The main claim of this talk is that mathematical physics and philosophy of physics are not different. This claim, so formulated, is obviously false because it is overstated; however, since no non-tautological statement is likely to be completely true, it is a meaningful question whether the overstated claim expresses some truth. I hope it does, or so I’ll argue. The argument consists of two parts: First I’ll recall some characteristic features of von Neumann’s work on mathematical (...) foundations of quantum mechanics and will claim that von Neumann’s motivation and results are essentially philosophical in their nature; hence, to the extent von Neumann’s work exemplifies what is considered to be mathematical physics, mathematical physics appears as formally explicit philosophy of physics. The second argument is based on a rather trivial interpretation of what mathematical physics is. That interpretation implies that mathematical physics shares some key characteristic features with philosophy of physics which make the two almost indistinguishable. (shrink)

The main claim of this talk is that mathematical physics and philosophy of physics are not different. This claim, so formulated, is obviously false because it is overstated; however, since no non-tautological statement is likely to be completely true, it is a meaningful question whether the overstated claim expresses some truth. I hope it does, or so I’ll argue. The argument consists of two parts: First I’ll recall some characteristic features of von Neumann’s work on mathematical (...) foundations of quantum mechanics and will claim that von Neumann’s motivation and results are essentially philosophical in their nature; hence, to the extent von Neumann’s work exemplifies what is considered to be mathematical physics, mathematical physics appears as formally explicit philosophy of physics. The second argument is based on a rather trivial interpretation of what mathematical physics is. That interpretation implies that mathematical physics shares some key characteristic features with philosophy of physics which make the two almost indistinguishable. (shrink)

This book highlights the emergence of a new mathematical rationality and the beginning of the mathematisation of physics in Classical Islam. Exchanges between mathematics, physics, linguistics, arts and music were a factor of creativity and progress in the mathematical, the physical and the social sciences. Goods and ideas travelled on a world-scale, mainly through the trade routes connecting East and Southern Asia with the Near East, allowing the transmission of Greek-Arabic medicine to Yuan Muslim China. The (...) development of science, first centred in the Near East, would gradually move to the Western side of the Mediterranean, as a result of Europe's appropriation of the Arab and Hellenistic heritage. Contributors are Paul Buell, Anas Ghrab, Hossein Masoumi Hamedani, Zeinab Karimian, Giovanna Lelli, Marouane ben Miled, Patricia Radelet-de Grave, and Roshdi Rashed. (shrink)

This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is (...) treated at some length are Kolmogorov, von Mises and de Finetti. The principal audience for the book comprises philosophers and historians of science, mathematicians concerned with probability and statistics, and physicists. The book will also interest anyone fascinated by twentieth-century scientific developments because the birth of modern probability is closely tied to the change from a determinist to an indeterminist world-view. (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)

This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is (...) treated at some length are Kolmogorov, von Mises and de Finetti. The principal audience for the book comprises philosophers and historians of science, mathematicians concerned with probability and statistics, and physicists. The book will also interest anyone fascinated by twentieth-century scientific developments because the birth of modern probability is closely tied to the change from a determinist to an indeterminist world-view. (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)

This article addresses the problem of Jacques Rohault’s Cartesianism. It aims to enrich the current portrayal of Rohault (1618–72) as a Cartesian natural philosopher concerned with experimentation. The modern evaluation of Rohault as an experimentalist can benefit from another explanatory layer, emphasizing the mathematical physics that shapes his natural philosophy. In order to argue for this complementary account, I focus on an early episode in Rohault’s career, represented by his reply to Fermat’s attacks against Descartes’s law of refraction (...) (1658). The source has been discussed, so far, solely as part of the dispute in optics between Fermat and the Cartesians. The reading endorsed here explores the source from a different angle, suggesting an alternative account grounded on a more contextual understanding of Rohault’s contribution to Cartesianism. More diverse dimensions of Rohault’s intellectual formation are revealed, which in turn provides a more complex picture of this strain of Cartesian natural philosophy. (shrink)

