphasis on its relational and structural aspects rather than restraining it to a simple «numerically valued» acception. Such a broadening would have immediate implications on our comprehension of «non classical» aspects of contemporary physics in the quantum area and in dynamical systems. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider (...) the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

Foundations of Algorithms, Fifth Edition offers a well-balanced presentation of algorithm design, complexity analysis of algorithms, and computational complexity. Ideal for any computer science students with a background in college algebra and discrete structures, the text presents mathematical concepts using standard English and simple notation to maximize accessibility and user-friendliness. Concrete examples, appendices reviewing essential mathematical concepts, and a student-focused approach reinforce theoretical explanations and promote learning and retention. C++ and Java pseudocode help students better understand (...) class='Hi'>complex algorithms. A chapter on numerical algorithms includes a review of basic number theory, Euclid's Algorithm for finding the greatest common divisor, a review of modular arithmetic, an algorithm for solving modular linear equations, an algorithm for computing modular powers, and the new polynomial-time algorithm for determining whether a number is prime. The revised and updated Fifth Edition features an all-new chapter on genetic algorithms and genetic programming, including approximate solutions to the traveling salesperson problem, an algorithm for an artificial ant that navigates along a trail of food, and an application to financial trading. With fully updated exercises and examples throughout and improved instructor resources including complete solutions, an Instructor's Manual and PowerPoint lecture outlines, Foundations of Algorithms is an essential text for undergraduate and graduate courses in the design and analysis of algorithms. Key features include: • The only text of its kind with a chapter on genetic algorithms • Use of C++ and Java pseudocode to help students better understand complex algorithms • No calculus background required • Numerous clear and student-friendly examples throughout the text • Fully updated exercises and examples throughout • Improved instructor resources, including complete solutions, an Instructor's Manual, and PowerPoint lecture outlines. (shrink)

In the range of his intellectual interests and the profundity of his mathematical thought Hermann Weyl towered above his contemporaries, many of whom viewed him with awe. This volume, the most ambitious study to date of Weyl's singular contributions to mathematics, physics, and philosophy, looks at the man and his work from a variety of perspectives, though its gaze remains fairly steadily fixed on Weyl the geometer and space‐time theorist. Structurally, the book falls into two parts, described in the (...) general introduction by the editor: Part 1 contains four essays on particular aspects of Weyl's work, highlighting ideas he developed in various editions of his classic Raum‐Zeit‐Materie. Part 2 presents a lengthy study by Robert Coleman and Herbert Korté covering nearly the whole gamut of Weyl's mathematical research, an impressive feat. Both in the introduction as well as in footnotes to the articles Erhard Scholz's editorial voice chimes in discreetly, helping tie all five studies together.Coleman and Korté begin chronologically with Weyl's early work in analysis and the modern theory of Riemann surfaces before turning to differential geometry, unified field theory, and the space problem, a topic they use as a springboard for a discussion of their own recent work on the foundations of space‐time. They then take up Weyl's shift to group representation theory and its applications to quantum mechanics, ending with his much earlier research on the structure of the continuum. All of these topics are well handled, but the authors' own agendas coupled with their penchant for overlooking chronology in order to package Weyl's work into neat little bundles leave one feeling rather stranded and far removed from the sources of Weyl's inspiration. Moreover, the narrative style makes this part of the volume read like a technical appendix, albeit a most informative one. Readers who tackle Scholz's far more contextualized essay will be amply rewarded by comparing his views with the opinions set forth by Coleman and Korté in Part 2.Scholz gives a masterful account of Weyl's intellectual journeys from 1917 to 1925 in a study that serves as a fulcrum for the entire volume. Drawing on a number of recently published studies, including his own, on the interplay between mathematics and physics inspired by Einstein's theory of general relativity, Scholz describes how Weyl responded to this challenge by developing a truly infinitesimal space‐time geometry that generalized classical Riemannian geometry. Although unconvinced by Einstein's critique of his unified field theory, Weyl shifted his focus from this realm to the classical space problem, analyzed earlier with more primitive techniques by Hermann Helmholtz and Sophus Lie. In this connection, it should be mentioned that Thomas Hawkins has given a probing analysis of Weyl's related work on the representation of Lie groups in his tour‐de‐force work, Emergence of the Theory of Lie Groups . Scholz argues that Weyl's struggle to tame his modernized version of the space problem stemmed from a deep‐seated belief in his geometrical ideas, which in turn were nourished by philosophical musings. By demonstrating the closely related conceptual links that motivated Weyl's research in infinitesimal geometry, space‐time physics, and the foundations of mathematics, Scholz nicely illuminates the underlying fabric of epistemological concerns that occupied Weyl's attention during this fertile period.The three remaining essays in Part 1 focus on other aspects of Weyl's work in mathematical physics and cosmology. Skuli Sigurdsson's “Journeys in Spacetime” offers a broad interpretation of Weyl's career, one that emphasizes Weyl's sensitivity to cultural tensions as reflected in his philosophical roots, which combined phenomenology with facets of German idealism. Shaken by the annihilation of cultural values in Nazi Germany, Weyl became deeply aware of the gulf that separated his earlier life in Göttingen and Zurich from the one he took up at Princeton's Institute for Advanced Study in 1933. He tried to adapt, but felt out of place in an Anglo‐American scientific culture openly hostile toward metaphysics and speculative philosophy. Sigurdsson stresses these tensions, contrasting the introspective, creative individual against the backdrop of the collective in the age of the machine, but without spelling out which collective were most important for him. Wolfgang Pauli thought he knew and, like Einstein before him, he had no compunction about bluntly telling Weyl he was a mathematician, not a physicist.Pauli's opinions notwithstanding, Weyl did far more than just dabble around the mathematical edges of the new physics. If Coleman and Korté perhaps press their case for his visionary accomplishments too far, Norbert Straumann's essay “Ursprünge der Eichtheorien” suggests why Weyl's reputation among physicists has risen steadily ever since the advent of Yang‐Mills theory in the 1950s. In the course of describing Weyl's adaptation of his gauge transformation formalism to Dirac's electron theory, Straumann sheds considerable light on Pauli's role as self‐appointed watchman guarding the disciplinary boundary that separated theoretical physics from physical mathematics . He further suggests that disciplinary jealousy was a major reason why Pauli dismissed Weyl's two‐component formalism for spinors out of hand.In the realm of cosmology, on the other hand, Weyl's work has long since passed into the dustbins of history, as Hubert Goenner remarks in recounting a fascinating chapter in the infancy of space‐time physics. While doing so, Goenner shows how initially Weyl almost slavishly adopted what Einstein called Mach's principle, which asserts that the metric structure of space‐time is solely determined by the distribution of matter in the universe. This notion was quickly challenged by Willem De Sitter, who showed that Einstein's matter‐free field equations admitted a global solution with non‐zero constant curvature. Both Einstein and Weyl tried to argue that invisible masses must be present just over the “spatial horizon” of De Sitter's world in order to account for its curvature. Goenner meticulously analyzes the physical and mathematical issues at stake in this debate, stressing how Weyl gradually moved away from a strong physical interpretation to one in which mathematics models rather than physics models simply reveal natural phenomena. He argues further that Weyl's cosmological principle arose as the final expression of his search for a deeper physical meaning.Given the quality of these essays, it is regrettable that this book contains so little about Weyl's professional career, a weakness the editor could have redressed at least partially in his general introduction. This omission is all the more unfortunate given the dearth of readily accessible information about Weyl's life available elsewhere. For however mundane his outward existence may have been, the reader cannot be expected to appreciate the interplay between the world Weyl knew and his creative responses to it without fairly detailed knowledge of his biography. Shorn from these contexts, it becomes difficult to form a flesh‐and‐blood image of Weyl beyond the cliché‐ridden stereotype that sees him as a “heroic thinker in the grand German tradition.” While none of the authors falls into this trap, the collective impression they leave suggests a most enigmatic figure. Either Weyl the man tends to get lost in the shadows of his collected scientific output or he appears as a mystic loner, an outcast who abhorred the machine age in which he lived. Closer attention to the people in his life would no doubt produce a very different picture of the man and his interests. This major lacuna notwithstanding, the present volume will surely remain an indispensable resource for any future investigations of Weyl's staggering intellectual achievements. (shrink)

Edmund T. Whittaker’s second edition of his A History of the Theories of Aether and Electricity is famous for his treatment of Einstein as an almost irrelevant character in the emergence of what he called “the relativity theory of Poincaré and Lorentz.” Historians of science have given a number of explanations, which include Whittaker’s scientific conservatism as an old classical physicist, his commitment to the ether, the pre-eminent role he attributed to mathematics over physics, and foundational philosophical disagreements, (...) to name a few. And in the background, often more implicit than forthright, the accusation of antisemitism looms over Whittaker. In this paper I intend to shed new light on this controversy by taking into consideration the abundant correspondence between Whittaker and his son preserved in the archives of the Fisher Library, University of Toronto. With it, we will get a more complex and personal view of the context in which his attempt at dethroning Einstein took place. Together with the abovementioned reasons, this correspondence shows that the problematic status quo of general relativity in the early 1950s, a period that has been described as the low-mark of general relativity, was very influential in the historical treatment he gave to Einstein. This is an aspect hardly mentioned in the historical work on this controversy and, from this correspondence, it appears to be central to understanding Whittaker at the time of drafting the new History. His possible antisemitic bias will also be addressed, though with the insufficient information on this subject the matter cannot be settled. (shrink)

