Depending on what one means by the main connective of logic, the -if..., then... -, several systems of logic result: classic and modal logics, intuitionistic logic or relevance logic. This book presents the underlying ideas, the syntax and the semantics of these logics. Soundness and completeness are shown constructively and in a uniform way. Attention is paid to the interdisciplinary role of logic: its embedding in the foundations of mathematics and its intimate connection with philosophy, in particular the (...) class='Hi'>philosophy of language. Set theory is presented both as a conditio sine qua non for logic and as a interesting exact ontology. The study of infinite sets yields perplexing results. Formalization of informal number theory results in formal number theory; Godel's incompleteness is treated. At appropriate places attention is paid to paradoxes, intuitionism, conditionals, the historical development of logic, to logic programming and automated theorem proving for classical logic.". (shrink)

This ground-breaking collection of essays tackles the philosophical issues at play in cosmology, physics, mathematics, and neuroscience. The book considers topics such as the presence of ontology in cosmological theory and physics. It also weighs the philosophical issues connected with mathematical method, the neuroscience of emotions, and evolutionary anthropology. The contributions look at structuralism in the Platonic philosophy of science and the issue of knowledge and faith. This book is a must read for anyone interested in the (...) philosophy of science and the increasingly blurred borders between the two. (shrink)

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, (...) and I discuss potential future directions of research for each side in the debate over the existence of abstract mathematical objects. (shrink)

This paper explores the question of Leibniz’s contribution to the rise of modern ‘science’. To be sure, it is now generally agreed that the modern category of ‘science’ did not exist in the early modern period. At the same time, this period witnessed a very important stage in the process from which modern science eventually emerged. My discussion will be aimed at uncovering the new enterprise, and the new distinctions which were taking shape in the early modern (...) period under the banner of the old Aristotelian terminology. I will argue that Leibniz begins to theorize a distinction between physics and metaphysics that tracks our distinction between the autonomous enterprise of science in its modern meaning, and the enterprise of philosophy. I will try to show that, for Leibniz, physics proper is the study of natural phenomena in mathematical and mechanical terms without recourse for its explanations to metaphysical notions. This autonomy, however, does not imply for Leibniz that physics can say on its own all that there is to be said about the natural world. Quite the opposite. Leibniz inherits from the Aristotelian tradition the view that physics needs metaphysical roots or a metaphysical grounding. For Leibniz, what is ultimately real is reached by metaphysics, not by physics. This is, in my view, Leibniz’s chief insight: the new mathematical physics is an autonomous enterprise which offers its own kind of explanations but does not exhaust what can (and should) be said about the natural world. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, (...) but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

This textbook is a logic manual which includes an elementary course and an advanced course. It covers more than most introductory logic textbooks, while maintaining a comfortable pace that students can follow. The technical exposition is clear, precise and follows a paced increase in complexity, allowing the reader to get comfortable with previous definitions and procedures before facing more difficult material. The book also presents an interesting overall balance between formal and philosophical discussion, making it suitable for both philosophy (...) and more formal/science oriented students. This textbook is of great use to undergraduate philosophy students, graduate philosophy students, logic teachers, undergraduates and graduates in mathematics, computer science or related fields in which logic is required. (shrink)

This short article pairs the realms of “Mathematics”, “Philosophy”, and “Poetry”, presenting some corners of intersection of this type of scientocreativity. Poetry have long been following mathematical patterns expressed by stern formal restrictions, as the strong metrical structure of ancient Greek heroic epic, or the consistent meter with standardized rhyme scheme and a “volta” of Italian sonnets. Poetry was always connected to Philosophy, and further on, notable mathematicians, like the inventor of quaternions, William Rowan Hamilton, or Ion (...) Barbu, the creator of the Barbilian spaces, have written appreciated poems. We will focus here on an avant-garde movement in literature, art, philosophy, and science, called Paradoxism, founded in Romania in 1980 by a mathematician, philosopher and poet, and on the laboured writing exercises of the Oulipo group, founded in Paris in 1960 by mathematicians and poets, both of them still in act. (shrink)

