Review of W "ust's 'Mathematik f "ur Physiker und Mathematiker (original) (raw)
Related papers
The Contributions of Logic to the Foundations of Physics: Foreword
Studia Logica, 2010
This special issue on The Contributions of Logic to the Foundations of Physics, is the result of the sixth International Studia Logica Conference Trends in Logic VI held in Brussels on December 11-12, 2008. The conference was organized by the Center for Logic and Philosophy of Science and the Center Leo Apostel at the Vrije Universiteit Brussel in cooperation with the Institute of Philosophy and Sociology of the Polish Academy of Sciences. The main goal of this conference was to present current trends situated at the interface of Logic and the Foundations of Physics. The conference brought together researchers from both fields, giving them a forum to present new developments, exchange ideas, explore and establish new connections between logic and physics. On the one hand, we hosted technical contributions on the use of new methods and techniques coming from logic, computation and information theory to axiomatize and model physical theories and to reason about their concepts, phenomena and/or applications. On the other hand, we hosted contributions coming from the foundations and philosophy of physics dealing with the general conceptual framework and with questions of interpretation. The conference was a success with 56 registered participants, 7 invited lectures and two full parallel sessions of contributed talks spread over two days.
Logical and historical review on theoretical physics
Journal of physics, 2022
Since it started about three centuries ago, theoretical physics went through a huge advancement and, particularly in the last century, the development was material. Its application to engineering brought a massive revolution in the way we humanity live now. Its interpretation opened up astoundingly deep understanding of our universe. One important research activity for the future is to further develop our theories and to further deepen our understanding of the universe. However, as Tomonaga said, when we are in a phase of looking for new paradigm, it is important to understand how our current theory was developed. The purpose of this paper is to present a logical and historic study of the conceptual development of theoretical physics. As the field of theoretical physics is so vast, we cannot cover all theories we have now. We will focus on the most fundamental theories of physics. As this field of physics is as deep and intricate as pure mathematics, if not more, it will be helpful to compare our challenge with that pure mathematicians are facing in the field of the foundations of mathematics. Such common ground will inevitably lead us to deeper philosophical issues. After all what we call physics started with Newton who developed both calculus and dynamics. He called it not physics but natural philosophy. So, it is naturally expected that philosophy, mathematics and theoretical physics develop hand in hand. It has been about a century since these fields started to develop separately and it is about time to restart the original interaction between these three intrinsic intellectual activities. Certainly this will help our timely search for a new paradigm. We must move forward.
viXra, 2016
This book proposes a review and, on some important points, a new interpretation of the main concepts of Theoretical Physics. Rather than offering an interpretation based on exotic physical assumptions (additional dimension, new particle, cosmological phenomenon,.. .) or a brand new abstract mathematical formalism, it proceeds to a systematic review of the main concepts of Physics, as Physicists have always understood them : space, time, material body, force fields, momentum, energy.. . and propose the right mathematical tools to deal with them, chosen among well known mathematical theories. After a short introduction about the place of Mathematics in Physics, a new interpretation of the main axioms of Quantum Mechanics is proposed. It is proven that these axioms come actually from the way mathematical models are expressed, and this leads to theorems which validate most of the usual computations and provide safe and clear conditions for their use, as it is shown in the rest of the book. Relativity is introduced through the construct of the Geometry of General Relativity, based on 5 propositions and the use of tetrads and fiber bundles, which provide tools to deal with practical problems, such as deformable solids. A review of the concept of momenta leads to the introduction of spinors in the framework of Clifford algebras. It gives a clear understanding of spin and antiparticles. The force fields are introduced through connections, in the, now well known, framework of gauge theories, which is here extended to the gravitational field. It shows that this field has actually a rotational and a transversal component, which are masked under the usual treatment by the metric and the Levy-Civita connection. A thorough attention is given to the topic of the propagation of fields with interesting results, notably to explore gravitation. The general theory of lagrangians in the application of the Principle of Least Action is reviewed, and two general models, incorporating all particles and fields are explored, and used for the introduction of the concepts of currents and energy-momentum tensor. Precise guidelines are given to find operational solutions of the equations of the gravitational field in the most general case. The last chapter shows that bosons can be understood as discontinuities in the fields. In this 4th version of this book, changes have been made :-in Relativist Geometry : the ideas are the same, but the chapter has been rewritten, notably to introduce the causal structure and explain the link with the practical measures of time and space;-in Spinors : the relation with momenta has been introduced explicitly-in Force fields : the section dedicated to the propagation of fields is new, and is an important addition.-in Continuous Models : the section about currents and energy-momentum tensor are new.-in Discontinuous Processes : the section about bosons has been rewritten and the model improved. 