Based on the ideas of Einstein and Minkowski, this concise treatment is derived from the author's many years of teaching the mathematics of relativity at the University of Michigan. Geared toward advanced undergraduates and graduate students of physics, the text covers old physics, new geometry, special relativity, curved space, and general relativity. Beginning with a discussion of the inverse square law in terms of simple calculus, the treatment gradually introduces increasingly complicated situations and more sophisticated mathematical tools. Changes in (...) fundamental concepts, which characterize relativity theory, and the refinements of mathematical technique are incorporated as necessary. The presentation thus offers an easier approach without sacrifice of rigor. Dover (2014) republication of the edition published by John Wiley & Sons, New York, 1950. See every Dover book in print at www.doverpublications.com. (shrink)

Complexified Liénard–Wiechert potentials simplify the mathematics of Kerr–Newman particles. Here we constrain them by fiat to move along Bohmian trajectories to see if anything interesting occurs, as their equations of motion are not known. A covariant theory due to Stueckelberg is used. This paper deviates from the traditional Bohmian interpretation of quantum mechanics since the electromagnetic interactions of Kerr–Newman particles are dictated by general relativity. A Gaussian wave function is used to produce the Bohmian trajectories, which are found to be (...) multi-valued. A generalized analytic continuation is introduced which leads to an infinite number of trajectories. These include the entire set of Bohmian trajectories. This leads to multiple retarded times which come into play in complex space-time. If one weights these trajectories by their natural Bohmian weighting factors, then it is found that the particles do not radiate, that they are extended, and that they can have a finite electrostatic self energy, thus avoiding the usual divergence of the charged point particle. This effort does not in any way criticize or downplay the traditional Bohmian interpretation which does not assume the standard electromagnetic coupling to charged particles, but it suggests that a hybridization of Kerr–Newman particle theory with Bohmian mechanics might lead to interesting new physics, and maybe even the possibility of emergent quantum mechanics. (shrink)

"Clifford was famous for his public lectures on physics and math and ethics because he explained complex things with easily understood, concrete examples. As you read through his clear, simple explanations of the true bases of number, algebra and geometry you will find yourself getting angry and saying "Why the hell wasn't I taught math this way?" and "Do math ed professors know so little mathematics that they have never heard of Clifford.?" Clifford was destined to be England's (...) Einstein until his untimely death at the age of 34, just 11 days before Einstein's birth. More than 30 years before the Special Theory of Relativity was proposed he had already concluded that the force of gravity was actually due to changes in the curvature of space. He gives explanatory examples in this book that middle school children can understand."--review on Amazon.com viewed July 6, 2020. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.2 (2001) 193-210 [Access article in PDF] Zeno Against Mathematical Physics Trish Glazebrook Galileo wrote in The Assayer that the universe "is written in the language of mathematics," and therein both established and articulated a foundational belief for the modern physicist. 1 That physical reality can be interpreted mathematically is an assumption so fundamental to modern physics that chaos and super-strings (...) are examples of physical theories developed on the basis of success in mathematics. The mathematics came first, then the physical theory. Quantum theory is an awkward and similar case in point. Despite the fact that there is much agreement among physicists about the mathematical theory, its physical interpretation remains a matter of controversy. There are several interpretations, all of which challenge our everyday assumptions about reality. This led Bohr in 1935 to call for "a radical revision of our attitude towards the problem of physical reality." 2 Arthur Fine, originally a staunch defender of realist interpretations of quantum theory, saw the success of Bohr's Copenhagen interpretation as the end of Einsteinian realism, and subsequently Fine declared that "realism is well and truly dead." 3 Mathematical operators simply do not correspond with real entities in any conceivable way.Ilkka Niiniluoto has argued convincingly, however, that realism is alive and well in quantum theory. 4 The problem is that Bohr is right: quantum theory can be considered to concern the real only upon radical revision of ordinary conceptions of reality. Quantum theory does not describe the real in any meaningful sense of "the real." Rather, it is mathematical projection: ideal rather than real. [End Page 193]Aristotle did not hold that the physicist deciphers nature mathematically. He distinguished the physicist from the mathematician at Physics 2.2 by analogy to the definitions of "snub nose" and "curved." 5 The definition of "snub nose" requires the mention of a nose, that is, of matter which is inseparable from motion, whereas "curved" and other mathematical concepts like " 'odd' and 'even', 'straight'... 'number,' 'line,' and 'figure,' " which are separable from matter and from motion, are investigated by the mathematician. 6 Aristotle claims that physics is different from mathematics because the mathematician investigates physical lines, but not qua physical, and the physicist investigates mathematical lines "but qua physical, not qua mathematical." 7A peculiar consequence of my argument is that it establishes a common principle between two thinkers whose beliefs are otherwise so fundamentally in opposition. Zeno's paradoxes are commonly read as a denial of motion. For Aristotle, that move is axiomatic to the physicist. At 185a15 he considers arguments against this assumption outside the realm of objections to which the physicist must reply. If I am right that Zeno's paradoxes of motion challenge the very notion of mathematical physics, then he and Aristotle agree, contra Galileo, that the universe is not written in the language of mathematics. They simply come to this point from opposite directions, Zeno from an Eleatic thesis that physical reality is illusion, Aristotle from the natural scientist's pragmatic commitment to a truth about nature.I intend to use Zeno's paradoxes of motion to argue that the application of mathematical concepts to the physical world results in paradox. Zeno's arguments are limited within mathematics to geometry. This paper is accordingly a beginning from which I hope more work can be done later. Furthermore, Zeno's paradoxes are standardly read as directed against anyone who disagrees with Parmenides' thesis that there is but one thing which is motionless and indivisible. The paradoxes achieve this end by means of a reductio against the possibility of dividing space and time. That there are four paradoxes is accordingly explicable by the fact that there are four possibilities for the divisibility of space and time. The Achilles argues against the possibility that space and time are both infinitely divisible; the Dichotomy, against the infinite divisibility of space and finite divisibility of time; the Arrow, against the infinite divisibility of time and the finite divisibility of space; and finally, the Stadium shows that time and space cannot both... (shrink)

"Clifford was famous for his public lectures on physics and math and ethics because he explained complex things with easily understood, concrete examples. As you read through his clear, simple explanations of the true bases of number, algebra and geometry you will find yourself getting angry and saying "Why the hell wasn't I taught math this way?" and "Do math ed professors know so little mathematics that they have never heard of Clifford.?" Clifford was destined to be England's (...) Einstein until his untimely death at the age of 34, just 11 days before Einstein's birth. More than 30 years before the Special Theory of Relativity was proposed he had already concluded that the force of gravity was actually due to changes in the curvature of space. He gives explanatory examples in this book that middle school children can understand."--review on Amazon.com viewed July 6, 2020. (shrink)

In this work, we develop and propose an ontological formal definition of time, based on a topological analysis of the formal mathematical description of time, coming from approaches to both quantum theories and Relativity; thus, being compatible with all physical epistemological theories. We find out a mathematical topological invariability, thus establishing a rigorous definition of time, as fundamental generic magnitude. Very preliminary analysis of physical epistemology is provided; likely highlighting a path towards a final (...) common vision between Quantum and Cosmology ontology and human feeling of time. (shrink)

The fact that physical laws often admit certain kinds of space-time symmetries is often thought to be problematic for substantivalism --- the view that space-time is as real as the objects it contains. The most prominent alternative, relationism, avoids these problems but at the cost of giving abstract objects (rather than space-time points) a pivotal role in the fundamental metaphysics. This incurs related problems concerning the relation of the physical to the mathematical. In this paper I will (...) present a version of substantivalism that respects Leibnizian theses about space-time symmetries, and argue that it is superior to both relationism and the more orthodox form of substantivalism. (shrink)

The facts that led to establishment of the special theory of relativity are reanalyzed. The analysis leads to the well-known formalism, involving, however, somewhat unusual notations. The object of the analysis is to start more closely from the directly observed experimental facts than is usually done; at the same time, great stress is laid on giving formulations independent of the representation in particular reference systems. A detailed analysis is given as to the actual physical methods involved when introducing three- (...) or four-dimensional reference systems. The orthogonal transformations and also the Lorentz transformations are introduced not so much as coordinate transformations but as operators reflecting physical properties of material systems. The principle of relativity is replaced by a mathematically equivalent principle denoted as theLorentz principle which reflects certain symmetries of the known physical laws. (shrink)

A square tabular array was introduced by R. C. Punnett in (1907) to visualize systematically and economically the combination of gametes to make genotypes according to Mendel’s theory. This mode of representation evolved and rapidly became standardized as the canonical way of representing like problems in genetics. Its advantages over other contemporary methods are discussed, as are ways in which it evolved to increase its power and efficiency, and responded to changing theoretical perspectives. It provided a natural visual decomposition of (...) a complex problem into a number of inter-related stages. This explains its computational and conceptual power, for one could simply “read off” answers to a wide variety of questions simply from the “right” visual representation of the problem, and represent multiple problems, and multiple layers of problems in the same diagram. I relate it to prior work on the evolution of Weismann diagrams by Griesemer and Wimsatt (What Philosophy of Biology Is, Martinus-Nijhoff, the Hague, 1989), and discuss a crucial change in how it was interpreted that midwifed its success. (shrink)

