This paper explores the relationship between Kant's views on the metaphysical foundations of Newtonian mathematicalphysics 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.
In this paper I argue three things: (1) that the interactionist view underlying Benacerraf's (1973) challenge to mathematical beliefs renders inexplicable the reliability of most of our beliefs in physics; (2) that examples from mathematicalphysics suggest that we should view reliability differently; and (3) that abstract mathematical considerations are indispensable to explanations of the reliability of our beliefs.
Fundamental notions Husserl introduced in Ideen I, such as epochè, reality, and empty X as substrate, might be useful for elucidating how mathematicalphysics concepts are produced. However, this is obscured in the context of Husserl’s phenomenology itself. For this possibility, the author modifies Husserl’s fundamental notions introduced for pure phenomenology, which found all sciences on the absolute Ego. Subsequently, the author displaces Husserl's phenomenological notions toward the notions operating inside scientific activities themselves and shows this using a (...) case study of the construction of noncommutative geometry. The perspective in Ideen I about geometry and mathematicalphysics includes points that are inappropriate to modern geometry and to modern physics, especially to noncommutative geometry and to quantum physics. The first point relates to the intuitive character of geometrical objects in Husserl. The second is linked to the notion of locality related to the notion of extension, by which Husserl characterizes the essence of physical things. The points show that the notion of empty X as a substrate, developed in “Phenomenology of Reason” in Ideen I, is helpful for considering the notions of physical reality and of geometrical space, especially reality in quantum physics and space in noncommutative geometry. The salient conclusions include the proposition that aphilosophical study of the relationship between the physical object X, which imparts a unity to what is given to sensibility, and the geometrical space X, which imparts a unity of sense to various mathematical operations, opens a reinterpretation of Husserl’s interpretation, supporting an epistemology of mathematicalphysics. (shrink)
This is a comprehensive discussion of complexity as it arises in physical, chemical, and biological systems, as well as in mathematical models of nature. Common features of these apparently unrelated fields are emphasised and incorporated into a uniform mathematical description, with the support of a large number of detailed examples and illustrations. The quantitative study of complexity is a rapidly developing subject with special impact in the fields of physics, mathematics, information science, and biology. Because of the (...) variety of the approaches, no comprehensive discussion has previously been attempted. This book will be of interest to graduate students and researchers in physics (nonlinear dynamics, fluid dynamics, solid-state, cellular automata, stochastic processes, statistical mechanics and thermodynamics), mathematics (dynamical systems, ergodic and probability theory), information and computer science (coding, information theory and algorithmic complexity), electrical engineering and theoretical biology. (shrink)
This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematicalphysics ...
Many arguments found in the physics literature involve concepts that are not well-defined by the usual standards of mathematics. I argue that physicists are entitled to employ such concepts without rigorously defining them so long as they restrict the sorts of mathematical arguments in which these concepts are involved. Restrictions of this sort allow the physicist to ignore calculations involving these concepts that might lead to contradictory results. I argue that such restrictions need not be ad hoc, but (...) can sometimes be justified by considering some of the metaphysical issues surrounding the question of the applicability of mathematics to physical reality. 1 Introduction 2 Rejecting inferential permissiveness 3 The agreement problem 4 Independent objections to the liberal view. (shrink)
A mathematical model of the natural origin of our universe is presented. The model is based only on well-established physics. No claim is made that this model uniquely represents exactly how the universe came about. But the viability of a single model serves to refute any assertions that the universe cannot have come about by natural means.
Causal claims in physics may have two familiar kinds of support: theoretical and experimental. This paper claims that a rigorous mathematical derivation in a realistic model is necessary, though not sufficient, for full theoretical support. The support is not provided by the derivation itself; but rather it comes from a detailed back-tracing through the derivation, matching the mathematical dependencies, point by point, with details of the causal story. This back-tracing is not enough to pick out the correct (...) causal story, however; a good deal of background causal knowledge is required as well. These claims are illustrated by a detailed example of what causes the Lamb dip in gas lasers. (shrink)
I outline an intrinsic (coordinate-free) formulation of classical particle mechanics, making no use of set theory or second-order logic. Physical quantities are accepted as real, but are constrained only by elementary axioms. This contrasts with the formulations of Field and Burgess, in which space-time regions are accepted as real and are assumed to satisfy second-order comprehension axioms. The present formulation is both logically simpler and physically more realistic. The theory is finitely axiomatizable, elementary, and even quantifier-free, but is provably empirically (...) equivalent to the standard coordinate formulations. (shrink)
If physicalism is true, everything is physical. In other words, everything supervenes on, or is necessitated by, the physical. Accordingly, if there are logical/mathematical facts, they must be necessitated by the physical facts of the world. In this paper, I will sketch the first steps of a physicalist philosophy of mathematics; that is, how physicalism can account for logical and mathematical facts. We will proceed as follows. First we will clarify what logical/mathematical facts actually are. Then, we (...) will discuss how these facts can be accommodated in the physicalist ontology. This might sound like immanent realism (as in Mill, Armstrong, Kitcher, or Maddy), according to which the mathematical concepts and propositions reflect some fundamental features of the physical world. Although, in my final conclusion I will claim that mathematical and logical truths do have contingent content in a sophisticated sense, and they are about some peculiar part of the physical world, I reject the idea, as this thesis is usually understood, that mathematics is about the physical world in general. In fact, I reject the idea that mathematics is about anything. In contrast, the view I am proposing here will be based on the strongest formalist approach to mathematics. (shrink)
Remembering Fermi MORREL H. COHEN Department of Physics and Astronomy, Rutgers University Frelinghuysen Road. Piscataway, NJ 08854-8019 USA and Department ...
