One of the chief concerns of the young Descartes was with what he, and others, termed “physico-mathematics”. This signalled a questioning of the Scholastic Aristotelian view of the mixed mathematical sciences as subordinate to natural philosophy, non explanatory, and merely instrumental. Somehow, the mixed mathematical disciplines were now to become intimately related to natural philosophical issues of matter and cause. That is, they were to become more ’physicalised’, more closely intertwined with natural philosophising, regardless of which species of (...) natural philosophy one advocated. A curious, short-lived yet portentous epistemological conceit lay at the core of Descartes’ physico-mathematics—the belief that solid geometrical results in the mixed mathematical sciences literally offered windows into the realm of natural philosophical causation—that in such cases one could literally “see the causes”. Optics took pride of place within Descartes’ physico-mathematics project, because he believed it offered unique possibilities for the successful vision of causes. This paper traces Descartes’ early physico-mathematical program in optics, its origins, pitfalls and its successes, which were crucial in providing Descartes resources for his later work in systematic natural philosophy. It explores how Descartes exploited his discovery of the law of refraction of light—an achievement well within the bounds of traditional mixed mathematical optics—in order to derive—in the manner of physico-mathematics—causal knowledge about light, and indeed insight about the principles of a “dynamics” that would provide the laws of corpuscular motion and tendency to motion in his natural philosophical system. (shrink)

This article deals with physico-mathematical approaches to anatomy in post-Harveyan physiology. But rather than looking at questions of iatromechanics and animal locomotion, which often attracted this approach, I look at the problem of how blood returned to the heart – a part of the circulation today known as venous return but poorly researched in the early modern period. I follow the venous return mechanisms proposed by lesser-known authors in the mechanization of anatomy, such as Jean Pecquet (1622–1674) and Nicolaus (...) Steno (1638–1686), alongside the more famous Giovanni Alfonso Borelli (1608–1679). Their mechanisms differed only in small details. Yet, these minor differences highlight significant aspects of the mechanization of the life sciences in the seventeenth century. First, they relied more on observations than hitherto acknowledged, even if only indirectly. Second, their mechanisms drew more from the physico-mathematical disciplines than from the trending corpuscularian philosophies of their time. Finally, these mechanisms led to a more accurate understanding of the circulation that remains valid today, thus revealing their cognitive benefits. In short, through the single problem of how blood returned to the heart, this article portrays the increasing complexity of anatomy in the early modern period. (shrink)

This book reconstructs key aspects of the early career of Descartes from 1618 to 1633; that is, up through the point of his composing his first system of natural philosophy, Le Monde, in 1629-33. It focuses upon the overlapping and intertwined development of Descartes’ projects in physico-mathematics, analytical mathematics, universal method, and, finally, systematic corpuscular-mechanical natural philosophy. The concern is not simply with the conceptual and technical aspects of these projects; but, with Descartes’ agendas within them and (...) his construction and presentation of his intellectual identity in relation to them. Descartes’ technical projects, agendas and senses of identity shifted over time, entangled and displayed great successes and deep failures, as he morphed from a mathematically competent, Jesuit trained graduate in neo-Scholastic Aristotelianism to aspiring prophet of a systematised corpuscular-mechanism, passing through stages of being a committed physico-mathematicus, advocate of a putative ‘universal mathematics’, and projector of a grand methodological dream. In all three dimensions—projects, agendas and identity concerns—the young Descartes struggled and contended, with himself and with real or virtual peers and competitors, hence the title ‘Descartes-Agonistes’. (shrink)

Following the general program, outlined in a previous paper on the Foundations of Mathematical Biophysics, we shall outline some physico-mathematical aspects of the Gestalt-problem. Faithful to the principles laid down before, we shall not attempt to give a theory of such and such psycho-physiological phenomena connected with our problem, but we shall investigate in a preliminary way the abstract conceptual basis, which must underlie the construction of any concrete theory. To make our aim quite clear and to avoid at (...) the same time unnecessary repetition, we refer the reader to our previous paper. Furthermore we shall give here only an outline of the physico-mathematical aspects; and of necessity we must leave a number of details untouched and unanswered. (shrink)

