Maths and Physics #1: Early Period

You can’t do physics without maths. It’s been that way since Newton. But has it gone too far, many have asked, to a point where physicists fall in love with “beautiful” mathematical theories and stop caring if it aligns with the real world? Phrases like “fairy tale physics”, “not even wrong”, and “lost in math” capture that sentiment.

 

Farmelo Graham’s book, The Universe Speaks in Numbers, traces the history of the relation between physics and maths. The story starts with Newton’s theory of gravity – the equations matched observations, but, complained the critics (even back then), it didn’t describe the physical mechanism behind gravity. This was also a case of Continental envy – the British worshipped Newton, while the Continent felt he was a mathematician, not a physicist.

 

A generation later, the roles reversed. Frenchmen like Laplace advocated and advanced physics via maths, while the British dismissed such an approach as “flowery regions of algebra”. Not just in gravity, Laplace argued, but all other phenomenon “were dancing to mathematical tunes”, from electricity, magnetism, heat, light and the flow of liquids.

 

Laplace, however, didn’t dismiss the need for a physical explanation of things. Others, however, didn’t care for physical explanations. Fourier, for example, came up with a purely mathematical description of heat flow – it used differential equations, and made no reference to atoms or the forces between them!

 

In other cases, physical ideas and maths fed into each other. Faraday had proposed that there is a “field” associated with a magnet that affects the whole of space. Theorists didn’t care for such vague “weird talk of fields”. Except for Maxwell, who believed that the field idea was correct, and when made mathematically precise, would be the key to a theory on electricity and magnetism (Experiments had already shown the two were connected).

 

It took a long while, but Maxwell finally found a way to make those fields mathematically precise. As a bonus of that maths wizardry, it emerged that electromagnetism and light (optics) were related. Such maths-only theories didn’t go down well with everyone – JJ Thompson hated theories that were so “dry and almost incomprehensible mathematics”. Maxwell was undeterred – he felt physical observations and maths complemented each other, and their union was one of “thought weds fact”. Even though:

“Maxwell never knew whether the waves he predicted were real or figments of his mathematically disciplined imagination. It took almost a decade before Hertz demonstrated the existence of electromagnetic waves.”

An oddity here about the 4 famous Maxwellian equations on electromagnetism - Maxwell didn’t come up with them! Later day mathematicians redid things in “a clearer, more concise and more accessible form”. It was as part of that exercise that Heaviside came up with the 4 “Maxwellian” differential equations.

 

The biologist, Thomas Huxley, sympathized with both sides. On the one hand, maths, he declared, “is that study which knows nothing of observations, nothing of experiment, nothing of induction, nothing of causation”. On the other, he could also see the beauty of maths theories, and lamented about “the great tragedy of science – the slaying of a beautiful hypothesis by an ugly fact”.