Within this 1963 text, Professor Watson writes as a physicist seeking to understand how it is that physics goes on at an ever increasing pace to reveal new ...
This volume is about searching for fundamental theory in physics which has become somewhat elusive in recent decades. Like a group of blind men investigating an elephant, one physicist postulates the trunk as a hose, another a leg as a tree, the body a wall or barrier, the tail a rope and the ears as a fan. The organizers of the Vigier series symposia strongly believe cross polination by exploring many avenues of seemingly disparate research is key to breakthrough (...) discovery and solicited papers on all areas of physics deemed pertinent in Astrophysics, Cosmology, nuclear physics, quantum theory, electromagnetism, thermodynamics, vacuum field theory and topology. (shrink)
The present paper examines the role of exact results in the theory of many‐body physics, and specifically the example of the Mermin‐Wagner theorem, a rigorous result concerning the absence of phase transitions in low‐dimensional systems. While the theorem has been shown to hold for a wide range of many‐body models, it is frequently ‘violated’ by results derived from the same models using numerical techniques. This raises the question of how scientists regulate their theoretical commitments in such cases, given that (...) the models, too, are often described as approximations to the underlying ‘full’ many‐body problem. (shrink)
Sir Arthur Eddington, the celebrated astrophysicist, made great strides towards his own 'theory of everything'in his last two books published in 1936 and 1946. Unlike his earlier lucid and authoritative works, these are strangely tentative and obscure - as if he were nervous of the significant advances that he might be making. This volume examines both how Eddington came to write these uncharacteristic books - in the context of the physics and history of the day - and what value (...) they have to modern physics. The result is an illuminating description of the development of theoretical physics, in the first half of the twentieth century, from a unique point of view: how it affected Eddington's thought. This will provide fascinating reading for scholars in the philosophy of science, theoretical physics, applied mathematics and the history of science. (shrink)
The notion of order as a universal and fundamental conceptual category is discussed as being based on sets of similar differences and different similarities. A discussion of relationships between order and disorder is followed by a proposal for a mathematical theory based on non-ordinality which could also have relevance for indistinguishables in physics.
This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of the laws (...) of physics governing these processes. In accordance with the physicalist understanding of mind, this is true even if the operations in question are executed in the head. A truth obtained through (mathematical) reasoning is, therefore, an observed outcome of a neuro-physiological (or other physical) experiment. Consequently, deduction is nothing but a particular case of induction. (shrink)
This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical (...) modelling, the formation of mathematics on the basis of internal mathematical principles and the mathematical theory of experiment. (shrink)
This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of the laws (...) of physics governing these processes. In accordance with the physicalist understanding of mind, this is true even if the operations in question are executed in the head. A truth obtained through (mathematical) reasoning is, therefore, an observed outcome of a neuro-physiological (or other physical) experiment. Consequently, deduction is nothing but a particular case of induction. (shrink)
Written by Sadri Hassani, the author of several mathematicalphysics textbooks, this work covers the essentials of modern physics, in a way that is as thorough ...
