When I next get around to teaching philosophy of science again I think I’ll have my students watch an episode or two from Feynman’s Messenger series of lectures. With the possibility of having studied only humanities subjects from the age of 16 it’s no surprise that there’s many a student with only the sketchiest idea of physics. Last time around when I asked the class to name any of Newton’s laws of motion, a long wait ensued before someone finally proffered an approximation to the first.

I was watching the second lecture yesterday, towards the end of which Feynman distinguishes between what he calls a ‘Babylonian’ approach and a ‘Greek’ one. His idea here is that mathematicians have a tendency to arrange their theories in the Greek style on an axiomatic basis, while this can’t work in physics, at least in a time of theoretical growth, because it is never clear which approach is basic. For example, one may view a physical theory in terms of forces or field potentials or conservation laws or paths of least action, and unpredictably any one of them might provide new insight as to how to link together physical facts.

I found myself torn between responding in two different ways:

The difference is only apparent. Multiple ways of thinking about the same piece of mathematics play the same role of allowing insight into the best way to proceed there. The difference is real, but mathematics can be done in the same ‘Babylonian’ way, and this should be encouraged.

As representatives of these two responses, let’s take a look at passages by William Thurston and Pierre Cartier.

In On proof and progress in mathematics Thurston writes:

People have very different ways of understanding particular pieces of mathematics. To illustrate this, it is best to take an example that practicing mathematicians understand in multiple ways, but that we see our students struggling with. The derivative of a function fits well. The derivative can be thought of as: Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function. Symbolic: the derivative of x n x^n is n x n − 1 n x^{n−1} , the derivative of sin ( x ) sin(x) is cos ( x ) cos(x) , the derivative of f ∘ g f \circ g is f ′ ∘ g * g ′ f' \circ g * g' , etc. Logical: f ′ ( x ) = d f'(x) = d if and only if for every ϵ \epsilon there is a δ \delta such that when 0 < | Δ x | < δ 0 \lt |\Delta x| \lt \delta , | f ( x + Δ x ) − f ( x ) Δ x − d | < ϵ . \left| \frac{f(x + \Delta x) − f(x)}{\Delta x} - d \right| \lt \epsilon. Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent. Rate: the instantaneous speed of f ( t ) f(t) , when t t is time. Approximation: The derivative of a function is the best linear approximation to the function near a point. Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power. This is a list of different ways of thinking about or conceiving of the derivative, rather than a list of different logical definitions. Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit definitions. I can remember absorbing each of these concepts as something new and interesting, and spending a good deal of mental time and effort digesting and practicing with each, reconciling it with the others. I also remember coming back to revisit these different concepts later with added meaning and understanding. The list continues; there is no reason for it ever to stop. A sample entry further down the list may help illustrate this. We may think we know all there is to say about a certain subject, but new insights are around the corner. Furthermore, one person’s clear mental image is another person’s intimidation: 37. The derivative of a real-valued function f f in a domain D D is the Lagrangian section of the cotangent bundle T * ( D ) T^{*}(D) that gives the connection form for the unique flat connection on the trivial ℝ \mathbb{R} -bundle D × ℝ D \times \mathbb{R} for which the graph of f f is parallel. (pp. 3-4)

Now for Cartier, where in Mathemagics (A Tribute to L. Euler and R. Feynman) he writes

The implicit philosophical belief of the working mathematician is today the Hilbert-Bourbaki formalism. Ideally, one works within a closed system: the basic principles are clearly enunciated once for all, including (that is an addition of twentieth century science) the formal rules of logical reasoning clothed in mathematical form. The basic principles include precise definitions of all mathematical objects, and the coherence between the various branches of mathematical sciences is achieved through reduction to basic models in the universe of sets… My thesis is: there is another way of doing mathematics, equally successful, and the two methods should supplement each other and not fight. This other way bears various names: symbolic method, operational calculus, operator theory… Euler was the first to use such methods in his extensive study of infinite series, convergent as well as divergent. The calculus of differences was developed by G. Boole around 1860 in a symbolic way, then Heaviside created his own symbolic calculus to deal with systems of differential equations in electric circuitry. But the modern master was R. Feynman who used his diagrams, his disentangling of operators, his path integrals… The method consists in stretching the formulas to their extreme consequences, resorting to some internal feeling of coherence and harmony. They are obvious pitfalls in such methods, and only experience can tell you that for the Dirac δ \delta -function an expression like x δ ( x ) x \delta(x) or δ ′ ( x ) \delta'(x) is lawful, but not δ ( x ) / x \delta(x)/x or δ ( x ) 2 \delta(x)^2 . (p. 3)

On reflection, the Thurston response is the more relevant. Feynman’s thought about the dangers of the ‘Greek’ approach wasn’t so much about concerns of rigour excluding ways of calculating, as about obstructing ways of thinking by ordering concepts as more or less basic, so reducing heuristically valuable ways of thinking to derivatives of more basic reasoning and thereby neglecting them. But this is just what Thurston warns against within mathematics itself: “Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit definitions.”