Euler, the Basel Problem, and the Art of Knowing More Than One Can Prove

Euler, the Basel Problem, and the Art of Knowing More Than One Can Prove

An interesting and genuinely stimulating issue: the hashtagBasel_Problem and the hashtagRiemann_zeta_function. By the way, and at the risk of sounding reflective rather than revisionist, my own view is this:

Despite its elegance, Euler’s treatment of the zeta function—and much early work that followed—lacked a unifying conceptual framework. He handled ζ(s) more as a pliant formal expression than as a structured mathematical object. Infinite series were freely rearranged, divergent sums tamed, and analytic continuation used long before anyone defined it. hashtagEuler often knew the answer so well he scarcely felt compelled to justify it.

Crucially, neither he nor his successors had
- a structural reason for the appearance of π in even zeta values,
- a geometric or spectral view of ζ(s),
- or a principled account of how arithmetic data mesh with analytic behaviour.

hashtagRigor eventually arrived with hashtagCauchy, hashtagRiemann, and others, but it merely tidied the theory rather than explaining its deeper architecture.

Further progress on the Basel Problem and the zeta function may require shifting attention from sums to spectra. The presence of π² hints at eigenvalues, boundary conditions, and hidden operators, much like the Laplacian on an interval. From this angle, ζ(2) becomes the trace of an operator on an appropriate space rather than a numerical curiosity.

A promising line of inquiry is to identify a canonical dynamical or geometric system whose spectral invariants encode the special values of ζ(s) across the complex plane. This would situate the zeta function in a natural mathematical setting, not one built from ad hoc analytic continuations. It would place the Basel Problem within broader themes such as spectral geometry, non‑commutative spaces, and the elusive bridge between arithmetic and analysis. Instead of asking how Euler reached the right answer, we ask what sort of mathematical universe renders his result inevitable.

To be clear: Euler’s hashtagsolution was not mistaken but premature—an inspired success achieved before its conceptual foundations existed. He delivered the destination; it has fallen to later mathematics to build the road and explain why the route must look as it does.

Greek: A Language That Refuses to Retire / On the occasion of World Greek Language Day – 9 February

 

Greek: A Language That Refuses to Retire 

On the occasion of World Greek Language Day – 9 February

I. An Ancient Tongue, Impertinently Alive

Greek belongs to that vanishingly small club of languages that have been spoken, written, argued over, sung, cursed and refined for more than three millennia without ever quite expiring. From Mycenaean tablets to modern street conversation, it has undergone reforms, simplifications and the occasional ideological skirmish, yet its structural spine remains unmistakably intact. As Georgios Babiniotis has repeatedly observed, Greek is not a “dead language with a good publishing agent” but a living continuum, whose vocabulary, syntax and semantic depth preserve an extraordinary internal coherence. Words coined for Homer still resonate—sometimes literally—in contemporary usage. Few languages can boast such diachronic audacity: changing just enough to survive, yet never enough to forget who they are.

II. A Language Built Like an Algorithm (Before Algorithms Were Fashionable)

Greek is unapologetically engineered. With its full set of vowels, grammatical genders, moods, voices and inflections, it operates less like a loose collection of words and more like a finely tuned system. Meaning is generated through structure, not merely placement—a fact Andrea Marcolongo delights in emphasising. Greek words are assembled with almost mathematical logic: prefixes, roots and suffixes combine to produce precise semantic outcomes. This “algorithmic” morphology makes Greek remarkably amenable to formal modelling, whether by linguists or modern programming languages. Long before code was compiled, Greek was already running elegant routines for causality, possibility and necessity—without ever crashing.

III. A Language That Invented the Things We Still Study

Armed with Greek, its speakers proceeded to invent an impressive portion of civilisation. Poetry and theatre, philosophy and historiography, mathematics and medicine were not merely expressed in Greek; they were conceptualised through it. The language proved capable of articulating abstraction with surgical precision, allowing Aristotle to dissect logic, Hippocrates the body, and Euclid space itself. As Babiniotis notes, Greek does not merely name ideas—it explains them from within. Even today, the sciences continue to borrow Greek terminology, a quiet admission that when precision is required, one returns to the original toolkit.

IV. UNESCO and a Modest Act of Recognition

In acknowledging this singular legacy, UNESCO designated 9 February as World Greek Language Day — a symbolic yet meaningful gesture. It is not nostalgia that is being honoured, but continuity: the rare phenomenon of a language that has never stopped thinking aloud. As Marcolongo wryly suggests, Greek does not demand admiration; it simply continues to function, centuries on, with calm confidence. One might say that celebrating Greek is less an act of reverence than a courteous acknowledgement that some foundations, once laid properly, do not require replacement.