Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can (...) be also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

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

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

The practice of mathematics applied to physics, more particularly mathematical physics, leads to demands for explicitness and rigor. The features of this practice are in effect unavoidable—although derivations of the same fact may go by different routes. Rigor matters substantively.

The 2022 Prizes from the International Mathematical Union featured two focusing on mathematical physics: a Fields Medal to Hugo Duminil-Copin, and the Carl Friedrich Gauss Prize for Applications of Mathematics to Elliott H. Lieb. Duminil-Copin (and his teacher, Stanislav Smirnov, 2010 Fields Medal) developed rigorous ways of thinking of a lattice at its critical local connection probability that leads to a path through the lattice (Fig. 2.1). They employed a discrete complex analysis. Then, they develop ingenious ways (...) of describing that lattice at the critical point, ways amenable to rigorous mathematical proof. In a lifetime of work, Lieb has employed mathematics as the usual physicists’ starting point to reveal the physics. For example, Lieb and collaborators (1964) showed how that Ising lattice might be thought of as a quantum field theory (with annihilation and creation operators). Themes are: Technique is physical. Rigor is revealing. Tricks are physical. And, mathematical physics often leads to deep mathematics. (shrink)

In the ten years following the publication of Galileo Galilei's Discorsi e dimostrazioni matematiche intorno a due nuove scienze , the new science of motion was intensely debated in Italy, France and northern Europe. Although Galileo's theories were interpreted and reworked in a variety of ways, it is possible to identify some crucial issues on which the attention of natural philosophers converged, namely the possibility of complementing Galileo's theory of natural acceleration with a physical explanation of gravity; the legitimacy of (...) Galileo's methods of proof and of his theory of the composition of continuous magnitudes; and the adequacy of the experimental evidence in favor of his theory.Through his published works and his correspondence, Marin Mersenne contributed in a significant way to each of these discussions. In the following, I shall provide a sketch of Mersenne's multiple engagement with the new science of motion, while trying to account for his growing skepticism concerning Galileo's law of free fall. Whereas in the 1630s, Mersenne still firmly believed in the validity of this law and tried to devise experimental and mathematical proofs in its favor, in the 1640s he developed serious doubts regarding the possibility of constructing an exact science of motion. As we shall see, Mersenne's evolving attitude sheds an interesting light on his views concerning the relation between physics and mathematics.---------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ----------------------------------------------------. (shrink)

Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this article, I (...) show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes' mathematical physics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematical physics. Thus, in Descartes' physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact, Descartes establishes the reality and limitlessness of the extension of the cosmos and, by extension, the actual nature of his indefinite world. This indefinite has a physical dimension, even if it is not measurable. La filosofía de Descartes contiene una noción intrigante de lo infinito, un concepto nombrado por el filósofo como indefinido. Aunque en varias ocasiones Descartes definió claramente este término en su correspondencia con sus contemporáneos y en sus Principios de filosofía, han surgido muchos problemas acerca de su significado a lo largo de los años. La mayoría de comentaristas rechaza la idea de que indefinido podría significar una cosa real y, en cambio, la identifica con un infinito potencial aristotélico. En la primera parte de este artículo muestro por qué no hay infinito numérico en las matemáticas cartesianas, en la medida en que tal concepto sería inconsistente con el principal atributo fundamental de los números: ser comparables entre sí. En la segunda parte analizo lo indefinido en el contexto de la física matemática de Descartes. Mi argumento es que, aunque no hay rastro de infinito en sus matemáticas, Descartes se refiere a un indefinido real a causa de sus aplicaciones al mundo material dentro del sistema de su física. Este hecho subraya una discrepancia entre sus matemáticas y su física de lo infinito, pero no implica ninguna dificultad en su física matemática. Así pues, en la física de Descartes, lo indefinido se refiere a una dimensión real del mundo más que a una infinitud potencial matemática aristotélica. De hecho, Descartes establece la realidad e infinitud de la extensión del cosmos y, por extensión, la naturaleza real de su mundo indefinido. Esta indefinición tiene una dimensión física aunque no sea medible. (shrink)

