On foundations of quantum physics

Some aspects of the interpretation of quantum theory are discussed. It is emphasized that quantum theory is formulated in the Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between 'canonically conjugated' coordinate and momentum operators leads to a wrong version of quantum mechanics. The origin of time is analyzed in detail by the example of atomic collision theory. It is shown that for a closed system like the three-body (two nuclei + electron) time-dependent Schroedinger equation has no physical meaning since in the high impact energy limit it transforms into an equation with two independent time-like variables; the time appears in the stationary Schroedinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system. Following the Einstein-Rozen-Podolsky experiment and Bell's inequality the wave function is interpreted as an actual field of information in the elementary form. The relation between physics and mathematics is also discussed.Comment: This article is extended version of paper: Solov'ev, E.A.: Phys.At.Nuc. v. 72, 853 (2009

Accounting for primitive terms in mathematics

The philosophical problem of unity and diversity entails a challenge to the rationalist aim to define everything. Definitions of this kind surface in various academic disciplines in formulations like uniqueness, irreducibility, and what has acquired the designation “primitive terms”. Not even the most “exact” disciplines, such as mathematics, can avoid the implications entailed in giving an account of such primitive terms. A mere look at the historical development of mathematics highlights the fact that alternative perspectives prevailed – from the arithmeticism of Pythagoreanism, the eventual geometrisation of mathematics after the discovery of incommensurability up to the revival of arithmeticism in the mathematics of Cauchy, Weierstrass, Dedekind and Cantor (with the later orientation of Frege, who completed the circle by returning to the view that mathematics essentially is geometry). An assessment of logicism and axiomatic formalism is followed by looking at the primitive meaning of wholeness (and the whole-parts relation). With reference to the views of Hilbert, Weyl and Bernays the article concludes by suggesting that in opposition to arithmeticism and geometricism an alternative option ought to be pursued – one in which both the uniqueness and mutual coherence between the aspects of number and space are acknowledged

Beyond deficit-based models of learners' cognition: Interpreting engineering students' difficulties with sense-making in terms of fine-grained epistemological and conceptual dynamics

Researchers have argued against deficit-based explanations of students' troubles with mathematical sense-making, pointing instead to factors such as epistemology: students' beliefs about knowledge and learning can hinder them from activating and integrating productive knowledge they have. In this case study of an engineering major solving problems (about content from his introductory physics course) during a clinical interview, we show that "Jim" has all the mathematical and conceptual knowledge he would need to solve a hydrostatic pressure problem that we posed to him. But he reaches and sticks with an incorrect answer that violates common sense. We argue that his lack of mathematical sense-making-specifically, translating and reconciling between mathematical and everyday/common-sense reasoning-stems in part from his epistemological views, i.e., his views about the nature of knowledge and learning. He regards mathematical equations as much more trustworthy than everyday reasoning, and he does not view mathematical equations as expressing meaning that tractably connects to common sense. For these reasons, he does not view reconciling between common sense and mathematical formalism as either necessary or plausible to accomplish. We, however, avoid a potential "deficit trap"-substituting an epistemological deficit for a concepts/skills deficit-by incorporating multiple, context-dependent epistemological stances into Jim's cognitive dynamics. We argue that Jim's epistemological stance contains productive seeds that instructors could build upon to support Jim's mathematical sense-making: He does see common-sense as connected to formalism (though not always tractably so) and in some circumstances this connection is both salient and valued.Comment: Submitted to the Journal of Engineering Educatio

Informal proof, formal proof, formalism

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened

Symbols and the bifurcation between procedural and conceptual thinking

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to d o mathematical problems and to think about mathematical relationships. In this presentation we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus and Brazil, with data collected in the USA, Malaysia and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail