Evariste GaloisBy Jimmy Tseng
Evariste Galois (1811 – 1832) was a mathematician whose genius engendered innovative ideas and inventive approaches. A tragic hero to mathematicians, Galois brief, chaotic, and dire life on the fringes of the mathematical and political establishments lends easily to the great myth of Galois as the genius oppressed by the hostile, ignorant power structure of his day. His untimely death in a duel, fought for reasons shrouded in the imbroglio of Galois last days, further lends a romantic aura to Galois the myth. Yet, the young man was also an incendiary and hothead whose radical political stances coupled with an acerbic personality, that, perversely, seemed to find comfort in conflict with the mathematical and political establishments, helped to create the conditions for his own death. What can be surely said about Galois is that his early death is a loss to mathematics.
The Myth
It is only a slight exaggeration to say that the myth of Galois is the first and primary story that novice mathematicians learn about being a mathematician. The values embodied in the myth are the values that the mathematical community hold dear: a sharp, lightning-like, supernatural genius, which engenders bolts of original and innovative ideas and methods out of the very air; a forced eremitism caused by the ignorance and hostility of the establishment, and youth. All three of these values conveyed by the myth are held in high esteem by mathematicians.
Beyond the mathematical community, the myth of Galois also holds appeal. It is the story of a young love-struck genius and revolutionary who dies defending the honor of his love in a duel. Yet, as fate would have it, his love is unrequited. Atlas, the young Galois proffers that last measure of devotion as a propitiatory sacrifice on the altar of Romantic Love, only to be so tragically spurned.
Childhood
Evariste was born in Bourg-La-Reine on the 25th of October in 1811 to Nicholas-Gabriel Galois and Adelaide-Marie Demante. Although well educated in the manner of the day, neither parent had any known propensity for mathematical thinking. Evariste, then, was taught by others, notably through the printed word of the masters and the personal encouragement of his secondary school teacher, Louis-Paul-Emile Richard. Abetted by an indefatigable persistence and marked focus, Evariste learned and mastered a huge library of mathematics in a very short period of time. It is no wonder that other portions of his education suffered. Yet, his devotion to mathematics proved fruitful even as early as 1829 when he published his first work on a topic in number theory called continued fractions.
Tragedies
Even as Galois started to contribute to mathematics, a series of tragedies were becoming manifest. In 1828, Galois failed the entrance examination to the vaunted École Polytechnique. In 1829, Galois’s father committed suicide to escape a scandal from which historians have exonerated the elder Galois. A few weeks afterwards, Galois, in his characteristically impetuous way, took the École Polytechnique’s entrance examination again even though his father’s death was a very heavy, recent blow. Again, he failed. These entrance examinations are supposed to predict a student’s true ability to create original mathematics in the future. Considering Galois’s true mathematical ability, are these failures an indictment of examinations or an excusable example of the limitations of examinations? Ironically, it is at this time, near his death, that Galois’s great mathematical abilities are at their pinnacle. Clearly, these two examinations failed their purpose of predicting Galois’s true abilities. Apparently, genius is harder to quantify than academicians, even those at the École Polytechnique, suspect. His true abilities notwithstanding, Galois’s examination failures resigned the genius to École Normale, a university of lesser reputation than the École Polytechnique. It was this condition of grief, humiliation, and exasperation mixed with Galois’s innate proclivity towards impetuous acts that sparked Galois’s incendiary acts and his death.
Last Days and Critical Times
In the scholarly Western mind, the prototypical hero is Achilles. Often, the Homeric poet describes Achilles as god-like. Yet, Achilles has a fatal weak spot that allowed an adversary, in collusion with a superhuman power, to kill him. Galois’s weak spot was his impetuous nature that allowed an adversary, in collusion with the superhuman power of society, to kill him. It was this impetuous nature that manifested itself in a published letter from Galois lambasting the director of the École Normale for locking students away from a pro-Republican riot. An angered director summarily expelled Galois.
After expulsion, Galois joined the Artillery of the National Guard, a republican bastion. Soon after joining the Guard, on the 9th of May in 1831, Galois impetuously made a public threat against the newly installed king, Louis-Philippe. Quickly, Galois was arrested, tried, and acquitted. His acquittal was based not on his innocence, but on what the judge deemed as his youthful foolishness. Indeed, Galois impetuously pilloried Louis-Philippe in open court and intimated assassination.
However, Galois was arrested again on Bastille Day 1831 for possessing a weapon and wearing the uniform of the Guard, which had been ordered disbanded by a fearful Louis-Philippe. This time Galois was not as fortunate as the last. For his crimes, he was sentenced to jail for many months. The last six weeks of Galois’s sentence were spent in a nursing home where Galois had an affair with the resident physician’s daughter, Stephanie du Motel. After Galois’s release, the affair apparently soured; a letter from du Motel pleads with Galois to end the affair. Could it be that du Motel was the reason for the infamous duel between Galois and fellow Republican, Pescheux d’Herbinville? History does not confirm or deny such an allegation. Could it be that the duel was a conspiracy between du Motel and d’Herbinville to assassinate Galois? Again, history remains silent.