I undertake to examine how Descartes understood the relationship between physics and mathematics. My thesis is that what distinguishes the objects of mathematics from those of physics on Descartes's view is that the former are considered in abstraction from a material substratum while the latter are considered as involving a material substratum. Since it has often been maintained that Descartes identified matter with extension, and hence rejected the notion of a material substratum, I attempt in the first part (...) of my paper to establish the textual basis for ascribing a substratum view to Descartes. In the second part of the paper, I present the case for my thesis concerning Descartes's account of the difference between the objects of mathematics and those of physics. (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)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.2 (2001) 193-210 [Access article in PDF] Zeno Against Mathematical Physics Trish Glazebrook Galileo wrote in The Assayer that the universe "is written in the language of mathematics," and therein both established and articulated a foundational belief for the modern physicist. 1 That physical reality can be interpreted mathematically is an assumption so fundamental to modern physics that chaos and (...) super-strings are examples of physical theories developed on the basis of success in mathematics. The mathematics came first, then the physical theory. Quantum theory is an awkward and similar case in point. Despite the fact that there is much agreement among physicists about the mathematical theory, its physical interpretation remains a matter of controversy. There are several interpretations, all of which challenge our everyday assumptions about reality. This led Bohr in 1935 to call for "a radical revision of our attitude towards the problem of physical reality." 2 Arthur Fine, originally a staunch defender of realist interpretations of quantum theory, saw the success of Bohr's Copenhagen interpretation as the end of Einsteinian realism, and subsequently Fine declared that "realism is well and truly dead." 3 Mathematical operators simply do not correspond with real entities in any conceivable way.Ilkka Niiniluoto has argued convincingly, however, that realism is alive and well in quantum theory. 4 The problem is that Bohr is right: quantum theory can be considered to concern the real only upon radical revision of ordinary conceptions of reality. Quantum theory does not describe the real in any meaningful sense of "the real." Rather, it is mathematical projection: ideal rather than real. [End Page 193]Aristotle did not hold that the physicist deciphers nature mathematically. He distinguished the physicist from the mathematician at Physics 2.2 by analogy to the definitions of "snub nose" and "curved." 5 The definition of "snub nose" requires the mention of a nose, that is, of matter which is inseparable from motion, whereas "curved" and other mathematical concepts like " 'odd' and 'even', 'straight'... 'number,' 'line,' and 'figure,' " which are separable from matter and from motion, are investigated by the mathematician. 6 Aristotle claims that physics is different from mathematics because the mathematician investigates physical lines, but not qua physical, and the physicist investigates mathematical lines "but qua physical, not qua mathematical." 7A peculiar consequence of my argument is that it establishes a common principle between two thinkers whose beliefs are otherwise so fundamentally in opposition. Zeno's paradoxes are commonly read as a denial of motion. For Aristotle, that move is axiomatic to the physicist. At 185a15 he considers arguments against this assumption outside the realm of objections to which the physicist must reply. If I am right that Zeno's paradoxes of motion challenge the very notion of mathematical physics, then he and Aristotle agree, contra Galileo, that the universe is not written in the language of mathematics. They simply come to this point from opposite directions, Zeno from an Eleatic thesis that physical reality is illusion, Aristotle from the natural scientist's pragmatic commitment to a truth about nature.I intend to use Zeno's paradoxes of motion to argue that the application of mathematical concepts to the physical world results in paradox. Zeno's arguments are limited within mathematics to geometry. This paper is accordingly a beginning from which I hope more work can be done later. Furthermore, Zeno's paradoxes are standardly read as directed against anyone who disagrees with Parmenides' thesis that there is but one thing which is motionless and indivisible. The paradoxes achieve this end by means of a reductio against the possibility of dividing space and time. That there are four paradoxes is accordingly explicable by the fact that there are four possibilities for the divisibility of space and time. The Achilles argues against the possibility that space and time are both infinitely divisible; the Dichotomy, against the infinite divisibility of space and finite divisibility of time; the Arrow, against the infinite divisibility of time and the finite divisibility of space; and finally, the Stadium shows that time and space cannot both... (shrink)

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

Die Arbeit schlägt eine beweistheoretische Analyse der mathematischen Physik im Gegensatz zu gegenwärtigen modelltheoretischen Ansätzen vor. Über eine oberflächliche Analogie hinaus haben beweistheoretische Techniken und Renormalisationsverfahren ein gemeinsames Ziel: die Ausschaltung von Unendlichkeiten in einer konsistenten Theorie. Die Geschichte der Renormalisation in Quantenfeldtheorien wird kurz skizziert und eine allgemeine These über die Natur und Justizfizierung von Theorien in der mathematischen Physik vorgeschlagen. Wir schließen mit den Grundlinien für ein Forschungsprogramm für eine physikalische Logik.

