Reflexionen über die Struktur der Physikalischen Sprache

The aim of this paper is to discuss how far physics differs frommathematics, and if a philosophy of science which uses mathematics or logicsas a model for physics would be unable to be aware of many importantfeatures of that natural science.Many functions in physics differ from those of mathematics in beingfunctional dependecies and in having a lawlike character.Physical quantities have the character of "`determinables"', sets ofspecial entities which are presupposed by physical theories.One may suspect that physics also could not be formulated in anextensional language. This cannot be true, however, since every language canbe translated into an extensional version. Neverthless the existence ofdeterminables in physics shows that physics does not only talk aboutconcrete entities like space, time, spacetime, and particles, but also aboutvalues of abstract sets like determinables, and that it thus acknowledgestheir existence.     

Throwing spatial light: on topological explanations in Gestalt psychology

Phenomenology and the Cognitive Sciences - It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it....     

A FORMAL CONSTRUCTION OF THE SPACETIME MANIFOLD

The spacetime manifold, the stage on which physics is played, is constructed ab initio in a formal program that resembles the logicist reconstruction of mathematics. Zermelo’s set theory extended by urelemente serves as a framework, to which physically interpretable proper axioms are added. From this basis, a topology and subsequently a Hausdorff manifold are readily constructed which bear the properties of the known spacetime manifold. The present approach takes worldlines rather than spacetime points to be primitive, having them represented by urelemente. Thereby it is demonstrated that an important part of physics is formally reducible to set theory.     

Historicism,Entrenchment, and Conventionalism

W. V. Quine famously argues that though all knowledge is empirical, mathematics is entrenched relative to physics and the special sciences. Further, entrenchment accounts for the necessity of mathematics relative to these other disciplines. Michael Friedman challenges Quine’s view by appealing to historicism, the thesis that the nature of science is illuminated by taking into account its historical development. Friedman argues on historicist grounds that mathematical claims serve as principles constitutive of languages within which empirical claims in physics and the special sciences can be formulated and tested, where these mathematical claims are themselves not empirical but conventional. For Friedman, their conventional, constitutive status accounts for the necessity of mathematics relative to these other disciplines. Here I evaluate Friedman’s challenge to Quine and Quine’s likely response. I then show that though we have reason to find Friedman’s challenge successful, his positive project requires further development before we can endorse it.     

Alessandro Piccolomini and the certitude of mathematics

This paper offers a reconstruction of Alessandro Piccolomini's philosophy of mathematics, and reconstructs the role of Themistius and Averroes in the Renaissance debate on Aristotle's theory of proof. It also describes the interpretative context within which Piccolomini was working in order to show that he was not an isolated figure, but rather that he was fully involved in the debate on mathematics and physics of Italian Aristotelians of his time. The ideas of Lodovico Boccadiferro and Sperone Speroni will be analysed. This paper demonstrates that Piccolomini's attack on the certitude of mathematics was a product of discussions between Aristotelians.     

Confucian Order at the Edge of Chaos: The Science of Complexity and Ancient Wisdom

Many academics extol chaos theory and the science of complexity as significant scientific advances with application in such diverse fields as biology, anthropology, economics, and history. In this paper we focus our attention on structure-within-chaos and the dynamic self-organization of complex systems in the context of social philosophy. Although the modern formulation of the science of complexity has developed out of late-twentieth-century physics and computational mathematics, its roots may extend much deeper into classical thinking. We argue here that the essential ideas and predictions of the science of complexity are found within the social ordering principle of li (the rites) in Confucius's Analects .     

Fundamentality,Effectiveness, and Objectivity of Gauge Symmetries

Much recent philosophy of physics has investigated the process of symmetry breaking. Here, I critically assess the alleged symmetry restoration at the fundamental scale. I draw attention to the contingency that gauge symmetries exhibit, that is, the fact that they have been chosen from an infinite space of possibilities. I appeal to this feature of group theory to argue that any metaphysical account of fundamental laws that expects symmetry restoration up to the fundamental level is not fully satisfactory. This is a symmetry argument in line with Curie’s first principle. Further, I argue that this same feature of group theory helps to explain the ‘unreasonable’ effectiveness of (this subfield of) mathematics in (this subfield of) physics, and that it reduces the philosophical significance that has been attributed to the objectivity of gauge symmetries.     

