The Emergence of First-Order Logic (Stanford Encyclopedia of Philosophy)

New Stanford Encyclopedia of Philosophy entry on the emergence of first-order logic by William Ewald:

For anybody schooled in modern logic, first-order logic can seem an entirely natural object of study, and its discovery inevitable. It is semantically complete; it is adequate to the axiomatization of all ordinary mathematics; and Lindström’s theorem shows that it is the maximal logic satisfying the compactness and Löwenheim-Skolem properties. So it is not surprising that first-order logic has long been regarded as the “right” logic for investigations into the foundations of mathematics. It occupies the central place in modern textbooks of mathematical logic, with other systems relegated to the sidelines. The history, however, is anything but straightforward, and is certainly not a matter of a sudden discovery by a single researcher. The emergence is bound up with technical discoveries, with differing conceptions of what constitutes logic, with different programs of mathematical research, and with philosophical and conceptual reflection. So if first-order logic is “natural”, it is natural only in retrospect. The story is intricate, and at points contested; the following entry can only provide an overview.

Probing the Great Unknown

[D]espite over 150 years of endeavor by the world’s greatest minds, the hunch that Riemann had in 1859, the same year that Darwin published On the Origin of Species, has not revealed the origins of what makes the primes tick. They are still part of the unknown. But I think all of the mathematicians that strive to prove this great theorem believe that it won’t remain unknown forever. But how can we be sure? How can we know that we will ultimately discover the answer. The unknown is the life blood of mathematics. The things we don’t know are what make it a living breathing subject not just a subject confined to the static journals on the library shelves of our great institutions. It is the unknown that thrills us as practicing mathematicians.  But what if there is no proof. What if there are things that there is no way we can know. It is the unknowable that fills us with terror.

Prison breakthrough

JOHN NASH arrived at Princeton University in 1948 to start his PhD with a one-sentence recommendation: “He is a mathematical genius”. He did not disappoint. Aged 19 and with just one undergraduate economics course to his name, in his first 14 months as a graduate he produced the work that would end up, in 1994, winning him a Nobel prize in economics for his contribution to game theory