ArgumentAccording to Hermann von Helmholtz, free mobility of bodies seemed to be an essential condition of geometry. This free mobility can be interpreted either as matter of fact, as a convention, or as a precondition making measurements in geometry possible. Since Henri Poincaré defined conventions as principles guided by experience, the question arises in which sense experiential data can serve as the basis for the constitution of geometry. Helmholtz considered muscular activity to be the basis on which the form of (...) space could be construed. Yet, due to the problem of illusion inherent in the subject’s self-assessment of muscular activity, this solution yielded new difficulties, in that if the manifold is abstracted from rigid bodies which serve as empirical justification of the geometrical notion of space, then illusionary bodies will produce fictive manifolds. The present article is meant to disentangle these difficulties. (shrink)

This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem (...) leads to the exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. (shrink)

A large part of the difficulty of writing "conceptual history"—to borrow a term from Reviel Netz —is that once an illuminating new conceptual framework is articulated, it begins to seem self-evident and commonsensical to later thinkers. The historian's task of problematizing the obvious, and showing us the moves by which commonsense came to be created historically, is an arduous and challenging one, requiring resources of imagination, patience, and attention to detail. Sattler displays all those qualities in this dense and demanding (...) study, and a review needs to begin by acknowledging the sheer... (shrink)

I begin by reviewing some recent work on the status of the geodesic principle in general relativity and the geometrized formulation of Newtonian gravitation. I then turn to the question of whether either of these theories might be said to ``explain'' inertial motion. I argue that there is a sense in which both theories may be understood to explain inertial motion, but that the sense of ``explain'' is rather different from what one might have expected. This sense of (...) explanation is connected with a view of theories---I call it the ``puzzleball view''---on which the foundations of a physical theory are best understood as a network of mutually interdependent principles and assumptions. (shrink)

Zeno’s paradoxes of motion, allegedly denying motion, have been conceived to reinforce the Parmenidean vision of an immutable world. The aim of this article is to demonstrate that these famous logical paradoxes should be seen instead as paradoxes of immobility. From this new point of view, motion is therefore no longer logically problematic, while immobility is. This is convenient since it is easy to conceive that immobility can actually conceal motion, and thus the proposition “immobility is (...) mere illusion of the senses” is much more credible than the reverse thesis supported by Parmenides. Moreover, this proposition is also supported by modern depiction of material bodies: the existence of a ceaseless random motion of atoms—the ‘thermal agitation’—in the scope of contemporary atomic theory, can offer a rational explanation of this ‘illusion of immobility’. Our new approach to Zeno’s paradoxes therefore leads to presenting the novel concept of ‘impermobility’, which we think is a more adequate description of physical reality. (shrink)

This paper examines the young Kant’s claim that all motion is relative, and argues that it is the core of a metaphysical dynamics of impact inspired by Leibniz and Wolff. I start with some background to Kant’s early dynamics, and show that he rejects Newton’s absolute space as a foundation for it. Then I reconstruct the exact meaning of Kant’s relativity, and the model of impact he wants it to support. I detail (in Section II and III) his polemic (...) engagement with Wolffian predecessors, and how he grounds collisions in a priori dynamics. I conclude that, for the young Kant, the philosophical problematic of Newton’s science takes a back seat to an agenda set by the Leibniz-Wolff tradition of rationalist dynamics. This results matters, because Kant’s views on motion survive well into the 1780s. In addition, his doctrine attests to the richness of early modern views of the relativity of motion. (shrink)

Philosophy -- Entry foundations -- Phenomenology of immediacy -- Polarized braids and little primoridal frames -- Emergence of primordial minkowski frames -- Majorana space-time spinors -- Color braids -- Motion and method -- Envisioned memory.