In recent years, the ideas of the mathematician Bernhard Riemann have come to the fore as one of Deleuze's principal sources of inspiration in regard to his engagements with mathematics, and the history of mathematics. Nevertheless, some relevant aspects and implications of Deleuze's philosophical reception and appropriation of Riemann's thought remain unexplored. In the first part of the paper I will begin by reconsidering the first explicit mention of Riemann in Deleuze's work, namely, in the second chapter of (...) Bergsonism. In this context, as I intend to show first, Deleuze's synthesis of some key features of the Riemannian theory of multiplicities is entirely dependent, both textually and conceptually, on his reading of another prominent figure in the history of mathematics: Hermann Weyl. This aspect has been largely underestimated, if not entirely neglected. However, as I attempt to bring out in the second part of the paper, reframing the understanding of Deleuze's philosophical engagement with Riemann's mathematics through the Riemann–Weyl conjunction can allow us to disclose some unexplored aspects of Deleuze's further elaboration of his theory of multiplicities and profound confrontation with contemporary science. This finally permits delineation of a correlation between Deleuze's plane of immanence and the contemporary physico-mathematical space of fundamental interactions. (shrink)

Book review of John Schuster: Descartes-agonistes: Physico-mathematics, method and corpuscular-mechanism 1618-33. (Studies in History and Philosophy of Science, Volume 27.) Dordrecht: Springer, 2013, xix + 631pp. Descartes-Agonistes is the magnum opus of John Schuster, formerly of the University of New South Wales, honorary fellow at the University of Sydney. Its roots go back to the dissertation he wrote 35 years ago under Thomas Kuhn at Princeton University. As Schuster correctly remarks, some regard his dissertation as an underground classic. (...) I count myself among them: Schuster’s work has been directional in my work on the history of early modern science. Schuster himself prefers to regard his dissertation as ‘a vestige’ of a history of science of the 1970s that he has since gone beyond. In the 1980s, he elaborated and published on some central insights in a series of seminal articles, but his work on Descartes really got into its stride again around the turn of the millennium (Schuster 1980, 1990; Gaukroger et al. 2000). This book is the acme of this work, and grand it is: clocking 600 pages (excluding appendices) it extends, deepens and reinforces his analysis of Descartes into a dense and .. (shrink)

"Reproduced from an original in the Libraries of Stanford University, Stanford, Calif.

One may discuss the role played by mechanical science in the history of scientific ideas, particularly in physics, focusing on the significance of the relationship between physics and mathematics in describing mathematical laws in the context of a scientific theory. In the second Newtonian law of motion, space and time are crucial physical magnitudes in mechanics, but they are also mathematical magnitudes as involved in derivative operations. Above all, if we fail to acknowledge their mathematical meaning, we fail to (...) comprehend the whole Newtonian mechanical apparatus. For instance, let us think about velocity and acceleration. In this case, the approach to conceive and define foundational mechanical objects and their mathematical interpretations changes. Generally speaking, one could prioritize mathematical solutions for Lagrange’s equations, rather than the crucial role played by collisions and geometric motion in Lazare Carnot’s operative mechanics, or Faraday’s experimental science with respect to Ampère’s mechanical approach in the electric current domain, or physico-mathematical choices in Maxwell’s electromagnetic theory. In this paper, we will focus on the historical emergence of mechanical science from a physico-mathematical standpoint and emphasize significant similarities and/or differences in mathematical approaches by some key authors of the 18th century. Attention is paid to the role of mathematical interpretation for physical objects. (shrink)

ArgumentThis paper investigates the conceptual treatment and mathematical modeling of force in Newton's Principia. It argues that, contrary to currently dominant views, Newton's concept of force is best understood as a physico-mathematical construct with theoretical underpinnings rather than a “mathematical construct” or an ontologically “neutral” concept. It uses various philosophical and historical frameworks to clarify interdisciplinary issues in the history of science and draws upon the distinction between axiomatic systems in mathematics and physics, as well as discovery patterns (...) in science. It also dwells on Newton's “philosophy” of mathematics, described here in terms of mathematical naturalism. This philosophy considers mathematical quantities to be physically significant quantities whose motions are best mapped by geometry. It then shows that to understand the epistemic status of force in the Principia, it is important to scrutinize both Newton's mathematical justificatory strategies and his background assumptions about force – without constructing, however, an overarching metaphysical framework for his science. Finally, the paper studies scientific attempts to redefine or eliminate force from science during the period between Newton and Laplace. From a philosophical standpoint, the paper implicitly suggests that questions about the reality of force be distinguished from questions about the validity of force, and that both sets of questions be distinguished from questions about the utility of the concept of force in science. (shrink)

