Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an explanatory one.

ABSTRACT Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an explanatory (...) one. (shrink)

THIS ARTICLE IS A DISCUSSION OF THE VARIOUS ALTERNATIVE CRITERIA THAT HAVE BEEN OFFERED FOR MIRACLE. (EDITED).

Batterman raises a number of concerns for the inferential conception of the applicability of mathematics advocated by Bueno and Colyvan. Here, we distinguish the various concerns, and indicate how they can be assuaged by paying attention to the nature of the mappings involved and emphasizing the significance of interpretation in this context. We also indicate how this conception can accommodate the examples that Batterman draws upon in his critique. Our conclusion is that ‘asymptotic reasoning’ can be straightforwardly accommodated within the (...) inferential conception. 1 Introduction2 Immersion, Inference and Partial Structures3 Idealization and Surplus Structure4 Renormalization and the Stability of Mathematical Representations5 Explanation and Eliminability6 Requirements for Explanation7 Interpretation and Idealization8 Explanation, Empirical Regularities and the Inferential Conception9 Conclusion. (shrink)

For an ‘explanation' of physical facts by laws of nature, we have to establish a relation between physical facts and laws of nature. It is an open question, whether the laws of nature govern the facts with necessity or whether the laws are related to the facts merely by supervenience. In addition, it is not quite clear, whether the known laws of physics describe only artificially simplified cases, e.g. isolated situations, or whether the laws of physics actually grasp (...) real facts. Known solutions of these problems refer to situations where laws of classical physics are applied to phenomena of classical physics. However, if the same laws were applied to matter of facts of the domain of modern physics, then in many cases there would be no ‘explanation' in the sense mentioned. These new problems can be treated either by additional ‘interpretations' of the theories in question, or by a radical change of the ontological preconditions of classical physics. (shrink)

In this paper I defend an intermediate position between the ‘bare mathematical results’ view and the ‘transmission’ view of mathematical explanations of physical phenomena (MEPPs). I argue that, in MEPPs, it is not enough to deduce the explanandum from the generalizations cited in the explanans. Rather, we must add information regarding why those generalizations obtain. However, I also argue that it is not necessary to provide explanatory proofs of the mathematical theorems that represent those generalizations. I illustrate this (...) with the bridges of Königsberg case. -/- . (shrink)

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...) issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)

Can mathematics contribute to our understanding of physical phenomena? One way to try to answer this question is by getting involved in the recent philosophical dispute about the existence of mathematical explanations of physical phenomena. If there is such a thing, given the relation between explanation and understanding, we can say that there is an affirmative answer to our question. But what if we do not agree that mathematics can play an explanatory role in science? Can (...) we still consider that the above question can have an affirmative answer? My main aim here is to give an account that takes mathematics, in some of the cases discussed in the literature, as contributing to our understanding of physical phenomena despite not being explanatory. (shrink)

The increase of our knowledge of nature and the multiplicity of phenomena to be studied and investigated has led to specialization in pure and applied science and in practical work. In many instances the individual branches of science became isolated from each other to a considerable degree.The establishment of the closest ties between the individual lines of knowledge, theoretical as well as practical, is one of the important aims of scientific thought. The analysis of the nature of the factors, (...) used for the quantitative expression of physical processes, shows that many of them have a striking similarity as far as energy is concerned. Such similarity, and even identity, leads one to the conclusion that the concept of energy can serve as the basis for the unified interpretation of physical and, possibly, other phenomena of nature. This paper, which deals with physical processes, is an attempt toward such unification. (shrink)

The methodology of physics is discussed. The limitations of the empirical method are exposed, and it is argued that these limitations are related to our sensory input. The limitations of mathematics and of the representation of physical theories by mathematical models are also examined. An alternative methodology, the establishing of physical models on neuropsychology, is suggested and demonstrated. A cognitive psychological model of the genesis and the termination of time is explored.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Thermodynamics and Statistical Mechanics are related to one another through the so-called "thermodynamic limit'' in which, roughly speaking the number of particles becomes infinite. At critical points (places of physical discontinuity) this limit fails to be regular. As a result, the "reduction'' of Thermodynamics to Statistical Mechanics fails to hold at such critical phases. This fact is key to understanding an argument due to Craig Callender to the effect that the thermodynamic limit leads to mistakes in Statistical Mechanics. I (...) discuss this argument and argue that the conclusion is misguided. In addition, I discuss an analogous example where a genuine physical discontinuity---the breaking of drops---requires the use of infinite idealizations. (shrink)

Duhem's 1908 essay questions the relation between physical theory and metaphysics and, more specifically, between astronomy and physics–an issue still of importance today. He critiques the answers given by Greek thought, Arabic science, medieval Christian scholasticism, and, finally, the astronomers of the Renaissance.

