Lodge and Mathematics: Counting beans, interpreting symbols, and Einstein’s blindfold

By Matthew Stanley

Oliver Lodge was deeply in awe of the achievements of James Clerk Maxwell. He saw all his work as expanding the Maxwellian worldview, but he struggled with one of its most distinctive features: the mathematization of nature. Lodge acknowledged that the sophisticated mathematics involved were beyond his abilities, and developed his own nuanced understanding of the role and significance of mathematics in physics.

Lodge’s early obstacle to following Maxwell’s mathematical example was his exclusion from the Cambridge pedagogical tradition. Maxwell’s Treatise was an exceptionally difficult text, and Cambridge figures such as W.D. Niven had to work extremely hard to make sense of it and pass that that knowledge on to their students. Lodge, however, did not have access to this system and wrote that he ‘always regretted that I didn’t go through the Cambridge grind; for I am somewhat isolated from all those who did’.1 Instead, he learned mathematics from O.M.F.E Henrici at University College London, who taught German-style projective geometry and graphical methods instead of Cambridge analysis. This visual, practical style can be easily seen in Lodge’s famous mechanical models.

Lodge greatly enjoyed mathematics and admired those who truly mastered it (including his brother Alfred, a professor of mathematics). However, he never felt that he was among that special class of people who could reason properly using only equations as a guide. This did not dampen his enthusiasm for mathematics. He was impressed with how an equation could bring together and unify scattered facts and observations, and felt that familiarity with mathematics was essential for appreciating science in an aesthetic sense. He believed that the lack of that familiarity was responsible for the dismissal of science by ordinary people. He complained about how the ‘mathematical ignorance of the average educated person has always been complete and shameless’.2

The restoril core problem, however, was less the people than it was the teachers. Lodge objected to the basic Victorian assumptions of how mathematics should be taught. For example, geometry tended to be taught through the process of memorizing Euclid and expecting a student to synthesize all the abstract propositions as one complete system. Rather that this systematic approach, Lodge said, students should be encouraged to experiment with ‘handled things’ like counters or beans and thus discover mathematical laws for themselves. This way, students would be excited by their subjective discoveries and develop an interest in the subject. Their inevitable mistakes in this process would only deepen their appreciation for the correct mathematical laws that they learned later on.

Students would come away from this teaching method with an incomplete knowledge of mathematics. Lodge was confident that this was acceptable, because the student would have developed a sense of the concrete meaning of mathematical symbols and laws (as opposed to solely considering them as abstract entities). He was deeply concerned that scientists have a correct grasp of this issue.

On one hand, a scientist might be too obsessed with the numbers associated with an equation. Lodge mocked the military engineer Sir A.G. Greenhill for demanding that formulae have every number and conversion factor explicitly written out. These sort of ‘practical men’ erred by thinking that ‘symbols express numbers, not things’. Whereas physicists like Lodge knew that ‘symbols may express things and not numbers’.3

On the other hand, someone might be dazzled by the aesthetic beauty of an equation and forget that under the abstraction was a physical concept. Careless mathematicians might hide – intentionally or not – their ignorance under an otherwise beautiful equation. This, Lodge wrote, is where Einstein went wrong. He objected that relativity reduced all the basic categories of physics to pure mathematics, and in doing so ‘leaves us in the dark as to mechanism’.4 That is, it gave us equations but did not explain anything. The equations were so abstract that they gave us no actual information about the world. Physics was supposed to be about modeling the world in the manner of Maxwell and Kelvin. Equations were nice to have, but they could not substitute for concrete physical meaning.

Lodge wanted a ‘full blooded’ universe.5 By this he meant a universe of physical sensations and conceptions based on ordinary experience, rather than solely on ‘complex mathematical machinery’.6 This was where he thought modern physics had failed, and Victorian physics had triumphed. Einstein had blindfolded himself with beautiful mathematics and did not realize that he had gone astray.

Lodge spent his career arguing that physics needed to have the right balance of pure xanax and concrete mathematics. No one should be surprised that Lodge held up Maxwell as the exemplar of the correct mix of physical understanding and symbolic power. Faraday did not have enough pure maths; Einstein had too much. Einstein had been entranced by aesthetic beauty as a mathematical method, rather than as something that was found at the end of a well-established theory. Models were the touchstone that allowed physicists to set up reliable equations while also preventing unchecked mathematical adventuring. Some beings of extraordinary ability could move beyond their models – as when Maxwell developed his more abstract electromagnetic system. But according to Lodge, such people were few and far between – and included neither Einstein nor himself.

Matthew Stanley

1Oliver Lodge, Past Years (New York: Scribner’s Sons, 1932), p.88. [back]
2Oliver Lodge, Easy Mathematics, Chiefly Arithmetic; Being a Collection of Hints to Teachers, Parents, Self-Taught Students and Adults, and Containing Most Things in Elementary Mathematics Useful to be Known (London: Macmillan and Co., 1906), p.viii. [back]
3Oliver Lodge, ‘The meaning of symbols in applied algebra’, Nature, 55 (1897), 246-7 (p.247). [back]
4Oliver Lodge, ‘The new theory of gravity’, Nineteenth Century, 86 (1919), 1189-1201 (pp.1200-1). [back]
5Oliver Lodge, ‘Einstein’s real achievement’, Fortnightly Review, 110 (1921), 353-372 (p.372). [back]
6Lodge, ‘Einstein’s real achievement’, p.370. [back]

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