This paper discusses Husserl’s views on physical theories in the first volume of his Logical Investigations, and compares them with those of his contemporaries Pierre Duhem and Henri Poincaré. Poincaré’s views serve as a bridge to a discussion of Husserl’s almost unknown views on physical geometry from about 1890 on, which in comparison even with Poincaré’s—not to say Frege’s—or almost any other philosopher of his time, represented a rupture with the philosophical tradition and were much more in (...) tune with the physical geometry underlying the Einstein-Hilbert general theory of relativity developed more than two decades later. (shrink)

Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron (...) research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his (...) influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

1. Introduction: The problems of time and consciousness What is time? St. Augustine remarked that when no one asked him, he knew what time was; however when someone asked him, he did not. Is time a process which flows? Is time a dimension in which processes occur? Does time actually exist? The notion that time is a process which "flows" directionally may be illusory (the "myth of passage") for if time did flow it would do so in some medium or (...) vessel (e.g. minutes per what?) [1]. But if time is a dimension in which processes occurred, e.g. as one component of a 4 dimensional spacetime, then why would processes occur unidirectionally in time? Yet we perceive time as an orderly, unidirectional process. An alternative explanation is that time does not exist as either a process or dimension, but that reality is a collage of discrete, disconnected and haphazardly arranged configurations of the universe, e.g. as described in Julian Barbour's "The end of time" [2]. In this view our perception of a unidirectional flow of time occurs because each moment, or "Now" as Barbour terms them, involves memory of other conceptually relevant moments, and the orderly flow of time is an illusion. Barbour's deconstruction of time contrasts the Newtonian reality of objects moving deterministically through 4 dimensional spacetime. Newton's contemporary (and rival) Leibniz [3] viewed the world in a manner consistent with Barbour (and with Mach's principle that the spatiotemporal structure of the universe is dependent on the distribution of mass, a foundation of Einstein's general relativity). According to Leibniz the world is to be understood not as matter/mass moving in a framework of space and time, but of more fundamental snapshot-like entities that momentarily fuse space and matter into single possible arrangements or configurations of the entire universe. Such configurations, which can be fabulously rich and complex considering the vastness of the universe, are the ultimate "things" of reality, which Leibniz termed "monads".. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Special and General theories of relativity may be considered as the most significant examples of integrative thinking. From these works we see that Albert Einstein attached great importance to how we understand geometry and dimensions. It is shown that physics powered by the new multidimensional elastic geometry is a reliable basis for science integration. Instead of searching for braneworlds (elastic membranes - EM) in higher dimensions we will start by searching them in our 3+1 dimensional world. The cornerstone of (...) the new philosophy is an idea that lower dimensional EMs are an essential component of the living matter, they are responsible for our perceptions, intellect, pattern recognition and high speed signal propagation. According to this theory each EM has both physical and perceptive (psychological) meanings: it exists as our Universe-like physical reality for its inner objects and at the same time it plays perceptive (psychological) role in the external bulk space-time. This philosophy may help us to build up a science which explains not only inanimate, unconscious phenomena, but consciousness as well. (shrink)

The standard mathematical account of the sub-metrical geometry of a space employs topology, whose foundational concept is the open set. This proves to be an unhappy choice for discrete spaces, and offers no insight into the physical origin of geometrical structure. I outline an alternative, the Theory of Linear Structures, whose foundational concept is the line. Application to Relativistic space-time reveals that the whole geometry of space-time derives from temporal structure. In this sense, instead of spatializing time, Relativity (...) temporalizes space. (shrink)