It remains to summarize the contributions which each of the three disciplines discussed here is making toward the development of a science of man. "Significs" makes a study of the effects on human behavior of the linguistic aspects of the evaluative process, the most distinctly human aspect of the behavior of the human organism. "Mathematical Biophysics" seeks to describe the events associated with evaluative processes in physico-mathematical terms. "Cybernetics" is discovering important invariants common to these processes and others, particularly (...) those observed in man-made machines and in situations which lend themselves to description in thermo-dynamic or statistical (order -- chaos) terms. (shrink)

It remains to summarize the contributions which each of the three disciplines discussed here is making toward the development of a science of man. "Significs" makes a study of the effects on human behavior of the linguistic aspects of the evaluative process, the most distinctly human aspect of the behavior of the human organism. "Mathematical Biophysics" seeks to describe the events associated with evaluative processes in physico-mathematical terms. "Cybernetics" is discovering important invariants common to these processes and others, particularly (...) those observed in man-made machines and in situations which lend themselves to description in thermo-dynamic or statistical terms. (shrink)

My title, of course, is an exaggeration. The world no more became mathematical in the seventeenth century than it became ironic in the nineteenth. Either it was mathematical all along, and seventeenth-century philosophers discovered it was, or, if it wasn’t, it could not have been made so by a few books. What became mathematical was physics, and whether that has any bearing on the furniture of the universe is one topic of this paper. Garber says, and I agree, that for (...) Descartes bodies are the things of geometry made real ( Ref). That is a claim about the world: what God created, and what we know in physics, is nothing other than res extensa and its modes. Others, including Marion, hold that in modern science, here represented at its origins by Descartes, representation displaces beings: the knower no longer confronts Being or beings but rather a system of signs, a “code” as Marion calls it, to which the knower stands in the relation of subject to object. The Meditations, or perhaps even the Regulæ, are the first step toward the transcendental idealism of Kant. Most of this paper will be devoted to a more concrete question. Physics in the seventeenth century increasingly became a matter of applying mathematical knowledge to the solution of physical problems. The “mixed sciences” of astronomy, optics, and music, sciences then distinct from physics, became models of understanding for all of natural philosophy. My interest here is in one aspect of that development: how particular physical situations are transformed into mathematical. I will look at the work of Descartes and Isaac Beeckman, contrasting their visions of “physico-mathematics”. (shrink)

In our previous writings we have outlined a mathematical theory of biological phenomena. In our systematic construction of “mathematical biology,” similar in its aims to mathematical physics, we have started with the fundamental unit,—the living cell. After having established a physico-mathematical theory of the fundamental properties of the cell, we have studied the interaction of several cells. This led us into two different fields. On the one hand we studied such interactions of cells, which determine the form of cellular (...) aggregates, constituting multicellular organisms. On the other hand we studied different types of functional interactions, which determine the reactions of the aggregate as a whole to different environmental changes. While the first field leads us eventually to the theory of organic form, the second brings us to the physico-mathematical theory of behavior. (shrink)