1 To be precise : assumptions are labeled "propositions", and the results which can be proven from these propositions are labeled "theorems". xi open, but I hope that their meaning will be clearer, leading the way to a better and stronger understanding of the real world. The first chapter is devoted to a bit of philosophy. From many discussions with scientists I felt that it is appropriate. Because the book is centered on the relation between Mathematics and Physics, it is necessary to have a good understanding of what is meant by physical laws, theories, validation by experiments, models, representations,...Philosophy has a large scope, so it deals also with knowledge : epistemology helps us to sort out the different meanings of what we call knowledge, the status of Science and Mathematics, how the Sciences improve and theories are replaced by new ones. This chapter will not introduce any new Philosophy, just provide a summary of what scientists should know from the works of professional philosophers. The second chapter is dedicated to Quantum Mechanics (QM). This is mandatory, because QM has dominated theoretical Physics for almost a century, with many disturbing and confusing issues. It is at the beginning of the book because, as we will see, actually QM is not a physical theory per se, it does not require any assumption about how Nature works. QM is a theory which deals with the way one represents the world : its axioms, which appear as physical laws, are actually mathematical theorems, which are the consequences of the use by Physicists of mathematical models to make their computations and collect their data from experiments. This is not surprising that measure has such a prominent place in QM : it is all about the measures, that is the image of the world that physicists build, and not about the world itself. And this is the first, and newest, example of how the use of Mathematics can be misleading. The third chapter is dedicated to the Geometry of the Universe. By this we do not mean how the whole universe is, which is the topic of Cosmology. Cosmology is a branch of Physics of its own, which raises issues of an epistemological nature, and is, from my point of view, speculative, even if it is grounded in Astrophysics. We will only evoke some points of Cosmology in passing in this book. By Geometry of the Universe I mean here the way we represent locations of points, components of vectors and tensors, and the consequences which follow for the rules in a change of representation. This will be done in the relativist framework, and more precisely in the framework of General Relativity. It is less known, seen usually as a difficult topic, but, as we will see, some of the basic concepts of Relativity are easier to understand when we quit the usual, and misleading, representations, and are not very complicated when one uses the right mathematical tools. We show that the concept of deformable solid can be transposed in GR and can be used practically in elaborate models. such as those necessary in Astrophysics. The fourth chapter addresses Kinematics, which, by the concept of moment, is the gate between forces and geometry. Relativity requires a brand new vision of these concepts, which has been engaged, but neither fully or consistently. Rotation in particular has a different meaning in the 4 dimensional space than in the usual euclidean space, and a revision of rotational moment requires the introduction of a new framework. Spinors are not new in Physics, we will see what they mean, in Physics and in Mathematics, with Clifford algebras. This leads naturally to the introduction of the spin, which has a clear and simple interpretation, and to the representation of particles by fields of spinors, which incorporates in a single quantity the motion, translational and rotational, and the kinematics characteristics of material objects, including deformable solids. The fifth chapter addresses Force Fields. After a short reminder of the Standard Model we will see how charges of particles and force fields can be represented, with the concept of connections on fiber bundles. We will not deal with all the intricacies of the Standard Model, but focus on the principles and main mechanisms. The integration of Gravity, not in a Great Unification Theory, but with tools similar to the other forces and in parallel with them, opens a fresh vision on important issues in General Relativity. In particular it appears that the common and exclusive use of the Levi-Civita connection and scalar curvature introduces useless complications xii INTRODUCTION but, more importantly, misses important features of the gravitational field. One of the basic properties of fields is that they propagate. This phenomenon is more subtle than it is commonly accepted. In a realist view of fields, that is the acceptance that a field is a physical entity which occupies a definite area in the universe, and experimentally checked assumptions, we deduce fundamental equations which can be used to explore the fields which are less well known, and notably gravitation. The sixth chapter is dedicated to lagrangians. They are the work horses of Theoretical Physics, and we will review the problems, physical and mathematical, that they involve, and how to deal with them. We will see why a lagrangian cannot incorporate explicitly some variables, and build a simple lagrangian with 6 variables, which can be used in most of the problems. The seventh chapter is dedicated to continuous models. Continuous processes are not the rule in the physical world, but are the simplest to represent and understand. We will see how the material introduced in the previous chapters can be used, how the methods of Variational Calculus, and its extension to functional derivatives, can be used in solving two models, for a field of particles and for a single particle. In this chapter we introduce the concept of currents and Energy-Momentum tensor and prove some important theorems. We give guidelines which can solve the equations for the gravitational field in the vacuum in the most general concept. The eighth chapter is dedicated to discontinuous processes. They are common in the real world but their study is difficult. From the concept of propagation of fields, we shall accept that this is not always a continuous process. Discontinuities of fields then appear as particles, which can be assimilated to bosons. We show how their known properties can be deduced from this representation.