The appearance of the time derivative of the acceleration in the equation of motion (EOM) of an electric charge is studied. It is shown that when an electric charge is accelerated, a stress force exists in the curved electric field of the accelerated charge, and in the case of a constant linear acceleration, this force is proportional to the acceleration. This stress force acts as a reaction force which is responsible for the creation of the radiation (instead of the (...) “radiation reaction force” that actually does not exist at low velocities). Thus the initial acceleration should be supplied as an initial condition for the solution of the EOM of an electric charge. (shrink)

This book is intended as a historical and critical study on the origin of the equations of motion as established in Newton's Principia. The central question that it aims to answer is whether it is indeed correct to ascribe to Galileo the inertia principle and the law of falling bodies. In order to accomplish this task, the study begins by considering theories on the motion of bodies from classical antiquity, and especially those of Aristotle. The theories developed during (...) the Middle Ages and the Renaissance are then reviewed, with careful analysis of the contributions of, for example, the Merton and Parisian Schools and Galileo's immediate predecessors, Tartaglia and Benedetti. Finally, Galileo's work is examined in detail, starting from the early writings. Excerpts from individual works are presented, to allow the texts to speak for themselves, and then commented upon. The book provides historical evidence both for Galileo's dependence on his forerunners and for the major breakthroughs that he achieved. It will satisfy the curiosity of all who wish to know when and why certain laws have been credited to Galileo. (shrink)

In this paper I argue for the following two related claims. First, the science of phoronomy from Kant's Metaphysical Foundations of Natural Science is grounded in the duplication of space. Second, this duplication is made possible through the self-localization of the subject, as Kant shows in the "Gegnden-Schrift". The thesis of this paper is that the self-localization transforms space into an object that can be cinematically moved and that this action sets the ground for a science of phoronomy, which (...) presupposes the existence of at least two spaces. In the end, the other endpoint of the Kantian determination of space is discussed, namely space understood as an idea of reason. (shrink)

Konstantin Pollok - Kant's Critical Concepts of Motion - Journal of the History of Philosophy 44:4 Journal of the History of Philosophy 44.4 559-575 Muse Search Journals This Journal Contents Kant's Critical Concepts of Motion Konstantin Pollok There are two significant places in Kant's Critical corpus where he discusses the concept of motion. The first is in the Critique of Pure Reason, where in the "Deduction of the Categories" Kant writes: Motion, as an act of the (...) subject , and therefore the synthesis of the manifold in space, first produces the concept of succession—if we abstract from this manifold and attend solely to the act through which we determine the inner sense according to its form. In this passage Kant simply refers to the concept of motion, and the immediate context in which this reference occurs reveals little about what he means by it. In the Metaphysical Foundations of Natural Science, however, motion plays a more significant role, and, concomitantly, it is in his writings on natural science that Kant most fully expresses his thoughts on the subject. For present purposes, the most relevant passage is paragraph 15 of the Preface of the Metaphysical Foundations, where Kant says that the concept of matter had … to be carried through all four of the indicated functions of the concepts of the understanding , where in each a new determination of this concept was added. The.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 44.4 (2006) 559-575 MuseSearchJournalsThis JournalContents[Access article in PDF]Kant's Critical Concepts of MotionKonstantin PollokThere are two significant places in Kant's Critical corpus where he discusses the concept of motion. The first is in the Critique of Pure Reason, where in the "Deduction of the Categories" Kant writes:Motion, as an act of the subject (not as a determination of an object†), and therefore (...) the synthesis of the manifold in space, first produces the concept of succession—if we abstract from this manifold and attend solely to the act through which we determine the inner sense according to its form.In this passage Kant simply refers to the concept of motion, and the immediate context in which this reference occurs reveals little about what he means by it. In the Metaphysical Foundations of Natural Science, however, motion plays a more significant role, and, concomitantly, it is in his writings on natural science that Kant most fully expresses his thoughts on the subject. For present purposes, the most relevant passage is paragraph 15 of the Preface of the Metaphysical Foundations, where Kant says that theconcept of matter had … to be carried through all four of the indicated functions of the concepts of the understanding (in four chapters), where in each a new determination of this concept was added. The basic determination of something that is [End Page 559] to be an object of the outer senses had to be motion, because only thereby can these senses be affected. The understanding traces back all other predicates of matter belonging to its nature to this, and natural science, therefore, is either a pure or applied doctrine of motion. The metaphysical foundations of natural science are therefore to be brought under four chapters. The first considers motion as a pure quantum in accordance with its composition, without any quality of the movable, and may be called phoronomy. The second takes into consideration motion as belonging to the quality of matter, under the name of an original moving force, and is therefore called dynamics. The third considers matter with this quality as in relation to another through its own inherent motion, and therefore appears under the name of mechanics. The fourth chapter, however, determines matter's motion or rest merely in relation to the mode of representation or modality, and thus as appearance of the outer senses, and is called phenomenology. 2The first passage from the first Critique (quoted above) reveals both that the concept of motion plays a central role in Kant's epistemology and that he distinguishes clearly between motion of the object, on the one hand, and action of the subject, on the other. The two concepts of motion are quite different—the former belonging to transcendental philosophy and geometry (to "pure science" in the Kantian sense), the latter to the metaphysical and empirical investigation of nature. By contrast, the second passage from the Metaphysical Foundations shows that the conceptual apparatus Kant developed in his transcendental philosophy is also relevant for understanding the empirical concept of motion. Since all empirical concepts are based on transcendental conditions for their possibility, this connection between elements of the Critical philosophy should come as no surprise. What remains unclear, however, is the sort of connection involved, and whether in the particular case of motion there might not be a more intimate relationship between the transcendental conditions and the empirical concept than one might at first suspect.Indeed, this is a possibility that has been entertained by a number of commentators. In his investigation of Kant's metaphysics of nature, Lothar Schäfer, for example, contends that "the original perspective on the unity of the sensibility in which nature reveals itself, and which belongs to the conception of nature, results in the original perspective on motion. Therefore, the laws of nature are laws of motion …." 3 Similarly, in her book on Kant's theory of natural science, Karen Gloy argues thatin the predicable 'motion' the connection of elements of the understanding with... (shrink)