Since the end of the last century, there has been several ambitious attempts to naturalize Husserlian phenomenology by way of mathematization. To justify themselves in view of Husserl’s adamant antinaturalism, many of these attempts appeal to the new physico-mathematical tools that were unknown in Husserl’s time and thus allegedly make his position outdated. This paper critically addresses these mathematization proposals and aims to show that Husserl had, in fact, sufficiently good arguments that make his antinaturalistic position sound even today. (...) The starting point of the discussion presented in this paper is the mathematization project introduced by Jean-Michel Roy, Jean Petitot, Bernard Pachoud, and Francisco Varela in their introduction to the book Naturalizing Phenomenology. This proposal was followed by a number of critiques but also by several alternative naturalization attempts clearly inspired by Roy et al.’s ambitious project. The review of some of Husserl’s important arguments often overlooked or misinterpreted by both the naturalization advocates and their critics leads the author of the paper to the twofold conclusion which, on the one hand, explores the deeper reasons for the impossibility of a physical and mathematical treatment of phenomenology, on the other hand, clarifies the sense in which such treatments are possible, namely by way of restriction of the variety of experiential aspects that undergo naturalization and substitution of the aspects amenable to the direct mathematization for the directly unmathematizable ones. In the fourth section of this paper, the author attempts to demonstrate that, contrary to widespread belief, Husserl’s arguments are not obsolete by the standards of the contemporary physico-mathematical approaches employed in the mathematization of phenomenology and indeed stand the test of time. (shrink)

This chapter argues that the standard conception of Spinoza as a fellow-travelling mechanical philosopher and proto-scientific naturalist is misleading. It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise (TTP) points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly sceptical (as specified below). Third, in the seventeenth-century debates over proper methods in the (...) sciences, Spinoza sided with those that criticized the aspirations of those (the physico-mathematicians, Galileo, Huygens, Wallis, Wren, etc) who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piece-meal methodology (see section 2). Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion (see section 3). (shrink)

Nach einer allgemeinen Diskussion des Zusammenhanges zwischen theoretischer und experimenteller Forschung, wird in Hinblick auf die vom Verfasser entwickelten physikalisch-mathematischen Grundlagen der Biologie, eine Reihe von Einzelproblemen betrachtet. Es wird an Hand von Kurvenmaterial gezeigt wie weit die mathematisch vorausgesagten Beziehungen mit den experimentellen Befunden übereinstimmen. Folgende Fragen werden besprochen: Zellatmung, Zellgrössen, deren Abhängigkeit von Stoffwechsel, Zellteilung, Protoplasmaströmungen, Nervenerregung, psychophysische Gesetze, Reaktion auf geometrische Gestalten.Après une mise au point générale de la relation entre les sciences théoriques et expérimentales, diverses questions (...) sont discutées, du point de vue des fondations physico-mathématiques de la biologie, développées récemment par l'auteur. On montre par la comparaison des courbes calculées à celles observées, comment les relations prédites mathématiquement se trouvent vérifiées. Les questions suivantes sont discutées: respiration cellulaire, dimensions cellulaires et leur rapport au métabolisme, division cellulaire, mouvements protoplasmiques, excitation nerveuse, les lois psychophysiques, perception des formes géométriques. (shrink)

In a long question, "Whether there is motion to quality," which became part of his Oxford Expositio omnium librorum Physicorum cum questionibus optime disputatis, composed before 1310, Walter Burley supported the succession-of-forms theory of qualitative change. After commenting on Peter Lombard's Sentences at Paris, Burley took part in disputations on controversial questions in the early 1320s, resulting in his De primo et ultimo instanti and his Tractatus Primus and Tractatus Secundus de intensione et remissione formarum. In these independent controversial works, (...) Burley advanced and defended the succession-of-forms theory with greater detail and precision. This paper examines the treatment of the mathematics of alteration in Burley's final Parisian Expositio in libros octo de physico auditu in light of this earlier controversial work. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, (...) there was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

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

The dependence on history of both present and future dynamics of life is a common intuition in biology and in humanities. Historicity will be understood in terms of changes of the space of possibilities as well as by the role of diversity in life’s structural stability and of rare events in history formation. We hint to a rigorous analysis of “path dependence” in terms of invariants and invariance preserving transformations, as it may be found also in physics, while departing from (...) the physico-mathematical analyses. The idea is that the invariant traces of the past under organismal or ecosystemic transformations contribute to the understanding of present and future states of affairs. This yields a peculiar form of unpredictability in biology, at the core of novelty formation: the changes of observables and pertinent parameters may depend also on past events. In particular, in relation to the properties of synchronic measurement in physics, the relevance of diachronic measurement in biology is highlighted. This analysis may a fortiori apply to cognitive and historical human dynamics, while allowing to investigate some general properties of historicity in biology. (shrink)

