Nothing has been more central to philosophy of mathematics than the distinction between mathematical and physical objects. Yet consideration of quantum particles shows the inadequacy of the popular spacetime and causal characterizations of the distinction. It also raises problems for an assumption used recently by Field, Hellman and Horgan, namely, that the mathematical realm is metaphysically independent of the physical one.

The distinction between mathematical and physical objects has probably played a greater role shaping the philosophy of mathematics than the distinction between observable and theoretical entities has had in defining the philosophy of science. All the major movements in the philosophy of mathematics may be seen as attempts to free mathematics of an abstract ontology or to come to terms with it. The reasons are epistemic. Most philosophers of mathematics believe that the abstractaess of mathematical objects (...) introduces special difficulties in accounting for our ability to know them, to refer to them and even to entertain beliefs about them. These difficulties—supposedly absent even in the case of the most theoretical physical objects—make mathematical objects especially problematic and philosophically unattractive.Few have questioned this epistemic thesis or the ontic distinction it presupposes. LaVerne Shelton (1980) challenged the abstract-concrete distinction some years ago in an unpublished APA address. (shrink)

In his De primo et ultimo instanti, Walter Burley paid careful attention to continuity, assuming that continua included and were limited by indivisibles such as instants, points, ubi, degrees of quality, or mutata esse. In his Tractatus primus, Burley applied the logic of first and last instants to reach novel conclusions about qualities and qualitative change. At the end of his Quaestiones in libros Physicorum Aristotelis, William of Ockham used long passages from Burley’s Tractatus primus, sometimes agreeing with Burley and (...) sometimes disagreeing. How may this interaction between Burley and Ockham be understood within its historical context? (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)

For more than a century the notion of a pre-established harmony between the mathematical and physical sciences has played an important role not only in the rhetoric of mathematicians and theoretical physicists, but also as a doctrine guiding much of their research. Strongly mathematized branches of physics, such as the vortex theory of atoms popular in Victorian Britain, were not unknown in the nineteenth century, but it was only in the environment of fin-de-siècle Germany that the idea of a (...) pre-established harmony really took off and became part of the mathematicians’ ideology. Important historical figures were in this respect David Hilbert, Hermann Minkowski and, somewhat later, Albert Einstein. Roughly similar ideas can be found also among British theorists, among whom Arthur Eddington, Arthur Milne, and Paul Dirac are singled out. Although largely limited to the period 1870–1940, the paper also considers Max Tegmark’s recent hypothesis of the universe being a one-to-one reflection of mathematical structures. (shrink)

Motivated by the analogy which holds within the context of discovery between mathematics and physics, we aim to show that there is a connection between two fields within the context of justification too. Based on the careful analysis of examples from science (especially within the domain of physics) we suggest that the logic of scientific research, which might appear as enumerative induction, is deduction, and we propose it to be universal generalization inference rule. Our main argument closely (...) follows the analysis of the structure of physical theory proposed by theoretical physicist Eugene P. Wigner. (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)

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the (...) mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

Foundational questions in logic, mathematics, computer science and physics are constant sources of epistemological debate in contemporary philosophy. To what extent is the transfinite part of mathematics completely trustworthy? Why is there a general 'malaise' concerning the logical approach to the foundations of mathematics? What is the role of symmetry in physics? Is it possible to build a coherent worldview compatible with a macroobjectivistic position and based on the quantum picture of the world? What account (...) can be given of opinion change in the light of new evidence? These are some of the questions discussed in this volume, which collects 14 lectures on the foundations of science given at the School of Philosophy of Science, Trieste, October 1989. The volume will be of particular interest to any student or scholar engaged in interdisciplinary research into the foundations of science in the context of contemporary debates. (shrink)

This paper examines physics and mathematics textbooks published in France at the end of the 1950s and at the beginning of the 1960s for children aged 11–15 years old. It argues that at this “middle school” level, textbooks contributed to shape cultural representations of both disciplines and their mutual boundaries through their contents and their material aspect. Further, this paper argues that far from presenting clearly delimited subjects, late 1950s textbooks offered possible connections between mathematics and (...) class='Hi'>physics. It highlights that such connections depended upon the type of schools the textbooks aimed at, at a time when educational organization still differentiated pupils of this age. It thus stresses how the audience and its projected aptitudes and needs, as well as the cultural teaching traditions of the teachers in charge, were inseparable from the diverse conceptions of mathematics and physics and their relationships promoted through textbooks of the time. (shrink)

This book offers a phenomenological conception of experiential justification that seeks to clarify why certain experiences are a source of immediate justification and what role experiences play in gaining (scientific) knowledge. Based on the author's account of experiential justification, this book exemplifies how a phenomenological experience-first epistemology can epistemically ground the individual sciences. More precisely, it delivers a comprehensive picture of how we get from epistemology to the foundations of mathematics and physics. The book is unique as it (...) utilizes methods and insights from the phenomenological tradition in order to make progress in current analytic epistemology. It serves as a starting point for re-evaluating the relevance of Husserlian phenomenology to current analytic epistemology and making an important step towards paving the way for future mutually beneficial discussions. This is achieved by exemplifying how current debates can benefit from ideas, insights, and methods we find in the phenomenological tradition. (shrink)