In Kant's Metaphysical Foundations of Natural Science are found a dynamist reduction of matter and an account of the communication of motion by impact. One would expect to find an analysis of the causal mechanism involved in the communication of motion between bodies given in terms of the fundamental dynamical nature of bodies. However, Kant's analysis, as given in the discussion of his third law of mechanics (an action-reaction law) is purely kinematical, invoking no causal mechanisms at (...) all, let alone dynamist mechanisms, in the explanation. It is argued that the reason for the non-incorporation of Kant's theory of matter in the account of impact is his overriding desire to provide a mathematical framework to explain the communication of motion, a provision which Kant felt to be impossible in the context of a metaphysical dynamism. (shrink)

Brukner and Dakić proposed a very simple axiom system as a foundation for quantum theory. It implies the qubit and quantum entanglement. Because this axiom system aims at the core of our understanding of nature, it must be brought to the forum of the philosophy of nature. For philosophical reasons, a completely denied champion of quantum theory, imaginarity i, is added into this axiom system. In relation to Bell’s inequality, this leads to a deeper ‘philosophical’ understanding of quantum nature based (...) on qubits and entanglement. Both opens a way as well as one can get to the fundamental Schrödinger equation of quantum mechanics with the help of a complex valued Brownian motion. (shrink)

In this paper I examine the foundations of Laplace's famous statement of determinism in 1814, and argue that rather than derived from his mechanics, this statement is based on general philosophical principles, namely the principle of sufficient reason and the law of continuity. It is usually supposed that Laplace's statement is based on the fact that each system in classical mechanics has an equation of motion which has a unique solution. But Laplace never proved this result, and in (...) fact he could not have proven it, since it depends on a theorem about uniqueness of solutions to differential equations that was only developed later on. I show that the idea that is at the basis of Laplace's determinism was in fact widespread in enlightenment France, and is ultimately based on a re-interpretation of Leibnizian metaphysics, specifically the principle of sufficient reason and the law of continuity. Since the law of continuity also lies at the basis of the application of differential calculus in physics, one can say that Laplace's determinism and the idea that systems in physics can be described by differential equations with unique solutions have a common foundation. (shrink)

A nonlocal equation of motion with damping is derived by means of a Mori-Zwanzig renormalization process. The treatment is analogous to that of Mori in deriving the Langevin equation. For the case of electrodynamics, a local approximation yields the Lorentz equation; a relativistic generalization gives the Lorentz-Dirac equation. No self-acceleration or self-mass difficulties occur in the classical treatment, although runaway solutions are not eliminated. The nonrelativistic quantum case does not exhibit runaways, however, provided one remains within a weak damping (...) approximation. The correspondence limit shows that a classical limit may be taken, again within the same approximation. (shrink)

These papers, arising from a 1983 conference on one of the last and most acute Neoplatonist commentators on Aristotle, a Christian later condemned for his monophysitism and tritheism, focus on the arguments in which he objects to tenets of Aristotle's philosophy of nature, notably on the eternity of the world and the natures of place and projectile motion.