continent. 1.2 (2011): 78-91. This article consists of three parts. First, I will review the major themes of Quentin Meillassoux’s After Finitude . Since some of my readers will have read this book and others not, I will try to strike a balance between clear summary and fresh critique. Second, I discuss an unpublished book by Meillassoux unfamiliar to all readers of this article, except those scant few that may have gone digging in the microfilm archives of the École normale (...) supérieure. The book in question is Meillassoux’s revised doctoral dissertation L’Inexistence divine (or The Divine Inexistence ), with its seemingly bizarre vision of a God who does not yet exist but might exist in the future. Without literally accepting this view, I will claim that it is philosophically interesting in ways that even a hardened sceptic might be able to appreciate. Third and finally, I will speculate on the possible future of Meillassoux’s speculative materialism itself. And here I mean its future development not by Meillassoux, but by those readers who might be inspired by his book. Plato could never have predicted the emergence of Aristotle’s philosophy, despite the obvious debt of the latter to the former. Nor could Descartes have predicted Spinoza and Leibniz, nor Kant the German Idealists, and neither could Husserl in 1901 have foreseen the later emergence of Heidegger. How are the works of interesting philosophers transformed by later thinkers of comparable importance? While it may seem that there are countless ways to do this, I think there are only two basic ways in which this happens: you can radicalize your predecessors, or you can reverse them. I will close this article with a few words about these two methods, and try to imagine how Meillassoux might be radicalized or reversed by some future admirer. My view is that the more important thinkers are, the easier they are to radicalize or reverse. This helps explain why the great philosophers of the West have so often appeared in clusters, succeeding one another at relatively brief intervals during periods of especial ferment. 1. After Finitude After Finitude is unusually short for such an influential book of philosophy: running to just 178 pages in the original French, and an even more compact 128 pages in the English version, despite the introduction of roughly eight pages of new material for the English edition. Rather than summarizing Meillassoux’s book in the order he intended, I will focus on six points that strike me as the pillars of his debut book. Along the way, I will offer a few criticisms as well. The first pillar of the book is Meillassoux’s own term “correlationism.”1 Although he introduces this term as the name for an enemy, it is striking that Meillassoux remains impressed by correlationism much more than his fellow speculative realists are. This continued appreciation for his great enemy influences the shape of his own ontology. Is there a world outside our thinking of it, or does the world consist entirely in being thought? Traditionally, this dispute between realism and idealism has been dismissed in continental philosophy as a “pseudo-problem,” in a strategy pioneered by Husserl and extended by Heidegger. We cannot be realists, since following Kant we have no direct access to things-in-themselves. But neither are we idealists, since the human being is always already outside itself, aiming at objects in intentional experience, deeply engaged with practical implements, or stationed in some particular world-disclosing mood. The centuries-old dispute between realism and idealism is dissolved by saying that we cannot think either real or ideal in isolation from the other. There is neither human without world nor world without human, but only a primordial correlation or rapport between the two. This is what “correlationism” means: philosophy trapped in a permanent meditation on the human-world correlate, trying to find the best model of the correlate: is it language, intentionality, embodiment, or some other form of correlation between human and world? Among other problems, this generates some friction between philosophy and the literal meaning of science. When cosmologists say that the universe originated 13.5 billion years ago, they do not mean “13.5 billion years ago for us ,” but literally 13.5 billion years ago, well before conscious life existed, and thus at a time when there was no such thing as a correlate. Meillassoux also coins the term “ancestrality” (10) for the reality that predated the correlate, and later expands this term to “dia-chronicity,” (112) to refer to events occurring after the extinction of human beings no less than to those occurring before we existed. Up to this point, Meillassoux’s focus on ancestral entities existing prior to consciousness might seem like a straightforward realist who wants to unmask correlationism as just another form of idealism. Yet Meillassoux also admires the correlationist maneuver, which can obviously be traced back to Kant. Unlike a thinker such as Whitehead, Meillassoux feels no nostalgia for the pre-critical realism that came before Kant: “we cannot but be heirs of Kantianism,” he says (29). What impresses Meillassoux about correlationism is something both simple and familiar. If we attempt to think a tree outside thought, this is itself a thought . Any form of realism which thinks it can simply and directly address the world the way it is fails to escape the correlational circle, since the attempt to think something outside of thought is itself nothing other than a thought, and thereby collapses back into the very human-world correlate that it pretends to escape. For Meillassoux this step, suggested by Kant but first refined by the ensuing figures of German Idealism, marks decisive forward progress in the history of philosophy that must not be abolished. Any attempt to break free from the correlate must first acknowledge its mighty intellectual power. Realist though he may seem, Meillassoux’s works are filled with praise of such figures as Fichte and Hegel, not of so-called “naïve realists.” It is also the case that for Meillassoux, not all correlationisms are the same. The second pillar of his book is a distinction between various positions that I have termed “Meillassoux’s Spectrum,” though of course he is never so immodest as to name it after himself. He distinguishes between at least six different possible positions, and perhaps we could add even subtler variations if we wished. But in its simplest form, Meillassoux’s Spectrum allows for just four basic outlooks on the question of realism vs. anti-realism. Three of these are easy to understand, since we have already been discussing them. At one extreme is so-called “naïve realism,” which holds that a world exists outside the mind, and that we can know this world. Meillassoux rejects this naïve realism as having been overthrown by Kant’s critical philosophy. At the other extreme is subjective idealism, in which nothing exists outside the mind. For to think a dog outside thought immediately turns it into a thought, and therefore there cannot be anything outside; the very notion is meaningless. In between these two is what we have called correlationism. And here comes a crucial moment for Meillassoux, since he distinguishes between the two forms of “weak” and “strong” correlationism, and chooses the strong form as the launching pad for his own philosophy. Weak correlationism is easy to explain, since we all know it from the philosophy of Kant. The things-in-themselves can be thought but not known. They certainly must exist, since there cannot be appearances without something that appears. And we can think about them, which idealism holds to be impossible. They are simply unknowable due to the finitude of human thought. Strong correlationism is the new position introduced by Meillassoux (though he sees it at work in numerous twentieth century thinkers), midway between weak correlationism and subjective idealism. The major difference between the three positions is as follows. Weak correlationism says: “The things-in-themselves exist, but we cannot know them.” The subjective idealist says: “This is a contradiction in terms, since when we think the things-in-themselves, we already turn them into thoughts.” But the strong correlationist says: “Just because ‘things-in-themselves’ is a meaningless notion does not mean that they cannot exist. No one has ever traveled to the world-in-itself and come back to make a report on it. Thus, the fact that we cannot think things-in-themselves without contradiction does not prove that they do not exist anyway. There may be things-in-themselves, we simply are not capable of thinking them without contradiction form within the correlational circle.” This step is crucial for Meillassoux, since strong correlationism is the position he attempts to radicalize into his own new standpoint: speculative materialism. As I see it, this step of the argument fails. Strong correlationism cannot avoid collapsing into subjective idealism, since the statements of the strong correlationist are rendered meaningless from within. All three of the other positions in the Spectrum make perfectly good sense even for those who disagree with them. The naïve realist says that things-in-themselves exist and we can know them; the meaning of this statement is clear. The weak correlationist can say that things-in-themselves exist but lie forever beyond our grasp; this too makes perfect sense, even though the German Idealists try to show a contradiction at work here. We can also understand the claim of the subjective idealist that to think anything outside thought turns it into a thought, and that for this reason we cannot think the unthought. The strong correlationist, alone among the four, speaks nonsense . This person says “I cannot think the unthought without turning it into a thought, and yet the unthought might exist anyway.” But notice that the final phrase “the unthought might exist anyway” is fruitless for this purpose. For we have already heard that to think any unthought turns it into a thought. But now the strong correlationist wants to do two incompatible things simultaneously with this unthought. On the one hand, he neutralizes the unthought by showing that it instantly changes into just another thought. But on the other hand, he wants to appeal to the unthought as a haunting residue that might exist outside thought, thereby undercutting the absolute status of the human-world correlate found in idealism. But this is impossible. If you accept the argument that thinking the unthought turns it into a thought, you cannot also add “but maybe there is something outside that prevents this conversion from being absolutely true,” because this “something outside” is immediately converted into nothing but a thought for us. In short, Meillassoux here seems to be offering a kind of Zen koan: his “strong correlationism” is reminiscent of the gateless gate, or the sound of one hand clapping, or the command to punch Hegel in the jaw when meeting him on the road. We cannot at the same time both destroy the realist challenge of the things-in-themselves in order to undercut realism and reintroduce that very realist sense in order to undercut idealism. In a world where everything is instantly converted into thought, we cannot claim that there might be something extra-mental anyway, because this “might be something” is itself converted into a thought by the same rules that condemned dogs, trees, and houses to the idealist prison. This brings us to the third pillar of Meillassoux’s argument, which is the key to all the rest: the necessity of contingency. His strategy is to transform our supposed ignorance of things-in-themselves into an absolute knowledge that they exist without reason, and that the laws of nature can change at any time for no reason at all. In this way the cautious agnosticism of Kantian philosophies is avoided, but so is the collapse of reality into thought as found in German Idealism. Meillassoux does try to prove the existence of things-in-themselves existing outside thought; he simply holds that they must be proven after passing through the rigors of the correlationist challenge, not just arbitrarily decreed to exist in the manner of naïve realism. As he puts it, “Everything could actually collapse: from trees to stars, from stars to laws, from physical laws to logical laws; and this not by virtue of some superior law whereby everything is destined to perish, but by virtue of the absence of any superior law capable of preserving anything, no matter what, from perishing” (53). If idealism thinks that the human-world correlate is absolute, for Meillassoux it is the facticity of the correlate that is absolute. He tries to show this with a nice brief dialogue between five separate characters (55-59) which is covered in detail in my forthcoming book,2 but which I will simplify here for reasons of time. In this simplified version, we first imagine a dogmatic realist arguing with a dogmatic idealist. The realist says that we can know the truth about the things-in-themselves; the idealist counters that we can only the truth about thought, since all statements about reality must be turned into statements concerning our thoughts about reality. Here the correlationist enters and proclaims that both of these positions are equally dogmatic. For although we have access to nothing but thoughts, we cannot be sure that these thoughts are all that exist; there could be a reality outside thought, there is simply no way to know for sure. And this latter position is the one that Meillassoux attempts to transform from an agnostic, skeptical point into an ontological claim about the contingency of everything. Consider it this way. How does the correlationist defeat the idealist? The idealist holds that the existence of anything outside thought is impossible. The correlationist, by contrast, holds that something might exist outside the human-world correlate. But this “something might” has to be an absolute possibility. It cannot mean that “something outside thought might exist for thought ,” because that is what the idealist already says. No, the correlationist must mean that something might exist outside thought quite independently of thought. In other words, the correlationist says that idealism might be wrong, and this means it is absolutely true that idealism might be wrong. Thus, correlationism is no longer just a skeptical position. It holds that all the possibilities of the world are absolute possibilities. We have absolute knowledge