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

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

We consider the answer to this question to be the central problem of the interpretation of Descartes’ scientific work. Schuster declared Descartes’ physics, because of its verbal character, to be natural philosophy, and excluded it from the tradition of mathematical physics, which he associates with Galileo and Newton. This assessment is mistaken, and the reason for the mistake is the vagueness of the concept of mathematical physics. Galileo’s physics, which completely lacks any description of (...) interaction, which attributes to bodies a natural tendency to accelerate in free fall, and which considers circular motion as inertial, is automatically taken to be mathematical physics. On the other hand Descartes who introduced the notion of interaction into physics, who clearly saw that the acceleration of falling bodies must be the result of interaction, and who regarded only uniform rectilinear motion to be inertial, is not included in mathematical physics. My aim is to correct this obvious error. (shrink)

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 mathematical physics 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 mathematical physics 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 mathematical physics 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 mathematical physics. (shrink)

Many philosophers, physicists, and mathematicians have wondered about the remarkable relationship between mathematics with its abstract, pure, independent structures on one side, and the wilderness of natural phenomena on the other. Famously, Wigner found the "effectiveness" of mathematics in defining and supporting physical theories to be unreasonable, for how incredibly well it worked. Why, in fact, should these mathematical structures be so well-fitting, and even heuristic in the scientific exploration and discovery of nature? This book argues that the effectiveness (...) of mathematics in physics is reasonable. The author builds on useful analogies of prime numbers and elementary particles, elementary structure kinship and the structure of systems of particles, spectra and symmetries, and for example, mathematical limits and physical situations. The two-dimensional Ising model of a permanent magnet and the proofs of the stability of everyday matter exemplify such effectiveness, and the power of rigorous mathematical physics. Newton is our original model, with Galileo earlier suggesting that mathematics is the language of Nature. (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)

Many mathematical or physical paper would seem to magically go from one line to the next, the reader unable to figure out the logic of the transition. Such legerdemain, whether it be magical or in doing physics, is no less impressive if you know how it is done, for you yourself would have to train extensively to actually perform these sleights of hand. What you might have ignored turns out to have needed informed careful attention. Yet, to be (...) struck by legerdemain you must have actually read the paper, so that the device or method would stop you cold. Where did that come from? How do you get from line A to line A + 1? (I should note that much of the discussion below will benefit from having those papers in front of you.). (shrink)

This book explains how limitations in the movement and perception of information constrain human behavior, cognition, interaction, and perspective. How fast can we learn? How much? Why are habits and biases unavoidable? Aspects considered include: how much information can one human absorb in a lifetime? How far does a process of perturbation propagate? How do specialization or generalization, critical thinking or belief, influence what people accomplish? It is aimed at general readers and scientists with an interest in how limitations of (...) the physical sciences affect society and human behavior. (shrink)

The Baconian method had few followers at first. The attention of philosophers was soon riveted on Galileo Galilei, who had trained the telescope on the heavens and discovered many surprising things there. He championed the Copernican system of the planets, describing the earth as a moving planet. He also proposed a new conception of geometry, which underlies modern metaphysics, describing a mathematical universe through and through that undercut the ancient conception of a qualitative and finite cosmos. Together with these (...) innovations, he proposed a new method of gaining knowledge, according to which we can know some mathematical principles as certainly as God knows them. However, our intensive knowledge is limited to a few Principles, whereas God’s is extensive and complete. In this chapter, we see how the problem of knowledge that Aristotle raised in his Posterior Analytics can be solved by a foundational method proposed by Galileo Galilei, using thought experiments and a reductio ad absurdum technique. Although Galileo Galilei’s remarks were suggestive and not systematically developed, they immensely influenced later natural philosophers, thus overshadowing Francis Bacon’s method for some time. (shrink)