A complete philosophy of mathematics must address Paul Benacerraf’s dilemma. The requirements of a general semantics for the truth of mathematical theorems that coheres also with the meaning and truth conditions for non-mathematical sentences, according to Benacerraf, should ideally be coupled with an adequate epistemology for the discovery of mathematical knowledge. Standard approaches to the philosophy of mathematics are criticized against their own merits and against the background of Benacerraf’s dilemma, particularly with respect to the (...) problem of understanding the distinction between pure and applied mathematics and the effectiveness of applied mathematics in the natural sciences and engineering. The evaluation of these alternatives provides the basis for articulating a philosophically advantageous Aristotelian inherence concept of mathematical entities. An inherence account solves Benacerraf’s dilemma by interpreting mathematical entities as nominalizations of structural spatiotemporal properties inhering in existent spatiotemporal entities. (shrink)

The contributions to volume 8 of the Internationales Jahrbuch des Deutschen Idealismus/International Yearbook of German Idealism pursue from various perspectives the multifarious relations of German Idealism to the natural sciences and mathematics. The concepts of nature and of the basis for mathematics develop complexly in German philosophy after Kant. At issue are: the foundation of mathematics; the relation of freedom to nature; the significance of philosophy to emerging research in biology, chemistry, and physics, and reconsideration of the thought (...) of Leibniz and Spinoza. Contributors: Gideon Freudenthal, Michael Friedman, Hans-Peter Neumann, Wolfgang Neuser, Konstantin Pollok, Sebastian Rand, Pirmin Stekeler-Weithofer, Thomas Sturm, Lars-Thade Ulrichs, Eric Watkins, John Zammito, Paul Ziche, Rachel Zuckert. (shrink)

We count apples and divide a cake so that each guest gets an equal piece; we weigh galaxies and use Hilbert spaces to make amazingly accurate predictions about spectral lines. It would seem that we have no difficulty in applying mathematics to the world; yet the role of mathematics in its various applications is surprisingly elusive. Eugene Wigner has gone so far as to say that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious (...) and that there is no rational explanation for it” (1960, p. 223). The issue is not much discussed under the heading “applied mathematics,” yet it is pivotal to several philosophical debates. In recent years three rather general questions have been central: (1) Just how does mathematics “hook onto” the world? This is the main concern of a rather technical branch of philosophy of science known as measurement theory (see measurement). (2) Are some of the objects referred to in various theories merely mathematical objects, or do they have some other status? This problem often comes up in the philosophy of the special sciences. For example, do space‐time and the quantum state exist in their own right, separate from their mathematical representations; or are they nothing but mathematical entities? (3) Is mathematics essential for science? Following Hartry Field's work, this has become a focal point in the debate between realists and nominalists in the philosophy of mathematics. (shrink)

Attempts to lay a foundation for the sciences based on modern mathematics are questioned. In particular, it is not clear that computer science should be based on set-theoretic mathematics. Set-theoretic mathematics has difficulties with its own foundations, making it reasonable to explore alternative foundations for the sciences. The role of computation within an alternative framework may prove to be of great potential in establishing a direction for the new field of computer science.Whitehead''s theory of reality is re-examined as (...) a foundation for the sciences. His theory does not simply attempt to add formal rigor to the sciences, but instead relies on the methods of the biological and social sciences to construct his world-view. Whitehead''s theory is a rich source of notions that are intended to explain every element of experience. It is a product of Whitehead''s earlier attempt to provide a mathematical foundation for the physical sciences and is still consistent with modern physics. (shrink)