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

In 1887 Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics, on the possibility of quantitative psychology, and on the meaning of temperature measurement. Late nineteenth-century scrutinisers of the foundations of mathematics made little of Helmholtz’s essay. Yet it inspired two mathematicians with an eye on physics, and a few philosopher-physicists. The aim of the present paper is to situate Helmholtz’s (...) contribution in this complex array of nineteenth-century philosophies of number, quantity, and measurement.Author Keywords: Helmholtz; Measurement; Arithmetic; P. Du Bois-Reymond; H. and R. Grassmann; J. von Kries. (shrink)

How does the physics we know today-- a highly professionalized enterprise, inextricably linked to government and industry-- link back to its origins as a liberal art in ancient Greece? What is the path that leads from the old philosophy of nature and its concern with humankind's place in the universe to modern massive international projects that hunt down fundamental particles and industrial laboratories that manufacture marvels? John Heilbron's fascinating history of physics introduces us to Islamic astronomers and (...) mathematicians, calculating the size of the earth whilst their caliphs conquered much of it; to medieval scholar-theologians investigating light; to Galileo, Copernicus, Kepler, and Newton, measuring, and trying to explain, the universe. We visit the 'House of Wisdom' in 9th-century Baghdad; Europe's first universities; the courts of the Renaissance; the Scientific Revolution and the academies of the 18th century; the increasingly specialized world of 20th and 21st century science. Highlighting the shifting relationship between physics, philosophy, mathematics, and technology-- and the implications for humankind's self-understanding-- Heilbron explores the changing place and purpose of physics in the cultures and societies that have nurtured it over the centuries. (shrink)

It is commonly thought that before the introduction of quantum mechanics, determinism was a straightforward consequence of the laws of mechanics. However, around the nineteenth century, many physicists, for various reasons, did not regard determinism as a provable feature of physics. This is not to say that physicists in this period were not committed to determinism; there were some physicists who argued for fundamental indeterminism, but most were committed to determinism in some sense. However, for them, determinism was often (...) not a provable feature of physical theory, but rather an a priori principle or a methodological presupposition. Determinism was strongly connected with principles of causality and continuity and the principle of sufficient reason; this thesis examines the relevance of these principles in the history of physics. Moreover, the history of determinism in this period shows that there were essential changes in the relation between mathematics and physics: whereas in the eighteenth century, there were metaphysical arguments which lent support to differential calculus, by the early twentieth century the development of rigorous foundations of differential calculus led to concerns about its applicability in physics. The thesis consists of six papers. In the first paper, "On the origins and foundations of Laplacian determinism", I argue that Laplace, who is usually pointed out as the first major proponent of scientific determinism, did not derive his statement of determinism directly from the laws of mechanics; rather, his determinism has a background in eighteenth century Leibnizian metaphysics, and is ultimately based on the law of continuity and the principle of sufficient reason. These principles also provided a basis for the idea that one can find laws of nature in the form of differential equations which uniquely determine natural processes. In "The Norton dome and the nineteenth century foundations of determinism", I argue that an example of indeterminism in classical physics which has attracted attention in philosophy of physics in recent years, namely the Norton come, was already discussed during the nineteenth century. However, the significance which this type of indeterminism had back then is very different from the significance which the Norton dome currently has in philosophy of physics. This is explained by the fact that determinism was conceived of in an essentially different way: in particular, the nineteenth century authors who wrote about this type of indeterminism regarded determinism as an a priori principle rather than as a property of the equations of physics. In "Vital instability: life and free will in physics and physiology, 1860-1880", I show how Maxwell, Cournot, Stewart and Boussinesq used the possibility of unstable or indeterministic mechanical systems to argue that the will or a vital principle can intervene in organic processes without violating the laws of physics, so that a strictly dualist account of life and the mind is possible. Moreover, I show that their ideas can be understood as a reaction to the law of conservation of energy and to the way it was used in physiology to exclude vital and mental causes. In "The nineteenth century conflict between mechanism and irreversibility", I show that in the late nineteenth century, there was a widespread conflict between the aim of reducing physical processes to mechanics and the recognition that certain processes are irreversible. Whereas the so-called reversibility objection is known as an objection that was made to the kinetic theory of gases, it in fact appeared in a wide range of arguments, and was susceptible to very different interpretations. It was only when the project of reducing all of physics to mechanics lost favor, in the late nineteenth century, that the reversibility objection came to be used as an argument against mechanism and against the kinetic theory of gases. In "Continuity in nature and in mathematics: Boltzmann and Poincaré", I show that the development of rigorous foundations of differential calculus in the nineteenth century led to concerns about its applicability in physics: through this development, differential calculus was made independent of empirical and intuitive notions of continuity and was instead based on mathematical continuity conditions, and for Boltzmann and Poincaré, the applicability of differential calculus in physics depended on whether these continuity conditions could be given a foundation in intuition or experience. In the final paper, "Determinism around 1900", I briefly discuss the implications of the developments described in the previous two papers for the history of determinism in physics, through a discussion of determinism in Mach, Poincaré and Boltzmann. I show that neither of them regards determinism as a property of the laws of mechanics; rather, for them, determinism is a precondition for science, which can be verified to the extent that science is successful. (shrink)

