Why I Teach 154-year-old Mathematical Physics
The rise in advocacy for education in Science, Technology, Engineering, and Mathematics (STEM) has centered around the importance of these fields to our nation’s economy and business competitiveness. Recently, however, Fareed Zakaria made a case for STEM education as a part of a liberal education. Zakaria placed particular emphasis on the humanities as being at least as important as the STEM fields.
Today’s guest post by Matthew Linck, a faculty member at St. John’s College in Annapolis, explains why science and mathematics are fundamental to a solid education.
Fareed Zakaria’s argument for serious engagement with science and mathematics as part of a liberal education is rather unexpected in the current climate of public opinion. But this is the view taken here at St. John’s. Our students spend nearly half of their class time on mathematics and science. We read and discuss primary texts or engage in direct experience with observable phenomena, duplicating the experiments that the authors themselves designed. Among these texts are Euclid’s Elements, the Conics of Apollonius, Newton’s Principia, Dedekind’s “Continuity and Irrational Numbers,” and Einstein’s original 1905 paper on special relativity.
What do we mean when we say that these studies are part of a liberal education? One thing it means is that they are worth knowing for their own sake. The study of mathematical physics is inherently interesting in a number of ways: for its conceptual foundations; for the phenomena it both attends to and brings to light; for the rather mysterious fact that physical phenomena can be captured in mathematical expressions; for the insights it offers into the workings of powerful minds; and for the discernment it engenders concerning the power and the limits of modern natural science.
James Clerk Maxwell’s work on magnetism (On Physical Lines of Force, 1861-62) and electricity (A Dynamical Theory of the Electromagnetic Field, 1865) can stand for the sort of things I’m talking about. To understand Maxwell, one must study the principal phenomena related to magnetic and electric bodies. Three of these phenomena have already been explored directly in our classrooms, through discussion and experiment, by the time our students and faculty members read Maxwell together.
First, William Gilbert’s (1544-1603) study of the lodestone introduces us to the strangeness of magnets and magnetic polarity. The existence of magnetic attraction and repulsion is easily seen, but the source of the power is deeply mysterious. Splitting a magnetized body and finding it retains a north and south pole is even more mysterious.
Second, Hans Christian Ørsted (1777-1851) reveals that circular magnetic lines are formed around a wire carrying an electric current. It is stunning to place a number of compasses around a wire and watch their needles form a circle when current is sent through the wire.
And third, Michael Faraday (1791-1867) shows that current can be induced in a wire by the mere proximity of another current-carrying wire. How this can happen at all is a marvel. Why it only happens for an instant when the inducing current is first turned on remains a nagging conundrum.
It turns out that each of these three phenomena has an exact mathematical formulation in Maxwell’s writings. That magnets always have a north and south pole is captured in this equation: dα/dx + dβ/dy + dγ/dz = 0. The Ørsted phenomenon is captured by equations of this form: dβ/dz – dα/dy = 1/c dP/dt. And the momentary induction of one current by another can be expressed by equations that look like this: dP/dy – dQ/dx = 1/c dγ/dt. To see how this is so in each case is enormously difficult! But to really see that it is so, even in one case, is to experience the strange fact that mathematical symbols can be lined up precisely with observable phenomena. In our classes, we make this discovery repeatedly by working through these equations for ourselves, following the steps in Maxwell’s papers.
Now Maxwell had a real problem in trying to develop his equations: there was no way to measure directly how a field of electromagnetic force might be acting. So he created fictional ways! Maxwell called these fictions “physical analogies”—tubes of flowing liquids, spinning fluid vortices, and tiny circulating particles. He could then determine quite precisely the quantities of force, velocity, momentum, and so on, of these fictional bodies using Newtonian mechanics. Maxwell was subsequently able to show that the equations he derived from these “fictions” exactly match all his observations of electric and magnetized bodies.
Following Maxwell’s account not only shows us a powerful and imaginative mind at work, but also raises many confounding questions. How can thinking about fictions yield equations that match real phenomena? Is the electromagnetic field itself real or fictional? Can equations alone explain physical reality? Such questions might incline us to temper our admiration for the power of mathematical physics with questions about the true nature of its results.
Taking up science as part of a liberal education means taking it up as something worth doing as its own end. We don’t need to ask what such study is good for. We only need to see that doing it is good.
Matthew Linck is a faculty member at St. John’s College in Annapolis.