What history does say is that Galois did die in that duel. On the 31st of May in 1832, Galois died. He was not yet 21. Unlike Socrates, the philosopher Achilles, who, according to Plato, drank hemlock to preserve his philosophy, Galois agreed to duel in spite of his mathematics. Certainly, Galois must have regretted leaving his mathematics. On the night before his duel, Galois wrote a long letter to a friend outlining his unpublished discoveries and leaving for others to sort out his legacy. Galois’s friend met with little success for many years in convincing established mathematicians to consider Galois’s work. Finally, the mathematician Liouville was the first to realize the importance of Galois’s work. In 1846, years after Galois’s death, Liouville had Galois’s work published. Soon afterwards, Galois’s work became standard mathematics.
Is the Myth Wrong?
Myths are old. They are created by generations of human beings to explain their environment and lives or to assuage their pain and guilt. The Blackfoot Native Americans of Montana, a hunter society, have the myth of the buffalo dance, a myth (and concomitant ritual dance) that expurgates the bloodguilt of eating buffalo flesh. This myth served the hunter society well. Yet, as societies embraced agriculture, their myths changed from one of hunting to one of planting. This change took place, scholars say, over many generations, but the change is clearly evident. Another Native American tribe, the Lakota, have the myth of the White Buffalo Calf Woman. In this myth, a goddess, in the form of a buffalo calf, gives the Lakota the first corn. Hence the change from hunting to agriculture is made. Myths, invaluable as guides, change as the societies that use them change. These changes are inevitable.
The myth of Galois is, by now, becoming out of date. Mathematics has advanced much in both depth and breath. No longer is it likely for a neophyte to grasp the full depth of a field of mathematics, including the necessary tools found in other fields of mathematics, without many concerted years of study. The young genius is becoming history, not news. Mathematicians might well become more like Möbius who, at the age of 67 or 68, discovered his famous namesake, the Möbius strip.
Another aspect of the myth of Galois is his forced eremitism. This aspect is merely an illusion. Galois had a good teacher who encouraged him. He published and contributed to the mathematical debate. He read and greatly benefited from the masters. He was not alone.
The third aspect of the myth of Galois is that his genius did not benefit from concerted, persistence work in the study of mathematics. Galois’s mathematics came, the myth suggests, as a bolt of lightening engendered by the very air. Clearly, this aspect is also false. That Galois was a genius is not in question. That Galois did not exhaust himself at mathematics is ridiculous. Even the greatest genius must learn the basic ideas of legions of mathematicians before him even if to consign their works to forgotten history. Galois’s mathematics came, bit by bit, through exhaustive work, attention to the mathematics of others, and devotion to the subject. After all, Galois is genius, but also man. No mathematician or genius can exempt himself from the limitations of man.
What should replace the myth of Galois? A first step would be to emphasize the historical Galois and recognize that the mythical Galois has been a combination of the historical Galois and the aspirations, fears, and idealizations of many generations of mathematicians. However, the next step is harder. Who knows how to proceed? Reassurance can be taken from the old myths. After many generations, did hunter myths become agricultural myths because the society changed. Likewise, a change has occurred in mathematics in the twentieth century. A huge library of mathematics has been erected and endowed with the collective study and work of legions. Any mathematician must master a least a significant portion of that library in order to create or, as the case may be, destroy parts of that library. No mathematician can master it all. Hence, a new period of cooperation among and ageing of practitioners along with the old ways of toil and exhaustion must be and is occurring. The new myths of mathematics and mathematicians will come out of this sea change. No, the myth of Galois is not completely wrong; it is simply outmoded.
Galois’s Mathematics
Besides his myth, Galois’s legacy is his contributions to mathematics. Galois named the mathematical concept group, a fundamental mathematical object defined in the major branch of mathematics called algebra. The era that started then has sometimes been called the era of the algebraization of mathematics. Groups constitute the bulk of this algebraization.
Besides naming and studying groups, Galois contributed to many different parts of mathematics such as number theory. However, his main contribution is to the connection between groups and another discipline of algebra called fields. This contribution, an intricate and exquisite gem of a theory, is called Galois theory. Using this theory, Galois gave an elegant proof of the mathematical theorem that states that the general quintic polynomial is not solvable, a problem of much interest in his day.
Galois’s mathematics may seem simple to modern experts. Yet, his grand ways of thought—perhaps his philosophy of how to think in mathematics is a better description—is his true genius. That these grand ways of thought are still influencing modern mathematicians is Galois’s greatest legacy. The details of his theorems, although great gems of mathematics, pale in comparison. Galois’s ways remain some of the best ways in mathematics today.
Coda
Galois is one of the most fascinating mathematicians in history. He led such a troubled life and had such a violent death that, to some, his mathematical genius and contributions may just merit a collateral mention at best. Yet, it is because he was a great mathematician that he is remembered at all. His myth has provided a guide to young mathematicians for well over a century. Now the myth must be refurbished because the way mathematics must be done has changed. Galois will survive, in the minds of mathematicians and historians, the refurbishing of his myth. His contributions to mathematics are genuine. His place in the pantheon is assured. Perhaps, he is more like Hercules than Achilles after all.
Copyright © 2003 by Jimmy Tseng
Book Resources On Evariste GaloisMathematics and Its History by John Stillwell
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