In this paper I discuss Mark Steiner’s view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question – a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? (...) The related epistemic question concerns humans’ cognitive ability to reach the formal structure of the physical world. Among other things, I explore the interaction of mathematical and non-mathematical cognition in physical discovery. (shrink)

The cultures here in question are those of mathematics and of physics that I shall interpret with the goal of exploring different modes of creativity. As case studies I will consider two scientists who were exemplars of these cultures, the mathematician Henri Poincaré (1854-1912) and the physicist Albert Einstein (1879-1955). The modes of creativity that I will compare and contrast are their notions of aesthetics and intuition. In order to accomplish this we begin by studying their introspections.

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)

It is commonly the case that a problem concerning a mathematical or physical system can be solved in two quite different ways--by an internal or an external approach. For example, the area of a triangle can be found by integration or by showing it to be half that of a certain rectangle. In general, the first approach is, to analyse the given system into component parts, and the second approach is to deal with the system as a whole. It seems (...) that even in cases where solutions to physical problems obtained according to these two approaches are equally valid, and are equally good as explanations, scientists prefer the solution obtained by the internal approach. The reasons for this preference are examined. And it is suggested that whatever the reasons, this preference may have been partly responsible for the 19th century preference for the Kinetic Theory rather than Thermodynamics. (shrink)

The quality of doctoral-level chemistry (N=145), computer science (N=58), geoscience (N=91), mathematics (N=115), physics (N=123), and statistics/biostatistics (N=64) programs at United States universities was assessed, using 16 measures. These measures focused on variables related to: program size; characteristics of graduates; reputational factors (scholarly quality of faculty, effectiveness of programs in educating research scholars/scientists, improvement in program quality during the last 5 years); university library size; research support; and publication records. Chapter I discusses prior attempts to assess quality in (...) graduate education, development of the study plans, and the selection of disciplines and programs to be evaluated. Chapter II discusses the methodology used, focusing on each of the assessment measures. Chapters III to VIII present, respectively, findings from the analyses of the chemistry, computer science, geoscience, mathematics, physics, and statistics/biostatistics programs. Chapter IX includes a summary of results, correlations among measures, several additional analyses, and suggestions for future studies. Among the findings reported are those indicating that mathematics programs had, on the average, the largest number of faculty (N=33) in December 1980 followed closely by physics (N=28) and chemistry (N=23), and that 80 percent of computer science students had job commitments by graduation. (Survey instruments and supporting documentation are included in appendices.) (JN). (shrink)

This book identifies the organizing concepts of physical and biological phenomena by an analysis of the foundations of mathematics and physics.

According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that (...) nevertheless intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)

This paper presents a model for the structure of universal frameworks in logic, mathematics, and physics that are closed to logical conclusion by the mechanism of paradox across a dualism of elements. The prohibition takes different forms defined by the framework of observation inherent to the structure. Forms include either prohibition to conclusion on the logical relationship of internal elements or prohibition to conclusion based on the existence of an element not included in the framework of a first (...) element. The model is applied to logical arguments in philosophy, mathematics, and physics and is initially a geometrical analysis of quantum theory and its application in experiment. Conclusion from the analysis is extended to give insight into the complexity of infinities above the two-dimensional boundary of the model. (shrink)

Hermann Weyl was one of the twentieth century's most important mathematicians, as well as a seminal figure in the development of quantum physics and general relativity. He was also an eloquent writer with a lifelong interest in the philosophical implications of the startling new scientific developments with which he was so involved. Mind and Nature is a collection of Weyl's most important general writings on philosophy, mathematics, and physics, including pieces that have never before been published in (...) any language or translated into English, or that have long been out of print. Complete with Peter Pesic's introduction, notes, and bibliography, these writings reveal an unjustly neglected dimension of a complex and fascinating thinker. In addition, the book includes more than twenty photographs of Weyl and his family and colleagues, many of which are previously unpublished. Included here are Weyl's exposition of his important synthesis of electromagnetism and gravitation, which Einstein at first hailed as "a first-class stroke of genius"; two little-known letters by Weyl and Einstein from 1922 that give their contrasting views on the philosophical implications of modern physics; and an essay on time that contains Weyl's argument that the past is never completed and the present is not a point. Also included are two book-length series of lectures, The Open World and Mind and Nature, each a masterly exposition of Weyl's views on a range of topics from modern physics and mathematics. Finally, four retrospective essays from Weyl's last decade give his final thoughts on the interrelations among mathematics, philosophy, and physics, intertwined with reflections on the course of his rich life. (shrink)