Depending on different positions in the debate on scientific realism, there are various accounts of the phenomena of physics. For scientific realists like Bogen and Woodward, phenomena are matters of fact in nature, i.e., the effects explained and predicted by physical theories. For empiricists like van Fraassen, the phenomena of physics are the appearances observed or perceived by sensory experience. Constructivists, however, regard the phenomena of physics as artificial structures generated by experimental and mathematical methods. (...) My paper investigates the historical background of these different meanings of phenomenon in the traditions of physics and philosophy. In particular, I discuss Newton’s account of the phenomena and Bohr’s view of quantum phenomena, their relation to the philosophical discussion, and to data and evidence in current particle physics and quantum optics. (shrink)

In 1974 perfect crystal interferometry has been developed and immediately afterwards the 4π-symmetry of spinor wave-functions has been verified. The new method opened a new access to the observation of intrinsic quantum phenomena. Spin-superposition, quantum state reconstruction and quantum beat effects are examples of such investigations. In this connection efforts have been made to separate and measure various dynamical and geometrical phases. Non-cyclic and non-adiabatic topological phases have been identified and their stability against various fluctuations and dissipative forces has (...) been investigated by means of ultra-cold neutrons. An entanglement between different degrees of freedom of a single neutron system demonstrated the contextuality feature of quantum mechanics. In its continuation this yields to Kochen-Specker theorem like investigations providing a new basis for information processing and for the understanding of quantum physics in general. All investigations show the equivalence of various phase spaces and show how physical phenomena are correlated by quantum laws. Some curiosa occurred during the experiments and some epistemological aspects will be discussed as well. (shrink)

There are lots of arguments for, or justifications of, mathematical theorems that make use of principles from physics. Do any of these constitute explanations? On the one hand, physical principles do not seem like they should be explanatorily relevant; on the other, some particular examples of physical justifications do look explanatory. In this article, I defend the idea that physical justifications can and do explain mathematical facts. 1 Physical Arguments for Mathematical Truths2 Preview3 Mathematical Facts4 Purity5 (...) Doubts about Purity: I6 Doubts about Purity: II7 How Physical Arguments Might Explain: I8 How Physical Arguments Might Explain: II9 Conclusion. (shrink)

The topic of this essay concerns the interest that the conception of nature in Schelling has aroused in the philosophical culture in the nineteenth and twentieth centuries. I propose to examine the retrieval of the philosophy of the nature of Schelling from the history of philosophy and history of science standpoints. Therefore, my paper will be divided into three parts. In particular, I will start by examining the state of the art on the reevaluation of Schelling in the context of (...) the contemporary philosophy of science; then I will pass by analyzing the relationship between Schelling’s thought and the quantum physics according to Slavoj Žižek and Karen Barad approach; finally, I will conclude by going back to the Kantian source of this branch of study. (shrink)

When we examine the history of astronomy up to the end of the seventeenth century by considering the relation between mathematical astronomy and natural philosophy, it has been argued that there were two groups of philosophers and astronomers: instrumentalists and realists. However, this classification is deficient when we consider attitudes toward the explanatory power of mathematics in determining astronomical theories. I offer the solution of dividing realists into two subcategories—mathematical realists and physical realists. Mathematical realists include those who thought (...) mathematics could provide the real motions of celestial bodies. The physical realists, who include Ibn Rushd (Averroës), believed mathematics could not provide us with the real structure of the heavens. To avoid contradictions between the physical and mathematical models, the physical realists held that a mathematical model of the heavens should be designed by considering physics. I argue, in this article, that Francis Bacon can be considered a physical realist. (shrink)

Newton's methodology emphasized propositions "inferred from phenomena." These rest on systematic dependencies that make phenomena measure theoretical parameters. We consider the inferences supporting Newton's inductive argument that gravitation is proportional to inertial mass. We argue that the support provided by these systematic dependencies is much stronger than that provided by bootstrap confirmation; this kind of support thus avoids some of the major objections against bootstrapping. Finally we examine how contemporary testing of equivalence principles exemplifies this Newtonian methodological theme.

