Leopold Kronecker

Leopold Kronecker: God Made the Integers

"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" — God made the integers, all else is the work of man. Heinrich Weber reported the line, and it has outlived nearly everything else its author said, because it is not a pious aside but a declaration of war. The man who uttered it spent his final decades trying to strip modern mathematics of the infinite objects his colleagues were busy inventing, and he made enough enemies doing it that one of the greatest analysts in Europe nearly quit Berlin to get away from him.

Liegnitz and a Teacher Named Kummer

Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia — now Legnica, Poland — into a wealthy Jewish family. His parents, Isidor and Johanna née Prausnitzer, engaged private tutors at home before sending him to the Liegnitz Gymnasium, where he ranged across science, history and philosophy and swam and did gymnastics with the same appetite. His younger brother Hugo became a distinguished physiologist. At the Gymnasium his mathematics teacher was Ernst Kummer, who spotted the talent and pushed it — a relationship that would shape the rest of Kronecker's life, since the two men's careers kept converging.

He enrolled at the University of Berlin in 1841, hedging between astronomy, philosophy and mathematics, spent the summer of 1843 studying astronomy at Bonn, then followed Kummer to Breslau for 1843-44 before returning to Berlin, where in 1845 he defended a dissertation in algebraic number theory under Peter Gustav Lejeune Dirichlet.

The Businessman Who Did Mathematics

What he did next was unusual for a man of his gifts: nothing academic at all. He went home to manage a large farming estate and attend to family business. He married his cousin Fanny Prausnitzer in 1848 and had six children. He was, for roughly a decade, a prosperous provincial businessman who happened to publish original mathematics on the side — including an 1853 memoir on the algebraic solvability of equations that extended Galois's work, and an 1850 treatment of the quintic using group theory (not by radicals, which Abel and Ruffini had already shown to be impossible).

The business succeeded well enough that in 1855 he could return to Berlin and simply live there as a private scholar with no post and no need of one. Election to the Berlin Academy in 1861 gave him the right to lecture at the university, which he began doing in 1862; he taught for two decades without a chair. In 1866 he declined the mathematics chair at Göttingen — the seat of Gauss and Dirichlet, arguably the most prestigious position in European mathematics — preferring Berlin. Only in 1883, when Kummer retired, did he take an ordinary professorship, succeeding the schoolmaster who had discovered him.

Number Theory's Dearest Dream

Kronecker's mathematics is the kind that seeds a discipline rather than closing it. He formulated in 1853 what became the Kronecker-Weber theorem, whose complete proof David Hilbert supplied later. He introduced the structure theorem for finitely generated abelian groups, one of the load-bearing results of modern algebra. He worked on elliptic functions and conjectured what he called his "liebster Jugendtraum" — dearest dream of youth — a vision of explicit class field theory that Hilbert later folded into his twelfth problem, where it remains a live research programme.

In algebraic number theory he built a theory of divisors as an alternative to Dedekind's theory of ideals, and he did so for philosophical rather than technical reasons: divisors could be handled by explicit construction, whereas ideals required infinite sets. Dedekind's approach won the century, and Kronecker's was widely treated as a curiosity — until the twentieth century revived it. His name is now attached to an improbable quantity of mathematical furniture: the Kronecker delta, the Kronecker product, the Kronecker symbol, the Kronecker limit formula, Kronecker's theorem, Kronecker's lemma, the Kronecker-Capelli theorem, Kronecker substitution.

The War on the Infinite

The philosophical position that generated the divisor theory generated everything else, including the enmity. Kronecker held that mathematics should be built from the integers by finite, explicit construction — a doctrine now called finitism, which made him a forerunner of intuitionism. He contributed to the concept of continuity and to the reconstruction of the irrational numbers in real analysis, but he rejected Weierstrass's continuous nowhere-differentiable function on philosophical grounds, and he attacked Georg Cantor's set theory, with its hierarchy of actual infinities, as illegitimate.

He did not confine these views to seminars. His campaign strained his relationship with Karl Weierstrass so badly that in 1888 Weierstrass nearly left the university. In the last year of his life Kronecker converted to Christianity. He died in Berlin on 29 December 1891 and was buried in the Alter St Matthäus Kirchhof cemetery in Schöneberg.

Why Leopold Is Called a Genius

The case rests on a specific and rare combination: an algebraist's instinct for structure paired with an almost obsessive demand for constructive content. Kronecker seems to have been unable to accept a mathematical object he could not, in principle, build — and this constraint, which looks like a limitation, functioned as a generator. It produced the divisor theory, the structure theorem for finitely generated abelian groups, the Kronecker-Weber theorem and the Jugendtraum, several of which reorganized their fields. His conjectures were also unusually well aimed: it says something that Hilbert spent significant effort proving one of them and enshrining another as a named problem.

The honest counter-case is that his judgment about mathematics as a whole was poor, and expensively so. He was wrong about Weierstrass's pathological function, which is now standard undergraduate material. He was wrong about Cantor, whose transfinite arithmetic became foundational. His preferred divisor theory was overshadowed by Dedekind's ideals for most of a century, which suggests his constructive scruples cost real expressive power at the time. And he pressed these convictions with a personal ferocity that nearly drove a colleague of Weierstrass's stature out of Berlin — conduct that historians have not been able to excuse as mere rigor. The fair summary is that Kronecker was a first-rank mathematician and a second-rank prophet: his theorems have aged far better than his opinions, and his philosophical objections turned out to be more useful as a permanent minority position within mathematics than as the programme he intended.

Legacy

He was elected to the Prussian Academy of Sciences in 1861, the French Academy of Sciences in 1868 and the Royal Society in 1884, and asteroid 25624 Kronecker carries his name. He supervised Kurt Hensel, Adolf Kneser, Mathias Lerch and Franz Mertens — Hensel's p-adic numbers being a direct descendant of the constructive impulse. The twentieth century's revival of his divisor theory, and the survival of intuitionism as a serious foundational alternative, mean that the man who lost the argument in his lifetime never entirely lost it.