Dr. Vukov analyzing patients with disorders of consciousness, proposed that medical well-regarded policy recommendations cannot be justified by looking solely to patients’ actual levels of consciousness (minimally conscious state – MCS versus vegetative state – VS), but that they can be justified by looking to patients’ potential for consciousness. One objective way to estimate this potential (actual physical possibility) is to consider a neurophysiologically informed strategy. Ideally such strategy would utilize objective brain activity markers of consciousness/unconsciousness. The Operational Architectonics (OA) (...) theory of brain-mind functioning is an example of such a strategy. Besides being mathematically simple, neurophysiologically accurate, and compatible with the cognitive/phenomenal perspectives on consciousness, application of OA strategy to quantitative EEG analysis of patients in VS and MCS revealed that the absence of consciousness in VS is paralleled by impairment of overall EEG operational architecture. Specifically, neuronal assemblies become smaller, their life span shortened, and they became highly unstable and functionally disconnected (desynchronized). At the same time, fluctuating (minimal) awareness in patients in MCS was paralleled by partial restoration of EEG operational architecture (increased size, life span, and stability of neuronal assemblies, together with an increased number and strength of functional connections among them), approaching the level found in healthy fully conscious participants. Moreover OA strategy to EEG analysis allowed predicting the emergence of consciousness in VS patients after six years following the brain trauma. Therefore, the OA methodology could reliably predict which patients may regain consciousness, and thus determine which patients may have the potential for consciousness. (shrink)

Constructibility and complexity play central roles in recent research in computer science, mathematics and physics. For example, scientists are investigating the complexity of computer programs, constructive proofs in mathematics and the randomness of physical processes. But there are different approaches to the explication of these concepts. This volume presents important research on the state of this discussion, especially as it refers to quantum mechanics. This `foundational debate' in computer science, mathematics and physics was already fully (...) developed in 1930 in the Vienna Circle. A special section is devoted to its real founder Hans Hahn, referring to his contribution to the history and philosophy of science. The documentation section presents articles on the early Philipp Frank and on the Vienna Circle in exile. Reviews cover important recent literature on logical empiricism and related topics. (shrink)

The target paper of Schoeller, Perlovsky, and Arseniev is an essential and timely contribution to a current shift of focus in neuroscience aiming to merge neurophysiological, psychological and physical principles in order to build the foundation for the physics of mind. Extending on previous work of Perlovsky et al. and Badre, the authors of the target paper present interesting mathematical models of several basic principles of the physics of mind, such as perception and cognition, concepts and emotions, instincts (...) and learning. Their conceptualization helps to clarify the distinction between conscious and unconscious aspects of mind that is often neglected and further provide a clear description of the mental hierarchy, which extends from physical objects in the physical world to abstract ideas in the mental/subjective realm. While we agree that identification of a few fundamental principles is a first step toward developing the physics of the mind, and we concur with the selection of those principles in the target review paper, we think that the theory of the physics of mind would much benefit from considering also the most basic principles that are common for the physics/matter/brain and the mind/subjectivity/cognition. In this respect, such basic principles as time and space, as well as criticality, self-organization, and emergence seem to be the most interesting. (shrink)

Even though mathematics and physics have been related for centuries and this relation appears to be unproblematic, there are many questions still open: Is mathematics really necessary for physics, or could physics exist without mathematics? Should we think physically and then add the mathematics apt to formalise our physical intuition, or should we think mathematically and then interpret physically the obtained results? Do we get mathematical objects by abstraction from real objects, or vice (...) versa? Why is mathematics effective into physics? These are all relevant questions, whose answers are necessary to fully understand the status of physics, particularly of contemporary physics. The aim of this book is to offer plausible answers to such questions through both historical analyses of relevant cases, and philosophical analyses of the relations between mathematics and physics. (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. (shrink)

Constructibility and complexity play central roles in recent research in computer science, mathematics and physics. For example, scientists are investigating the complexity of computer programs, constructive proofs in mathematics and the randomness of physical processes. But there are different approaches to the explication of these concepts. This volume presents important research on the state of this discussion, especially as it refers to quantum mechanics. This `foundational debate' in computer science, mathematics and physics was already fully (...) developed in 1930 in the Vienna Circle. A special section is devoted to its real founder Hans Hahn, referring to his contribution to the history and philosophy of science. The documentation section presents articles on the early Philipp Frank and on the Vienna Circle in exile. Reviews cover important recent literature on logical empiricism and related topics. (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)

This paper examines the thoughts and early career of the astrophysicist and cosmologist E. A. Milne. Although Milne only turned to cosmology in 1932, many of the ideas that characterised his heterodox system of world physics can be traced back to his works from the 1920s. Contrary to what has been stated in the literature, we argue that Milne was familiar with and interested in cosmology even before 1932. The relationship between mathematics and physics, an important topic (...) in Milne's cosmophysics, was the subject of an unpublished paper he gave in 1922. We place the paper in its historical context and comment on its relevance for Milne's later work. An annotated transcript of the address is included as an accompanying paper under the title 'The Relations of Mathematics to Science'. (shrink)