The author presents Proclus’ philosophy of mathematics, and its relation to the other sciences. The first half of the chapter collects the main questions and problems concerning the philosophy of mathematics as raised by Plato and Aristotle and taken up again by Proclus; the sources Proclus took recourse to in his addressing these issues; and the texts composed by Proclus in which these issues are primarily addressed. The second part, in which Proclus’ position is presented, first discusses the (...) nature of mathematical objects, and their relation to soul; the distinction and relation between the mathematical sciences ; the mathematical methods and their relation to dialectic; and the relation between mathematics and metaphysics and physics. Finally, the author briefly addresses Proclus’ relation to the mathematics of his day. (shrink)

Recent advances in neuroscience lead to a wider realm for philosophy to include the science of the Darwinian-evolved computational brain, our inner world producing organ, a non-recursive super- Turing machine combining 100B synapsing-neuron DNA-computers based on the genetic code. The whole system is a logos machine offering a world map for global context, essential for our intentional grasp of opportunities. We start from the observable contrast between the chaotic universe vs. our orderly inner world, the noumenal cosmos. So (...) far, philosophy has been rehearsing our thoughts, our human-internal world, a grand painting of the outer world, how we comprehend subjectively our experience, worked up by the logos machine, but now we seek a wider horizon, how humans understand the world thanks to Darwinian evolution to adapt in response to the metaphysical gap, the chasm between the human animal and its environment, shaping the organism so it can deal with its variable world. This new horizon embraces global context coded in neural structures that support the noumenal cosmos, our inner mental world, for us as denizens of the outer environment. Kant’s inner and outer senses are fundamental ingredients of scientific philosophy. Several sections devoted to Heidegger, his lizard debunked, but his version of the metaphysical gap & his doctrine of the logos praised. Rorty and others of the behaviorist school discussed also. (shrink)

Mathematics plays a central role in modern science. However, it is very common an instrumental and utilitarian view of math leading to the underestimation of its scientific nature. We propose some consideration to emphasize a different idea on the fundamental role of math in modern science.

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

In the sense in which astronomy or botany are sciences, philosophy is not a science. Philosophers have theories, but their theories do not enable them to make predictions; they can not be empirically confirmed or refuted in the way that scientific theories can. But, it will be objected, this is not true of all the sciences. Palaeontologists do not make predictions: in pure mathematics there is no appeal to experience. But even if they are not predictive the propositions (...) which figure in the historical sciences are at any rate empirically testable: and even if the propositions of pure mathematics are not confutable by observation, they are subject to recognized methods of proof. There are standard procedures for deciding whether they are true or false. But where in philosophy are such procedures to be found? If anywhere, in formal logic which has come very close to mathematics. Nowadays, indeed, it is hardly possible to draw a line between them. But by the very process of becoming a science, formal logic detaches itself from philosophy. Philosophers do indeed make use of formal logic. They employ deductive arguments; sometimes they are able to take advantage of the economy and precision of logical symbolism. But the premises on which they reason, the propositions which the use of logical symbolism helps them to state more clearly, are not themselves drawn from formal logic. The truths which can be established by formal logic alone are not philosophical. (shrink)

Les auteurs de cet ouvrage ont cherché à approfondir les fondements de la rationalité en philosophie des sciences pour montrer ce qu'une pensée rationnelle peut apporter à la démarche scientifique, pour mieux analyser un raisonnement ou les résultats d'une expérience. Les deux grands épistémologues et philosophes des sciences, Gaston Bachelard et Alexandre Koyré, les ont beaucoup inspirés dans leurs travaux, notamment sur la question du déterminisme. La pensée logique héritée d'Aristote, réduite au modus ponens, ne suffit pas aujourd'hui pour comprendre (...) l'avancée séculaire et fulgurante des connaissances en sciences. Les auteurs présentent à cet effet une étude des différentes méthodes de raisonnements. L'histoire des mathématiques, du point de vue de son épistémologie, fait l'objet d'un chapitre riche en citations montrant l'évolution de la pensée logique, d'Euclide à la théorie des ensembles, outil nécessaire à la progression des connaissances scientifiques. La mise en oeuvre de mesures expérimentales tenant compte des erreurs de mesure, suppose l'exploitation d'outils mathématiques et statistiques, illustrés par des exemples. Par leur approche zététique et épistémologique des connaissances, les auteurs ont souhaité que la pensée rationnelle ne s'efface pas sous le déferlement médiatique de certaines dérives et perversions de l'esprit scientifique propres aux altersciences, comme le scientisme, le positivisme ou le relativisme. (shrink)