This book explores the persistence of Pythagorean ideas in theoretical physics. It shows that the Pythagorean position is both philosophically deep and scientifically interesting. However, it does not endorse pure Pythagoreanism; rather, it defends the thesis that mind and mathematical structure are the grounds of reality. The book begins by examining Wigner's paper on the unreasonable effectiveness of mathematics in the natural sciences. It argues that, whilst many issues surrounding the applicability of mathematics disappear upon examination, there are (...) some core issues to do with the effectiveness of mathematics in fundamental physics which remain. The core issues are the existence of the laws of nature and our minds ability to fathom them, the use of formal mathematics in discovering things about the quantum world, the fact that deep mathematics is needed to describe fundamental physics, and the asymptotic nature of the quest for knowledge. These issues are the focus of the book. The author seeks to explain them within a metaphysical framework that takes mind and mathematical structure as its fundamental principles. The framework - called quantum monadology - combines ideas from Leibnizian monadology, set theory, and consistent histories quantum theory. (shrink)

Our understanding of nature, and in particular of physics and the laws governing it, has changed radically since the days of the ancient Greek natural philosophers. This book explains how and why these changes occurred, through landmark experiments as well as theories that - for their time - were revolutionary. The presentation covers Mechanics, Optics, Electromagnetism, Thermodynamics, Relativity Theory, Atomic Physics and Quantum Physics. The book places emphasis on ideas and on a qualitative presentation, rather than on (...) mathematics and equations. Thus, although primarily addressed to those who are studying or have studied science, it can also be read by non-specialists. The author concludes with a discussion of the evolution and organization of universities, from ancient times until today, and of the organization and dissemination of knowledge through scientific publications and conferences. (shrink)

How does the physics we know today-- a highly professionalized enterprise, inextricably linked to government and industry-- link back to its origins as a liberal art in ancient Greece? What is the path that leads from the old philosophy of nature and its concern with humankind's place in the universe to modern massive international projects that hunt down fundamental particles and industrial laboratories that manufacture marvels? John Heilbron's fascinating history of physics introduces us to Islamic astronomers and (...) mathematicians, calculating the size of the earth whilst their caliphs conquered much of it; to medieval scholar-theologians investigating light; to Galileo, Copernicus, Kepler, and Newton, measuring, and trying to explain, the universe. We visit the 'House of Wisdom' in 9th-century Baghdad; Europe's first universities; the courts of the Renaissance; the Scientific Revolution and the academies of the 18th century; the increasingly specialized world of 20th and 21st century science. Highlighting the shifting relationship between physics, philosophy, mathematics, and technology-- and the implications for humankind's self-understanding-- Heilbron explores the changing place and purpose of physics in the cultures and societies that have nurtured it over the centuries. (shrink)