A new scheme is proposed in order to deduce an equation of motion for a spinless charged point particle leading to an equivalent Landau–Lifshitz equation of motion. Consequently Larmor’s formula must be substituted by a new expression for the large distance radiation rate of energy. A constraint appears on the applicability of the Maxwell electromagnetic tensor. The particular case of a sudden force is analyzed in order to show the physical results predicted by the new model. A geometrical (...) rearrangement of the energy explains the balance. (shrink)

In his Metaphysical Foundations of Natural Science, Kant accounts for the possibility of an acting-at-a-distance gravitational force, demonstrates the infinite divisibility of matter, and derives analogues to Newtonian laws of motion. The work is his major statement in philosophy of science, and was especially influential in German-speaking countries in the nineteenth century. However, this complex text has not received the scholarly attention it deserves. The chapters of this Critical Guide clarify the accounts of matter, motion, the mathematization (...) of nature, space, and natural laws exhibited in the Metaphysical Foundations; elucidate the relationship between its metaphysics of nature and Kant's critical philosophy; and describe the historical context for Kant's account of natural science. The volume will be an invaluable resource for understanding one of Kant's most difficult works, and will set the agenda for future scholarship on Kant's philosophy of science. (shrink)

The author has recently shown that a mathematical question regarding the fundamental constituents of hardrons cannot be resolved unless the classical axioms of nonfinite mathematics are revised in such a way as to produce a new theory of particle motion in continuous space-time. Under this new theory, the instantaneous position of a moving object has a magnitude that is increasing as the object's velocity. The purpose of this paper is to show that, quite apart from the question of Cantorian (...) axiomatics, the new theory of motion is more consistent with what physicists really believe than the traditional theory of motion. (shrink)

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties (...) of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND. (shrink)

Some unusual relations between stress tensors, conservation and equations of motion are briefly reviewed.

It is shown that the following three common understandings of Newton’s laws of motion do not hold for systems of infinitely many components. First, Newton’s third law, or the law of action and reaction, is universally believed to imply that the total sum of internal forces in a system is always zero. Several examples are presented to show that this belief fails to hold for infinite systems. Second, two of these examples are of an infinitely divisible continuous body with (...) finite mass and volume such that the sum of all the internal forces in the body is not zero and the body accelerates due to this non-null net internal force. So the two examples also demonstrate the breakdown of the common understanding that according to Newton’s laws a body under no external force does not accelerate. Finally, these examples also make it clear that the expression ‘impressed force’ in Newton’s formulations of his first and second laws should be understood not as ‘external force’ but as ‘exerted force’ which is the sum of all the internal and external forces acting on a given body, if the body is infinitely divisible. (shrink)

The Addendum of this note presents a brief perspective and an additional development pertaining to a previous paper. The even forceK n as well as the odd forceK 0 of the time-inversion-covariant (TIC) equation of motion in the presence of a magnetic field are derived from the results of the previous paper by a hint of generalization in classical physics. Then, by following identical steps as in the previous paper, the Addendum completes the derivation of the stochastic Hamilton-Jacobi and (...) the Schrödinger equations in the electromagnetic field. The Erratum of our note points out minor typographical errors plus one clarification of a definition. (shrink)