If the allusive stratagems can claim to define a new type of systematicity, it is because they give access to a space where the singularity, the diagram and the metaphor may interlace, to penetrate further into the physico-mathematic intuition and the discipline of the gestures which precede and accompany ‘formalisation’. This interlacing is an operation where each component backs up the others: without the diagram, the metaphor would only be a short-lived fulguration because it would be unable to operate: (...) without the metaphor, the diagram would only be a frozen icon, unable to jump over its bold features which represent the images of an already acquired knowledge; without the subversion of the functional by the singular, nothing would come to oppose the force of habit. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

I present an account of deterministic chance which builds upon the physico-mathematical approach to theorizing about deterministic chance known as 'the method of arbitrary functions'. This approach promisingly yields deterministic probabilities which align with what we take the chances to be---it tells us that there is approximately a 1/2 probability of a spun roulette wheel stopping on black, and approximately a 1/2 probability of a flipped coin landing heads up---but it requires some probabilistic materials to work with. I contend (...) that the right probabilistic materials are found in reasonable initial credence distributions. I note that, with some normative assumptions, the resulting account entails that deterministic chances obey a variant of Lewis's 'principal principle'. I additionally argue that deterministic chances, so understood, are capable of explaining long-run frequencies. (shrink)

This paper draws attention to the crucial importance of a new kind of precisely defined law of nature in the Scientific Revolution. All explanations in the mechanical philosophy depend upon the interactions of moving material particles; the laws of nature stipulate precisely how these interact; therefore, such explanations rely on the laws of nature. While this is obvious, the radically innovatory nature of these laws is not fully acknowledged in the historical literature. Indeed, a number of scholars have tried to (...) locate the origins of such laws in the medieval period. In the first part of this paper these claims are critically examined, and found at best to reveal important aspects of the background to the later idea, which could be drawn upon for legitimating purposes by the mechanical philosophers. The second part of the paper argues that the modern concept of laws of nature originates in René Descartes's work. It is shown that Descartes took his concept of laws of nature from the mathematical tradition, but recognized that he could not export it to the domain of physico-mathematics, to play a causal role, unless he could show that these laws were underwritten by God. It is argued that this is why, at an early stage of his philosophical development, Descartes had to turn to metaphysics. (shrink)

The principle of energy conservation is widely taken to be a se- rious difficulty for interactionist dualism (whether property or sub- stance). Interactionists often have therefore tried to make it satisfy energy conservation. This paper examines several such attempts, especially including E. J. Lowe’s varying constants proposal, show- ing how they all miss their goal due to lack of engagement with the physico-mathematical roots of energy conservation physics: the first Noether theorem (that symmetries imply conservation laws), its converse (that (...) conservation laws imply symmetries), and the locality of continuum/field physics. Thus the “conditionality re- sponse”, which sees conservation as (bi)conditional upon symme- tries and simply accepts energy non-conservation as an aspect of interactionist dualism, is seen to be, perhaps surprisingly, the one most in accord with contemporary physics (apart from quantum mechanics) by not conflicting with mathematical theorems basic to physics. A decent objection to interactionism should be a posteri- ori, based on empirically studying the brain. (shrink)

I provide a critical commentary regarding the attitude of the logician and the philosopher towards the physicist and physics. The commentary is intended to showcase how a general change in attitude towards making scientific inquiries can be beneficial for science as a whole. However, such a change can come at the cost of looking beyond the categories of the disciplines of logic, philosophy and physics. It is through self-inquiry that such a change is possible, along with the realization of the (...) essence of the middle that is otherwise excluded by choice. The logician, who generally holds a reverential attitude towards the physicist, can then actively contribute to the betterment of physics by improving the language through which the physicist expresses his experience. The philosopher, who otherwise chooses to follow the advancement of physics and gets stuck in the trap of sophistication of language, can then be of guidance to the physicist on intellectual grounds by having the physicist’s experience himself. In course of this commentary, I provide a glimpse of how a truthful conversion of verbal statements to physico-mathematical expressions unravels the hitherto unrealized connection between Heisenberg uncertainty relation and Cauchy’s definition of derivative that is used in physics. The commentary can be an essential reading if the reader is willing to look beyond the categories of logic, philosophy and physics by being ‘nobody’. (shrink)