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist (...) perspective, the “unreasonable effectiveness of mathematics in philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate. (shrink)

Many centuries ago, at the very beginning of the systematic development of philosophy, Plato declared that the thinker's domain comprises “the wholeness of things;” and indeed, the earlier thinkers took all knowledge for their province and did not hesitate to discuss problems now referred to art, psychology, economics, mathematics, or physics. Since then the meaning of philosophy has appreciably changed, however, and the intellectual descendants of the great founder of the Academy no longer claim the monopoly of all (...) fields of study. For there appeared in the meantime a mighty competitor—or should we say a partner?—in the pursuit of truth, namely, science. (shrink)

Proclus, head of the Philosophy School at Athens for fifty years, was one of the leading philosophical figures in Late Antiquity. Lucas Siorvanes here introduces Proclus to English-language readers, discussing his metaphysics and theory of knowledge and focusing in particular on his Neo-Platonism. Proclus lived in the turbulent fifth century A.D., a time of struggles among Christians, Jews, and pagans, the invasion of Attila the Hun, the fall of the Western Roman Empire, and the rise of the Eastern Roman (...) Empire in Byzantium. While Late Antiquity has been regarded as a time of superstition and forbiddingly complex philosophies, recent scholarship has shown it to be full of cultural and intellectual vigor. During Proclus's tenure as head of the Philosophy School, he systematized Neo-Platonism as the summit of ancient Greek thought, brought it to its peak of influence, and became responsible for the form in which it was transmitted to the Byzantine, Western European, and Islamic civilizations. Siorvanes's extensive original research presents Proclus as much more than just a metaphysician. He surveys all of Proclus's philosophical interests—including religion, physics, astronomy, mathematics, and poetry—revealing the philosopher's central concern with the problems of being and knowledge and relating the ideas of Proclus to those of such other major thinkers as Plato, Aristotle, and Ptolemy. To help the newcomer, Siorvanes supplies more than 200 quotations from Proclus's works. He also traces the impact of Proclus's Neo-Platonism across cultures, religions, and centuries, in such diverse areas as Christian and Islamic theologies, Renaissance art, Kepler's astronomy, Romantic poetry, Emerson's thought, modern philosophy and science, and current popular phrases. (shrink)

This chapter chronicles the complex relationship between philosophy and science throughout history. It illustrates how they have mutually influenced each other in modern times. Philosophy and science are thought to be polar opposites, but they are not as different as they seem to be. Philosophy is considered part of the humanities and not the sciences. However, it can be argued that schools of science branched off from the domain of philosophy. Scientific studies start (...) as or are inspired by philosophical ideas. Mathematics, the very paradigm of a logical and rational discipline that much of science embodies, is a lot like philosophy in that it involves ideas that have no empirical basis and uses logic. (shrink)

The first task of the philosophy of nature -- The problem of elementarity -- The philosophical myth of creation : the Platonic philosophy of nature -- Aristotle's Physics -- Aristotle's method of cosmological speculation -- Descartes' mechanism -- Isaac Newton and the mathematical principles of natural philosophy -- The world of Leibniz : the best of all possible worlds -- Immanuel Kant : the a priori conditions of the sciences -- The romantic philosophy of nature (...) -- The cosmology of Whitehead: the universe as process -- Popper's open universe -- Science as philosophy -- Problems and methods of the philosophy of nature. (shrink)