In the first part of chapter 2 of book II of the Physics Aristotle addresses the issue of the difference between mathematics and physics. In the course of his discussion he says some things about astronomy and the ‘ ‘ more physical branches of mathematics”. In this paper I discuss historical issues concerning the text, translation, and interpretation of the passage, focusing on two cruxes, the first reference to astronomy at 193b25–26 and the reference to the more physical branches at (...) 194a7–8. In section I, I criticize Ross’s interpretation of the passage and point out that his alteration of has no warrant in the Greek manuscripts. In the next three sections I treat three other interpretations, all of which depart from Ross's: in section II that of Simplicius, which I commend; in section III that of Thomas Aquinas, which is importantly influenced by a mistranslation of, and in section IV that of Ibn Rushd, which is based on an Arabic text corresponding to that printed by Ross. In the concluding section of the paper I describe the modern history of the Greek text of our passage and translations of it from the early twelfth century until the appearance of Ross's text in 1936. (shrink)

In this paper, we discuss the history of the concept of function and emphasize in particular how problems in physics have led to essential changes in its definition and application in mathematical practices. Euler defined a function as an analytic expression, whereas Dirichlet defined it as a variable that depends in an arbitrary manner on another variable. The change was required when mathematicians discovered that analytic expressions were not sufficient to represent physical phenomena such as the vibration (...) of a string and heat conduction. The introduction of generalized functions or distributions is shown to stem partly from the development of new theories of physics such as electrical engineering and quantum mechanics that led to the use of improper functions such as the delta function that demanded a proper foundation. We argue that the development of student understanding of mathematics and its nature is enhanced by embedding mathematical concepts and theories, within an explicit–reflective framework, into a rich historical context emphasizing its interaction with other disciplines such as physics. Students recognize and become engaged with meta-discursive rules governing mathematics. Mathematics teachers can thereby teach inquiry in mathematics as it occurs in the sciences, as mathematical practice aimed at obtaining new mathematical knowledge. We illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework by two examples of student-directed, problem-oriented project work following the Roskilde Model, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science. (shrink)

I would like to start with a historical question or, more precisely, a question pertaining to the history of science itself. It is a widely accepted idea that Aristotelism has been an obstacle to the emergence of modern physical science, and this was for at least two reasons. The first one is the cognitive role Aristotle is supposed to have attributed to perception. Instead of considering perception as an origin of error, Aristotle thinks that our senses provide us with (...) a reliable image of the external world. The perceptive knowledge is a kind of knowledge in its own right, and the theoretical knowledge is, in fact, the continuation of the perceptive knowledge in some way. The second reason is the presumed inability of the Aristotelian philosophers to apply mathematics to the physical world. This was a formidable obstacle because modern physics came to be but as a mathematical physics. Aristotelianism had therefore to be, so to speak, superseded by the Platonic movement that originated in Florence around Ficino in order to give modern physics the conditions of its appearance. Galileo had to say that “Nature is written with mathematical letters” and Descartes that “our senses do not teach us what things are, but to what extent they are useful or harmful to us”. Alexandre Koyré is right to consider Galilean physics to be basically Platonic. The theoretical justification Aristotle offers for the impossibility of a convergence between mathematics and physics seems to be based on some fundamental features of his philosophy, i.e. he rejects the Platonic conception of a unique science, encompassing all things, and replaces it with the doctrine of the incommunicability of genera, whose corollary is that there is but one science for each genus. (shrink)

I characterize Bishop's constructive mathematics as an alternative to classical mathematics, which makes use of the actual infinity. From the history an accurate investigation of past physical theories I obtianed some ones - mainly Lazare Carnot's mechanics and Sadi Carnot's thermodynamics - which are alternative to the dominant theories - e.g. Newtopn's mechanics. The way to link together mathematics to theoretical physics is generalized and some general considerations, in particualr on the geoemtry in theoretical physics, are obtained.that.