Physical dimensions like “mass”, “length”, “charge”, represented by the symbols [M], [L], [Q], are not numbers, but used as numbers to perform dimensional analysis in particular, and to write the equations of physics in general, by the physicist. The law of excluded middle falls short of explaining the contradictory meanings of the same symbols. The statements like “m tends to 0”, “r tends to 0”, “q tends to 0”, used by the physicist, are inconsistent on dimensional grounds because “m”, “r”, (...) “q” represent quantities with physical dimensions of [M], [L], [Q] respectively and “0” represents just a number—devoid of physical dimension. Consequently, due to the involvement of the statement “q tends to 0'', where q is the test charge” in the definition of electric field leads to either circular reasoning or a contradiction regarding the experimental verification of the smallest charge in the Millikan–Fletcher oil drop experiment. Considering such issues as problematic, by choice, I make an inquiry regarding the basic language in terms of which physics is written, with an aim of exploring how truthfully the verbal statements can be converted to the corresponding physico-mathematical expressions, where “physico-mathematical” signifies the involvement of physical dimensions. Such investigation necessitates an explanation by demonstration of “self inquiry”, “middle way”, “dependent origination”, “emptiness/relational existence”, which are certain terms that signify the basic tenets of Buddhism. In light of such demonstration I explain my view of “definition”; the relations among quantity, physical dimension and number; meaninglessness of “zero quantity” and the associated logico-linguistic fallacy; difference between unit and unity. Considering the importance of the notion of electric field in physics, I present a critical analysis of the definitions of electric field due to Maxwell and Jackson, along with the physico-mathematical conversions of the verbal statements. The analysis of Jackson’s definition points towards an expression of the electric field as an infinite series due to the associated “limiting process” of the test charge. However, it brings out the necessity of a postulate regarding the existence of charges, which nevertheless follows from the definition of quantity. Consequently, I explain the notion of undecidable charges that act as the middle way to resolve the contradiction regarding the Millikan–Fletcher oil drop experiment. In passing, I provide a logico-linguistic analysis, in physico-mathematical terms, of two verbal statements of Maxwell in relation to his definition of electric field, which suggests Maxwell’s conception of dependent origination of distance and charge ) and that of emptiness in the context of relative vacuum. This work is an appeal for the dissociation of the categorical disciplines of logic and physics and on the large, a fruitful merger of Eastern philosophy and Western science. Nevertheless, it remains open to how the reader relates to this work, which is the essence of emptiness. (shrink)

Karim Thébault has argued that for ontic structural realism to be a viable ontology it should accommodate two principles: physico-mathematical structures it deploys must be firstly consistent and secondly substantial. He then contends that in geometric quantization, a transitional machinery from classical to quantum mechanics, the two principles are followed, showing that it is a guide to ontic structure. In this article, I will argue that geometric quantization violates the consistency principle. To compensate for this shortcoming, the deformation quantization (...) procedure will be offered. (shrink)

Baruch Spinoza’s (1632–77)Tractatus theologico-politicus (1669 or 1670) caused outrage across the Dutch Republic, for it obliterated the carefully installed separation between philosophy and theology. The posthumous publication of Spinoza’s Ethica, which is contained in his Opera posthuma (1677), caused similar consternation. It was especially the mathematical order in which the Ethica was composed that caused fierce opposition, for its mathematical appearance gave the impression that Spinoza’s heretical teachings were established demonstratively. In this essay, I document how the Dutch physician, local (...) politician, and amateur mathematician and experimenter Bernard Nieuwentijt (1654–1718) attempted to physico-mathematically and methodologically counter the threats posed by Spinoza’s program. Nieuwentijt tried to defend the authority of the scriptures in times at which they came under attack by the new philosophy and by the emerging sciences that were gradually winning terrain. The crux of his defense consisted in delineating a modest epistemology, a “learned ignorance,” that would cure Spinoza’s followers of their pansophical aspirations, on the one hand, and remove the conflict between the Bible and reason, on the other. The specific way in which he sought to accomplish this distinguished him from Dutch Reformed thought, or so I will argue. (shrink)

