# Has the heyday of mathematical formulae ended?

I have a strong emotional reaction when I read the works of Euler. I have seen many extremely beautiful and intriguing identities in the notebook of Ramanujan, so much so that I think he is indeed a Euler of th 20th century.

But what I notice is that mathematics have changed since the 19th century. Math seems to develop more theory and become much more rigorous due to the development of logic (think of Gottlob Frege). I realize that math during the day of Euler, Lagrange and d'Alembert is very calculational, computational. They trust algebraic symbols and are much more formal.

From Littlewood's Miscellany (page 95, 96):

But the great day of formulae seems to be over. No-one, if we are again to take the highest standpoint, seems to be able to discover a radically new type, though Ramanujan comes near in his work of partition series; it is futile to multiply examples in the sphere of Cauchy's theorem and elliptic function theory, and some general theory dominates, if in a less degree, every other field. A hundred years or so ago, his power would have had ample scope. Discoveries alter the general mathematical atmosphere and have very remote effects, and we are not prone to attach great weight to rediscoveries, however independent they seem. How much are we allow for this, how great a mathematician might Ramanujan have been 100 or 150 years ago; and what would have happened if he had come in touch with Euler at the right moment?

In the article "Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem", the author also wrote that:

From the mid-18th century to the mid-19th century, the style of mathematical research in analysis underwent changes from a formula-centered approach epitomized by L. Euler (1707–1783) to a concept-centered style presented in works of G. P. L. Dirichlet (1805–1859) and G. F. B. Riemann (1826–1866). The transition manifested itself in multiple aspects of the mathematical enterprise including notations, questions, results, methods, and techniques. It was felt by the active and creative mathematicians of the nineteenth century who spotted a difference between the computational machinery associated with the formula-centered approach and the decidedly mental analysis belonging to the concept-centered approach. We find this distinction seized upon for instance in Dirichlet's obituary of C. G. J. Jacobi (1804–1851) where Dirichlet noticed1

[…] the constantly increasing tendency of the new analysis to put thoughts in the place of calculations, […],2

in the methodological principle attributed by D. Hilbert (1862–1943) to Riemann near the end of the century, I have tried to avoid the large computational apparatus of Kummer such that also here Riemann's principle should be observed, according to which one should conquer proofs not by computations but solely through thoughts,3

or in what H. Minkowski (1864–1909) called the “second Dirichlet principle” heralding the modern times in mathematics, according to which problems should be conquered with a minimum of blind calculations and a maximum of enlightening thoughts.4

Is it true that formulae is no longer in its heyday as compared to the 18th, early 19th century? After all, we have Roger Apéry proving that $$\zeta(3)$$ is an irrational number and probably cannot be expressed in closed form (I think I am wrong here).

• Not even close. If you want formulas take a look at the bulk of current papers on enumerative geometry, representation theory or combinatorics. And if you want less rigorous look at any of that being applied to string theory. Here is an example from last year. Feb 14 at 21:12

There are plenty of formulas in the new maths. Littlewood is right that the great "heyday" of certain types of formulas is over. But what this really means that mathematical taste has both changed and have widened.

One formula that I like is Stokes' theorem. It's a formula of vector analysis. By finding the right context for it, that is on manifolds, the formula is now valid for any curved space (the original formulas was only valid in flat 3d space), and for curved spaces in higher dimensions as well as lower. In fact, in 1d the formula becomes the fundamental theorem of calculus first shown by Newton. Moreover, the formula is systematic.

One aim of the new maths is to make sure that all the formulas found by Euler, Littlewood et al. find similar contexts that explain why they take the form they do and relate them.

Another aim is that proofs become mathematised into formulae. This is one of the great successes of category theory and why it's talked about as a 'language' of mathematics. It helps discern proof strategies in theories and makes formulae out of them.

For example, Emily Riehl, in one of her books on category theory, showed that singular homology is the composition of five functors. This is as much of a formula as the identities beloved by Euler or Littlewood.

What's not to like about that either?

It seems, you are talking about new formulas in analysis and algebra. I think, the formula age is not yet over there, and yes, I think, many of new formulas will be related to zeta function and differentiated gamma functions.

We still do not understand zeta function well.

Regarding $$\zeta(3)$$ is not closed form, I disagree with you. How closer do you want it? As an elementary function of $$3$$? It is already as much closed form as closed form can be.