The beginning of the journey -- What this book is about : using ideas from mathematics, economics, and physics to tackle the big questions in philosophy : what is real? what can we know? what is the difference between right and wrong? and how should we live? -- Reality and unreality -- On what there is -- Why is there something instead of nothing? the best answer I have : mathematics exists because it must and everything else exists because (...) it is made of mathematics, with an excursion into artificial intelligence -- Unfinished business -- Unfinished business from chapter two : the nature and purpose of economic models -- How Richard Dawkins got it wrong -- Why Dawkins's argument against intelligent design can't be right and a mathematical analysis of the arguments for the existence of God -- Belief -- Daydream believers -- Most beliefs are ill-considered because most false beliefs are costless to hold -- The next several chapters will explore the consequences of this observation before we return to the question of where our beliefs and knowledge come from -- Unfinished business -- Unfinished business from the preceding chapter : how color vision works, sound, and water waves, the sheer craziness of economic protectionism -- Do believers believe? -- Our ill-considered beliefs about religion : why I believe that almost nobody is deeply religious -- On what there obviously is -- Our ill-considered beliefs about free will, ESP, and life after death -- Diogenes's nightmare -- How is legitimate disagreement possible if you're arguing with someone who is as intelligent and informed as you are, shouldn't you put just as much weight on your opponent's arguments as your own? -- The fact that we persist in disagreeing is strong evidence that we don't really care what's true -- Knowledge -- Knowing your math -- Where mathematical knowledge comes from and logic and why evidence and logic are not enough -- Unfinished business -- Unfinished business from the preceding chapter : the tale of hercules and the hydra, with an excursion into the lore of very large numbers -- Incomplete thinking -- Godel's incompleteness theorem and what it doesn't say about the limits of human knowledge -- The rules of logic and the tale of a Potbellied pig -- The power of logical thought, with excursions into the most counterintuitive theorem in all of mathematics and the tale of a potbellied pig -- The rules of evidence -- What we can and can't learn from evidence, with excursions into the value of preschool and how internet porn prevents rape -- The limits to knowledge -- What physics does and doesn't tell us about what we can and cannot know -- Understanding Heisenberg's uncertainty principle -- Unfinished business -- The oddness of the quantum world and why it matters to game theorists -- Right and wrong -- Telling right from wrong -- Some hard questions about right and wrong and about life and death -- The economist's golden rule -- A rule of thumb for good behavior -- How to be socially responsible -- Putting the rule of thumb into practice -- On not being a jerk -- Goofus and gallant on immigration policy -- The economist on the playground -- Our ill-considered beliefs about fairness in the market place and in the voting booth, contrasted with our carefully considered beliefs about fairness on the playground -- Unfinished business -- How ancient talmudic scholars anticipated modern economic theory -- The life of the mind -- How to think -- Some basic rules for clear thinking, mostly about economics, but also about arithmetic, neurobiology, sin, and eschewing blather -- What to study -- Advice to college students : stay away from the English department and approach the philosophy department with caution, with an excursion into the remarkable life of Frank Ramsey. (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 (Grassmann / Du Bois), on the possibility of quantitative psychology (Fechner / Kries, Wundt / Zeller), and on the meaning of temperature measurement (Maxwell, Mach). Late nineteenth-century scrutinisers of the foundations of mathematics (Dedekind, Cantor, Frege, Russell) made little of Helmholtz's essay. Yet it inspired two mathematicians with an eye on (...)physics (Poincare and Holder), and a few philosopher-physicists (Mach, Duhem, Campbell). 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. (shrink)
The author focuses on the tension "realism - idealism" in the philosophy of mathematics, but he does that from the perspective of a theoretical physicist. It is not only that one's standpoint in the philosophy of mathematics determines our understanding of the effectiveness of mathematics in physics, but also the fact that mathematics is so effective in physical sciences tells us something about the nature of mathematics.
Making Sense of Inner Sense 'Terra cognita' is terra incognita. It is difficult to find someone not taken abackand fascinated by the incomprehensible but indisputable fact: there are material systems which are aware of themselves. Consciousness is self-cognizing code. During homo sapiens's relentness and often frustrated search for self-understanding various theories of consciousness have been and continue to be proposed. However, it remains unclear whether and at what level the problems of consciousness and intelligent thought can be resolved. Science's greatest (...) challenge is to answer the fundamental question: what precisely does a cognitive state amount to in physical terms? Albert Einstein insisted that the fundamental ideas of science are essentially simple and can be expressed in a language comprehensible to everyone. When one thinks about the complexities which present themselves in modern physics and even more so in the physics of life, one may wonder whether Einstein really meant what he said. Are we to consider the fundamental problem of the mind, whose understanding seems to lie outside the limits of the mind, to be essentially simple too? Knowledge is neither automatic nor universally deductive. Great new ideas are typically counterintuitive and outrageous, and connecting them by simple logical steps to existing knowledge is often a hard undertaking. The notion of a tensor was needed to provide the general theory of relativity; the notion of entropy had to be developed before we could get full insight into the laws of thermodynamics; the notice of information bit is crucial for communication theory, just as the concept of a Turing machine is instrumental in the deep understanding of a computer. To understand something, consciousness must reach an adequate intellectual level, even more so in order to understand itself. Reality is full of unending mysteries, the true explanation of which requires very technical knowledge, often involving notions not given directly to intuition. Even though the entire content and the results of this study are contained in the eight pages of the mathematical abstract, it would be unrealistic and impractical to suggest that anyone can gain full insight into the theory that presented here after just reading abstract. In our quest for knowledge we are exploring the remotest areas of the macrocosm and probing the invisible particles of the microcosm, from tiny neutrinos and strange quarks to black holes and the Big Bang. But the greatest mystery is very close to home: the greatest mystery is human consciousness. The question before us is whether the logical brain has evolved to a conceptual level where it is able to understand itself. (shrink)
Universally recognized as bringing about a revolutionary transformation of the notions of space, time, and motion in physics, Einstein's theory of gravitation, known as "general relativity," was also a defining event for 20th century philosophy of science. During the decisive first ten years of the theory's existence, two main tendencies dominated its philosophical reception. This book is an extended argument that the path actually taken, which became logical empiricist philosophy of science, greatly contributed to the current impasse over realism, (...) whereas new possibilities are opened in revisiting and reviving the spirit of the more sophisticated tendency, a cluster of viewpoints broadly termed transcendental idealism, and furthering its articulation. It also emerges that Einstein, while paying lip service to the emerging philosophy of logical empiricism, ended up siding de facto with the latter tendency. Ryckman's work speaks to several groups, among them philosophers of science and historians of relativity. Equations are displayed as necessary, but Ryckman gives the non-mathematical reader enough background to understand their occurrence in the context of his wider philosophical project. (shrink)
We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence (...) and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic. (shrink)
We give a overview of the main areas in theoretical physics, with emphasis on their relation to Lagrangian formalism in classical mechanics. This review covers classical mechanics; the road from classical mechanics to Schrodinger's quantum mechanics; electromagnetism, special and general relativity, and (very briefly) gauge field theory and the Higgs mechanism. We shun mathematical rigor in favor of a straightforward presentation.
Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified (...) presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment - such as visualization, explanation, and understanding - can nonetheless be subjected to philosophical analysis. -/- The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representation systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematicalphysics. (shrink)
The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions with (...) the related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematicalphysics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)
Wigner famously referred to the `unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of group theory to nuclear physics, I hope to indicate that this effectiveness can be seen to be not so unreasonable if attention is paid to the various idealising moves undertaken. The overall framework for analysing this relationship between mathematics and physics is that of da Costa's partial structures programme.
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 article is devoted to the nature of science. To what extent are science and mathematics affected by the society in which they are developed? Philosophy of science has accepted the social influence on science, but limits it only to the context of discovery (a "locational" approach). An opposite "attributive" approach states that any part of science may be so influenced. L. Graham is sure that even the mathematical equations at the core of fundamental physical theories may display social (...) attributes. He has used the investigations of the famous Soviet physicist V. Fock on the General Theory of Relativity which were under the influence of Marxism. The Goal of the article is to demonstrate: 1) Why Soviet science is not an appropriate subject-matter for testing the thesis of social constructivism, 2) That differnt levels of science and different stages in the development of science undergo social influences in different degrees ranging from very significant and unavoidable to absolutely trivial and easy eliminated. (shrink)
This paper examines the problem of extending the programme of mathematical constructivism to applied mathematics. I am not concerned with the question of whether conventional mathematicalphysics makes essential use of the principle of excluded middle, but rather with the more fundamental question of whether the concept of physical infinity is constructively intelligible. I consider two kinds of physical infinity: a countably infinite constellation of stars and the infinitely divisible space-time continuum. I argue (contrary to Hellman) that (...) these do not. pose any insuperable problem for constructivism, and that constructivism may have a useful new perspective to offer on physics. (shrink)
Recently, a number of philosophers of biology (e.g., Matthen and Ariew 2002; Walsh, Lewens, and Ariew 2002; Pigliucci and Kaplan 2006; Walsh 2007) have endorsed views about random drift that, we will argue, rest on an implicit assumption that the meaning of concepts such as drift can be understood through an examination of the mathematical models in which drift appears. They also seem to implicitly assume that ontological questions about the causality (or lack thereof) of terms appearing in the (...) models can be gleaned from the models alone. We will question these general assumptions by showing how the same equation – the simple (p + q)2 = p2 + 2pq + q2 – can be given radically different interpretations, one of which is a physical, causal process and one of which is not. This shows that mathematical models on their own yield neither interpretations nor ontological conclusions. Instead, we argue that these issues can only be resolved by considering the phenomena that the models were originally designed to represent and the phenomena to which the models are currently applied. When one does take those factors into account, starting with the motivation for Sewall Wright’s and R.A. Fisher’s early drift models and ending with contemporary applications, a very different picture of the concept of drift emerges. On this view, drift is a term for a set of physical processes, namely, indiscriminate sampling processes (Beatty 1984; Hodge 1987; Millstein 2002, 2005). (shrink)
It is proposed that the physical structure of an observer in quantum mechanics is constituted by a pattern of elementary localized switching events. A key preliminary step in giving mathematical expression to this proposal is the introduction of an equivalence relation on sequences of spacetime sets which relates a sequence to any other sequence to which it can be deformed without change of causal arrangement. This allows an individual observer to be associated with a finite structure. The identification of (...) suitable switching events in the human brain is discussed. A definition is given for the sets of sequences of quantum states which such an observer could occupy. Finally, by providing an a priori probability for such sets, the definitions are incorporated into a complete mathematical framework for a many-worlds interpretation. At a less ambitious level, the paper can be read as an exploration of some of the technical and conceptual difficulties involved in constructing such a framework. (shrink)
I point out that conceptions of particles as mathematical, or quasi mathematical, entities have a longer history than Resnik notices. I argue that Resnik's attack on the distinction between mathematical and physical entities is not deep enough. The crucial problem for this distinction finds its locus in the numerical indeterminancy of elementary particles. This problem, traced by Heisenberg, emerges from the discovery of antimatter.
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)
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)
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)
A green philosopher's peripeteia.--Physics and metaphysics of music.--The roots of arithmetic.--Critique of new geometrical abstractions.--The philosophical value of science.