The aim of the paper is this: Instead of presenting a provisional and necessarily insufficient characterization of what mathematical physics is, I will ask the reader to take it just as that, what he or she thinks or believes it is, yet to be prepared to revise his opinion in the light of what I am going to tell. Because this is precisely, what I intend to do. I will challenge some of the received or standard views about (...) mathematical physics and replace them by a more sophisticated picture, which takes into account the methodological and philosophical roots of mathematical physics in Göttingen. (shrink)

This “fun, brain-twisting book... will make you think” as it explores more than 75 paradoxes in mathematics, philosophy, physics, and the social sciences (Sean Carroll, New York Times–bestselling author of Something Deeply Hidden) Paradox is a sophisticated kind of magic trick. A magician’s purpose is to create the appearance of impossibility, to pull a rabbit from an empty hat. Yet paradox doesn’t require tangibles, like rabbits or hats. Paradox works in the abstract, with words and concepts and symbols, to (...) create the illusion of contradiction. There are no contradictions in reality, but there can appear to be. In Sleight of Mind, Matt Cook and a few collaborators dive deeply into more than 75 paradoxes in mathematics, physics, philosophy, and the social sciences. As each paradox is discussed and resolved, Cook helps readers discover the meaning of knowledge and the proper formation of concepts—and how reason can dispel the illusion of contradiction. The journey begins with “a most ingenious paradox” from Gilbert and Sullivan’s Pirates of Penzance. Readers will then travel from Ancient Greece to cutting-edge laboratories, encounter infinity and its different sizes, and discover mathematical impossibilities inherent in elections. They will tackle conundrums in probability, induction, geometry, and game theory; perform “supertasks”; build apparent perpetual motion machines; meet twins living in different millennia; explore the strange quantum world—and much more. (shrink)

In "On the Notion of Cause," Bertrand Russell expressed an eliminativist view about causation driven by an examination of the contents of mathematical physics. Russell's primary reason for thinking that the notion of causation is absent in physics was that laws of nature are mere "functional dependencies" and not "causal laws." In this paper, I show that several ordinary notions of causation can be found within the functional dependencies of physics. Not only does this show that (...) Russell's eliminitivism was misguided, but it shows that Russell's opponents, such as Nancy Cartwright, who think that mere functional dependenciescannot capture causal claims, also underestimate the causal content of such equations. (shrink)

On December 10th, 1947, John von Neumann wrote to the Spanish translator of his Mathematical Foundations of Quantum Mechanics: 1Your questions on the nature of mathematical physics and theoretical physics are interesting but a little difficult to answer with precision in my own mind. I have always drawn a somewhat vague line of demarcation between the two subjects, but it was really more a difference in distribution of emphases. I think that in theoretical physics the (...) main emphasis is on the connection with experimental physics and those methodological processes which lead to new theories and new formulations, whereas mathematical physics deals with the actual solution and mathematical execution of a theory which is assumed to be correct per se, or assumed to be correct for the sake of the discussion. In other words, I would say that theoretical physics deals rather with the formation and mathematical physics rather with the exploitation of physical theories. However, when a new theory has to be evaluated and compared with experience, both aspects mix. (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.

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

In 1730, the Dutch cartographer and meteorological observer Nicolaas Samuel Cruquius constructed a spectacular map of the river Merwede. Cruquius’s map is celebrated as one of the earliest to use lines of equal depth—or indeed any type of contour lines. So far, however, the secondary literature has paid no attention to why Cruquius created these lines or to the knowledge involved in his innovation. This essay makes three related points. First, Cruquius intentionally used lines representing equal depth in an entirely (...) new way: whereas earlier maps had used similar lines to represent a river’s navigability, for Cruquius they visualized river flow. Second, different technical and epistemic practices grounded the making of the map: Cruquius built on his expertise in surveying but also on practices from both natural history and mathematical physics, with which he was familiar through contacts at Leiden University and the Royal Society of London. Third, Cruquius’s case does not fit with the way work cutting across different fields of expertise is usually conceptualized. Cruquius did not just straddle different fields. Instead, he gave new meaning to lines of equal depth by integrating specific practices from different fields of expertise and omitting others; through this integration, Cruquius created a way to visualize water flows and gave shape to possible new ways of acting on the river and its environment. (shrink)