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)

A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such as the advent of the computer, (...) chaos theory, topology, mathematical physics, and the solution of Fermat's Last Theorem.Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises will stretch the more advanced reader. (shrink)

The concept of similar systems arose in physics, and appears to have originated with Newton in the seventeenth century. This chapter provides a critical history of the concept of physically similar systems, the twentieth century concept into which it developed. The concept was used in the nineteenth century in various fields of engineering, theoretical physics and theoretical and experimental hydrodynamics. In 1914, it was articulated in terms of ideas developed in the eighteenth century and used in nineteenth (...) century mathematics and mechanics: equations, functions and dimensional analysis. The terminology physically similar systems was proposed for this new characterization of similar systems by the physicist Edgar Buckingham. Related work by Vaschy, Bertrand, and Riabouchinsky had appeared by then. The concept is very powerful in studying physical phenomena both theoretically and experimentally. As it is not currently part of the core curricula of STEM disciplines or philosophy of science, it is not as well known as it ought to be. (shrink)

A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such as the advent of the computer, (...) chaos theory, topology, mathematical physics, and the solution of Fermat's Last Theorem. Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises will stretch the more advanced reader. (shrink)

In my dissertation I analyze the structure of fundamental physical theories. I start with an analysis of what an adequate primitive ontology is, discussing the measurement problem in quantum mechanics and theirs solutions. It is commonly said that these theories have little in common. I argue instead that the moral of the measurement problem is that the wave function cannot represent physical objects and a common structure between these solutions can be recognized: each of them is about a clear three-dimensional (...) primitive ontology that evolves according to a law determined by the wave function. The primitive ontology is what matter is made of while the wave function tells the matter how to move. One might think that what is important in the notion of primitive ontology is their three-dimensionality. If so, in a theory like classical electrodynamics electromagnetic fields would be part of the primitive ontology. I argue that, reflecting on what the purpose of a fundamental physical theory is, namely to explain the behavior of objects in three--dimensional space, one can recognize that a fundamental physical theory has a particular architecture. If so, electromagnetic fields play a different role in the theory than the particles and therefore should be considered, like the wave function, as part of the law. Therefore, we can characterize the general structure of a fundamental physical theory as a mathematical structure grounded on a primitive ontology. I explore this idea to better understand theories like classical mechanics and relativity, emphasizing that primitive ontology is crucial in the process of building new theories, being fundamental in identifying the symmetries. Finally, I analyze what it means to explain the word around us in terms of the notion of primitive ontology in the case of regularities of statistical character. Here is where the notion of typicality comes into play: we have explained a phenomenon if the typical histories of the primitive ontology give rise to the statistical regularities we observe. (shrink)

There is general agreement in mathematics about what continuity is. In this paper we examine how well the mathematical definition lines up with common sense notions. We use a recent paper by Hud Hudson as a point of departure. Hudson argues that two objects moving continuously can coincide for all but the last moment of their histories and yet be separated in space at the end of this last moment. It turns out that Hudson’s construction does not deliver mathematically (...) continuous motion, but the natural question then is whether there is any merit in the alternative definition of continuity that he implicitly invokes. (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)

We analyze the effects of the introduction of new mathematical tools on an old branch of physics by focusing on lattice fluids, which are cellular automata -based hydrodynamical models. We examine the nature of these discrete models, the type of novelty they bring about within scientific practice and the role they play in the field of fluid dynamics. We critically analyze Rohrlich's, Fox Keller's and Hughes' claims about CA-based models. We distinguish between different senses of the predicates “phenomenological” (...) and “theoretical” for scientific models and argue that it is erroneous to conclude, as they do, that CA-based models are necessarily phenomenological in any sense of the term. We conversely claim that CA-based models of fluids, though at first sight blatantly misrepresenting fluids, are in fact conservative as far as the basic laws of statistical physics are concerned and not less theoretical than more traditional models in the field. Based on our case-study, we propose a general discussion of the prospect of CA for modeling in physics. We finally emphasize that lattice fluids are not just exotic oddities but do bring about new advantages in the investigation of fluids' behavior. (shrink)

Mark Wilson argues that the standard categorizations of "Theory T thinking"— logic-centered conceptions of scientific organization (canonized via logical empiricists in the mid-twentieth century)—dampens the understanding and appreciation of those strategic subtleties working within science. By "Theory T thinking," we mean to describe the simplistic methodology in which mathematical science allegedly supplies ‘processes’ that parallel nature's own in a tidily isomorphic fashion, wherein "Theory T’s" feigned rigor and methodological dogmas advance inadequate discrimination that fails to distinguish between explanatory structures (...) that are architecturally distinct. One of Wilson's main goals is to reverse such premature exclusions and, thus, early on Wilson returns to John Locke's original physical concerns regarding material science and the congeries of descriptive concern insofar as capturing varied phenomena (i.e., cohesion, elasticity, fracture, and the transmission of coherent work) encountered amongst ordinary solids like wood and steel are concerned. Of course, Wilson methodologically updates such a purview by appealing to multiscalar techniques of modern computing, drawing from Robert Batterman's work on the greediness of scales and Jim Woodward's insights on causation. (shrink)