The so-called “Scientific Revolution” is the result of a complex interaction between the world of ideas and that of concrete human activity with the aim of discovering the mysteries of nature. Not only books but also notebooks mediate this dialectical relationship: in this way, the complex features of a theoretical system can coexist with the detailed observations of everyday natural phenomena (like water drops, or burning candles), in order to test the foundations of a whole philosophy of nature in the (...) micro-world. Bacon himself suggested leaving general observations aside in order to reach closer to phenomena: Isaac Beeckman, in the isolation of his Journal, notes in chronological order his own laboratory and reading experiences, together with his meditations, producing an intellectual account of great cultural and scientific interest, discovered by Cornelis De Waard in 1905, in the Provincial Library of Zeeland [Middelburg]. Beeckman also owned some of Francis Bacon’s major works: from some of the notebook annotations presented in this article it is possible to see that Bacon’s explanations of simple natural phenomena did not always agree with the emerging physico-mathematical turn. Bacon is blamed for the use of old-fashioned categories, like sympathies and occult qualities, which need to be replaced by a fully corpuscular, mathematical and geometrical mechanical philosophy. (shrink)

El tratado hidrostático fue uno de los primeros textos de Descartes, fruto de su decisivo encuentro con Isaac Beeckman. En este artículo, analizaremos cómo fue concebido y los motivos que llevaron a Descartes a cuestionar alguno de los aspectos fundamentales de la física matemática de Beeckman. Este episodio está íntimamente relacionado con la independencia de las disciplinas matemáticas y su aplicación a cuestiones propias de la filosofía natural.Descartes’ hydrostatic treatise was one of his first text, fruit of his crucial meeting (...) with Isaac Beeckman. In this article, we will analyse how it was conceived and the reasons which leaded Descartes to question some fundamental aspects of Beeckman’s Physico-mathematics. This subject is closely connected with theautonomy and implementation of mathematical sciences to typical matters of natural philosophy. (shrink)

RésuméLa probabilité présente, en Physique classique, des caractères paradoxaux; relative à un état de connaissance, et n'apportant de renseignements que sur un ensemble nombreux d'essais, elle semble pourtant se définir comme un caractère objectif de l'événement isolé; elle suppose qu'on doive répondre » peut‐ětre « à certaines questions, alors que la Logique classique n'admet d'autre réponse que » oui « ou » non «. Utilisée par la micro‐physique, elle brise l'unité des fondements physico‐mathématiques qui se manifestait dans la Logique (...) bivalente, conçue comme théorie de l'objet, de l'événement, de la théorie quelconques.L'analyse de ces caractères paradoxaux, ainsi que des notions d'objet et d'événement, montre que la Physico‐mathématique retrouve son unité de fondements si l'on substitue à la Logique bivalente, la Logique trivalente de la Composition , ou la Logique des Attitudes qui la généralise . Cette revision des notions fondamentales à laquelle la science en progrès s'oblige elle‐měme est une expérience de l'idonéisme en action.ZusammenfassungDie Wahrscheinlichkeit zeigt in der klassischen Physik paradoxe Züge; obwohl sie vom Stande des Wissens abhängt und nur über eine grosse Anzahl von Versuchen Auskünfte gibt, scheint sie als eine objektive Eigenschaft des Einzelereignisses definiert zu sein; sie nimmt an, dass gewisse Fragen mit » vielleicht « beantwortet werden müssen, obgleich die klassische Logik keine andere Antwort als » ja « oder » nein « zulässt. Durch die Mikrophysik nutzbar gemacht, durchbricht sie die Einheit der physikalisch‐mathematischen Fundamente, die sich kundtat in der bivalenten Logik, verstanden als Theorie des beliebigen Objektes, des beliebigen Ereignisses, der beliebigen Theorie. Die Analyse dieser paradoxen Züge, sowie der Begriffe des Objektes und des Ereignisses zeigt, dass Mathematik und Physik ihre einheitlichen Grundlagen wiederfinden, wenn man die bivalente Logik durch die trivalente der Komposition ersetzt oder durch die Logik der Einstellungen , welche sie verallgemeinert . Diese Revision der grundlegenden Begriffe, zu der sich die fortschrittliche Wissenschaft selbst verpflichtet, ist ein Beispiel des in die Tat umgesetzten Idoneismus.Probability shows paradoxical characteristics in classical physics; although dependent upon a state of knowledge, and yielding information only as the total result of numerous trials, it seems nevertheless to be definable as an objective characteristic of the isolated event; it assumes that one must reply » perhaps « to certain questions, whilst classical logic would allow no answer but » yes « or » no «. When used by micro‐physics, it breaks the unity of the physico‐mathematical foundations which became apparent in bivalent logic, in its conception as a theory of any object, any event or any theory.Analysis of these paradoxical characteristics and of the concepts » object « and » event « shows physics and mathematics regain their fundamental unity if bivalent logic is replaced by the trivalent logic of synthesis or the logic of attitudes, which generalises it . This revision of fundamental concepts which progressive science brings upon itself, is an example of the doctrine of suitability in action. (shrink)