The emergence of the term ‘Plebanski’ as a topic trend in the scientific literature is studied as a significant communication event resulting from its use by authors to refer to the relevant aspects of Jerzy Plebanski scientific work in the area of mathematical physics. We searched the ‘Plebanski’ topic included in the titles, abstracts and key words of the papers registered in five databases: ADS/NASA, MathSciNet, SCOPUS, SPIRES and Web of Science. Our results clearly show the evolution of (...) the JP’s scientific work from a cited reference to a recognized term in the indexes of the published literature, then to a nodal concept in the scientific communication process and finally as an eponym in the Latin American scholar literature. (shrink)

The dispute over the viability of various theories of relativistic, dissipative fluids is analyzed. The focus of the dispute is identified as the question of determining what it means for a theory to be applicable to a given type of physical system under given conditions. The idea of a physical theory's regime of propriety is introduced, in an attempt to clarify the issue, along with the construction of a formal model trying to make the idea precise. This construction involves a (...) novel generalization of the idea of a field on spacetime, as well as a novel method of approximating the solutions to partial-differential equations on relativistic spacetimes in a way that tries to account for the peculiar needs of the interface between the exact structures of mathematical physics and the inexact data of experimental physics in a relativistically invariant way. It is argued, on the basis of these constructions, that the idea of a regime of propriety plays a central role in attempts to understand the semantical relations between theoretical and experimental knowledge of the physical world in general, and in particular in attempts to explain what it may mean to claim that a physical theory models or represents a kind of physical system. This discussion necessitates an examination of the initial-value formulation of the partial-differential equations of mathematical physics, which suggests a natural set of conditions---by no means meant to be canonical or exhaustive---one may require a mathematical structure, in conjunction with a set of physical postulates, satisfy in order to count as a physical theory. Based on the novel approximating methods developed for solving partial-differential equations on a relativistic spacetime by finite-difference methods, a technical result concerning a peculiar form of theoretical under-determination is proved, along with a technical result purporting to demonstrate a necessary condition for the self-consistency of a physical theory. (shrink)

Edited book on the prospects of non-Platonist realism in the philosophy of mathematics. Physicalism holds that mathematics studies properties realised or realisable in the physical world. This collection of papers has its origin in a conference held at the University of Toronto in June of 1988. The theme of the conference was Physicalism in Mathematics: Recent Work in the Philosophy of Mathematics. At the conference, papers were read by Geoffrey Hellman (Minnesota), Yvon Gauthier (Montreal), Michael Hallett (McGill), Hartry Field (USC), (...) Bob Hale (Lancaster & St Andrew's), Alasdair Urquhart (Toronto) and Penelope Maddy (Irvine). This volume supplements updated versions of six of those papers with contributions by Jim Brown (Toronto), John Bigelow (La Trobe), John Burgess (Princeton), Chandler Davis (Toronto), David Papineau (Cambridge), Michael Resnik (North Carolina at Chapel Hill), Peter Simons (Salzburg) and Crispin Wright (St Andrews & Michigan). Together they provide a vivid, expansive snapshot of the exciting work which is currently being carried out in philosophy of mathematics. (shrink)

Et his principiis via sternitur ad majora. And by these principles the road is open to higher things. (Newton, quoted by Riemann in 1861)1 With Dirichlet and Riemann, Göttingen has remained the plantation of the most profoundly philosophical orientation in mathematical research that it became with Gauss. (Wilhelm Weber)2 There is no doubt that Bernhard Riemann was one of the main architects of modern mathematics, a visionary planner who delineated new outlines for quarters like complex analysis or abstract geometry, (...) and designed magnificent modern avenues to link mathematics with physics. Riemann’s work emerged from a most noteworthy interaction between the three disciplines of physics, mathematics, and philosophy – the ‘magic triangle,’ as Sánchez Ron has aptly put it in a paper about Einstein. In fact, the evolution of Riemann’s ideas affords a better example of the magic triangle at work than Einstein’s in 1905–1916. We shall examine this in the relatively localized domain of mathematical and physical geometry, but also at the more global level of Riemann’s epistemology of mathematics. (shrink)

