Stories and Numbers: How a Romantic Tale of Geographical Exploration Transformed Mathematics

January 2004 · Historically Speaking 13 Stories and Numbers: How a Romantic Tale of Geographical Exploration Transformed Mathematics Amir Alexander Evariste Galois was a young mathematical genius, who was beginning to make a name for himselfin European mathematical circles in die early 19th century . He was also a political radical, and an enthusiastic participantin French revolutionary politics. At the young age oftwenty-two he was challenged to a duel over his radical political beliefs. Knowing that he may not survive the morning, he spent the night before the encounterwriting down his latest mathematical insights. Tragically, he was indeed killed in the duel. His hastily jotted notes, however, bequeathed to mathematics an entire new field ofinquiry: the Theory of Groups. I cannot vouch for the accuracy of this story, which I heard repeatedly from mymathematics professors , but then, it was not told for its historicalveracity. Itwas told as a morality tale: the young Galois, we are left to surmise, would have presented the world with many more mathematical wonders if he had not been lured awayfrom his true vocation by politics and violence. The passions and turmoil ofhuman history appear as nothing more than senseless strife when viewed against the beauty and coherence ofeternal mathematics. One may argue—and most historians would—with the value judgments ofhistory and mathematics implicitin this tale. Butthe story does bring out the fundamental problem ofa history ofmathematics: mathematics deals with unchanging Platonic universale ; history deals with unexpected earthly contingencies. How, then, can one write a meaningful contingent history ofuniversal mathematics? The problem is shared, to a degree, with other areas of the history of science, where historians have had to deal with such stubbornlyahistorical factors as universal gravitation and the laws ofthermodynamics. Nevertheless , over the past thrity years historians ofscience have been remarkably successful in providinginsightful and culturallyrich historical accounts of crucial episodes in the development ofscience. Not so historians of mathematics, who have remained surprisingly unperturbed by the intellectual tempests raging in adjacent fields. Many historians of mathematics, though by no means all, remain thoroughly satisfied with the "Galois" model of history and are contentto contrasthistoryand madiematics rather than connecting them. In their viewhistorymight be the "theater" ofmathematics —since different mathematicians live in different historical times and places and Traditional mathematics sought to determine the nature ofthe world through rigorous deduction; Cavalieri andhis colleagues sought to learn ofit by exploring the secrets hidden within existing objects. I make no effort here to argue for this method in the abstract; rather, I will simply demonstrate how it is done. This, then, is the story ofhow a romantic tale of exploration and discoverychanged the face ofmathematics and opened thewayto the development of the calculus. • · · In 1539 GeorgJoachim Rheticus, professor of mathematics atthe UniversityofWittenberg, set out for the Prussian town ofFrauenburg to visit the reclusive astronomer Nicholas Copernicus. Rheticus had heard rumors of Copernicus's heliocentric system, and he was determined to investigate the truth and validity ofthis radical cosmology at first hand. Under the old canon's tutelage, Rheticus quickly became an ardent Copernican. He ended up stayinginFrauenburgfortwo years, often impressing upon his host the significance ofhis discoveries, and the importance of making them available to a wider audience. A few months after his arrival he composed a short treatise, summarizing the main points ofthe Copernican theory. It each gets a glimpse ofa part ofthe realm of was published in 1540 and became known as mathematics. History is not, however, the the Narratio Prima, the first published dimension of mathematics, since in these accounts mathematics trulyresides in its own insular Platonic sphere. What, then, can be done? How can a historical account be given of a field as insular and self-contained as mathematics? My suggestion here is to lookfor the narratives that structure mathematical systems, the underlyingstories thatguide mathematicians' work and help them construct their intricate systems . Most important for historians, these narratives are historically specific: different stories at different times can and do produce different types ofmathematical practices. account ofCopernicus's views. At the end of the treatise, Rheticus reflected upon the hospitality offered him by narrating the adventures ofthe philosopherAristippus ofCyrene: The story is frequently told of the shipwrecked Aristippus, which they say occurred...