Explicit concepts and sufficiently precise definitions are the basis for further advance of a science beyond a given level. To move toward a situation where the whole population has access to the authentic results of science (italics mine) requires making explicit some general philosophical principles which can help to guide the learning, development, and use of mathematics, a science which clearly plays a pivotal role regarding the learning, development and use of all the sciences. Such philosophical principles (...) have not come from speculation but from studying and concentrating the development of actual mathematical subjects such as algebraic geometry, functional analysis, continuum mechanics, combinatorics, etc. The simplifications in the conceptual relations revealed by such philosophical/mathematical advances are of great value, not only in: (1) clarifying and guiding further research in the mathematical sciences, but also in (2) reforming pedagogy and popularization in ways that will not prove deceptive. (shrink)

a Mathematicians, like Proust and everyone else, are at their best when writing about their first lovea (TM) a ] They are among the very best we have; and their best is very good indeed. a ] One approaches this book with high hopes. Happily, one is not disappointed. a ]In paperback it might well have become a best seller. a ]read it. From The Mathematical Intelligencer Mathematics is shaped by the consistent concerns and styles of powerful minds a (...) three of which are represented here. Kaca (TM)s work is marked by deep commitment and breadth of inquiry. Rota is the easiest of these authors to reada ]a delight: witty and urbane, with a clear and interesting agenda and an astonishing intellectual range. To read him is to be a part of a pleasant and rewarding conversation. Jacob T. Schwartz attacks problems ina ]computer science, mathematical economics, computer and instructional a ] No matter how slippery the problem, Schwartz manages to capture a substantial chunk of it in his mathematical net. a Mathematics Magazine This is a volume of essays and reviews that delightfully explore mathematics in all its moods a from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, Heidegger among others. (shrink)

This book reports on cutting-edge concepts related to Bourbaki’s notion of structures mères. It merges perspectives from logic, philosophy, linguistics and cognitive science, suggesting how they can be combined with Bourbaki’s mathematical structuralism in order to solve foundational, ontological and epistemological problems using a novel category-theoretic approach. By offering a comprehensive account of Bourbaki’s structuralism and answers to several important questions that have arisen in connection with it, the book provides readers with a unique source of information (...) and inspiration for future research on this topic. (shrink)

In this paper, I argue that in spite of suggestions to the contrary, Merleau-Ponty defends a positive account of the kind of abstract thought involved in mathematics and natural science. More specifically, drawing on both the Phenomenology of Perception and his later writings, I show that, for Merleau-Ponty, abstract thought and perception stand in the two-way relation of “foundation,” according to which abstract thought makes what we perceive explicit and determinate, and what we perceive is made to appear by (...) abstract thought. I claim that, on Merleau-Ponty's view, although this process can sometimes lead to falsification, it can also be carried out in such a manner that allows mathematics and natural science to articulate what we perceive in a way that is non-distortive and in keeping with the demands of perception itself. (shrink)

Mathematics is often taken to play one of two roles in the empirical sciences: either it represents empirical phenomena or it explains these phenomena by imposing constraints on them. This article identifies a third and distinct role that has not been fully appreciated in the literature on applicability of mathematics and may be pervasive in scientific practice. I call this the “bridging” role of mathematics, according to which mathematics acts as a connecting scheme in our explanatory reasoning about why and (...) how two different descriptions of an empirical phenomenon relate to each other. I discuss two bridging roles appearing in biological and physical explanations. (shrink)

Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous.

I construe mathematical philosophy not in the narrow sense of philosophy of mathematics but in a broad indefinite sense of different manners of giving mathematics a privileged place in the study of philosophy. For example, in one way or another, mathematics plays an important part in the philosophy of Plato, Descartes, Spinoza, Leibniz, and Kant. In contrast, history plays a central role in the philosophy of Vico, Hegel, and Marx. In more recent times, Frege, (...) Husserl, Russell, Ramsey, and Gödel all began as mathematicians. One way of viewing Kant’s system of philosophy might be to stress that he was struck by the synthetic a priori character of mathematical propositions. He went on to offer a remarkable account of this fact and also look for and propose synthetic a priori foundations of physics, morality, and esthetics. (shrink)

