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Jacques Hadamard: Prime Numbers, Poincaré’s Seat and the H Gate

Jacques Hadamard: Prime Numbers, Poincaré's Seat and the H Gate — Hadamard

Jacques Hadamard (1865–1963) was a French mathematician whose most important result was a proof of the prime number theorem in 1896. Today his name also appears in the Hadamard gate used in quantum circuits, in Hadamard matrices and in the Walsh–Hadamard transform, which is why many developers first meet him through code rather than through number theory.

This article focuses on what can be stated with confidence from the sources: his life, his career and his best-documented contributions. It also corrects several errors that circulate online, including some from an earlier version of this page.

Who was Jacques Hadamard? Quick facts

The short version, drawn mainly from the MacTutor biography of Jacques Hadamard and the Dictionary of Scientific Biography entry:

  • Born: 8 December 1865, Versailles, France
  • Died: 17 October 1963, Paris, aged 97
  • Education: École Normale Supérieure, 1884–1888; doctorate 1892
  • Best-known result: proof of the prime number theorem, 1896
  • Main posts: Bordeaux, the Sorbonne, Collège de France (1909–1937), École Polytechnique (1912–1937), École Centrale (1920–1937)
  • Académie des Sciences: elected December 1912, to the seat left vacant by Henri Poincaré
  • Other ideas named after him: well-posed problems; his name is also attached to the Hadamard gate, Hadamard matrices and the Walsh–Hadamard transform

Early life, the École Normale Supérieure and the 1892 thesis

Hadamard’s father, Amédée, taught at the Lycée Impérial in Versailles. According to MacTutor, the family moved to Paris when Jacques was three and lived through the Prussian siege of Paris in 1870.

He was not an obvious mathematical prodigy. MacTutor notes that he was weak in arithmetic at first and excelled at Greek and Latin. That changed during his schooling: in 1884 he placed first in the entrance examinations for both the École Polytechnique and the École Normale Supérieure, and chose the ENS. His fellow students included Pierre Duhem and Paul Painlevé.

He studied at the ENS from 1884 to 1888 and then taught at the Lycée Buffon from 1890 to 1893, according to the DSB entry. In 1892 he received his doctorate for a thesis on functions given by their Taylor series, Essai sur l’étude des fonctions données par leur développement de Taylor, as recorded by the Fondation Hadamard. Wikipedia lists his doctoral advisors as Émile Picard and Jules Tannery.

The 1892 Grand Prix

The thesis turned out to matter far beyond its original subject. MacTutor reports that Hadamard realised his results applied to zeta functions, and his paper on entire functions and zeta functions won first prize in the 1892 Grand Prix des Sciences Mathématiques. That work laid the groundwork for the result he is best known for.

Jacques Hadamard and the 1896 prime number theorem

The prime number theorem describes how prime numbers thin out: the number of primes up to a large number x is approximately x divided by the natural logarithm of x. Mathematicians had suspected this for a long time, but a rigorous proof was missing.

In 1896 Jacques Hadamard proved it using complex function theory, the same toolkit he had developed for his work on entire functions and zeta functions. MacTutor calls the prime number theorem his “most important result”.

What “independently of de la Vallée Poussin” means

In the same year, the Belgian mathematician Charles-Jean de la Vallée Poussin also proved the theorem. As Wikipedia’s Hadamard article notes, the two proofs were found independently. They did not collaborate, and the phrase “in collaboration with”, which appears in some popular accounts, is wrong. Both men are credited with the proof, and both proofs relied on complex analysis.

Career: Bordeaux, the Sorbonne, the Collège de France and Poincaré’s seat

After his doctorate Hadamard moved through several of France’s leading institutions. The DSB gives the following sequence:

Years Position
1893–1897 Lecturer, University of Bordeaux
1897–1909 Lecturer, the Sorbonne
1909–1937 Professor, Collège de France
1912–1937 Professor, École Polytechnique
1920–1937 Professor, École Centrale

His Collège de France appointment, to the chair of analytical and celestial mechanics, is confirmed by IdRef and Persée. He gave his opening lecture there on 18 May 1909.

Successor to Poincaré

Henri Poincaré died in 1912. MacTutor records that by the end of that summer Hadamard had written two major surveys of Poincaré’s work. In December 1912 he was elected to the Académie des Sciences, taking the seat left empty by Poincaré’s death, as stated in his Times obituary and by the École Polytechnique library.

His reputation extended abroad. He was elected a corresponding member of the Russian Academy of Sciences in 1922 and a foreign member of the USSR Academy of Sciences in 1929, according to a paper on arXiv citing the biography by Maz’ya and Shaposhnikova.