It is argued that the evolution of fundamental microphysics throughout the twentieth century is characterised by two interrelated developments. On the one hand, the experimental signatures which confirm theoretical statements are moving towards the fringes of the phenomenal world and, at the same time, leave increasingly wide spaces for entirely theoretical reasoning with little or no empirical interference. On the other hand, assessments of limitations to scientific underdetermination gain importance within the theoretical process.

It is argued that the evolution of fundamental microphysics throughout the twentieth century is characterised by two interrelated developments. On the one hand, the experimental signatures which confirm theoretical statements are moving towards the fringes of the phenomenal world and, at the same time, leave increasingly wide spaces for entirely theoretical reasoning with little or no empirical interference. On the other hand, assessments of limitations to scientific underdetermination gain importance within the theoretical process.

How can be interchangeable with mirroring, which is an automatic response that, from a first-person perspective, enters awareness only after the act? The correspondence between perception of another's action and execution of one's similar action may be an example of a general perception-motor interface that maps perception onto behaviour or disposition towards action, without the need for simulation.

The existence of various physical phenomena stems from the concept called asymptotic emergence, that is, they seem to be exclusively reserved for certain limiting theories. Important examples are spontaneous symmetry breaking (SSB) and phase transitions: these would only occur in the classical or thermodynamic limit of underlying finite quantum systems, since for finite quantum systems, due to the uniqueness of the relevant states, such phenomena are excluded by Theory. In Nature, however, finite quantum systems describing real materials (...) clearly exhibit such effects. In this paper we discuss these apparently “paradoxical” phenomena and outline various ideas and mechanisms that encompass both theory and reality, from physical and mathematical points of view. (shrink)

The concept of similar systems arose in physics, and appears to have originated with Newton in the seventeenth century. This chapter provides a critical history of the concept of physically similar systems, the twentieth century concept into which it developed. The concept was used in the nineteenth century in various fields of engineering, theoretical physics and theoretical and experimental hydrodynamics. In 1914, it was articulated in terms of ideas developed in the eighteenth century and used in nineteenth century mathematics and (...) mechanics: equations, functions and dimensional analysis. The terminology physically similar systems was proposed for this new characterization of similar systems by the physicist Edgar Buckingham. Related work by Vaschy, Bertrand, and Riabouchinsky had appeared by then. The concept is very powerful in studying physical phenomena both theoretically and experimentally. As it is not currently part of the core curricula of STEM disciplines or philosophy of science, it is not as well known as it ought to be. (shrink)

One of the distinctive features of George Smith’s work on celestial mechanics is his emphasis on the role of what he calls “second-order phenomena” in the production of high-quality evidence. On Smith’s view, these gaps between theoretical predictions and observations can, under certain circumstances, be a source of evidence far stronger than that achievable through the hypothetico-deductive method. The practice of examining gaps between predictions and observations for the purposes of discovery and testing is commonplace in certain sciences such (...) as seismology, and has played an important role in their development. I use the term reasoning from residuals as a general term for this practice. I think it is worth investigating examples of this set of practices from the history of science, in order to understand the different ways in which reasoning from residuals is done, under what situations it is done, and how it contributes to the growth of knowledge in certain sciences. (shrink)