that any of the possibilities about the existence or non-existence of things-in-themselves might be true, and this means that correlationism flips into Meillassoux’s own position: speculative materialism. As Meillassoux sees it, there are only two options here. Option A is to absolutize the human-world correlate, which is what the idealist does: there absolutely cannot be anything outside thought. Option B, by contrast, is to absolutize the facticity of the correlate: its character of simply being given to us, without any inherent necessity. The correlationist cannot have it both ways by saying: “there absolutely might be something outside thought, yet maybe this is absolutely impossible.” In other words, once we escape dogmatism we can only be idealists or speculative materialists, not correlationists. The human-world correlate is merely a fact, not an absolute necessity. But this facticity itself cannot be merely factical: it must be absolute. Here Meillassoux coins the French neologism factualité , which has been suitably translated into the English neologism “factiality.” (7, 122-3) Factialty means that for everything that exists, it is absolutely possible that it might be otherwise, not just that we cannot know whether or not it might be otherwise. Just as Kant transformed philosophy into a meditation on the categories governing human finitude, Meillassoux wishes to turn philosophy into a meditation on the necessary conditions of factiality, which he calls “figures”—a new technical term for him. (80) One such figure is that the law of non-contradiction must be true, and for an unusual reason. Since everything is proven to be contingent, nothing that exists can be contradictory, for whatever is contradictory has no opposite into which it might be transformed, and thus contingency would be impossible.3 Another such figure is that there must be something rather than nothing: for since contingency exists, something must exist in order to be contingent. It is a daring high-wire act, one that sacrifices realism to the correlational circle in order to rebuild it from out of its own ashes. Some might conclude that the lack of reason in things is a byproduct of the ignorance of finite humans, Meillassoux is making precisely the opposite point. For in fact, the doctrine of finitude usually leads directly to belief in a hidden reason. The fact that it lies beyond human comprehension merely increases our belief in this arbitrarily chosen concealed ground. By defending anew the concept of absolute knowledge Meillassoux evacuates the world of everything hidden. The reason for things having no reason is not that the reason is hidden, but that no reason exists. Thus, even while insisting on the necessity of non-contradiction, he rejects the other Leibnizian principle: sufficient reason. Everything simply is what it is, in purely immanent form, without deeply hidden causes. Or as Meillassoux puts it: “There is nothing beneath or beyond the manifest gratuitousness of the given—nothing but the limitless and lawless power of its destruction, emergence, or persistence” (63).The world is a “hyper-chaos”(64). But this is not the same thing as flux. For the chaos of the world is such that stability might occur just as easily as constant, turbulent change. Let’s now digress a bit, and return to the question of ancestrality, which Meillassoux transforms later in the book into “dia-chronicity.” Correlationism holds that all talk of a world outside the correlate is immediately recuperated by the correlate. The phrase “13.5 billion years ago” becomes “13.5 billion years ago for us ,” and the phrase “the universe following the extinction of humans” becomes “the universe following the extinction of humans for humans .” But notice that whether we talk about the world before or after humans, in both cases it is time that is used to challenge the correlate. Meillassoux has no interest in challenges that might be posed by space. For example, what about a vase in a lonely country house that topples to the floor and smashes when no one is there to watch it? Isn’t this also a challenge to correlationism, no less than the Big Bang or the heat death of the universe long after humans have vanished? In an eight-page supplement to the English translation of After Finitude ,4 possibly in response to my own 2007 review of the French original,5 Meillassoux bluntly denies that space is of any relevance to the question. Spatial distance is a merely harmless challenge to the human-world correlate. After all, even though no one is there in the lonely country house to witness the shattering of the vase, we can say that had there been an observer , that observer would have witnessed the toppling and destruction of the vase. For this event still occurs in a world in which the human-world correlate already exists, whereas the diachronicity of events both before and after the existence of humans makes it impossible to say that had there been an observer they would have witnessed the Big Bang occurring in such and such a fashion. However, it seems to me that Meillassoux merely asserts that the temporal simultaneity of our existence with that of the vase in the lonely country house is enough to render it harmless. It is true that the house does not exist prior to the correlate, but nonetheless it exists outside the correlate, and that is enough to make the same challenge. It is difficult to see why the “had there been an observer” maneuver succeeds in the case of a vase in the countryside in April 2011 but fails in the case of the Big Bang. This is not just a matter of nitpicking Meillassoux’s argumentative style: the fact that he bases his argument on time has at least two important consequences for his position. For in the first place, even though Meillassoux insists that the laws of nature are absolutely contingent, this turns out to be true only in a temporal sense. That is to say, it is a paradoxical feature of Meillassoux’s philosophy that he does allow for the existence of laws of nature, and simply believes that they can change at any moment without reason. Within any given moment, laws of nature do exist. He never suggests that different parts of the universe can have different laws at the same time, nor does he have any interest in the laws of part/whole composition that take place within any given instant. Could it be the case that rather than being made of gold atoms, a small chunk of gold could be made of silver atoms, cotton, horses, or that this same small piece of gold could be made of gigantic vaults filled with even more gold? These are not topics that draw Meillassoux’s attention, since he is focused solely on how the laws of nature might change or endure from one moment to the next . Another implication for Meillassoux’s system is that his concept of things-in-themselves turns out to be to be inadequate. For when he proves that things-in-themselves can exist without humans, this turns out to be true only in a temporal sense as well. Namely, things-in-themselves existed ten billion years ago, and they will continue to exist after all humans have succeeded in exterminating themselves. However, being able to exist before our births and after our deaths is just one small part of what it means to be a thing-in-itself. The more important part is that even if a thing is sitting on a table right now, in front of me, even if I stroke it lovingly or press my face up against it directly, I am still dealing only with a phenomenal version of the thing; the thing-in-itself continues to withdraw from all access. Yet no such thing is acknowledged by Meillassoux. For him finitude is a disaster, and absolute knowledge is in fact possible. Meillassoux’s thing-in-itself exists in independence only of the human lifespan , not of human knowledge. The fifth pillar of Meillassoux’s argument is his use of Cantor’s transfinite mathematics to show that even if the laws of nature are contingent, they need not be unstable, and thus we cannot use the apparent stability of nature to disprove his metaphysics of absolute contingency. What Cantor showed is that there are different sizes of infinity, and that all these infinities cannot be totalized in a single infinite number of infinities. Meillassoux sees this as crucial, since it allows him to discredit any “probabilistic” argument against his theory. The probabilistic argument (as defended quite clearly by Jean-René Vernes)6 would say this: given that the laws of nature seem so stable, it it is extremely improbable that there is no hidden reason for their remaining so stable. As Meillassoux sees it, probability is of value only when we can index an accessible total of cases. These can even be infinite: for example, there are an infinite number of points where a rope can break when stretched tight, but this does not stop us from calculating probabilities for various sections of the rope to break. By contrast, there is no way to sum up the number of possible laws of nature. For here there is no way to totalize; we cannot stand outside of nature and calculate the possible number of laws so as to determine the probabilities that any one of them might change. Therefore, although we can speak of probability when dealing with intraworldly events such as elections, horse races, and coin-flips, we cannot use the words “probable” or “improbable” when describing alterations at the level of nature as a whole. Rather than commenting on the validity of this argument and its use of Cantor, let me simply note that it once again creates a dualistic ontology. We already saw that Meillassoux treats time differently from space. In analogous fashion, he now treats the level of world differently from that of intraworldly events. The emergence of worlds is purely contingent and virtual and governed by no probability at all, while events within the world necessarily follow laws (even if these laws can change at any moment without reason), and thus their probabilities can be calculated. It is a strategy deeply reminiscent of Badiou’s (2005) own dualism between the normal “state of the situation” and the rare and intermittent “event.” The sixth and final pillar of Meillassoux’s book can be dealt with briefly, since we have already touched on it elsewhere. It comes at the very beginning of the book, when Meillassoux says that we must revive the distinction between primary and secondary qualities, and that the primary qualities are the ones that can be mathematized. He admits that he has not yet published a proof of this idea, though in fact it is already known as one of his primary doctrines. And here we encounter the familiar problem with Meillassoux’s inadequate conception of things-in-themselves. “Primary qualities” refers to those qualities that a thing has independently of its relations with us or anything else. But if the primary qualities can be mathematized, this means that they are not entirely independent of us, since our knowledge can get right to the bottom of them. The mathematized qualities of things are independent of us only in Meillassoux’s sense that they will still have those qualities even when all humans are dead. But to repeat, autonomy from the human lifespan is not the same as autonomy from human access. Here once more Meillassoux is concerned only with independence from the human-world correlate across time, not in any given instant. 2. L’Inexistence divine In 1997, the same year in which he turned thirty years old, Meillassoux earned his doctorate at the École normale supériuere with a brazen dissertation entitled L’Inexistence divine ( The Divine Inexistence ). The work was substantially revised in 2003. But even then, with typical fastidiousness, Meillassoux decided that the work was not yet ready for press. It has now been scrapped in favor of some future, multi-volume work bearing the same title. While writing my book on Meillassoux for Edinburgh University Press, I was permitted to translate excerpts from this unpublished work for use as an appendix in my own book; in total, the appendix contains approximately twenty percent of Meillassoux’s 2003 manuscript, the first time any of it will be published in any language. Nonetheless, a portion of the argument was already tested in the article “Spectral Dilemma,” published in English in the journal Collapse (2008: 261-75). There the philosophical motives for the virtual God are already made clear. What troubles us most are early deaths, brutal deaths, deaths of especial injustice– the sorts of deaths in which the brutal twentieth century was so abundant. And here, neither the atheist nor the believer can help us. The atheist can offer nothing but a sad and cynical resignation when reflecting on the victims of these terrible crimes. The believer does little better, being unable to explain how God could have allowed such things to happen, due to the famous intractability of the problem of evil. The solution offered to this dilemma by Meillassoux is bold, and all the more so given that he emerges from such a deeply Leftist, materialist, and unreligious background. His solution is that God does not yet exist, and therefore is not blameworthy for these catastrophes. Given that everything is contingent in Meillassoux’s philosophy, this God and divine justice might never exist, but they can at least exist as an object of hope. Let’s begin by jumping to the end of L’Inexistence divine , where the alternatives are laid out so nicely. There are four basic attitudes that humans can have towards God, Meillassoux says. First, we can believe in God because he exists. This is the classical theist attitude, rejected for the simple reason that it would be amoral and blasphemous to believe in a God who allows children to be eaten by dogs, to use Dostoevsky’s example. Second, we can disbelieve in God because he does not exist: the classical atheist attitude. But this leads to sadness, cynicism, and a sneering contempt for the greatness of human capacity. The third option, rather more complex, is to disbelieve in God because he does exist: in other words, to exist in rebellion against God as the one who must be blamed for the evils of the earth. The examples here might range from Lucifer himself, to the more human figure of Captain Ahab in Melville’s Moby-Dick , to Werner Herzog’s even more recent catchphrase, “Every man for himself, and God against all.” That leaves only the fourth option: believing in God because he does not exist. Meillassoux closes his book by saying that the fourth option has now been tried (namely, in the course of his own book), and that now that all