Stimulating, thought-provoking analysis of a number of the most interesting intellectual inconsistencies in mathematics, physics and language. Delightful elucidations of methods for misunderstanding the real world of experiment (Aristotle’s Circle paradox), being led astray by algebra (De Morgan’s paradox) and other mind-benders. Some high school algebra and geometry is assumed; any other math needed is developed in text. Reprint of 1982 ed.
We are used to talking about the “structure” posited by a given theory of physics. We say that relativity is a theory about spacetime structure. Special relativity posits one spacetime structure; different models of general relativity posit different spacetime structures. We also talk of the “existence” of these structures. Special relativity says the world’s spacetime structure is Minkowskian: it posits that this spacetime structure exists. Understanding structure in this sense seems important for understanding what physics is telling us (...) about the world. But it is not immediately obvious just what this structure is, or what we mean by the existence of one structure, rather than another. The idea of mathematical structure is relatively straightforward. There is geometric structure, topological structure, algebraic structure, and so forth. Mathematical structure tells us how abstract mathematical objects t together to form different types of mathematical spaces. Insofar as we understand mathematical objects, we can understand mathematical structure. Of course, what to say about the nature of mathematical objects isn’t easy. But there seems to be no further problem for understanding mathematical structure. Modern theories of physics are formulated in terms of these mathematical structures. In order to understand “structure” as used in physics, then, it seems we must simply look at the structure of the mathematics that is used to state the physics. But it is not that simple. Physics is supposed to be telling us about the nature of the world. If our physical theories are formulated in mathematical language, using mathematical objects, then this mathematics is somehow telling us about the physical make-up of the world. What is.. (shrink)
The physicist's conception of space-time underwent two major upheavals thanks to the general theory of relativity and quantum mechanics. Both theories play a fundamental role in describing the same natural world, although at different scales. However, the inconsistency between them emerged clearly as the limitation of twentieth-century physics, so a more complete description of nature must encompass general relativity and quantum mechanics as well. The problem is a theorists' problem par excellence. Experiment provide little guide, and the inconsistency mentioned (...) above is an important problem which clearly illustrates the intermingling of philosophical, mathematical, and physical thought. In fact, in order to unify general relativity with quantum field theory, it seems necessary to invent a new mathematical framework which will generalise Riemannian geometry and therefore our present conception of space and space-time. Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new kind of geometry may enter physics, and space-time itself can be reinterpreted as an approximate, derived concept. The main purpose of this article is to show the great significance of space-time geometry in predetermining the laws which are supposed to govern the behaviour of matter, and further to support the thesis that matter itself can be built from geometry, in the sense that particles of matter as well as the other forces of nature emerges in the same way that gravity emerges from geometry. Scientific research is not a process of steady accumulation of absolute truths, which has culminated in present theories, but rather a much more dynamic kind of process in which there are no final theoretical concepts valid in unlimited domains. (David Bohm). (shrink)
Depending on different positions in the debate on scientific realism, there are various accounts of the phenomena of physics. For scientific realists like Bogen and Woodward, phenomena are matters of fact in nature, i.e., the effects explained and predicted by physical theories. For empiricists like van Fraassen, the phenomena of physics are the appearances observed or perceived by sensory experience. Constructivists, however, regard the phenomena of physics as artificial structures generated by experimental and mathematical methods. My (...) paper investigates the historical background of these different meanings of phenomenon in the traditions of physics and philosophy. In particular, I discuss Newton’s account of the phenomena and Bohr’s view of quantum phenomena, their relation to the philosophical discussion, and to data and evidence in current particle physics and quantum optics. (shrink)
Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question (...) impacts on broader issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)
A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...) Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)
Classical physics is a theory of nature that originated with the work of Isaac Newton in the seventeenth century and was advanced by the contributions of James Clerk Maxwell and Albert Einstein. Newton based his theory on the work of Johannes Kepler, who found that the planets appeared to move in accordance with a simple mathematical law, and in ways wholly determined by their spatial relationships to other objects. Those motions were apparently independent of our human observations of (...) them. (shrink)
Within the philosophy of science, the realism debate has been revitalised by the development of forms of structural realism. These urge a shift in focus from the object oriented ontologies that come and go through the history of science to the structures that remain through theory change. Such views have typically been elaborated in the context of theories of physics and are motivated by, first of all, the presence within such theories of mathematical equations that allow straightforward representation (...) of the relevant structures; and secondly, the implications of such theories for the individuality and identity of putative objects. My aim in this talk is to explore the possibility of extending such views to biological theories. An obvious concern is that within the context of the latter it is typically insisted that we cannot find the kinds of highly mathematised structures that structural realism can point to in physics. I shall indicate how the model-theoretic approach to theories might help allay such concerns. Furthermore, issues of identity and individuality also arise within biology. Thus Dupre has recently noted that there exists a ‘General Problem of Biological Individuality’ which relates to the issue of how one divides ‘massively integrated and interconnected’ systems into discrete components. In response Dupre advocates a form of ‘Promiscuous Realism’ that holds, for example, that there is no unique way of dividing the phylogenetic tree into kinds. Instead I shall urge serious consideration of those aspects of the work of Dupre and others that lean towards a structuralist interpretation. By doing so I hope to suggest possible ways in which a structuralist stance might be extended to biology. (shrink)