Indifference, Necessity, and Descartes's Derivation of the Laws of Motion BLAKE D. DUTTON WHILE WORKING ON Le Monde, his first comprehensive scientific treatise, Des- cartes writes the following to Mersenne: "I think that all those to whom God has given the use of this reason have an obligation to employ it principally in the endeavor to know him and to know themselves. This is the task with which I began my studies; and I can say that I would not (...) have been able to discover the foundations of physics if I had not looked for them along that road" . ' As the letter makes clear, knowl- edge of the foundations of Cartesian physics is inextricably linked to the knowledge of God. Unfortunately, Descartes never explains why this is the case, and the relation in which his theistic doctrine stands to his physics remains unspecified. In what follows I wish to clarify that relation by examining some of the problems surrounding Descartes's attempt to locate the metaphysical founda- tions of his physics in his doctrine of God. I begin my analysis with a discussion of the doctrine of divine indifference and argue that this doctrine is of great importance to any interpretation of the foundations of Cartesian physics, insofar as it provides Descartes with a rationale for dismissing the appeal to final causation in scientific explanation. Insofar as this is the case, it supplies an important piece of his justification for mechanism... (shrink)

Accepted quantum description is stochastic, yet history is nonstochastic, i.e., not representable by a probability distribution. Therefore ordinary quantum mechanics is unsuited to describe history. This is a limitation of the accepted quantum theory, rather than a failing of mechanics in general. To remove the limitation, it would be desirable to find a form of quantum mechanics that describes the future stochastically and the past nonstochastically. For this purpose it proves sufficient to introduce into quantum mechanics, by means of a (...) perfected formal correspondence, certain analogs of the classical initial-condition constants. Through the restoration of such parameters at the quantum level one accomplishes a natural accommodation of time anisotropy, wave-function reduction, and “event” description by quantum mechanical equations of motion alone, without the need for extra postulates (e.g., a projection postulate). This requires a complete restructuring of quantum measurement theory. (shrink)

This paper focuses on the mathematisation of mechanics in the seventeenth century, specifically on how the representation of compounded rectilinear motions presented in the ancient Greek Mechanica found its way into Newton’s Principia almost two thousand years later. I aim to show that the path from the former to the latter was optical: the conceptualisation of geometrical lines as paths of reflection created a physical interpretation of dia­grammatic principles of geometrical point-motion, involving the kinematics and dynamics of light reflection. (...) Upon the atomistic conception of light, the optical interpretation of such geometrical principles entailed their mechanical generalisation to local motion; rectilinear motion via the physico-mathemat­ics of reflection and the Mechanica’s parallelogram rule; circular motion via the physico-mathematics of reflection, the Archimedean squaring of the circle and the Mechanica’s extension of the parallelogram rule to centripetal motion. This appeal to the physico-mathematics of reflection forged a realist founda­tion for the mathematisation of motion. Whereas Aristotle’s physics rested on motions which had their source in the nature of the elements, early modern thinkers such as Harriot, Descartes, and Newton based their new principles of mechanical motion upon selected elements of the mechanics of light motion, projected upon the geometry of the parallelogram rule for rectilinear and, ultimately, circular motion. (shrink)

Presents outstanding translations of the Greek philosopher's works by leading British and American scholars of the last century.

The relation of the special and the general principle of relativity to the principle of covariance, the principle of equivalence and Mach's principle, is discussed. In particular, the connection between Lorentz covariance and the special principle of relativity is illustrated by giving Lorentz covariant formulations of laws that violate the special principle of relativity: Ohm's law and what we call “Aristotle's first and second laws.” An “Aristotelian” universe in which all motion is relative to “absolute space” is considered. The (...) first law: a free particle is at rest. The second law: force is proportional to velocity. Ohm's law: the current density is proportional to the electrical field strength. Neither of these laws fulfills the principle of relativity. The examples illustrate, in the context of Lorentz covariance and special relativity, Kretschmann's critique of founding Einstein's general principle of relativity on the principle of general covariance. A modification of the principle of covariance is suggested, which may serve as a restricted criterium for a physical law to satisfy Einstein's general principle of relativity. Other objections that have been raised to the validity of Einstein's general principle of relativity are based upon the preferred state of inertial frames in the general, as well as in the special theory, the existence of tidal effects in “true” gravitational fields, doubts as to the validity of Mach's principle, whether electromagnetic phenomena obey the principle, and, finally, the anisotropy of the cosmic background radiation. These objections are reviewed and discussed. (shrink)