An Inferential Conception of the Application of Mathematics

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called “the mapping account”. According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation of that system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception.     

The step to rationality: the efficacy of thought experiments in science, ethics, and free will

Examples from Archimedes, Galileo, Newton, Einstein, and others suggest that fundamental laws of physics were—or, at least, could have been—discovered by experiments performed not in the physical world but only in the mind. Although problematic for a strict empiricist, the evolutionary emergence in humans of deeply internalized implicit knowledge of abstract principles of transformation and symmetry may have been crucial for humankind's step to rationality—including the discovery of universal principles of mathematics, physics, ethics, and an account of free will that is compatible with determinism.     

Can Constructive Mathematics be Applied in Physics?

The nature of modern constructive mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.     

Hermann Weyl on Minkowskian Space–Time and Riemannian Geometry

Hermann Weyl as a founding father of field theory in relativistic physics and quantum theory always stressed the internal logic of mathematical and physical theories. In line with his stance in the foundations of mathematics, Weyl advocated a constructivist approach in physics and geometry. An attempt is made here to present a unified picture of Weyl’s conception of space–time theories from Riemann to Minkowski. The emphasis is on the mathematical foundations of physics and the foundational significance of a constructivist philosophical point of view. I conclude with some remarks on Weyl’s broader philosophical views.     

Boltzmann on Mathematics

Boltzmann’s lectures on natural philosophy point out how the principles of mathematics are both an improvement on traditional philosophy and also serve as a necessary foundation of physics or what the English call “Natura Philosophy”, a title which he will retain for his own lectures. We start with lecture #3 and the mathematical contents of his lectures plus a few philosophical comments. Because of the length of the lectures as a whole we can only give the main points of each but organized into a coherent study. Behind his mathematics stands his support of Darwinian evolution interpreted in a partly Lamarckian way. He also supported non-Euclidean geometry. Much of Boltzmann’s analysis of mathematics is an attempt to refute Kant’s static a priori categories and his identification of space with “non-sensuous intuition”. Boltzmann’s strong attention toward discreteness in mathematics can be seen throughout the lectures. Part II of this paper will touch on the historical background of atomism and focus on the discrete way of thinking with which Boltzmann approaches problems in mathematics and beyond. Part III briefly points out how Boltzmann related mathematics and discreteness to music. This revised version was published online in June 2006 with corrections to the Cover Date.     

The physiological and psychological grounds of Ptolemy's visual theory: Some methodological considerations

Until fairly recently, Ptolemy's Optics has been regarded as an exercise in what would today be called physical optics, its focus purportedly upon ray-geometry. In terms of methodology, therefore, the Optics has generally been regarded as a model of applied mathematics. The purpose of this essay is to show that this textbook interpretation is not only excessively restrictive but fundamentally misguided. As will become clear in the course of this essay, in fact, Ptolemy's primary goal in the Optics was to frame a comprehensive and coherent account of visual perception, not to explain the physics of radiation. © 1998 John Wiley & Sons, Inc.     

Does Poor Understanding of Physical World Predict Religious and Paranormal Beliefs?

Although supernatural beliefs often paint a peculiar picture about the physical world, the possibility that the beliefs might be based on inadequate understanding of the non‐social world has not received research attention. In this study (N = 258), we therefore examined how physical‐world skills and knowledge predict religious and paranormal beliefs. The results showed that supernatural beliefs correlated with all variables that were included, namely, with low systemizing, poor intuitive physics skills, poor mechanical ability, poor mental rotation, low school grades in mathematics and physics, poor common knowledge about physical and biological phenomena, intuitive and analytical thinking styles, and in particular, with assigning mentality to non‐mental phenomena. Regression analyses indicated that the strongest predictors of the beliefs were overall physical capability (a factor representing most physical skills, interests, and knowledge) and intuitive thinking style. Copyright © 2016 John Wiley & Sons, Ltd.     