The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible (...) with successful representation, scientists often rely on the existence of a ‘mature mathematical formalism’, where the latter refers to a—mathematically formulated and physically interpreted—notational system of locally applicable rules that derive from (but need not be reducible to) fundamental theory. As mathematical formalisms undergo a process of elaboration, enrichment, and entrenchment, they come to embody theoretical, ontological, and methodological commitments and assumptions. Since these are enshrined in the formalism itself, they are no longer readily obvious to either the novice or the proficient user. At the same time as formalisms constrain what may be represented, they also function as inferential and interpretative resources. (shrink)
The success of particle detection in high energy physics colliders critically depends on the criteria for selecting a small number of interactions from an overwhelming number that occur in the detector. It also depends on the selection of the exact data to be analyzed and the techniques of analysis. The introduction of automation into the detection process has traded the direct involvement of the physicist at each stage of selection and analysis for the efficient handling of vast amounts of (...) data. This tradeoff, in combination with the organizational changes in laboratories of increasing size and complexity, has resulted in automated and semi-automated systems of detection. Various aspects of the semi-automated regime were greatly diminished in more generic automated systems, but turned out to be essential to a number of surprising discoveries of anomalous processes that led to theoretical breakthroughs, notably the establishment of the Standard Model of particle physics. The automated systems are much more efficient in confirming specific hypothesis in narrow energy domains than in performing broad exploratory searches. Thus, in the main, detection processes relying excessively on automation are more likely to miss potential anomalies and impede potential theoretical advances. I suggest that putting substantially more effort into the study of electron–positron colliders and increasing its funding could minimize the likelihood of missing potential anomalies, because detection in such an environment can be handled by the semi-automated regime—unlike detection in hadron colliders. Despite virtually unavoidable excessive reliance on automated detection in hadron colliders, their development has been deemed a priority because they can operate at currently highest energy levels. I suggest, however, that a focus on collisions at the highest achievable energy levels diverts funds from searches for potential anomalies overlooked due to tradeoffs at the previous energy thresholds. I also note that even in the same collision environment, different research strategies will opt for different tradeoffs and thus achieve different experimental outcomes. Finally, I briefly discuss current searches for anomalous process in the context of the previous analysis. (shrink)
The rationality of scientific concept formation in theory transitions, challenged by the thesis of semantic incommensurability, can be restored by theChains of Meaning approach to concept formation. According to this approach, concepts of different, succeeding theories may be identified with respect to referential meaning, in spite of grave diversity of the mathematical structures characterizing them in their respective theories. The criterion of referential identity for concepts is that they meet a relation ofsemantic embedding, i.e. that the embedding concept can (...) be substituted by the embedded one in classical limit situations. Three case studies from contemporary physics theories will be used to show that the Chains of Meaning approach not only yields meaning comparisons for already established concepts (as for Newtonian and Schwarzschild mass) but is also well suited to characterize actual scientific strategies of concept formation in yet open cases such as black hole entropy or relativistic thermodynamics. (shrink)
Symmetries in physics are most commonly recognized and discussed in terms of their function in the mathematical formalism of the theories. Discussion of the observation of symmetries in nature is less common. This paper analyses the observation of particular symmetries such as Lorentz and gauge symmetries, distinguishing between direct observation of the symmetry itself and indirect evidence, the latter being the observation of some consequence of the symmetry are, in an important sense, directly observed, while local symmetries such (...) as gauge symmetry are not. (shrink)
: Some of the great physicists' belief in the existence of a connection between the aesthetical features of a theory (such as beauty and simplicity) and its truth is still one of the most intriguing issues in the aesthetics of science. In this paper I explore the philosophical credibility of a version of this thesis, focusing on the connection between the mathematical beauty and simplicity of a theory and its truth. I discuss a heuristic interpretation of this thesis, attempting (...) to clarify where the appeal of this Pythagorean view comes from and what are the arguments favoring its acceptance or rejection. Along the way, I sketch the historical context in which this heuristic interpretation gained credibility (the quantum crisis in physics in the 1920s), as well as the more general implications of this thesis for physicists' metaphysical outlook. (shrink)
A personal account is presented for the present status of mathematical chemistry, with emphasis on non-numerical applications. These use mainly graph-theoretical concepts. Most computational chemical applications involve quantum chemistry and are therefore largely reducible to physics, while discrete mathematical applications often do not. A survey is provided for opinions and definitions of mathematical chemistry, and then for journals, books and book series, as well as symposia of mathematical chemistry.
