In the last chapter the subject of our investigations is mathematics and its role as an auxiliary discipline for physics. It is undenied, I presume, that mathematics as a pure construction of the human mind has foundations of its own independent of physics and indeed of any other discipline. Many would, however, say that the reverse does not hold — that the physics of our day cannot exist without mathematics applied to and indeed, as it were, embodied in it. But even this has been denied: there are attempts at an elimination of mathematics from physics to a certain extent. 1 The other extreme was Kant’s position. For Kant mathematics and physics are at least partially identical, at any rate in geometry. Elimination of geometry from physics would then be impossible without destroying the latter. In the papers of the present chapter it is assumed without discussion that for the formulation of physical theories the use of mathematics is at least very expedient if not indispensable. There is then, of course, the question of which status one allows mathematics to have by itself and how it occurs as such when applied to physical science.

Mathematics and physics are closely related as disciplines. Their histories are intertwined, sometimes in the person of a single figure such as Newton. Each has helped to give form and emphasis to the other. The practice of physics requires a broad knowledge of mathematics, and the mathematician who seeks wider understanding in his or her own field does well to become familiar with the classical areas of physics.

In this conversation I hold that Mathematics Informs Physics and Physics Performs Mathematics accordingly. The two appear to be indifferent to and ignorant of the Human Condition (consciousness/selfawareness). And this Ignorance seems Invincible. I hope to illustrate some of these with a simple example taken from the mathematics of finite part of infinity (or, more precisely a regularization) based on Ramanujan, and the physics of the experimentally measured dispersion force (between two plane-parallel metallic plates) based on the experiment of Casimir: This measured Casimir force turns out to be in conformity with that calculated from Ramanujan’s highly counter-intuitive finite part of infinity. Behold! the Unreasonable Effectiveness of Mathematics in Natural Sciences that Wigner had spoken of. But clearly, the first-person singular CONSCIOUSNESS is missing from this discourse which lives in the public domain.

The author proposes a new integrated Physics course in physics and mathematics which would be suitable for A level students whose main interest is in biology or arts.

This chapter describes a case study of the role of mathematics in physics textbooks and in associated teacher-led lessons. The theoretical framework (Hansson L, Hansson O, Juter K, Redfors A. Sci Edu 24:615–644, 2015) used in the analysis focuses on relations communicated between three entities: Theoretical models, Mathematics and Reality. Previously the framework has been used for analysing classroom situations. In this chapter, the framework is further developed and refined and for the first time used to analyse physics textbooks. The case study described here is a synchronised analysis of a physics textbook and associated classroom communication during teacher-led lessons and contributes with an in-depth description of relations made between Theoretical models, Mathematics and Reality. With the starting point in this case, we discuss future uses of the analysis framework. We also raise questions for further research concerning how physics textbooks support and not support a meaningful physics teaching with respect to the role of mathematics and how relations between Theoretical models, Mathematics and Reality are communicated.

Orthodox physics makes extensive use of number relation mathematics such as mapping, probability, and infinite series. This mathematics is devoid of causative relations. Other scientific disciplines such as medicine and chemistry use causative models. Using causative models would advance physics. Causation should be overtly stated in the mathematics. Causation is linked with emergence philosophy and not reductionism.

This article discusses the major and minor problems in the organization of education and their solutions, the interdisciplinary relationship in the study of physics, in particular, the ideas of mathematics in physics.

The year 1908 saw the publication for the first time of Felix Klein’s lectures on elementary mathematics, under the title Elementarmathematik vom Hoheren Standpunkte Aus.17† In this memorable work, Klein expresses himself repeatedly on fundamental questions of mathematics and their relation to mechanics and physics. In this connection, he writes that physics does not need exact mathematics, or Prazisionsmathematik, in itself and can amply get along with an Approximationsmathematik, because the measurement data of physics that underlie any theoretical mathematical treatment are necessarily afflicted by uncertainties—called forth by the inescapable existence of limits on the precision with which measurements can be performed.

The present study identified university students’ conceptions of mathematics and physics as well as students’ conceptions of mathematics and physics learning. Furthermore, the study elicited university students’ conceptions of the relationship between mathematics and physics and the relationship between mathematics and physics learning. As a final, the study compared university students’ conceptions of mathematics and physics as well as students’ conceptions of mathematics and physics learning. A purposeful sample of university students (pre-service elementary science and mathematics teachers) who had taken required mathematics and physics courses participated in the study. Open-ended questionnaires and interviews were used to collect data. The analysis found that most students believed that mathematics was closely related to physics and that it was difficult to succeed in physics without understanding mathematics. In addition, more students exhibited coherent conceptions of physics and higher-level conceptions of physics learning compared to the number of the students with coherent conceptions of mathematics and higher-level conceptions of mathematics learning. How the nature of the disciplines of mathematics and physics affects the development of student conceptions is discussed. Key words: conceptions, conceptions of learning, mathematics, physics, qualitative study.

In this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for this view is to explain how mathematical theories can become increasingly abstract and develop their own internal structure, whilst still maintaining an appropriate empirical tether that can explain their later use in physics. In order to address this, I offer a theory of mathematical theory-building based on the idea that human knowledge has the structure of a scale-free network and that abstract mathematical theories arise from a repeated process of replacing strong analogies with new hubs in this network. This allows mathematics to be seen as the study of regularities, within regularities, within ..., within regularities of the natural world. Since mathematical theories are derived from the natural world, albeit at a much higher level of abstraction than most other scientific theories, it should come as no surprise that they so often show up in physics. This version of the essay contains an addendum responding to Slyvia Wenmackers' essay and comments that were made on the FQXi website.