At this stage in his education, before undertaking research for his doctorate, Couturat published a paper on Zeno of Elea's paradox of Achilles and the Tortoise in the Revue philosophique. For his doctorate he worked on two theses, one being a Latin thesis which made a scientific study of the myths of Plato in the Dialogues, the other being a mathematical thesis on infinity. We will say more below about the content of this thesis, written at a time when Cantor's ideas of the infinite were causing vigorous arguments and discussions among mathematicians and philosophers. Both of his theses were completed by May 1894 but it was two years later before he defended his theses at the Sorbonne and was awarded his doctorate with the highest distinction. The mathematical thesis was published as De l'Infini mathématique Ⓣ (1896).
In this work Couturat argued strongly in favour of the actual infinite. Dedekind, Kronecker, and Helmholtz were already strong advocates of formalist theories so Couturat took a stand against major established figures - a brave move in a doctoral thesis. For him the actual infinite was a generalisation of number, in the same way that negative numbers, fractions, irrational numbers and complex numbers had all been seen at extending the concept of number. Couturat argued that all of these generalisations had at first encountered strong opposition, but had become accepted in the end because they were suitable for representing new magnitudes and they allowed a calculus of operations which was impossible before their introduction. Infinite numbers, he claimed, were necessary in order to maintain the continuity of magnitudes.
Couturat became professor at the University of Toulouse in 1895 and taught philosophy there, lecturing on Lucretius and Plato. He obtained unpaid leave of absence and was able to undertake further study in Paris at the time when he defended his theses there. Of course for Couturat to be able to take unpaid leave he had to have independent means. Indeed he was well off and had no need to work to earn his living. During this leave of absence he attended lectures by Edmond Bouty and Victor Robin at the Sorbonne. In 1897 he moved to the University of Caen, taking up his appointment on 27 October, and there he taught mathematical philosophy. He remained in Caen until 1899 when he moved to Paris, again taking leave of absence, to continue his research on the logic of Leibniz. During 1900-01 he worked in Hanover studying the unpublished works of Leibniz in the Royal Library. He published La Logique de Leibniz Ⓣ (1901) in which he wrote that Leibniz's:-
... logic was not only the heart and soul of his system, but the centre of his intellectual activity, the source of all his discoveries, ... the obscure or at least concealed hearth from which sprang so many flashes of lightning.Couturat published many previously unpublished manuscripts of Leibniz in Opuscules et fragments inédits de Leibniz Ⓣ (1903). This work on Leibniz brought Couturat into contact with Russell, who had published A Critical Exposition of the Philosophy of Leibniz (1900). Couturat produced, in 1905, an edition of Russell's Principia Mathematica with a commentary on contemporary works on the subject. In fact the Couturat-Russell correspondence began in 1897, the first letter being from Couturat to Russell concerning the Russell's An essay in the foundations of geometry. Couturat published a review of this essay of Russell in 1898 and the review, and Russell's reply to it, attracted Poincaré's attention to Russell's work. In  198 letters and postcards between Couturat and Russell, found in the 1970s, are described. The topics covered in the correspondence include: the foundations of geometry, extension versus intension in logic, the Russell paradox, the axiom of choice, the controversies with Poincaré, logic, Leibniz, Peano, Kant, arithmetical induction, mathematical existence, politics, international language, and some personal matters.
In 1905 Couturat became Henri-Louis Bergson's assistant at the Collège de France, working on the history of logic during the academic year 1905-06. [Bergson was the first philosopher to propose what is today called a process philosophy, that is one which rejected static values in favour of values of motion, change, and evolution.] The article  discusses Couturat's inaugural lecture at the Collège de France in which he argued against the irrationalism of his time and firmly rejected all attempts to subordinate philosophy to psychology, sociology, or religion. He argued that philosophy must remain true to itself, and affirm its freedom from other disciplines. Following his year at the Collège de France, Couturat took no further employment and devoted himself to his researches. F C Kreiling writes:-
Outside of France he is generally known for his Leibniz studies, but he was distinctly a philosopher in his own right, with a central interest in mathematical logic.Leibniz had proposed a calculus of reason which would allow the mind to think directly of things themselves. He wanted this calculus of reason to be supported by a logical universal language. Couturat, following his hero Leibniz, became a prime mover in the development of the international language Ido based on Esperanto :-
On 1 October 1907 delegates from 310 societies throughout the world met and elected a committee to modify Esperanto. Couturat and Léau were the secretaries. With the collaboration of the Akademie di la Lingue Internaciona Ido, created in 1908, Couturat constructed the complete vocabulary of Ido, a language derived from Esperanto with reforms growing out of scientific linguistic principles. Couturat stood firmly for the application of his own logical principles, despite opposition from many quarters to changes in the already established forms of Esperanto.His other work includes L'Algèbre de la logique Ⓣ (1905) and Les Principes des Mathematiques Ⓣ (1905).
Couturat was killed in a car accident, his car being in hit by the car carrying the orders for mobilisation of the French army the day World War I broke out. Ironically he was a noted pacifist.
Article by: J J O'Connor and E F Robertson
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