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, ( I ) the first reference to astronomy at 193b25–26 and ( II ) (...) 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 ( I ) 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 ( II ), 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. (Published Online August 10 2006) Footnotes1 This paper was prepared as the basis of a presentation at a conference entitled “Writing and rewriting the history of science, 1900–2000,” Les Treilles, France, September, 2003, organized by Karine Chemla and Roshdi Rashed. I have compared Aristotle's and Ptolemy's views of the relationship between astronomy and physics in a paper called “Astrologogeômetria and astrophysikê in Aristotle and Ptolemy,” presented at a conference entitled “Physics and mathematics in Antiquity,” Leiden, The Netherlands, June, 2004, organized by Keimpe Algra and Frans de Haas. For a discussion of Hellenistic views of this relationship see Ian Mueller, “Remarks on physics and mathematical astronomy and optics in Epicurus, Sextus Empiricus, and some Stoics,” in Philippa Lang (ed.), Re-inventions: Essays on Hellenistic and Early Roman Science, Apeiron 37, 4 (2004): 57–87. I would like to thank two anonymous readers of this essay for meticulous corrections and thoughtful suggestions, almost all of which I readily adopted. (shrink)
An influential position in the philosophy of biology claims that there are no biological laws, since any apparently biological generalization is either too accidental, fact-like or contingent to be named a law, or is simply reducible to physical laws that regulate electrical and chemical interactions taking place between merely physical systems. In the following I will stress a neglected aspect of the debate that emerges directly from the growing importance of mathematical models of biological phenomena. My main aim is (...) to defend, as well as reinforce, the view that there are indeed laws also in biology, and that their difference in stability, contingency or resilience with respect to physical laws is one of degrees, and not of kind. In order to reach this goal, in the next sections I will advance the following two arguments in favor of the existence of biological laws, both of which are meant to stress the similarity between physical and biological laws. (shrink)
Co-authored by a mathematical physicist and a philosopher of science, this book is a welcome addition to the growing literature in the foundations of thermodynamics and statistical mechanics. A large and inter-disciplinary book, it contains an impressive range of information about the history, philosophy, and mathematics of thermostatistical physics. Fourteen chapters of physics and history of physics are sandwiched between two more philosophical chapters on the nature of theories and models. Throughout these middle chapters the authors (...) describe, more or less in historical order, a range of topics, including thermodynamics, kinetic theory, probability theory, ergodicity, phase transitions and more. The concentration is on the mathematical models used to describe physical systems. Usually the models are set in their historical context before being described. The necessary mathematical facts are either introduced or relegated to one of the five appendices. References for further reading are copious: there are almost sixty pages of references. (shrink)
Summary Some case studies in elementary particle physics are presented in this work, that can be used for the critical appraisal of specific criteria which were proposed to account for the development of Heisenberg's work. It is attempted to define the philosophical problems associated with and emerging from the structures of theories, rather than analyse the philosophical aspects of concepts used in elementary particle physics. This necessitates the discussion of the relationship between theory and experiment, and the role (...) of the mathematical framework being used. The question of elementarity is studied in this connection. Popper's tetradic schema and his concept of world three are modified, and it is shown that the new schema can accommodate the development of certain research programs in a much more satisfactory manner. As a result of the systematic treatment of specific research programs, the principle of contextual re-interpretation is expanded by specifying the different kinds of interpretations at the various stages of development of theories. It is also shown that principles such as those of simplicity and beauty can be understood in terms of the mathematical structure of the theories, and their methodological importance is emphasized. (shrink)
Self-organized criticality (SOC) is based upon the idea that complex behavior can develop spontaneously in certain multi-body systems whose dynamics vary abruptly. This book is a clear and concise introduction to the field of self-organized criticality, and contains an overview of the main research results. The author begins with an examination of what is meant by SOC, and the systems in which it can occur. He then presents and analyzes computer models to describe a number of systems, and he explains (...) the different mathematical formalisms developed to understand SOC. The final chapter assesses the impact of this field of study, and highlights some key areas of new research. The author assumes no previous knowledge of the field, and the book contains several exercises. It will be ideal as a textbook for graduate students taking physics, engineering, or mathematical biology courses in nonlinear science or complexity. (shrink)
Classical physics states that physical reality is local--a point in space cannot influence another point beyond a relatively short distance. However, In 1997, experiments were conducted in which light particles (photons) originated under certain conditions and traveled in opposite directions to detectors located about seven miles apart. The amazing results indicated that the photons "interacted" or "communicated" with one another instantly or "in no time." Since a distance of seven miles is quite vast in quantum physics, this led (...) physicists to an extraordinary conclusion--even if experiments could somehow be conducted in which the distance between the detectors was half-way across the known universe, the results would indicate that interaction or communication between the photons would be instantaneous. What was revealed in these little-known experiments in 1997 is that physical reality is non-local--a discovery that Robert Nadeau and Menas Kafatos view as "the most momentous in the history of science." In The Non-Local Universe, Nadeau and Kafatos offer a revolutionary look at the breathtaking implications of non-locality. They argue that since every particle in the universe has been "entangled" with other particles like the two photons in the 1997 experiments, physical reality on the most basic level is an undivided wholeness. In addition to demonstrating that physical processes are vastly interdependent and interactive, they also show that more complex systems in both physics and biology display