Yet the marriage between math and science may be merely a flirtation or friendship. The bond and similarity between the two is not as strong as is commonly supposed.
The thinkers who began the enterprises of modern mathematics and modern science saw them as two very different activities. The home of a priori rational certainty was found in mathematics, while approximation and tentativeness lived in the sciences.
The quadratic formula is the same in textbooks printed today or two hundred years ago — and will be the same centuries into the future. By contrast, the atomic weight of various isotopes of lead or gold, when calculated out to large numbers of significant figures, might be revised or refined over the years as successive editions of textbooks are printed for chemistry and physics classes.
There is a danger in overemphasizing the importance of mathematics in the natural sciences — and here one can also mean the observational sciences and empirical sciences. The non-mathematical aspects of scientific activity risk being ignored: the intuitive aspects.
There is an objection, of course, to say that seemingly non-mathematical properties like color and shape can in fact be reduced to mathematics: color is a wavelength which can be represented by a number; shape can be captured in an algebraic equation.
To this objection, one might respond by positing that a color is more than the number of its wavelength and a shape is more than its corresponding algebraic equation. As the human mind seeks correlations and systemic connections, a color or a shape functions differently, and is treated by the mind differently, than a number or an equation. It is in this seeking that scientific discoveries are made and insights gained.
Two scientists who were champions of mathematics — James Clerk Maxwell and Ludwig Boltzmann — nonetheless praised the intuitive and conceptual work of Michael Faraday. Faraday’s discoveries and descriptions of electromagnetic phenomena were shockingly free of mathematics, as historian Alan Hirschfeld writes:
Maxwell, the consummate mathematician, nonetheless understood the power of mathematics to mislead when not anchored in experiment or observation. In Faraday’s Researches, he encountered science in its purest form, “untainted” by mathematical manipulation. Here, he decided, would be the entry point for his own investigations into electricity and magnetism. In a later reflection, Maxwell sounds almost relieved that Faraday had stuck to his particular brand of investigation, thereby blazing a trail that Maxwell himself could follow.
Boltzmann explains one of Faraday’s many intuitive, i.e. non-mathematical, breakthroughs:
While the older system had held the centers of force to be the only realities, and the forces themselves to be mathematical conceptions, Faraday saw distinctly the continuous working of the forces from point to point in the intermediate space. The potential, which had hitherto been only a formula for lightening the work of calculation, was for him the bond really existing in space, the cause of the action of force.
Boltzmann uses the word ‘saw’ in the text above, emphasizing Faraday’s visual technique. Faraday’s work with magnetic fields was largely the work of observing patterns, e.g., the movement of iron filings. “By the light of his own clear conceptions,” Boltzmann writes, Faraday made “such great discoveries.” Boltzmann also explains Faraday’s effect on Maxwell:
Maxwell also, when he undertook the mathematical treatment of Faraday’s ideas, was from the very outset impelled by their influence into a new path.
Maxwell himself praises Faraday’s non-mathematical approach:
It was perhaps for the advantage of science that Faraday, though thoroughly conscious of the fundamental forms of space, time, and force, was not a professed mathematician. He was not tempted to enter into the many interesting researches in pure mathematics which his discoveries would have suggested if they had been exhibited in a mathematical form, and he did not feel called upon either to force his results into a shape acceptable to the mathematical taste of the time, or to express them in a form which mathematicians might attack. He was thus left at leisure to do his proper work, to coordinate his ideas with his facts, and to express them in natural, untechnical language.
Faraday was talented at drawing diagrams and sketches of magnetic fields and various configurations which were part of his experiments. The etymology of the word ‘intuition’ arises from verbs meaning to ‘see’ or to ‘look’ and Immanuel Kant chose the word ‘intuition’ — or rather, the German equivalent, Anschauung — which might prompt the reader to investigate any similarities between Kant’s thought and Faraday’s thought.
In any case, Faraday filled notebooks with drawings and illustrations, and it was in them and through them that he founded the modern science of electromagnetism and made his many significant discoveries: it was in and through the images and illustrations, not by means of equations and formulas.
Aside from Faraday’s visual method, there was a second feature of his thought which may have shaped his investigations and conclusions. His entire adult life was spent working in an organization known as the “Sandemanian” or “Glasite” church, as Alan Hirschfeld writes:
The Sandemanian church continued to hold a central place in Faraday’s life. He attended services and ritual feats, enjoyed the sense of community and, with rare exception, clung to its precepts.
The worldview of this faith shaped his exploration of electromagnetism. In the same way that a sort of spiritual humility caused Augustine to recognize the limits of human reason and caused Francis Bacon to formulate the sources of experimental error, so also Faraday was motivated to caution in his hypothesizing and to thoroughness in observation and experimentation.
Faraday understood the laws and principles of electromagnetism as being the products of God’s thought. He wrote: “for the book of nature, which we have to read is written by the finger of God.” Faraday understood lawlike phenomena, and nature’s laws, to be deliberately and rationally planned: “God has been pleased to work in his material creation by laws.”
There is a connection between Faraday’s intuitive approach to electromagnetism and his understanding of God. Verbs of sight were both metaphor and literally truth for Faraday in his investigations of electromagnetism: he was “looking” into God’s work and “saw” the rationality of it.
