The philosophy of science, can not avoid being viewed historically, which leads us to learn how mathematics literally took hold, from physics, to chemistry and biology. The application of mathematics to chemistry provides many and ongoing insights:
'Curiously enough, Kant’s own student, Jeremias Benjamin Richter (1762-1807), would only a few years later disprove his professor with his PhD dissertation On the use of the mathematical method in chemistry (Richter 1789). It laid the groundwork for stoichiometry as an algebraic approach to chemistry, including what was later called the ‘law of constant proportion’ that John Dalton used for his atomism on chemical grounds (Richter 1792-3, Dalton 1808). Moreover, at the time of Kant’s writing, experimental philosophy was taking over most of the centers of European research to become the mainstream methodology of modern science, which would later denounce the ideal of a priori knowledge in science as ‘mere’ metaphysics. Although some philosophers of mathematical physics still adhere to that ideal today, Kant was a late partisan in the struggle for the methodological priority of mathematics in the study of nature as it was exemplified by the old field of ‘rational mechanics’. Yet, his view on science became marginalized as much as his verdict on chemistry, that it would be alien to mathematics, was refuted.'
The pitfalls presented and Schummer's 2.4 A methodological suggestion for defining mathematical chemistry are helpful.
Joachim Schummer. Why Mathematical Chemistry Cannot Copy Mathematical Physics and How to Avoid the Imminent Epistemological Pitfalls. HYLE--International Journal for Philosophy of Chemistry, Vol. 18, No.1 (2012). pp. 71-89. https://www.hyle.org/journal/issues/18-1/schummer.htm
Maths is filled with tragedy, but the personal manifestation in my lack of ability, pales to insignificance to the many scientific and human tragedies history reveals. Consider Galois, and Lavoisier:
‘Only a moment to cut off that head and a hundred years may not give us another like it,’ lamented the 18th-century French mathematician Joseph Louis Lagrange. The head in question had belonged to the French aristocrat and chemist Antoine Laurent Lavoisier, who was executed at the hands of French revolutionaries on 8 May 1794. Lavoisier’s ideas changed the face of chemistry. He is best remembered for overthrowing the phlogiston theory, but perhaps his greater and more lasting achievement was to impose order on the language and symbolism that have shaped the thoughts of chemists.'
Paul Board, The Aristocrat who revolutionised chemistry, New Scientist. May 7, 1994 (Volume 142, Issue 1924). https://www.newscientist.com/article/mg14219243-300/
In an essay in the Charles Coulston Gillispie's essay in the history of scientific ideas, Chapter VI. The Rationalization of Matter, describes how Lavoisier sought to develop an algebraic approach:
'But what is most interesting is Lavoisier's mode of representing these results, so correct in quantity, so wrong in principle. In order to sec what he was about, one has to follow him into some detail. It is obvious, he writes, that acidification of a metal involves many variables-heat, concentration, chemical affinities, etc. each of which is a force acting with characteristic energy. Therefrom results a problem complex and difficult of solution: [my emphasis]
"Better to exhibit the state of the question in this respect, and in order to show at a glance the result of what happens in metallic solutions, I have constructed formulas of a sort, which could at first be taken for algebraic formulas, but which have not the same object, and do not derive from the same principles . . ."'
Gillispie, Charles Coulston. The Edge of Objectivity: An Essay in the History of Scientific Ideas. Princeton, N.J.: Princeton University Press, 1960.
Gillispie quotes at length to demonstrate Lavoisier's approach*. There is a more accessible (for me, here) source in -
Stefano Zambelli (2012). Chemical Kinetics, an Historical Introduction, Chemical Kinetics, Dr Vivek Patel (Ed.),
ISBN: 978-953-51-0132-1, InTech, Available from:
http://www.intechopen.com/books/chemicalkinetics/chemical-kinetics-an-historical-introduction
Please see page 6: 3. Chemical equilibrium conception: The law of mass action.
To close, there is reading and advice to be carried forward in:
Guillermo Restrepo & José L. Villaveces, "Mathematical Thinking in Chemistry". Special Issue: Chemistry and Mathematics, Part 1. HYLE--International Journal for Philosophy of Chemistry, Vol. 18, No.1 (2012). pp. 3-22. https://www.hyle.org/journal/issues/18-1/restrepo-villaveces.htm
Gillispie concludes his illustrated exploration of Lavoisier:
'Is this not the most tantalizing memoir in the history of chemistry, and in Lavoisier's the most appealing? Here alone, one gets a sense of modesty. He never claims for these expressions the dignity of algebra. "We are still very far from being able to introduce mathematical precision into chemistry, and I beg, therefore, that no one consider these formulas . . . as more than simple annotations, of which the object is to ease the labors of the mind." (But compare this disclaimer with what he says of algebra itself in the Method of Chemical Nomenclature: "Algebra is the analytical method par excellence: it was invented to facilitate the labors of the mind, to compress into a few lines what would take pages to discuss, and to lead, finally, in a more convenient prompt and certain manner to the solution of very complicated questions.")' p.245.
*in - Lavoisier, A.L. (1782). Considerations sur la dissolution des metaux dans les acides,
Mémoires de l’Académie des sciences 1782, pp. 492-527, Available from
http://www.lavoisier.cnrs.fr/ice/ice_book_detail-fr-text-lavosier-Lavoisier-49-
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