QUANTUM PHYSICS AND THEOLOGY: JOHN POLKINGHORNE ON THOUGHT EXPERIMENTS

Abstract Thought experimentation is part of accepted scientific practice, and this makes it surprising that philosophers of science did not seriously engage with it for a very long time. The situation changed in the 1990s, resulting in a highly intriguing debate over thought experiments. Initially, the discussion focused mostly on thought experiments in physics, philosophy, and mathematics. Other disciplines have since become the subject of interest. Yet, nothing substantial has been said about the role of thought experiments in nonphilosophical theology. This paper discusses the role of thought experiments in Christian theology in comparison to their role in quantum physics, as mentioned by John Polkinghorne in Quantum Physics and Theology. We first look briefly at the history of the inquiry into thought experiments and then at Polkinghorne's remarks about the role of thought experimentation in quantum physics and Christian eschatology. To determine the actual importance of thought experiments in Christian theology a number of new examples are introduced in a third step. In the light of these examples, in a fourth step, we address the question of what it is that explains the cognitive efficacy of thought experiments in quantum physics and Christian theology.     

Formal systems as physical objects: A physicalist account of mathematical truth

This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of the laws of physics governing these processes. In accordance with the physicalist understanding of mind, this is true even if the operations in question are executed in the head. A truth obtained through (mathematical) reasoning is, therefore, an observed outcome of a neuro-physiological (or other physical) experiment. Consequently, deduction is nothing but a particular case of induction.     

形式化量子力学不必使用量子逻辑

一般认为,标准量子力学需要使用一套它自己的逻辑系统,即量子逻辑。量子逻辑采用与一般逻辑系统不同的语义规则,因此和古典逻辑无法兼容。此篇文章将呈现一套量子力学的严格形式基础,它是对古典二值逻辑之保守扩充;保守扩充意指比原先之逻辑系统强,但较强的原因为它有较多之词汇。此套逻辑为三值逻辑。古典逻辑中为真的句子仍然为真。古典逻辑中为假的句子将被区分为强性假与中性。第三个真值一中性一考虑了非本征态情况中之观察句。本文详列了物理的公理并显示它们具有一个模型。此提案的可行性说明了量子逻辑是不必要的,并且存在一个共同的逻辑架构可提供给数学、非量子物理及量子力学使用。     

The logic of arithmetic

Conclusion The arithmetic developed along the lines indicated by Cantor has up until now found no application in the real, as opposed to the mathematical, world. It is to be hoped that the changes suggested in this paper bring mathematics closer to human thought and allow it to be of increasing benefit in its service to mankind.     

Intuitive physics and cognitive algebra: A review

IntroductionIntuitive physics explores how people without a formal instruction in physics intuitively understand physical phenomena. After a general overview of the topics of current research in intuitive physics and a discussion of current debates, this paper provides an introduction to Information Integration Theory (IIT).ObjectiveBy means of examples, it is shown how IIT can be used to directly compare the algebraic structure of physical laws and the algebraic structure of cognitive representations of these laws.MethodThe review considers about 40 years of research on the application of IIT in the field of intuitive physics. Occasionally, reference is also made to intuitive physics studies outside this theoretical framework.ResultsThe reviewed studies highlight four main factors that affect the degree of consistency between physical laws and cognitive algebraic laws: the participants’ age, their familiarity with the event under study, the type of task, and possible learning processes.ConclusionThe last part of the article discusses the implications of the results of the reviewed studies for the two main current hypotheses on the nature of intuitive physics, namely, that intuitive physics may be based on sub-optimal heuristics or may be based on the internalization of physical laws.