four have been specified, we must choose. The first reaction to this theory of the inexistent God will be laughter. Few readers will ever be literally convinced by it, and probably none will immediately be convinced by it. But if we ask ourselves why we laugh, the answer is because it sounds so improbable that an inexistent God might suddenly emerge and resurrect the dead. It obviously sounds more like a gullible theology than a rigorous piece of philosophical work. Yet two things need to be kept in mind. First, Meillassoux’s theories are hardly more unlikely than those of great philosophers of the past such as Plato, Plotinus, Avicenna, Malebranche, Spinoza, Leibniz, Nietzsche, or Whitehead. We read the great philosophers not because their systems are plausible in commonsense terms that can be measured by the laws of probability. Instead, we read them precisely because they shatter the existing framework of common sense and open up new window on the universe. Second, and even more importantly, Meillassoux has already rejected probability as a valid measuring stick in philosophy. Or rather, he accepts probability in the intra-worldly realm (where it is linked with potentiality), and rejects it at the level of the world itself (where potentiality is replaced with what he calls virtuality). The virtual God can appear at any moment for no reason at all, just as any other new configuration of laws of nature can appear: in a manner that the laws of probability cannot calculate. Responding to those who might ridicule the idea of a sudden emergence of God and a resurrection of the dead, Meillassoux cites Pascal, who asserts that the resurrection of the dead would be far less incredible than the fact that we were born in the first place. This shifts philosophy onto new ground. Rather than concerning ourselves with what is likely to happen in the world as we know it, we focus instead on the most important things that could happen. For this reason, the expected objection that a virtual God is no more likely to appear than a virtual unicorn or a virtual flying spaghetti monster misses the point. Unicorns and spaghetti monsters could also appear, just like any other non-contradictory thing. But these would just be novel bizarre entities among others, not the heralds of completely new worlds. For Meillassoux, the emergence of matter, life, and thought have been the three truly amazing advents of the world so far, each of them dependent on the advent(s) preceding them. As he sees it, there can be no greater intraworldly entity than the human beings who already exist, since nothing in the world is better than the absolute knowledge of which humans alone are capable. This means that the next great advent must be something that perfects human beings rather than superseding them. And this can only be the world of justice, in which the dead are resurrected and their horrible deaths partially cancelled (Meillassoux never considers the possibility of a God who would literally erase the pre-divine past so that it never happened at all). The only immortality worth having is an immortality of this life, not an existence in some ill-defined afterworld. Human existence, he holds, must always be governed by a “symbol” that gives us the “immanent and comprehensible inscription of values in a world.” And just as cosmic history made the three great contingent leaps of matter, life, and thought, with a leap to justice as the only one still to come, a similar structure occurs within human culture and its symbols, which consist so far of the cosmological, naturalistic, and historical symbols, with a “factial” symbol still to come. We can review each of these symbols briefly. The cosmological symbol refers to the ancient dualism between the terrestrial and celestial spheres. Here below everything is conflict, corruption, and decay; but in the heavens nothing is perishable, all movement is circular, and everything is arranged in mutual harmony. This symbol is ended by modern physics when Galileo discovers such blemishes as sunspots and craters on the moon, and when Newton integrates both celestial and terrestrial movement into a single gravitational law. Next comes the naturalistic or romantic symbol, in which perfection comes not from the sky but from nature itself. The world is filled with pretty flowers (Meillassoux claims that the ancients never discussed the beauty of flowers until Plotinus in the third century) and with living creatures naturally moved by pity, at least until society corrupts them. This symbol collapses in the face of reality as we know it, since pity is no more common than war, corruption, and violence. This brings us to the historical symbol, which only now is passing away. Bad things may happen, but history has an inner logic of its own, such that everything works out in the end. The ultimate form of the historical symbol is the economic symbol, whether in a Marxist or neo-liberal form. Just as the Marxist holds that the inner economic logic of the capitalists will inexorably lead them to self-destruction, the neo-liberal assumes that the sum total of individual selfish actions will lead, in the long run, to the greatest possible good. We worship the economy and let it guide history for us, just as the ancients worshipped celestial bodies and held them to be free from blemish. The final remaining symbol is the factial symbol, which Meillassoux hopes will now emerge. Factiality, we recall, is his term for the absolute contingency of everything that exists. Once we have grasped this absolute contingency, we are free to expect the dramatic advent of the coming fourth World: the world of justice, inaugurated by a virtual God and even mediated by a messianic human figure. There is the added feature, however, that this messiah must abandon all claims to special status once the messianic realm of justice is achieved. The messianic figure will then be no more special than any person on the street, since a reign of human equality will have arisen. Although this focus on human being might seem like a return to standard humanism, Meillassoux holds that human pre-eminence has never truly been maintained. Previously, humans have been treated as special only because they contemplate the Good, because they resemble their omnipotent creator, or because they happen to be the temporary victors in a cruel Darwinian death-match between millions of living species. For Meillassoux, by contrast, humans have value because they know the eternal. But it is not the eternal that is important, since this merely represents the blind, anonymous contingency of each thing. What is important is not knowledge of the eternal , but knowledge of the eternal. We should not admire Prometheus for stealing fire from the gods; Prometheus is simply as bad as all the gods, no matter how much he increased our power. Feuerbach and Marx were wrong to say that God is a projection of the human essence, since for Meillassoux the usual concept of God represents the degradation of the human essence. If the traditional God was allowed to inflict plagues and tsunamis on the human race, the Promethean human of the twentieth century simply assumes the right to inflict death camps and atomic fireballs instead. In this respect, we have simply begun to imitate the degradation of humanity that was formerly invested in an omnipotent and arbitrary God. In response to charges that absolute contingency might lead to political quietism, Meillassoux counters that the World of justice would mean nothing unless we had already hoped for it beforehand. A World of justice that came along at random would merely be an improved third World of thought: indeed, a perfect one. But it would have satisfied no craving, and would therefore have no redemptive power. For this reason, we must actively hope for the fourth World of justice for such a fourth World ever to arise. Not only justice, but beauty is dependent on such hope: for Meillassoux, who is here somewhat dependent on Kant, beauty means an accord between our human symbolization and the actual world, which could never be present in a World of the blessed any more than justice could. And just as a messianic figure is needed to incarnate our hope and then abandon power once the World of justice is realized, it is the figure of the child whose fragile contingency shows us a dignity and a demand for justice beyond all power. 3. Meillassoux Radicalized or Reversed Given the promising reception of Meillassoux’s first book, it would not be groundless to engage in early speculation about what it might take to earn him a place in the history of philosophy. Maybe this will never happen—who knows?—but quite possibly it will: his lucid argumentative methods and sheer philosophical imagination at least make him a good candidate to be read well into the future, especially following further elaboration in print of his mature system. Philosophy is often practiced as thought it were nothing more than the amassing of “knockdown arguments.” But this is no more insightful than saying that good architecture is the amassing of steal beams. It is true that poorly constructed building cannot stand for long, but sound construction is merely the first, indispensable step in building. In fact, I am inclined to say that what really makes a philosopher important is not being right, but being wrong . I mean this in a very specific sense. I once heard the interesting remark about twentieth century culture that “you have to remember that the sixties really happened in the seventies.” That is to say, it was in the 1970’s rather than the more honored 1960’s that civil rights, free love, long hair, and the rock and roll drug culture really took root. With respect to the history of philosophy, we might just as easily say: “you have to remember that Plato really happened in Aristotle,” that “Kant really happened in Hegel” or “Hume really happened in Kant,” or that “Husserl’s phenomenology first achieved its truth in Heidegger.” One becomes an important philosopher not by being right, but by attracting rebellious admirers who tell you that you are wrong , even as their own careers silently orbit around your own. To recruit faithful disciples may be comforting and flattering, but the greatest thinkers have generally had to experience refutation at the hands of their most talented heirs. For this reason, I would propose that we size up the magnitude of living thinkers not by deciding how many times they are right and wrong, but by asking instead: who would take the trouble to refute this author? For this reason I do not ask: “Is Meillassoux right?”, since I do not believe in the virtual God myself, nor am I convinced by any important aspect of Meillassoux’s philosophy. Instead, I ask if there are interesting ways to overturn him. Only by being overturned, by no longer remaining a contemporary, does one become a classic. Let’s begin with a simple model of refutation, which can be refined further at a later date once the basic point is established. One kind of refutation simply consists in saying: “This author is a complete idiot.” The refuter now walks away in celebration, and no link between the present and the future is built; all is reduced to rubble. But this sort of mediocre triumphalism is generally practiced by those who achieve little of their own, and is not especially interesting. Much more interesting is the sort of refutation that does not take its target to be a complete idiot. I would like to suggest that there are just two basic ways in which this can be done: radicalization and reversal . It has not escaped my notice that this is a fairly good match for the Deleuzian distinction between irony and humor. Whereas irony critiques and adopts the opposite principle of what it attacks, humor accepts what it confronts but pushes it into highly exaggerated form. The ironist is like the worker who sows chaos by rebeling and contradicting the boss, while the humorist is like the worker who follows orders to an absurdly literal degree, with equally chaotic results. Let’s start with a few examples. In Aristotle’s treatment of Plato, and Heidegger’s of Husserl, we find reversal. Plato’s eidei are transformed by Aristotle into mere secondary substances, and the individual worldly things despised by Plato become what is primary. For Husserl what is primary is whatever is present to consciousness, while for Heidegger this is precisely what is secondary, since the primary stuff of the world withdraws from any form of presence at all. As for radicalization, it is most easily found in the transformation of Kant by German Idealism: “Kant was right to wall off the things-in-themselves from human access, and simply should have realized that the thought of the Ding an sich is also a thought, and thereby the noumena are just special cases of the phenomena,” with much following from this discovery. It would also be easy to read Spinoza as a radicalizer of Descartes, and Berkeley and Hume as radicalized versions of Locke. Perhaps the distinction is now sufficiently clear. Admiring refutations are not those that say “Professor X is an idiot,” which is merely the flip side of the eager disciple’s fruitless “Professor X got everything right.” Instead, it will be some variant of one of the following two options: “Professor X is important, but got it backwards,” or “Professor X is important, but didn’t push things far enough.” In the history of philosophy these two latter cases have often been painful in purely human terms: Aristotle expresses sadness at refuting Plato, Kant is openly annoyed at Fichte, and Husserl feels betrayed and used by Heidegger. Rude handling from later figures almost seems to be the sine qua non of being a great philosopher. Now, it has already been claimed that Meillassoux is an emerging philosopher of the first importance, and by no less a figure than Alain Badiou: “It would be no exaggeration to say that Quentin Meillassoux has opened up a new path in the history of philosophy…” (Preface, vii). But rather than taking Badiou’s word for it, or rejecting his word, we might experiment by asking how Meillassoux could be radicalized or reversed. Are there interesting ways of doing this that might launch whole new schools of philosophy, unexpected or even condemned by Meillassoux himself? While no one can see the future, the present is poor when it is not riddled with virtual futures. The relation between philosophers and their predecessors and successors is always somewhat complicated, of course. But generally there is one central divergence at stake, which might