Born in Wolsztyn, Poland, he moved to France in his early twenties, and gained French citizenship in 1800. In 1803, he joined the observatory in Marseille, and began developing an enormously complex theory of the structure and origin of the universe. During this period he took up a correspondence with nearly all the major scientists and mathematicians of his day, and was well-respected at the observatory. In 1810 he published the results of his research in a massive tome, which he advocated as a new foundation for all of science and mathematics. His theories were strongly Pythagorean, holding numbers and their properties to be the fundamental underpinning of essentially everything in the universe. His claims were met with rather little acceptance, and his research and theories were generally dismissed as grandiose rubbish. His previous correspondence with major figures led to it gaining more attention than a typical crackpot theory might, even earning a review from the great mathematician Joseph Louis Lagrange (which turned out to be exceedingly disfavorable). In the ensuing controversy, he was forced to leave the observatory.
Immediately thereafter he turned his focus towards applying philosophy to mathematics (his critics charged that this meant dispensing with mathematical rigor in favor of generalities). In 1812 he published a paper purporting to show that every equation has an algebraic solution, directly contradicting results that had just been published by Paolo Ruffini; however, Ruffini turned out to be correct.
Thereafter he turned his attention to disparate and largely unsuccessful pursuits. He developed a fantastical design for caterpillar-like vehicles which he intended to replace railroad transportation, but did not manage to persuade anyone to give the design serious attention. In 1819 he went to England to try to gain a grant from the Board of Longitude to build a device to determine longitude at sea. After initial difficulties, he was given an opportunity to address the Board, but his grandiose address, On the Longitude, contained much philosophizing and generalities, but no specific plans for a working device, and thus failed to gain him support from the Board. He remained in England several years, publishing an introductory text on mathematics in London in 1821, which moderately improved his financial situation. He returned to France in 1822, and again took up a combination of mathematics and fantastical pursuits, largely in poverty and scorned by intellectual society. Along with his continuing Pythagorean obsession, he spent much time working on several notoriously futile endeavors, including attempts to build a perpetual motion machine, to square the circle, and to build a machine to predict the future (which he dubbed the prognometre). In 1852, shortly before his death, he did find a willing audience for his ideas: the occultist Eliphas Levi met Wronski and was greatly impressed and influenced by his work and dedication. Wronski died in 1853 in Neuilly-sur-Seine, France, on the outskirts of Paris.
Though during his lifetime nearly all his work was dismissed as nonsense, some of it has come in later years to be seen in a more favorable light. Although nearly all his grandiose claims were in fact unfounded, his mathematical work contains flashes of deep insight and many important intermediary results. Most significant was his work on series. He had strongly criticized Lagrange's use of infinite series, introducing instead a novel series expansion for a function. His criticisms of Lagrange were for the most part unfounded, but the coefficients in Wronski's new series were after his death found to be important, forming the determinants now known as the Wronskians (the name was given them by Thomas Muir in 1882).
- O'Connor, John J., and Edmund F. Robertson. "Josef HoënéWroński". MacTutor History of Mathematics archive.