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics, together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

What was the basis for the adoption of mathematics as the primary mode of discourse for describing natural events by a large segment of the philosophical community in the seventeenth century? In answering this question, this book demonstrates that a significant group of philosophers shared the belief that there is no necessary correspondence between external reality and objects of human understanding, which they held to include the objects of mathematical and linguistic discourse. The result is a scholarly reliable, but (...) accessible, account of the role of mathematics in the works of Galileo, Kepler, Descartes, Newton, Leibniz, and Berkeley. This impressive volume will benefit scholars interested in the history of philosophy, mathematical philosophy and the history of mathematics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Proclus: Neo-Platonic Philosophy and Science by Lucas SiorvanesP.A. MeijerLucas Siorvanes. Proclus: Neo-Platonic Philosophy and Science. New Haven: Yale University Press, 1996. Pp. xv+ 340. Cloth, $35.00.This book will be welcomed by scholars of Proclus and by readers unfamiliar with Proclus alike. There are not many introductory books on Proclus. And Siorvanes presents in an interesting way the latest developments in scholarship. [End Page 160]Siorvanes (...) gives an account of Proclus’s life and times, his position in the Athenian Neoplatonic school, and his influence and offers a panoramic view of Proclus’s philosophy in the chapters: “General Metaphysics,” “Knowledge and the levels of Being,” “Physics and Metaphysics,” and “Stars and Planets.” He presents not only Proclus’s views on physics and astronomy, but also his poetics (189).Siorvanes deals competently with Proclus’s metaphysics, analyzing it in its own right and as the indispensable background of his scientific views. His summaries are on the whole adequate and instructive. But a scheme or outline of levels of the multi-layered Proclean system of Gods (Henads), such as that offered by H. D. Saffrey and L. G. Westerink in Proclus, Théologie Platonicienne (Paris 1968) for example, would have facilitated understanding.In an impressive description of Proclus’s historical influence, Siorvanes establishes connections between Proclus’s ideas and modern views in metaphysics, mathematics, physics, logic and astronomy. Yet a warning would be appropriate in this context. There is, for instance, the strikingly modern Proclean theory that planets have satellites. Perhaps Siorvanes is too much enamoured with these modern Proclean theories. Though he recognizes that the theory about the satellites is really a consequence of Proclus’s metaphysics (268–271), he fails to emphasize that Proclus’s modern-seeming theories are no less speculative than other ancient theories. Neither Proclus’s metaphysics nor his mathematics can replace evidence as used in modern science. Thus, the rather triumphal tones Sioravanes sounds in this regard are somewhat out of place. The incidental similarities in theoretical structures and patterns that emerge in the course of history are interesting to note, but we should beware of becoming too enthusiastic about their significance. The so-called indirect influence may be due precisely to the structure of problems and the (limited) range of possible solutions.I would raise some objections to Siorvanes’s manner of translating and interpreting the Greek word for “to be” (einai) or “Being” (on). As is often done, especially in the Anglo-Saxon world, Siorvanes alternatively uses “being,” “existence” and “reality,” without being aware of the fact that in ancient philosophy the Greek words “on/ousia” (“Being”) generally do not admit of the translation “existence.” Such translations are anachronistic and wrong. “Reality” is entirely inappropriate as a translation. As an essentially medieval expression, the word “reality” (realitas) as such is nowhere to be found in antiquity. When Proclus presents his multilevel system of Being, there are no degrees of reality in the sense that one transcendent level is more real than another. There are only degrees of Being. The problematic nature of the issue stands out even more when one speaks about not being. “Being” interpreted as “reality” suggests that what is not being, has no reality. But that is not what Proclus meant to say. The translation “existence” is bound to bring about similar misunderstandings. The notion “existence” is seldom in the focus of the interest when an ancient philosopher uses “einai.” Thus, an innocent reader of Siorvanes’s book may despair when he encounters “real existence” as a translation for “ousia” (124). Do the levels of being other than the one called “ousia” exist or do they not exist? If they do not, Proclus’s entire philosophy would be incoherent, and our world would not even exist or possess “reality.” But this is not what Proclus or Siorvanes mean. To avoid misunderstandings, we should use only “Being” (and “to be”) as translations. [End Page 161]There are also problems with some aspects of Siorvanes’s description of participation. In picturing the connection of the entity that is participated (methekton), what is given to the participant (metechomenon) and the participant (metechon), the terms “predication” and... (shrink)

