Robyn Arianrhod (2005). Einstein's heroes. Imagining the world through the language of mathematics
Annotated by Ben Wilbrink
Robyn Arianrhod (2005). Einstein's heroes. Imagining the world through the language of mathematics. Oxford University Press.
Maxwell extended this opinion to philosophical or physical speculation: in the development of his theory of electromagnetism he based his mathematics exclusively on known physical phenomena.
p. 9: "—he found that the use of pure mathematical language was the best way to avoid bringing everyday beliefs about reality to the creation of new physical theories."
The interesting idea here is that mathematical modeling should be restricted to the empirical facts, there is no merit in trying to model intuitions and fantasies.
Robyn Arianrhod most of the time speaks of electricity in terms of current, flow, etcetera, as if electricity is a kind of substance, such as blood flowing in veins. It is a pity she has not been more careful in the use of language decribing electricity, after all the use of language, and mathematics is a language also, is the main subject of her book. Contrast this with the following quote from Slotta and Chi (2005).
"Note that the flow of blood in veins is a substance-based concept, as compared to the flow of electrons in wires – which is the very misconceptions that we are reporting about in this paper. Electrons do not flow in a wire in the same way that blood flows in a vein – although people easily think of them flowing that way. Instead, there is little to no movement of electrons (although they are all moving very quickly in random directions, with a small statistical drift in one direction). Then net result is a complex “emergent process” – which makes electric current one of the most challenging concepts to teach, in comparison to the circulatory system, which is quite easy to instruct. The contribution of our paper is that ontological commitments provide an account for why these two concepts differ in their level of difficulty for students and teachers. Chi (2005) provides much more detailed descriptions of the subtle differences between the different ontological classes. "
Note 4 in James D. Slotta and Michelene T. H. Chi (2006). Helping students understand challenging topics in science through ontology training. Cognitive Science, 24, 261-289. pdf
Michelene T. H. Chi (2005). Common sense conceptions of emergent processes: Why some misconceptions are robust. Journal of the Learning Science, 14, 161-199. pdf
Maxwell publishes appr. 1856 a paper on the rings of Saturn, predicting them to be made up of particles of stone or ice. The prediction is based on mathematics only, using Newton's laws, and only confirmed more than a century later—in the Voyager's probe of the rings of Saturn. He won Cambridge's Adams prize for his paper. This prediction is in the same class as that of the existence and location of Neptune, based on the disturbances observable in the orbit of Uranus, ten years earlier in 1846 by Urbain Leverrier.
Arianrhod is rather superficial in attributing important results to Galileo Galilei.
"Als vanzelfsprekend vindt men verder vaak de meening vermeld, als zou GALILEI langs experimenteelen weg, dus door systematische metingen van afgelegde wegen en de daarvoor benoodigde tijden, het verband van weg en tijd in de verlangzaamde valbeweging op hellende vlakken hebben gevonden en daaruit tot de geldigheid van dit verband, ook in vrijen val, hebben geconcludeerd. Bewijsgronden voor deze meening ontbreken echter. In de Discorsi
worden experimenten op een hellend vlak weliswaar vermeld, maar GALILEI gebruikt ze daar, om de mathematisch afgeleide quadratenwet te verifieeren; nergens echter blijkt, dat hij deze wet oorspronkelijk op deze wijze gevonden heeft."
E. J. Dijksterhuis (1924). Val en worp. Een bijdrage tot de geschiedenis der mechanica van Aristoteles tot Newton. Groningen: Noordhoff. p. 239 (noot 97 p. 294 voegt daar nog een opmerking aan toe, betreffende de Frammenti)