Pierre Duhem and Jacques Maritain, influenced by positivist philosophies of science that prevailed in the late nineteenth and early twentieth centuries, adopted markedly non-realist views about the mathematical theories of the modern physical sciences. The philosophies of science they developed were a hybrid of Thomism and positivism. This paper argues that the ideas of Duhem and Maritain about the relation of the mathematical theories of modern physics to physical reality are inadequate in light of the insights modern (...) physics has yielded about the physical world. (shrink)

Pierre Duhem and Jacques Maritain, influenced by positivist philosophies of science that prevailed in the late nineteenth and early twentieth centuries, adopted markedly non-realist views about the mathematical theories of the modern physical sciences. The philosophies of science they developed were a hybrid of Thomism and positivism. This paper argues that the ideas of Duhem and Maritain about the relation of the mathematical theories of modern physics to physical reality are inadequate in light of the insights modern (...) physics has yielded about the physical world. (shrink)

The author claims to have developed interactional expertise in gravitational wave physics without engaging with the mathematical or quantitative aspects of the subject. Is this possible? In other words, is it possible to understand the physical world at a high enough level to argue and make judgments about it without the corresponding mathematics? This question is empirically approached in three ways: anecdotes about non-mathematical physicists are presented; the author undertakes a reflective reading of a passage of (...) class='Hi'>physics, first without going through the maths and then after engaging with it and discusses the difference between the experiences; the aforementioned exercise gives rise to a table of Levels of Understanding of mathematics, and physicists are asked about the level mathematical understanding they applied when they last read a paper. Each phase of empirical research suggests that mathematics is not as central to gaining an understanding of physics as it is often said to be. This does not mean that mathematics is not central to physics, merely that it is not essential for every physicist to be an accomplished mathematician, and that a division of labour model is adequate. This, in turn, suggests that a stream of undergraduate physics education with fewer mathematical hurdles should be developed, making it easier to train wider groups of people in physical science comprehension.Keywords: Physics; Mathematics; Interactional expertise; Physics education; Mathematical literacy; Scientific literacy. (shrink)

This book is about the methods used for unifying different scientific theories under one all-embracing theory. The process has characterized much of the history of science and is prominent in contemporary physics; the search for a 'theory of everything' involves the same attempt at unification. Margaret Morrison argues that, contrary to popular philosophical views, unification and explanation often have little to do with each other. The mechanisms that facilitate unification are not those that enable us to explain how (...) or why phenomena behave as they do. A feature of this book is an account of many case studies of theory unification in nineteenth- and twentieth-century physics and of how evolution by natural selection and Mendelian genetics were unified into what we now term evolutionary genetics. (shrink)

InCelebratingthisDIUMessayHistoriographywe describeIEMNTorricelli’s lifeLille Universityand worksUdine University in their scientific context, including the ArchimedeanArchimedean heritage in Torricelli’s works. We analyse the changes of Torricelli’s works and explain the novelties of the edition we are offering. Then, we provide a picture of the most significant results obtained by Torricelli (1608–1647), particularly in mechanicsMechanicsand geometryGeometry. Furthermore, we also focus on the Torricelli’s methodology, specifying how he provedProved two of his achievements, given their novelty and mathematical meaning: (a) the volumeVolume of the “solido (...) acutissimo”; (b) the geometryGeometryof theLogarithmic Spirallogarithmic spiralGeometric Spiral. AnContent ad hoc ad hocAd hocContentContent of the Opera geometricaOpera geometrica is proposed as well. It is based on the original chronological sequences of the contentsContent wrote by Torricelli. This is important to evaluate the birth of the scientific matters presented by Torricelli and the editorial process of the publicationPublication. (shrink)