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, (...) in the Aufbau, contended that this endeavor can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

Most disciplines make use of thought experiments, but physics and philosophy lead the pack with heavy dependence upon them. Often this is for conceptual clarification, but occasionally they provide real theoretical advances. In spite of their importance, however, thought experirnents have received rather little attention as a topic in their own right until recently. The situation has improved in the past few years, but a mere generation ago the entire published literature on thought experiments could have been mastered in (...) a long weekend. Now the subject is beginning to flourish. Given the relative newness of the field, it might be useful to have several examples at one’s finger tips, so a number of great ones will be described. Attention will also be drawn outside physics and philosophy. In mathematics there is something analogous to thought experiments -- visual reasoning and picture proofs. I will look briefly at this class of thought experiments and try using them to make a case for possibly settling the continuum hypothesis. After this, I will return to thought experiments in the sciences and propose an account of how they work. Finally, I will end with a sketch of a topic I am currently working on, a kind of progress report which, I hope, will be an inducement to others. (shrink)

We have undertaken to discuss the nature of mathematical systems, the way in which they are discovered, and the uses to which they are put in the empirical sciences. The empirical sciences employ mathematical systems in framing final formulations, but adopt mathematical techniques long before reaching that stage. It is a prerequisite that the mathematics they employ has been developed separately and within its own domain. We shall return to the relation between mathematics and the empirical sciences (...) before we are done, but in the meanwhile we shall be constrained to begin by discussing the situation in mathematics in isolation from any empirical considerations. (shrink)

The work addresses the problem of the relationship between science and philosophy in the work of Hermann Weyl. The author begins by discussing Weylls Gottingen tradition. Contrary to standard accounts of this tradition, Edmund Husserl and Georg Cantor are included. The influence of this tradition on Weyl is then illustrated by an examination of Weyl's early philosophy of mathematics. Here Weyl attempts to use Husserl's early phenomenology to amalgamate the thought of Felix Klein, David Hilbert and Cantor. (...) Weyl's "phenomenological period," in which he wrote The Continuum and Space-Time-Matter, is then discussed in detail. The work reveals that for Weyl, philosophy and science, guided by the same telos to construct an objective picture of the world, are involved in a dialectic in which each serves as a corrective and limitation of the other. (shrink)

During his stay in Padua ca. 1592–1610, Galileo Galilei (1564–1642) was a lecturer of mathematics at the University of Padua and a tutor to private students of military architecture and fortifications. He carried out these activities at the Academia degli Artisti. At the same time, and in relation to his teaching activities, he began to study the equilibrium of bodies and strength of materials, later better structured and completed in his Dialogues Concerning Two New Sciences of 1638. This paper examines (...) important details of four works dating to the Paduan period: Breve instruzione dell’architettura militare; Trattato di Fortificazione; Le Mecaniche; Le operazioni del compasso geometrico et militare. The two works on military architecture and fortifications were compiled from notes taken by students, and are not by Galileo’s hand, but are still illustrative of his work and thinking at the time. (shrink)

his paper explores the relationship between Kant’s views on the metaphysical foundations of Newtonian mathematical physics and his more general transcendental philosophy articulated in the Critique of pure reason. I argue that the relationship between the two positions is very close indeed and, in particular, that taking this relationship seriously can shed new light on the structure of the transcendental deduction of the categories as expounded in the second edition of the Critique.Author Keywords: Kant; Mathematical physics; Transcendental (...) deduction. (shrink)