# Dutilh Novaes on Maxwell's equations

I've been kind of busy the last couple of weeks and also had a bunch of migraines. Looking at a screen for long periods is still a bit dodgy so I've been getting out of the habit of notebook posting. I want to get some momentum back but need a simple format I can manage with my brain not working properly, so I'll steal David MacIver's idea of picking some text from a book and saying a few things about it.

I opened up Catarina Dutilh Novaes's *Formal Languages in Logic*, and there's an interesting-looking section about Maxwell and electromagnetism in Chapter 6. This chapter is about desemantification, how theories start off grounded in some physical context and then some part of the structure is abstracted out from that. In this case, Maxwell was starting from Faraday's writeups of hundreds of experiments on electromagnetic phenomena.

Interestingly, there is not a single mathematical formula in his writings; Faraday was a self-taught physicist, a briliiant experimentalist but with no formal mathematical training.

So, what was the path from a big pile of experiments to Maxwell's equations? Faraday already had an idea of 'lines of force', expressed in words rather than equations. Maxwell reinterpreted this using fluid flow (which already had a long history of mathematical modelling) as a guide:

Having fluid flow in mind was a heuristic device allowing Maxwell to formulate a mathematical formalism which in fact could receivedifferentphysical interpretations, not only Faradayan lines of force.

Maxwell had three main papers, 'On Faraday's Lines of Force', 'On Physical Lines of Force', 'A Dynamical Theory of the Electromagnetic Field'. Not too much detail on them in this short sketch but the titles sort of tell a story of increasing abstraction from context by themselves. The last one is the one that strips out specific physical properties of the ether that the waves were thought to travel through.

More generally, Maxwell's theory of electromagnetism represents a historical turning point for what is to count as a physical theory; up to him, a physical theory was essentially a physical model, which could be then described mathematically. With Maxwell's mature theory, however, the mathematical formalism - the set of equations -isthe theory, which can then be instantiated in different physical models. This trend was further consolidated in the work of Lorentz, who explicitly rejected visualisation as a reliable method in physics.

I'd like to know more about this turning point in physics, and its later influence on positivism etc. That strand of thinking seemed to definitively win out against the more phenomenological one in the twentieth century. The Lorentz thing could be worth looking up for a start.

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