It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born Rule is postulated rather than derived from the first principles. In this work the Born Rule is derived from the time-reversal symmetry of quantum equations of motion. The derivation is based on a simple functional equation that takes into account properties of probability, as well as the linearity and time-reversal symmetry of quantum equations of motion. The derivation presented in this (...) work also allows to determine certain limits to applicability of the Born Rule. (shrink)

It is argued that time's arrow is present in all equations of motion. But it is absent in the point particle approximations commonly made. In particular, the Lorentz-Abraham-Dirac equation is time-reversal invariant only because it approximates the charged particle by a point. But since classical electrodynamics is valid only for finite size particles, the equations of motion for particles of finite size must be considered. Those equations are indeed found to lack time-reversal invariance, thus ensuring an arrow of (...) time. Similarly, more careful considerations of the equations of motion for gravitational interactions also show an arrow of time. The existence of arrows of time in quantum dynamics is also emphasized. (shrink)

Philosophical interest in intentional action has flourished in recent decades. Typically, writers in the field of action theory seek necessary and sufficient conditions for a movement's being an action, conditions derived from a conceptual analysis of everyday action ascriptions. However, only a naturalistic account of intentional action, an account whose methods and aims are continuous with those of the empirical sciences, will truly help further our understanding of action as a biological phenomenon. ;Action is naturalized as a species of movement (...) that is both purposive and self-controlled, using these terms in novel senses. Various psychological and physiological models of purposiveness have been offered, including negative feedback, motor program, and dynamical systems models. However, while such non-historical models offer valuable insights into the nature of how organisms control their movements, they fail as general models of purposiveness because consideration of a movement's selectional history is required in order to classify movements as unitary behaviors rather than merely accidental motions. Non-historical models also fail to cover instances of unsuccessful purposive movements and cases of "functionally emergent behaviors." For such reasons, purposiveness must be seen as an historical property of movements; a movement is purposive insofar as it has a Millikanian proper function. ;An agent's capacity to exert a high degree of control over its movements marks the second key underlying property of action. I highlight an aspect of behavior-control which empirical theorists have largely ignored, namely the extent to which agents "could have done otherwise" than how they have in fact acted upon particular occasions. Self-control in this "could have done otherwise" sense is naturalized in terms of "modulatory interconnectivity." Modulatory interconnectivity involves the extent to which an agent's internal behavior-control modules facilitate behavioral interruptability via the interaction of informational links of inhibition and/or suppression. The greater the extent to which an agent's behavior-control modules have the capacity to modulate one another's activities, the more the agent has the capacity for self-control in the "could have done otherwise" sense. This view is elucidated and supported with the aid of models from robotics and neurophysiology. (shrink)

This work builds on the Volterra series formalism presented in Dreisigmeyer and Young to model nonconservative systems. Here we treat Lagrangians and actions as ‘time dependent’ Volterra series. We present a new family of kernels to be used in these Volterra series that allow us to derive a single retarded equation of motion using a variational principle.

Kant famously distinguishes between the methods of mathematics and of metaphysics, holding that metaphysicians err when they avail themselves of the mathematical method. Nonetheless, in the Metaphysical Foundations of Natural Science, he insists that mathematics and metaphysics must jointly ground ‘proper natural science’. This article examines the distinctive contributions and unity of mathematics and metaphysics to the foundations of the science of body. I argue that the two are distinct insofar as they involve distinctive sorts of grounding relations (...) – mathematics pertains to formal grounding, while metaphysics concerns material grounding – while they are unified insofar as they treat motion, the fundamental determination of the science of body. (shrink)

We show that particle-antiparticle exchange and covariant motion reversal are two physically different aspects of the same mathematical transformation, either in the prequantal relativistic equation of motion of a charged point particle, in the general scheme of second quantization, or in the spinning wave equations of Dirac and of Petiau-Duffin-Kemmer. While, classically, charge reversal and rest mass reversal are equivalent operations, in the wave mechanical case mass reversal must be supplemented by exchange of the two adjoint equations, implying (...) ψ ⇄ $\bar \psi$ .Denoting by M the rest mass reversal, P the parity reversal, T the Racah time reversal, and Z the ψ ⇄ $\bar \psi$ exchange, the connection with the usual scheme of charge conjugation, parity reversal, and Wigner motion reversal, is with, of course. (shrink)