Bridging the gap between the world of science and the realm of the spiritual, B. Alan Wallace introduces a natural theory of human consciousness that has its roots in contemporary physics and Buddhism. Wallace's "special theory of ontological relativity" suggests that mental phenomena are _conditioned_ by the brain, but do not _emerge_ from it. Rather, the entire natural world of mind and matter, subjects and objects, arises from a unitary dimension of reality that is more fundamental than these dualities, (...) as proposed by Wolfgang Pauli and Carl Jung. To test his hypothesis, Wallace employs the Buddhist meditative practice of _samatha_, refining one's attention and metacognition, to create a kind of telescope to examine the space of the mind. Drawing on the work of the physicist John Wheeler, he then proposes a more general theory in which the participatory nature of reality is envisioned as a self-excited circuit. In comparing these ideas to the Buddhist theory known as the Middle Way philosophy, Wallace explores further aspects of his "general theory of ontological relativity," which can be investigated by means of _vipasyana_, or insight, meditation. Wallace then focuses on the theme of symmetry in reference to quantum cosmology and the "problem of frozen time," relating these issues to the theory and practices of the Great Perfection school of Tibetan Buddhism. He concludes with a discussion of the general theme of complementarity as it relates to science and religion. The theories of relativity and quantum mechanics were major achievements in the physical sciences, and the theory of evolution has had an equally deep impact on the life sciences. However, rigorous scientific methods do not yet exist to observe mental phenomena, and naturalism has its limits for shedding light on the workings of the mind. A pioneer of modern consciousness research, Wallace offers a practical and revolutionary method for exploring the mind that combines the keenest insights of contemporary physicists and philosophers with the time-honored meditative traditions of Buddhism. (shrink)

Stoic physics, based entirely on the continuum concept, is one of the great original contributions in the history of physical systems. Building on The Physical World of the Greeks, the author describes the main aspects of the Stoic continuum theory, traces its origins back to pre-Stoic science and philosophy, and shows the attempts of the Stoics to work out a coherent system of thought that would explain the essential phenomena of the physical world by a few (...) basic assumptions. Originally published in 1987. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

Fluctuation phenomena are the ''tip of the iceberg'' revealing the existence, behind even the most quiescent appearing macroscopic states, of an underlying world of agitated, ever-changing microscopic processes. While the presence of these fluctuations can be ignored in some cases, e.g. if one is satisfied with purely thermostatic description of systems in equilibrium, they are central to the understanding of other phenomena, e.g. the nucleation of a new phase following the quenching of a system into the co-existence region. (...) This volume, originally published several years ago, has stood the test of time well, as interest in fluctuation phenomena continues to grow. It contains a collection of review articles, some of them updated, written by experts in the field, on the subject of fluctuation phenomena. Some of the articles are of a very general nature discussing the modern mathematical formulation of the problems involved, while other articles deal with specific topics such as kinetics of phase transitions and conductivity in solids. The juxtaposition of the variety of physical situations in which fluctuation phenomena play an important role is novel and should give the reader a comprehensive and deeper insight into this fascinating subject. (shrink)

This paper explores the scientific viability of the concept of causality—by questioning a central element of the distinction between “fundamental” and non-fundamental physics. It will be argued that the prevalent emphasis on fundamental physics involves formalistic and idealized partial models of physical regularities abstracting from and idealizing the causal evolution of physical systems. The accepted roles of partial models and of the special sciences in the growth of knowledge help demonstrate proper limitations of the concept of fundamental physics. (...) We expect that a cause precedes its effect. But in some tension with this point, fundamental physical law is often held to be symmetrical and all-encompassing. Physical time, however, has not only measurable extension, as with spatial dimensions, it also has a direction—from the past through the present into the future. This preferred direction is time’s arrow. In spite of this standard contrast of time with space, if all the fundamental laws of physics are symmetrical, they are indifferent to time’s arrow. In consequence, excessive emphasis on the ideal of symmetrical, fundamental laws of physics generates skepticism regarding the common-sense and scientific uses of the concept of causality. The expectation has been that all physical phenomena are capable of explanation and prediction by reference to fundamental physicals laws—so that the laws and phenomena of statistical thermodynam- ics —and of the special sciences—must be derivative and/or secondary. The most important and oft repeated explanation of time’s arrow, however, is provided by the second law of thermodynamics. This paper explores the prospects for time’s arrow based on the second law. The concept of causality employed here is empirically based, though acknowledging practical scientific interests, and is linked to time’s arrow and to the thesis that there can be no causal change, in any domain of inquiry, without physical interaction. (shrink)