The origin of mathematics

Archive for History of Exact Sciences -     

The role of a posteriori mathematics in physics

The calculus that co-evolved with classical mechanics relied on definitions of functions and differentials that accommodated physical intuitions. In the early nineteenth century mathematicians began the rigorous reformulation of calculus and eventually succeeded in putting almost all of mathematics on a set-theoretic foundation. Physicists traditionally ignore this rigorous mathematics. Physicists often rely on a posteriori math, a practice of using physical considerations to determine mathematical formulations. This is illustrated by examples from classical and quantum physics. A justification of such practice stems from a consideration of the role of phenomenological theories in classical physics and effective theories in contemporary physics. This relates to the larger question of how physical theories should be interpreted.     

The heuristic role of mathematics in the initial development of superconductivity theory

During the first twenty-four years after the discovery of superconductivity many attempts to derive an adequate theory failed, mainly because the problem was not formulated quite correctly. In this paper we investigate certain questions related to the heuristic role of mathematics in the appropriate formulation of the problem that had to be solved and the development of a theory which was hindered by theoretical superstitions.     

The power of images: mathematics and metaphysics in Hobbes's optics

This paper deals with Hobbes's theory of optical images, developed in his optical magnum opus, ‘A Minute or First Draught of the Optiques’ (1646), and published in abridged version in De homine (1658). The paper suggests that Hobbes's theory of vision and images serves him to ground his philosophy of man on his philosophy of body. Furthermore, since this part of Hobbes's work on optics is the most thoroughly geometrical, it reveals a good deal about the role of mathematics in Hobbes's philosophy. The paper points to some difficulties in the thesis of Shapin and Schaffer, who presented geometry as a ‘paradigm’ for Hobbes's natural philosophy. It will be argued here that Hobbes's application of geometry to optics was dictated by his metaphysical and epistemological principles, not by a blind belief in the power of geometry. Geometry supported causal explanation, and assisted reason in making sense of appearances by helping the philosopher understand the relationships between the world outside us and the images it produces in us. Finally the paper broadly suggests how Hobbes's theory of images may have triggered, by negative example, the flourishing of geometrical optics in Restoration England.     

Structuralism and the conformity of mathematics and nature

Structuralists typically appeal to some variant of the widely popular ‘mapping’ account of mathematical representation to suggest that mathematics is applied in modern science to represent the world’s physical structure. However, in this paper, I argue that this realist interpretation of the ‘mapping’ account presupposes that physical systems possess an ‘assumed structure’ that is at odds with modern physical theory. Through two detailed case studies concerning the use of the differential and variational calculus in modern dynamics, I show that the formal structure that we need to assume in order to apply the mapping account is inconsistent with the way in which mathematics is applied in modern physics. The problem is that a realist interpretation of the ‘mapping’ account imposes too severe of a constraint on the conformity that must exist between mathematics and nature in order for mathematics to represent the structure of a physical system.     

Augustus De Morgan and the propagation of moral mathematics

In the early nineteenth century, Henry Brougham endeavored to improve the moral character of England through the publication of educational texts. Soon after, Brougham helped form the Society for the Diffusion of Useful Knowledge to carry his plan of moral improvement to the people. Despite its goal of improving the nation’s moral character, the Society refused to publish any treatises on explicitly moral or religious topics. Brougham instead turned to a mathematician, Augustus De Morgan, to promote mathematics as a rational subject that could provide the link between the secular and religious worlds. Using specific examples gleaned from the treatises of the Society, this article explores both how mathematics was intended to promote the development of reason and morality and how mathematical content was shaped to fit this particular view of the usefulness of mathematics. In the course of these treatises De Morgan proposed a fundamentally new pedagogical approach, one which focused on the student and the role mathematics could play in moral education.     

The structures of ambrosin and damsin

Zusammenfassung Es wird gezeigt, dass die vonAbu-Shady undSoine isolierten Sesquiterpenlaktone «Ambrosin» und «Damsin» vermutlich Gemische von isomeren Substanzen darstellen. Die Interpretation bereits bekannter Tatsachen über die beiden Naturstoffe und der Vergleich mit verwandten Produkten ermöglicht die Aufstellung provisorischer Strukturformeln.

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Terpenes VIII. Part. VII,G. Büchi andW. S. Saari, J. Amer. chem. Soc. (in press).

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On leave of absence from Farmitalia, Milano, Italy.     

The structures of isofusicoccin and allofusicoccin

Riassunto L'allofusicoccina e l'isofusicoccina, due isomeri della fusicoccina isolati dai brodi di coltura diFusicoccum amygdali Del., differiscono dalla fusicoccina solamente per la posizione del gruppo acetossilico sul residuo del glucosio; nella prima questo è sul C-2 e nella seconda sul C-4. Fusicoccina, allofusicoccina e isofusicocina, nonchè i loro 19-deacetilderivati ed i sei corrispondenti diidroderivati, si interconvertono a pH leggermente alcalino a temperatura ambiente.     

Outline of a dynamical inferential conception of the application of mathematics

We outline a framework for analyzing episodes from the history of science in which the application of mathematics plays a constitutive role in the conceptual development of empirical sciences. Our starting point is the inferential conception of the application of mathematics, recently advanced by Bueno and Colyvan (2011). We identify and discuss some systematic problems of this approach. We propose refinements of the inferential conception based on theoretical considerations and on the basis of a historical case study. We demonstrate the usefulness of the refined, dynamical inferential conception using the well-researched example of the genesis of general relativity. Specifically, we look at the collaboration of the physicist Einstein and the mathematician Grossmann in the years 1912–1913, which resulted in the jointly published “Outline of a Generalized Theory of Relativity and a Theory of Gravitation,” a precursor theory of the final theory of general relativity. In this episode, independently developed mathematical theories, the theory of differential invariants and the absolute differential calculus, were applied in the process of finding a relativistic theory of gravitation. The dynamical inferential conception not only provides a natural framework to describe and analyze this episode, but it also generates new questions and insights. We comment on the mathematical tradition on which Grossmann drew, and on his own contributions to mathematical theorizing. The dynamical inferential conception allows us to identify both the role of heuristics and of mathematical resources as well as the systematic role of problems and mistakes in the reconstruction of episodes of conceptual innovation and theory change.