Évariste Galois' Preface written in Sainte Pélagie
Two memoirs in pure analysis, by É Galois.
First, the second page of this work is not encumbered with the names, forenames, forms of address, honours and eulogies to some avaricious prince whose purse will be opened by the smoke of incense but with the threat of closure when the censer becomes empty. Nor will be seen there, in characters three times as large as the text, respectful homage to some person with a high position in science, to a scientific protector, a matter perhaps indispensable (I was going to say inevitable) for anyone who wishes to write at twenty years of age. I say to no-one that I owe to his counsel or to his encouragement all that is good in my work. I do not say it because it would be a lie. If I were to have to address something to the leading men of the world or to the leading men of science (and at the present time the distinction between these two classes of people is imperceptible), I swear that it would certainly not be thanks. I owe it to the one group to have caused the first of the two memoirs to appear so late, to the other to have written it all in prison. Why and how I am kept in prison is not my subject, a stay that one would be wrong to consider as a place of contemplation, and where I have often found myself astonished by my carefreeness at not closing my mouth to my stupid Zoïles; and I believe I can use this word Zoïle without compromising my humility, so low are my adversaries to my mind.
But I must relate how manuscripts disappear most often in the files of the members of the Institute even though in truth I do not understand such carelessness on the part of men who have the death of Abel on their conscience. It is sufficient for me, who do not wish to compare myself with this illustrious mathematician, to say that the substance of my memoir on the theory of equations was deposited with the Academy of Science in the month of February 1830, that extracts from it had been sent there in 1829, that no report on it followed, and that it was for me impossible to see the manuscripts again. There are some rather curious anecdotes of this kind but it would be bad grace for me to relate them, because a similar accident, other than the loss of my manuscripts, has not befallen me. Happy traveller, my nasty appearance has saved me from the jaws of wolves. I have said enough to make the reader understand why, whatever my good intentions had once been, it has been absolutely impossible for me to adorn or disfigure, as the reader wishes, my work with a dedication.
In the second place, the memoirs are short and in no way proportionate to their titles and then there is at least as much French as algebra, to the point where the printer, when the manuscripts were carried to him, believed in good faith that it was an introduction. In this matter I am completely inexcusable; it would have been so easy to review a whole theory from its beginnings under the pretext of presenting it in a form necessary for the understanding of the work, or perhaps better, without more ado to interlard a branch of knowledge with two or three new theorems, without indicating which they are. Again, it would have been so easy to substitute successively all the letters of the alphabet into each equation, numbering them in order so as to be able to recognise to which combination of letters subsequent equations belong; which would have multiplied infinitely the number of equations, if one reflects that after the Latin alphabet there is still the Greek, that, once the latter was exhausted there remain German characters, that nothing stops one from using Syriac letters, and if need be Chinese letters! It would have been so easy to transform each sentence ten times, taking care to precede each transformation with the solemn word theorem; or indeed to get by our analysis to results known since the good Euclid; or finally precede and follow each proposition with a redoubtable line of particular examples. And of all these means I have not thought to choose a single one!
In the third place, the first memoir is not new to the eye of the master. An extract sent in 1831 to the Academy of Science was submitted for refereeing to M Poisson, who just said at a meeting that he had not understood it. Which to my eyes, fascinated by the author's self-confidence, proves simply that M Poisson did not want or was not able to understand it; but certainly proves in the eyes of the public that my book means nothing.
Everything combines therefore to make me think that in the scientific world the work that I am submitting to the public would be received with a smile of compassion; that the most indulgent would accuse me of awkwardness; and that after some time I would be compared with Wronski or these indefatigable men who every year find a new solution to the squaring of the circle. Above all I would have to tolerate the laughter of the examiners of candidates for the École Polytechnique (who, by the way, I am astonished not to see, each one of them, occupying an armchair in the Academy of Science, for they certainly have no place in posterity), and who, having a tendency to monopolise the printing of mathematics books, will not understand without it being put to them formally, that a young man twice failed by them might also have the pretension to write not teaching books, it is true, but books of doctrine. I have said all that precedes to prove that it is knowingly that I expose myself to the laughter of fools.
If, with so little chance of being understood, I publish, in spite of everything, the fruit of my nightly labours it is in order to establish the date of my research and it is in order that the friends I have made in the world before I was entombed behind bars should know that I am alive and well. It is perhaps also in the hope that this research could fall into the hands of people whom a stupid morgue will not prevent them reading it, and, I think, direct them into a new path which must, according to me, follow analysis into its highest branches. One should know that I speak here only about pure analysis; my assertions transferred to the most direct applications of mathematics would become false.
I shall say a few words now about the new direction which I foresee that Analysis must follow.
Long algebraic calculations were at first hardly necessary for progress in Mathematics; the very simple theorems hardly gained from being translated into the language of analysis. It is only since Euler that this briefer language has become indispensable to the new extensions which this great mathematician has given to science. Since Euler calculations have become more and more necessary but more and more difficult, at least insofar as they are applied to the most advanced objects of science. Since the beginning of this century computational procedures have attained such a degree of complication that any progress had become impossible by these means, except with the elegance with which new modern mathematicians have believed they should bring to their research, and by means of which the mind promptly and with a single glance comprehends a large number of operations.
It is clear that such elegance, so vaunted and so aptly named, has no other goal. From the well established fact that the efforts of the most advanced mathematicians have elegance as their object, one may therefore conclude with certainty that it becomes more and more necessary to embrace several operations at once because the mind does not have the time any more to stop at details.
Thus I believe that the simplifications produced by elegance of calculations (intellectual simplifications, of course; there are no material ones) have their limits; I believe that the time will come when the algebraic transformations foreseen by the speculations of analysts will find neither the time nor the place for their realisation; at which point one will have to be content with having foreseen them. I would not wish to say that there is nothing new for analysis without this rescue; but I believe that without this one day all will run out.
Go to the roots of these calculations, put operations into groups, classify them according to their complexities and not according to their appearances; that is, according to me, the mission of future mathematicians, that is the path that I have begun to take in this work.
Here is nothing like that; here analysis of analysis is done: here the highest calculations executed up to now are considered as particular cases which have been useful, indispensable to deal with, but which it would be fatal not to abandon for broader research. It will be time to carry out calculations foreseen by this high analysis and classified according to their complexities, though not specified by their appearance, when the details of a question require them.
The opinion that I express here should not be confused with the affectation that certain people apparently have of avoiding any kind of calculation, translating into very long sentences what may be expressed very briefly by algebra, and thus adding to the boredom of their operations the boredom of a language which is not designed to express them. Such persons are a hundred years behind the times.
The general thesis that I am advancing cannot be well understood except when my work, which is an application of it, is read attentively: it is not that this theoretical point of view has preceded applications, but I ask myself, my book being finished, what renders it so strange to the majority of readers, and, looking into myself, I believe I observe a tendency in my mind to avoid calculations in the subjects that I treat, and moreover, I have recognised an insurmountable difficulty for whoever would wish to effect them generally in the matters that I have treated.
One who, treating such new subjects, taking a chance on such a strange road, pretty often difficulties presented themselves that I was unable to overcome. Even in these two memoirs, and especially in the second which is the more recent, the formula "I do not know" will often be found. The class of readers of whom I have spoken at the beginning will not fail to find something laughable there. Unhappily one cannot doubt that the most precious book of the greatest scientist will be that in which he tells us everything that he does not know; one cannot doubt that an author never betrays his readers so much as when he hides a difficulty. When competition, that is to say egoism reigns no more in science, when one gets together to study, instead of sending to the academies closed packets, one will hasten to publish one's least observations for their little novelty, and one will add: "I do not know the rest".
From Sainte Pelagie December 1831.
JOC/EFR November 2017
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