NOTES ON PHYSICS–MATHEMATICS RELATIONSHIP Epistemological notes
Nowadays it is unthinkable to learn and teach the scientific sense of physical and mathematical sciences without deepening its intellectual and cultural background, e.g. history and its foundations. In my talk several case–studies on the relationship between physics and mathematics were presented. Here, for brevity's sake, I will only discuss some of them.
The modes of physical properties in the logical foundations of physics
Logic and Logical Philosophy, 2005
We present a conceptual analysis of the notions of actual physical property and potential physical property as used by theoretical physicists/mathematicians working in the domain of operational quantum logic. We investigate how these notions are being used today and what role they play in the specified field of research. In order to do so, we will give a brief introduction to this area of research and explain it as a part of the discipline known as "mathematical metascience". An in depth analysis of Aristotle's use of the notions of "actuality" and "potentiality" is presented in order to point out exactly how much of the Aristotelian connotations are embedded in the contemporary use of the concepts under investigation. Although we will not focus in depth on all the drawbacks in the early historical development of physics due to the overwhelming influence of Aristotle's writings, our analysis does touch upon some aspects of the Aristotelian theory of movement that are often overthrown nowadays.
Some Aspects of the Relationship between Mathematical Logic and Physics. I
Journal of Mathematical Physics, 1970
Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state &eegr; where &eegr; does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [QsΤ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement ∀QsΤ [U(QsΤ) defined⇒E(H,U(QsΤ),&psgr;QsΤ)], where U(QsΤ) is the probability measure U associates with the infinite instruction string QsΤ and &psgr;QsΤ is the outcome sequence obtained by carrying out QsΤ. E(H, U(QsΤ),&psgr;QsΤ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(QsΤ) are true for &psgr;QsΤ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(QsΤ) corresponds to a nontrivial product probability measure and U H-agrees with experiment, then the outcome sequence &psgr;QsΤ is H-random. H-randomness is defined here in terms of the statement E(H, P, &phgr;). Another property of a valid physical theory, which is defined here, is that, for some QsΤ, U(QsΤ) must be determinable on much of AH from &psgr;QsΤ. Sufficient conditions for this property to hold are given. AH is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, &phgr;) are given. Among other things, it is proved that, if E(H, P, &phgr;) holds and P is a nontrivial probability measure on A, then &phgr; is not definable in H.
Mathematical Axiomatic Physics (Analytical form)
This paper shows that the concept of mathematical axiomatic physics is about the study of physics through the use of mathematics. The mechanisms of physics could be explained from the view of mathematics giving transitions to the field of physics. These are the two options of the same issue in the structure of things. This work is used philosophical concepts, mathematical and functional analysis, and contemporaneously some physical elements. This thesis aims to represent the commonplace of epistemic reasoning of philosophy, physics, and mathematics.
A Match Not Made In Heaven On the Applicability of Mathematics in Physics
In his seminal 1960 paper, the physicist Eugene Wigner formulated the question of the applicability of mathematics in physics in a way nobody had before. This formulation has been (almost) entirely overlooked due to an exclusive concern with (dis)solving Wigner's problem and explaining the effectiveness of mathematics in the natural sciences, in one way or another. Many have attempted to attribute Wigner's unjustified conclusion– that mathematics is unreasonably effective in the natural sciences– to his (dogmatic) formalist views on mathematics. My goal is to show that this reading misses out on Wigner's highly original formulation of the problem which is presented throughout his body of work in physics as well as in philosophy. This formulation, as I will show, leads us in a new direction in solving the applicability problem. This paper is published in Synthese 2016, pp. 1-23: http://link.springer.com/article/10.1007/s11229-016-1171-4?wt\_mc=Internal.Event.1.SEM.ArticleAuthorOnlineFirst