emergent properties and/or behaviors that cannot be explained in the terms of the sum of parts. One of the most startling implications of non-locality in human terms, claim the authors, is that there is no longer any basis for believing in the stark division between mind and world that has preoccupied much of western thought since the seventeenth century. And they also make a convincing case that human consciousness can now be viewed as emergent from and seamlessly connected with the entire cosmos. In pursuing this groundbreaking argument, the authors not only provide a fascinating history of developments that led to the discovery of non-locality and the sometimes heated debate between the great scientists responsible for these discoveries. They also argue that advances in scientific knowledge have further eroded the boundaries between physics and biology, and that recent studies on the evolution of the human brain suggest that the logical foundations of mathematics and ordinary language are much more similar than we previously imagined. What this new knowledge reveals, the authors conclude, is that the connection between mind and nature is far more intimate than we previously dared to imagine. What they offer is a revolutionary look at the implications of non-locality, implications that reach deep into that most intimate aspect of humanity--consciousness. (shrink)
This book successfully achieves to serve two different purposes. On the one hand, it is a readable physics-based introduction into the philosophy of science, written in an informal and accessible style. The author, himself a professor of physics at the University of Notre Dame and active in the philosophy of science for almost twenty years, carefully develops his metatheoretical arguments on a solid basis provided by an extensive survey along the lines of the historical development of physics. (...) On the other hand, this book supplies one long argument for Cushing´s own attitude in the philosophy of science. While former studies of the author, from which this book draws in part, focused each on one special episode in the history of science, this book gathers case material from many different parts of physics and epochs. The main goal of this book is ”to impress upon the reader the essential and ineliminable role that philosophical considerations have played in the actual practice of science” (p. xv). The book is beautifully edited and produced; it contains a wealth of illustrative figures, well-chosen short quotations from original sources and contemporary commentators (some longer quotations are relegated in an appendix at the end of a chapter) and does not dispense with insightful mathematical arguments in the main text (some advanced deductions are, however, relegated in the appendices). It contains nine parts, whereas only the first and the last one are exclusively devoted to philosophical issues. The seven remaining parts, each subdivided into three chapters, centre around one major episode (a theory, a world view, etc.) in the history of physics. The author presents this material in a clear and philosophically unbiased way so that also readers who do not share Cushing’s subsequent philosophical conclusions will find this inspiring book extremely useful. Part 1 (”The scientific enterprise”) discusses some traditional (”objectivist”) views concerning the status of scientific knowledge, ”the” scientific method, and the relation.... (shrink)
Quantum physics is believed to be the fundamental theory underlying our understanding of the physical universe. However, it is based on concepts and principles that have always been difficult to understand and controversial in their interpretation. This book aims to explain these issues using a minimum of technical language and mathematics. After a brief introduction to the ideas of quantum physics, the problems of interpretation are identified and explained. The rest of the book surveys, describes and criticises a (...) range of suggestions that have been made with the aim of resolving these problems; these include the traditional, or 'Copenhagen' interpretation, the possible role of the conscious mind in measurement, and the postulate of parallel universes. This new edition has been revised throughout to take into account developments in this field over the past fifteen years, including the idea of 'consistent histories' to which a completely new chapter is devoted. (shrink)
Modern economics, with its use of advanced mathematical methods, is often looked upon as the physics of the social sciences. It is here argued that deductive analyses are more important in economics than in physics, because the economists more seldom can confirm phenomenological laws directly. The economist has to use assumptions from fundamental theory when trying to bridge the gap between observations and phenomenological laws. Partly as a result of the difficulties of establishing phenomenological laws, analyses of (...) idealized 'model-economies' play a more important, but mainly heuristic role, in economics. (shrink)
This paper explores the work of Nicolas Rashevsky, a Russian émigré theoretical physicist who developed a program in "mathematical biophysics" at the University of Chicago during the 1930s. Stressing the complexity of many biological phenomena, Rashevsky argued that the methods of theoretical physics -- namely mathematics -- were needed to "simplify" complex biological processes such as cell division and nerve conduction. A maverick of sorts, Rashevsky was a conspicuous figure in the biological community during the 1930s and early (...) 1940s: he participated in several Cold Spring Harbor symposia and received several years of funding from the Rockefeller Foundation. However, in contrast to many other physicists who moved into biology, Rashevsky's work was almost entirely theoretical, and he eventually faced resistance to his mathematical methods. Through an examination of the conceptual, institutional, and scientific context of Rashevsky's work, this paper seeks to understand some of the reasons behind this resistance. (shrink)
This book is the first to offer a systematic account of the role of language in the development and interpretation of physics. An historical-conceptual analysis of the co-evolution of physics and mathematics leads to the classical/quantum interface. Bohr's interpretation is analyzed and extended to the interpretation of the standard model of particle physics.
The two main points of this contribution are the following: (1) Applied mathematical theories might complement physical theories in an essential way; some applied mathematical theories allow us to understand phenomena we are unable to explain by resorting to physical theories alone, (2) In the case of social sciences it might be necessary to account for examined phenomena by resorting to the idea of goal-oriented activity (the causal approach typical for natural science might be unsatisfactory). Weinberg's idea of (...) grand reductionism ignores the two above mentioned facts and hence overestimates the foundational role of physics and its methodology. (shrink)