be taken as the key to all the others. On this basis we could say that new thinkers primarily radicalize or primarily reverse the main ideas of their chief philosophical forerunner. There may be specific historical conditions and perhaps even personality traits connected with these two types, but this question can be left aside for now. More important for us is that radicalizers will generally be followed by reversers, and vice versa. Consider the textbook example of a reversal in the history of philosophy: Kant’s Copernican Revolution, which inverts the so-called dogmatic tradition that addresses the world itself, and makes the world revolve instead around the conditions by which it is known. While it is not completely impossible that Kant’s successors might have re-reversed this principle back into a new and stronger dogmatic realism, conditions were premature for such a move. Anyone doing this too early would likely have been an angry anti-Kantian reactionary rather than an original thinker in command of a genuinely new realist principle. The far more likely outcome is the one that actually happened: Kant’s reversal of his predecessors was viewed as incomplete, or as retaining lamentable bits of the traditional view, which despite his admirable breakthrough he was unable to shake off. This was the view of German Idealism, anyway. In similar fashion, Spinoza could also be viewed as a radicalizer of Descartes, who is equally accused of preserving various Scholastic dogmas in an otherwise radical project of philosophical reversal. The point is this: reversals in the history of thought tend to be followed soon thereafter by radicalizations of those reversals. The same may hold true in reverse: radicalizations might generally be followed by reversals, given that it is not always possible to be more radical than the radicals have already been. Consider the case of Husserl, who radicalizes Brentano’s early vagueness about what lies beyond immanent objectivity, and Twardowski’s assertion that there must be an external object lying outside the intentional content, by collapsing everything into the intentional sphere: there is no difference between the Berlin in my consciousness and the actual Berlin that is home to millions of people. It is difficult to see how one could be even more radical than Husserl’s idealist turn here. And thus the road is paved to Heidegger’s reversal of classical phenomenology, in which the key point is what lies deeper than any presence to consciousness: the Sein whose power and obscurity cannot be made exhaustively present, but only sends itself in historical epochs. In similar fashion we might also read Leibniz as a reverser of Spinoza’s radicalization, retrieving a strong sense of individual substance and a certain validity of what the Scholastics had said. Returning to Meillassoux, we might ask which kind of philosopher he is: a radicalizer or a reverser? At present, Meillassoux looks to me like a radicalizer (though for now his future remains shrouded in mist). He takes the correlationist tradition, which allows us to speak only of the relation between human and world, and tries to raise it into an even more extreme claim about the absolute contingency of everything. But whereas German Idealism did this by trying to collapse the distinction between thought and world entirely into the “thought” side, Meillassoux does it by trying to shift the non-absolute contingency of the thought-world correlate from epistemology to ontology. It is no longer a question of the inability of human knowledge to know what lies outside the correlate, but the inability of reality itself to be rooted in any definite laws. Furthermore, if we look at the various features of Meillassoux’s philosophy identified earlier tonight, all but one are already so radical that there is no obvious way to push them further. The one exception would be his claim that the world as a whole can change for no reason at any moment, coupled with the inconsistent claim that within a given world there are laws of nature that everything must follow. If gravitational attraction between all masses is a current law in our world, then for Meillassoux there can be no exceptions to this law for as long as it remains in force. A toppled vase will fall to the floor every time for sure,unless there is a cosmic change by which the laws of nature as a whole have altered. (This is reminiscent of the late medieval distinction between the absolute and ordained power of God, according to which God has the power to set or change the laws of nature, but not to contravene those laws locally once they are set.) On this point, to radicalize Meillassoux would simply be to say: there are no laws of nature even in the local sense. Everything that happens, even in the world here and now, is purely contingent and not governed by even a trace of law. And while this would be a more consistent development of Meillassoux’s thoughts on contingency, it is difficult to see how it could lead to a new philosophy. Instead, the admiring successors of Meillassoux are more likely to reverse one of his already sufficiently radical points. At least four candidates come to mind: *First, we have seen that Meillassoux thinks correlationism is challenged by a time before or after consciousness, but not by a space lying outside it. Perhaps this could be reversed into saying that spatial exteriority is the really crucial point. The arguments on this point are perhaps the least convincing in After Finitude (and do not even occur in the original French edition), and therefore it might be a candidate for the “blind spot” of which no philosopher is ever free. *Second, Meillassoux uses Cantor to claim that the contingency of laws of nature would not entail that they are unstable. A successor of Meillassoux might claim that it does make them unstable, and celebrate this fact. This person would then have to explain why common sense seems to encounter a relatively stable world despite its truly rampant instability. Whereas Meillassoux’s problem is to show how stability might exist despite contingency, this successor’s problem would be slightly different: to show why actual, full-blown instability might have the appearance of stability. *Third, Meillassoux claims that the primary qualities of things are those that can be mathematized. He might be reversed by a successor who says the opposite: the mathematizable qualities are the secondary ones, and the primary ones are those that elude symbolic formulation. While this is a perfectly valid possible objection to Meillassoux, it is one that is made in advance by some of his predecessors and is still made by some of his peers, making it less interesting for futurology than some of his other points. *Fourth and finally, whereas Meillassoux claims that God does not exist but might exist in the future, a successor might argue even more bizarrely that God has always existed but might vanish in the future. Let’s arbitrarily select the first of these possibilities, and imagine briefly where it might lead, if pursued in the future by admiring detractors. Meillassoux comes from the circle of Badiou, and some of Badiou’s most ardent admirers are found in Latin America. So, let’s imagine that towards mid-century some ingenious reversers of Meillassoux emerge in that portion of the Spanish-speaking world. Just for fun, let’s call them Castro and Chávez. And in order to avoid any confusion with the present-day politicians of those names, we will stipulate that Meillassoux’s great successors are both women. The philosopher Castro (we will suppose she comes from Peru) reverses Meillassoux’s argument that the ancestral or diachronic are what most threaten the human-world correlate. Instead, she claims that the diachronic does not threaten the correlate at all, and that we must instead look at space as what ruins the correlate and demands a strange new realism. What would such a philosophy look like? In order to determine this, we might ask what price Meillassoux pays for doing it the opposite way. As I see it, he pays in two separate ways. One is that laws of nature for him are contingent over time . The laws of nature apply to the universe as a whole at any given moment, and would be changed globally if they are ever changed at all. The second price he pays is that Meillassoux has no mereology , or theory of parts and wholes. Everything for him is on the level of the given, or immanent in experience, with the sole proviso that the laws governing this immanence might change without notice at any given moment. In reversing Meillassoux, Castro makes the following claims in the preface to her stunning debut book of 2045, The Cosmos and its Neighborhoods , rapidly translated from Spanish into all the languages of the world: Despite his brilliant analysis of the contingency of laws of nature over time, Meillassoux gets two important assumptions wrong. First, he allows for only one set of contingent laws to govern nature as a whole. Second, he allows laws to govern only the world that is immanent in experience, and thereby fails to explore the contingency among part-whole relations. In this book I will argue, first, that the laws of nature vary in any given instant between one region of the universe and the next; and second, that the world is made up of layers of parts and wholes that are also contingent with respect to one another. Those are the words of Castro. This may sound like a hopeless free-for-all of chaos, yet the book somehow succeeds in drawing some compelling deductions about how laws must vary from one place or level in the world to the next. Trapped in the limited horizon of 2011, and not yet inspired by the heavily balkanized political and technological situation of 2050 that somehow lends additional credence to Castro’s vision, we can only vaguely grasp what such a philosophy might look like. After this reversal of Meillassoux by Castro, the usual pattern leads us to expect a radicalization by Chávez, a young Argentine student of Castro. How could the already strange theories of Castro be radicalized? Perhaps as follows, in a disturbing new book entitled The Implosion of the Neighborhoods , which argues as follows: Castro was right to shift the Meillassouxian framework of contingency from time to space. However, in this respect she retained a surprisingly traditional opposition between the two. In this book I will show that time and space collapse into one another. This may sound too much like the discredited four-dimensional block universe of twentieth century physics and philosophy. However, the four-dimensional universe is a model biased in favor of space, merely adding an extra dimension to the commonsense spatial continuum while stipulating that the serial passage of time is an illusion. In this book I will argue instead for a one-dimensional space-time modeled after our experience of time, in which there is no simultaneous co-existence at all between different parts of the universe, or ‘neighborhoods’ as my esteemed teacher Castro has called them. Instead, the various portions of the universe link to one another by succession rather than by coexistence. Buenos Aires, New York, and Amsterdam do not exist simultaneously in the same landscape, but one after the other in the mind of some observer, and this observer can only be an observer much larger than any human. Against Meillassoux’s notion of a virtual God that does not exist now but might exist in the future, I will argue for an actual God that surveys the universe in sequence, thereby generating the illusion of spatial diversity and even the illusion of individual minds located within that diversity. Once this divine observer dies, the universe as a whole must perish. Again, these ideas are so bizarre that we of 2011 can barely comprehend them, just as Aristotle would have had a difficult time grasping the theories of Descartes. We could then perhaps imagine a further reversal of this theory, emanating from the intellectually resurgent Philippines of the twenty-second century. The Filipino School might argue that the universe is already dead, given the collapse of its spatial richness into the serial observations of a flimsy and mortal God. The virtual universe does not yet exist, but might exist in fully spatial form in the future, and this would require the death of God and the resulting liberation of God’s succession of images as independent, spatially situated realities. With a bit of sharpening, we might be able to make all of these imaginary thinkers more intuitively clear. Along with the history of philosophy, there might arise a new discipline generating imaginary futures for philosophy. The richness of Meillassoux’s system comes not from the fact that he is plausibly right about so many things, but because his philosophy offers such a treasury of bold statements ripe for being radicalized or reversed. He is a rich target for many still-unborn intellectual heirs, and this is what gives him the chance to be an important figure. NOTES 1. Quentin Meillassoux, After Finitude . Trans. R Brassier. (London: Continuum, 2008.) Page 5. The word “correlationism” does not appear in his doctoral thesis. As Meillassoux informed me in an email of February 8, 2011, he first coined this term in 2003 or 2004, while editing for publication a lecture he had given at the École normale supérieure on a day devoted to the theme of “Philosophy and Mathematics,” an event including Alain Badiou as one of the participants. 2. Graham Harman, Quentin Meillassoux: Philosophy in the Making . (Edinburgh: Edinburgh University Press, forthcoming 2011). 3. In an email of December 6, 2010, Meillassoux clarifies that in After Finitude he only deduces the impossibility of a “universal contradiction,” not of a determinate contradiction. In the same email he suggests that he can also prove the latter, though the proof is somewhat lengthier than the one found in After Finitude . 4. After Finitude (18-26), in the passage falling between the two sets of triple asterisks. These pages were sent by Meillassoux to translator Ray Brassier (in French) during the translation process, and do not appear in the original French version of the book. 5. Graham Harman, “Quentin Meillassoux: A New French Philosopher.” Philosophy Today 51.1 (2007): Pages 104-117. The passage where I raise the question of space can be found in the first column of page 107. 6. Vernes is first cited on p. 95 of After Finitude . See Jean-René Vernes, Critique de la raison aléatoire, ou Descartes contra Kant . (Paris: Aubier, 1982).  . (shrink)