Wikipedia lists Maurice Fréchet, Paul Lévy, Szolem Mandelbrojt, André Weil and Marc Krasner among his doctoral students, and records the Prix Poncelet (1898) and the CNRS Gold Medal (1956) among his awards. These details come from a secondary source and are best treated as such.

Well-posed problems and The Psychology of Invention

Outside number theory, one of Hadamard’s most widely used ideas is the notion of a well-posed problem. As summarised by Wolfram MathWorld, a problem is well-posed if:

  1. a solution exists,
  2. the solution is unique, and
  3. the solution depends continuously on the data.

Problems that fail these conditions are called ill-posed, a concept central to the study of inverse problems. The history is slightly more nuanced than the textbook version: according to a historical paper on arXiv, his first formulation in 1902 stressed existence and uniqueness, and he emphasised continuous dependence on the data in later work (1923).

How mathematicians think

Late in life Hadamard turned to the question of how mathematical ideas arise. The Psychology of Invention in the Mathematical Field was published by Princeton University Press in 1945. It grew out of lectures he gave in 1943 at the École Libre des Hautes Études in New York and was inspired by Poincaré’s lecture on mathematical invention, according to MacTutor and PhilPapers.

The Dreyfus affair, human rights and later life

Hadamard was connected to the Dreyfus affair both personally and politically. The Encyclopaedia Judaica entry says he took an active interest in the case. Alfred Dreyfus’s wife was Lucie Dreyfus, née Hadamard, so Hadamard was related to the Dreyfus family by marriage.

The exact relationship is reported inconsistently. The Encyclopaedia Judaica calls Hadamard Dreyfus’s “brother-in-law”, but the Wikipedia article on Lucie Dreyfus lists her brother as Paul Hadamard. Until a stronger source settles it, “related by marriage” is the accurate description.

The same encyclopedia entry states that he sat on the central committee of the Ligue des Droits de l’Homme, founded in 1898, for 60 years. He lived to 97, and his Times obituary describes him as the doyen, or most senior member, of the Académie des Sciences and of the whole Institut de France.

How is Hadamard linked to the H gate and quantum computing?

Many readers arrive here because of the “H” gate in quantum programming. The Hadamard gate, Hadamard matrices and the Walsh–Hadamard transform all carry his name, and these objects appear throughout quantum computing and coding theory. If you work with quantum algorithms, for example when reading about what Grover’s algorithm can and cannot do against SHA-256, you will meet his name constantly.

We have deliberately kept the technical history out of this article. Details such as the origin of his determinant inequality, earlier constructions of the matrices now named after him, the current status of the Hadamard conjecture and the exact path by which the name reached quantum gates and error-correcting codes require their own verified sources. We will cover them separately rather than repeat claims we have not checked.

For quantum topics that are already documented on this blog, see our explainers on how the BB84 quantum key distribution protocol works and on the EPR paradox and Bell tests.

Common myths about Jacques Hadamard, corrected

An earlier version of this article contained several errors that also appear elsewhere online. Here is what the sources actually support:

Common claim What the sources say
He proved the prime number theorem “in collaboration with” de la Vallée Poussin They proved it independently in 1896
He was elected to the Académie des Sciences in 1916 December 1912, to Poincaré’s seat
He taught at the Collège de France from 1920 to 1937 1909 to 1937
He studied under Poincaré and Hermite His listed doctoral advisors were Picard and Tannery
He won the Bolyai Prize Not confirmed by our sources
He “developed” the Hadamard transform for cryptography Not supported; the transform bears his name, but this framing is unsourced

FAQ

Why is the quantum Hadamard gate named after Jacques Hadamard?

The gate, like Hadamard matrices and the Walsh–Hadamard transform, carries his name in honour of his mathematical work. Hadamard himself died in 1963 and did not work on quantum computing; the precise naming history deserves a separately sourced treatment.

Did Hadamard prove the prime number theorem alone, or with de la Vallée Poussin?

Both proved it in 1896, independently of each other. Hadamard used complex function theory building on his earlier work on entire functions and zeta functions. It was not a collaboration.

When was Hadamard elected to the Académie des Sciences?

In December 1912, to the seat left vacant by Henri Poincaré’s death. The date of 1916 found in some sources is incorrect.

Was Jacques Hadamard related to Alfred Dreyfus?

Yes, by marriage: Dreyfus’s wife was Lucie Dreyfus, née Hadamard. Some references call Jacques Dreyfus’s brother-in-law, but Lucie’s listed brother was Paul Hadamard, so the exact relationship is uncertain.

Sources