The standard treatment of fundamental constants rests on two separate domains: physical constants such as α⁻¹ = 137.036 are measured experimentally and classified by CODATA, while mathematical constants such as π and e are defined through analytic or geometric necessity. No structural criterion distinguishes derivable constants from contingent ones, and the total number of fundamental constants has no theoretical bound. -/- This separation creates a deep asymmetry. Physicists treat α⁻¹ as a contingent empirical fact with no known reason for (...) its specific value; mathematicians treat π as logically necessary. Yet both are dimensionless, scale-invariant, and protocol-independent. The question of why they belong to different ontological categories has no answer within the standard paradigm. -/- The present paper resolves this within Operatiology by introducing Operational Invariants (操作遍数, Sousa Hensuu), dimensionless constants derivable from the common foundation C⁽³⁾_Πd realised as M₃(ℂ). An ontological trichotomy partitions all quantities into Class I (Operational Invariants), Class II (dimensional constants encoding unit system choice), and Class III (running coupling constants). A Class I Completeness Theorem is proved without circularity. -/- The Euler-Mascheroni constant γ_E is established as M₃(ℂ)-grounded as the unique structural invariant measuring the discrepancy between the discrete factorial structure of U(n) (Tier 2) and its continuous analytic extension (Tier 3), and as the compression residue of the relational closure of M₃(ℂ) into finite algebraic invariants. This enables identification of the BCS gap ratio κ_BCS = 2π/e^γ_E = 3.52775... as a new Operational Invariant. -/- Version 2 extended the framework by defining the set S of Operational Invariants without restriction to the candidate space F_ext: S = {κ | M₃(ℂ)-grounded and A4-irreducible}. S properly contains F_ext; confirmed elements of S \ F_ext include e, ln 2, α⁻¹, μ, and ξ². Each element of S is assigned a structural category: (A) M₁-algebraic; (B) M₁-condition/M₃-value (πᵏ, e); (C) projection-connection invariant (γ_E); (D) M₃-operation on M₁-element (ln 2); (E) composite. The constants √2 = γ⁻¹/² and √3 = K = √2·δ satisfy Class I observational conditions but are A4-redundant and excluded from S. The finiteness of S was proved in full, covering both S ∩ F_ext and S \ F_ext. -/- Version 3 completes the grounding structure. M₃(ℂ)-groundedness is now established via three structurally independent and jointly exhaustive conditions: (G1) volumetric grounding, governing πᵏ; (G2) normalization grounding, governing γ_E; and (G3) exponential unit grounding, newly introduced, governing the natural logarithm base e as the value of the Lie algebra exponential map at the multiplicative unit of the A2-selected field ℝ. The uniqueness of this evaluation point and the exhaustiveness of the three conditions are each proved as independent results. A new section establishes, paper by paper, the citation integrity of the companion derivations of e, μ, π, and α⁻¹ against the present version, resolving a prior citation gap with the companion paper on e and confirming the absence of impact elsewhere. -/- The distinction between physical and mathematical constants is dissolved: π, e, √2, √3, γ_E, and α⁻¹ are ontologically equivalent Class I Operational Invariants, all projectives of the common foundation C⁽³⁾_Πd. This Version 3 supersedes Version 2 (DOI: 10.5281/zenodo.20647095) in all respects. -/- Published on June 18, 2026 DOI: 10.5281/zenodo.20747691. (shrink)

Perspectival realists often appeal to the methodology of science to secure a realist account of the retention and continued success of scientific claims through the progress of science. However, in the context of modern physics, the retention and continued success of scientific claims are typically only definable within a mathematical framework. In this article, I argue that this concern leaves the perspectivist open to Cassirer’s neo-Kantian critique of the applicability of mathematics in the natural sciences. To support this (...) criticism, I present a case study on the conservation of energy in modern physics. (shrink)