Examples of Classical Approaches Being Potentially More Preferable than Modern Approaches

Question

It is a well-known result that given a field $F$, the set of elements algebraic over $F$ is also a field. When presenting this result in class, my professor proved it two ways.

• What he called the "classical" way: Suppose $a$ and $b$ are algebraic. Using their minimal polynomials and some properties of elementary symmetric polynomials, one can explicitly find polynomials certifying that $ab$ and $a+b$ are algebraic as well.

• What he called the "modern" way: One can show that $[F[a,b]:F]$ is finite and that if $g$ is in a finite field extension of $F$, then $g$ is algebraic over $F$. These facts combine to give the result.

He mentioned he specifically wanted to talk about the classical proof because it is constructive, and introduces elementary symmetric polynomials in a useful context. Since I've done some self-study in algebra, I've only seen the proof done the modern way.

So here is my question:

What are examples of classical approaches to subjects being potentially more preferable than modern approaches? In which way are these examples potentially more preferable (e.g. in the above example the classical approach is potentially more preferable because it is constructive)?

This question came to mind because I recently found an old complex analysis Dover book from the 40's, and while the mathematics is likely all correct, the perspective and presentation is undoubtedly different from a modern text. Certainly it's a fun piece to own, but I was curious if I had a reason to crack it open.

Answer 1

A standard example is Spivak's books on differential geometry (and calculus). As I understand, his perspective was that even though modern (invariant, coordinate-free) treatments of differential geometry are often more concise and more streamlined, they have a tendency to obscure the geometric intuition behind the definitions. Here is a relevant excerpt from his A Comprehensive Introduction to Differential Geometry, Vol.1, 3e (pp.ix-x)

For many years I have wanted to write the Great American Differential Geometry book. Today a dilemma confronts any one intent on penetrating the mysteries of differential geometry. On the one hand, one can consult numerous classical treatments of the subject in an attempt to form some idea how the concepts within it developed. Unfortunately, a modern mathematical education tends to make classical mathematical works inaccessible, particularly those in differential geometry. On the other hand, one can now find texts as modern in spirit, and as clean in exposition, as Bourbaki's Algebra. But a thorough study of these books usually leaves one unprepared to consult classical works, and entirely ignorant of the relationship between elegant modern constructions and their classical counterparts. Most students eventually find that this ignorance of the roots of the subject has its price — no one denies that modern definitions are clear, elegant, and precise; it's just that it's impossible to comprehend how any one ever thought of them. And even after one does master a modern treatment of differential geometry, other modern treatments often appear simply to be about totally different subjects.

Of course, these remarks merely mean that no matter how well some of the present day texts achieve their objective, I nevertheless feel that an introduction to differential geometry ought to have quite different aims. There are two main premises on which these notes are based. The first premise is that it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc. , which was developed precisely in order to rigorize the concepts of classical differential geometry. Rephrasing everything in more elementary terms involves incredible contortions which are not only unnecessary, but misleading. The work of Gauss, for example, which uses infinitesimals throughout, is most naturally rephrased in terms of differentials, even if it is possible to rewrite it in terms of derivatives. For this reason, the entire first volume of these notes is devoted to the theory of differentiable manifolds, the basic language of modern differential geometry. This language is compared whenever possible with the classical language, so that classical works can then be read.

The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature. I personally felt that I could never acquire a satisfactory understanding of differentiable geometry until I read the original works. [...]

Similarly, in his Calculus on Manifolds (pp.44-45) there is a digression on notation, where he compares the modern notations for partial derivatives and the classical notations (which are still taught in introductory calculus classes). It seems to me modern notation is more useful when trying to make sense, whereas the classical notation is more useful when one really needs to calculate things.

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Here is a more specific example; it comes from Katok's paper "Smooth Non-Bernoulli $K$-Automorphisms". Let $M$ be a compact Riemannian $C^\infty$ manifold, $\pi:\widetilde{M}\to M$ be its universal cover and define $\mathscr{C}(M)$ to be the collection of all continuous functions $\phi:\widetilde{M}\to \mathbb{R}$ with the property that for any sufficiently small open subset $U$ of $M$ and for any lifts $\widetilde{U_1},\widetilde{U_2}\subseteq \widetilde{M}$ of $U$, the function $\phi|_{\widetilde{U_1}}\circ\pi^{-1}|_U-\phi|_{\widetilde{U_2}}\circ\pi^{-1}|_U:U\to\mathbb{R}$ is constant. $\mathscr{C}(M)$ is the space of candidate solutions to a certain functional equation (called the cohomological equation in dynamics). It is straightforward that if we have a continuous function $\psi:M\to \mathbb{R}$, then $\psi\circ\pi\in \mathscr{C}(M)$; let's call such elements of $\mathscr{C}(M)$ trivial. It is then noted in passing that any element in $\mathscr{C}(M)$ defines a (Čech) cohomology class in $H^1(M;\mathbb{R})$; the discrepancy of the map $\mathscr{C}(M)\to H^1(M;\mathbb{R})$ is precisely the trivial elements. Beyond some basic properties this connection is not useful for the rest of the paper. This can be considered as a classical approach to Čech cohomology (in degree $1$); at the expense of being seemingly ad hoc, this definition bypasses describing (or requiring as a prerequisite for the paper) Čech cohomology, which is appropriate for the intended audience of the paper.