In the history of the scientific revolution, Descartes is often considered as the mechanical philosopher par excellence, and opposed as such to the founder of mechanical science, that is to say, Galileo: this cliché is not without foundation, but it must not make us forget that Descartes was himself a practitioner of mechanical science. In the article "Cartesian Mechanics" I detail the meaning and reach of "mechanics" in the Cartesian corpus, and do so in three steps. 1. I begin by (...) explaining the genesis of the thesis which states that there is no difference between the physical and the mechanical; this thesis is so famous that it is often imagined that it is constituted in Descartes's first writings. But we can in fact show that the first works of Descartes and Beeckman arise from a "physico-mathematical" practice which does not necessarily imply a complete reworking of traditional physics, even if it might have favored the Cartesian ambition for a physics as certain as geometry; it was only in the late 1630s that Descartes began to systematically affirm the identity of physics and mechanics, or of "rules of motion", of "laws of Nature" and of "laws of mechanics". I then analyze the consequences of this affirmation. 2. In the second part of this article I provide a step-by-step commentary of the response proposed by Descartes in a letter to Mersenne in July 1638 to the question raised by Beaugrand's Geostatice (...) dissertatio mathematica (1636), that is to say the question of whether a body weighs more when it is farther from the center of Earth than when it is close. Compared with other texts of the Cartesian corpus, this sample of mechanics has been little analyzed (to my knowledge, the only analyses are those of P. Duhem, P. Costabel, A. Gabbey and D. Garber). This text is nonetheless extremely interesting: Descartes responds to the geostatic question in turn in terms of physics, then mathematics; even more interesting, the examination of the details of his procedures allows us to confront Cartesian statics to other writings, whether those of Guidobaldo del Monte, Galileo, Stevin, Mersenne or Roberval. 3. In the third and final section, I analyze the reasons that lead Descartes to exclude velocity from his statics, I examine the difficulties that his explanation of gravity created for statics, and conclude with the confrontation between mechanical philosophy and mechanical science in the case of Descartes. (shrink)

The Cambrian explosion is characterized by the sudden outburst of organized animal plans, which occurred circa 530 M years ago. Around that time, many forms of animal life appeared, including several which have since disappeared. There is no general consensus about “why” this happened, and why it had any form of suddenness. However, all organized animal plans share a common feature: they are triploblastic, i.e., composed of 3 layers of tissue, endoderm, ectoderm and mesoderm. I show here that, within simple (...) hypotheses, the formation of the mesoderm has intrinsically a physical exponential dynamics, leading rapidly to triploblastism, and eventually, to animal formation. A novel physico-mathematical framework including epithelium-mesenchyme transition, visco-elastic constitutive equations, and conservation laws, is presented which allows one to describe gastrulation as a self-wetting phenomenon of a soft solid onto itself. This phenomenon couples differentiation and migration during gastrulation, and leads in a closed form to an exponential scaling law for the formation of the mesoderm. Therefore, the Cambrian explosion might have started, actually, by a true visco-elastic “explosion”: the exponential run-away of mesenchymal cells. (shrink)