This is not a textbook in mathematical physics—excepting for one chapter one need not possess much more than geometry and elementary algebra—rather it is a philosophically reflective examination of the cardinal features of special relativity theory. Throughout the book Bohm is not merely doing physics, but thinking about doing physics as well. This metatheoretical reflexion appears in chapters concerning pre-Einsteinian notions of relativity, attempts to save the aether theories, the "ambiguity" of space-time measurements in the new cosmology, "common (...) sense" notions of space and time, and the falsification of scientific theories. There is a long appendix dealing with physics and perception—the relation between scientific objects and perceptual processes. One will, however, by working through the text learn plenty of physics in a rigorous and concise fashion. The author wisely does not attempt to cover the mathematically far more difficult and philosophically more profound General Theory of Einstein. In the area chosen, Bohm has written clearly and felicitously; this should serve a model for others who like to take their physics with a dollop of philosophy.—P. J. M. (shrink)

The volume starts with the paper of Lynn Maurice Ferguson Arnold, former Premier of South Australia and former Minister of Education of Australia, concerning the Exposition Internationale des Arts et Techniques dans la Vie Moderne (International Exposition of Art and Technology in Modern Life) that was held from 25 May to 25 November 1937 in Paris, France. The organization of the world exhibition had placed the Nazi German and the Soviet pavilions directly across from each other. Many papers are devoted (...) to the interpretation of this opposition. Arnold’s paper considers the differences in the two totalitarian states’ architectural and design discourse styles. Although each of them communicated a totalitarian language of purposes, permissions, and boundaries, they essentially differed in the styles of discourse represented by the architecture and design of their respective pavilions. They were opposites of each other and the liberal ideals they contested. The internationalist viewpoint reflecting the multi-ethnic mix of the USSR is contrasted to the ein Volk homogeneity represented in the German pavilion. The Soviet pavilion opted for a utopian future to be arrived at by “benign” leadership, whereas the German pavilion anchored itself in the myth of Teutonic history with the nostalgic pride protected by the swastika-bearing eagle. Jean-Yves Béziau is best known as a logician and founder of Universal Logic. However, in this paper, he poses a new philosophical question: why philosophical discourse does not use images? The paper begins with a general analysis of different types of philosophical discourses. Then, he focuses on why images have been and are still rejected in philosophical discourse. He explains various ways to use images in a fruitful way to develop philosophical thinking and discourse and illustrates his view by providing several examples. He concludes with a promising programmatic declaration about founding a new journal entitled World Journal of Pictorial Philosophy to stimulate the usage of images for developing philosophical thinking. Although complaints about the obscurity of many philosophers’ discourse are widespread, Tatiana Denisova, ex-professor of the Surgut University and a research associate of the University of the Aegean undertakes a positive attitude to this problem. She explains that the reasons for the obscurity of philosophical texts, the subsequent complexity in communicating philosophers’ meanings and their eventual incomplete understanding is not a sign of their inferiority. On the contrary, it is a sign of the fruitfulness of philosophical discourse, which can generate new meanings. Thus, the darkness of philosophical discourse is like the life-giving chaos, and the obscurity that it inevitably contains can be the keeper of implicit meanings and even their generator. Katarzyna Gan-Krzywoszyńska and Piotr Leśniewski, authors of a monograph on Polish philosopher Kazimierz Ajdukiewicz (1890 – 1963), make a comparative analysis of his educational style with that of Paulo Reglus Neves Freire (1921 – 1997). The authors highlight that these scholars have distinctly different backgrounds. The former was an educator, philosopher and a leading advocate of critical pedagogy connected with the dialogical Latin-American tradition. The latter was a philosopher and logician, a notable representative of the Lwów–Warsaw school of logic with significant contributions to semantics, model theory and the philosophy of science. Despite their apparent difference, the authors identify some striking similarities regarding their attitudes towards education; notably, their approaches are essentially dialogical. Gilah Yelin Hirsch is a multidisciplinary artist who works as a painter, writer, curator, educator, and filmmaker. In her experiential paper, he explains how and why she uses four channels of communication in her creative expression: writing, painting, filmmaking, and teaching, as dialogic inquiry. She considers that these channels compose a continuum of discourse styles, each reaching a different facet of kaleidoscopic consciousness. She claims that her choice of medium is prompted by a need to communicate her insights profoundly. Thus, for instance, she created painted allegories to express narratives based on dreams and visions emerging from her subconscious. Art and healing is another Tibetan-rooted type of discourse practised by the author. Creating and observing a healing image produces a positive psychophysiological change in both the artist and the viewer. The discourse here is tripartite between creator, image, and viewer. Filmmaking is another type of discourse she practices between the filmmaker/artist as shaman or healer and the viewer as respondent or participant. These films are meant to be experienced frame by frame, physically and emotionally in both the body and mind. Jocelyn Ireson-Paine is a cartoonist and programmer. His paper offers an original attempt to use category theory in cartooning. Category theory is a branch of mathematics that provides concepts and methods to describe general abstract structures in terms of a labelled directed graph called category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms or transformations or mappings). The author explores different ways to define “style” in art using category theory and a relational view of art. Moreover, he examines how “translation” (transformation) between styles could be defined and reveals the difficulties of how it could be implemented in a computer using special software. Jens Lemanski is a philosopher who has revived research in Arthur Schopenhauer’s legacy by exploring his work on mathematical evidence, logic diagrams, and problems of semantics. In this paper, he advances a new approach to eristic as an art of protecting oneself from the one who deliberately violates norms of discourse ethics to win an argument. The author attributes the origins of this view to Schopenhauer, suggesting a new reading of his work. According to the author, eristic is a prohibitive technique that takes effect when the norms of discourse ethics are transgressed and violated. Thus, eristic is viewed as a discipline of Enlightenment philosophy and a correlate of discourse ethics. Roshdi Rashed is an authority on the history of Arabic mathematical sciences. Proceeding from Gilles-Gaston Granger’s definition of mathematical style, he studies the question of whether a mathematical work can be characterized by a single style or by a multitude of styles. This question is explored within the style of a single work, namely Menelaus’s Sphaerica, and through the study of the development of a single problem over time, namely the isoperimetric problem. He indicates Menelaus’s divergence from the Euclidean style of (plane and stereometric) geometry caused by the drop of the Parallel Postulate, which associated it with hyperbolic geometry. Hence, the author concludes that Menelaus’s Sphaerica incorporates the well-known Euclidean style with a variation of the non-Euclidean one. On the other hand, examining the isoperimetric problem reveals another interesting historical picture. A succession of different styles (cosmological, geometric, infinitesimalistic, style of the calculus of variations, of synthetic geometry) can be observed due to the transformation of the research object over time. Boris Shalyutin, a Russian social philosopher, and Ombudsman for Human Rights in the Kurgan Region, explores the origins of discourse in combination with the birth of society and law. He claims that legal discourse marks the historical emergence of discourse in general. Homo Juridicus generated Homo Sapiens, which have created new spheres of discourse, notably moral discourse and further philosophical, political, and scientific discourse. Petros Stefaneas, a logician, computer scientist and novelist, suggests an alternative to the traditional narratology approach for studying (interactive) social media discourse. He claims that style describes how the parts of a narrative are blended into a whole. The author relies upon Goguen’s and Harrel’s concept of style as a choice of blending principles and transfers to the study of the social media narratives elements from the methodology of studying Web-based collaborative search for mathematical proofs like those implemented within the Polymath project. The last paper belongs to Ghil‘ad Zuckermann, a linguist, language revivalist, and proponent of a model of the emergence of Israeli Hebrew, according to which Hebrew and Yiddish were the primary sources of Modern Hebrew. This paper explores the fascinating and multifaceted Yiddish language and its survival in Israeli. Yiddish is characterized by a unique style that embeds psycho-ostensive expressions throughout its discourse. The author highlights the cross-fertilization between Hebrew and Yiddish as it manifests itself in any aspect of the Israeli language. He claims that Yiddish survives beneath Israeli phonetics, phonology, discourse, syntax, semantics, lexis, and even morphology, although traditional and institutional linguists have been most reluctant to admit it. (shrink)

The purpose of the present paper is to reply to a misleading paper by M. Sachs entitled “Einstein's later view of the Twin Paradox” (TP) (Found. Phys. 15, 977 (1985)). There, by selecting some passages from Einstein's papers, he tried to convince the reader that Einstein changed his mind regarding the asymmetric aging of the twins on different motions. Also Sachs insinuates that he presented several years ago “convincing mathematical arguments” proving that the theory of relativity does not predict (...) asymmetrical aging in the TP. Here we give a definitive treatment to the clocks problem showing that Sachs' “convincing mathematical arguments” are non sequitur. Also, by properly quoting Einstein, we show that his later view of the TP coincides with the one derived from the rigorous theory of time developed in this paper. (shrink)