Galois's Analysis of Analysis


The following is a (very) rough English translation of an essay by Evariste Galois on mathematical abstraction, which he wrote as a forward for his papers on "Galois Theory", apparently while he was in prison for being a political agitator. I've been told by people fluent in French that this translation is very poor, so it should be taken with a grain of sel.



The Scientific Foreword with the Theory of Galois


Although with little chance of acceptance, I publish, despite everything, the product of my researches, so that the friends who knew me before my imprisonment may know that I am well, and perhaps also in the hope that these researches may fall into the hands of someone other than the dead-stupid people who cannot comprehend, someone who may proceed along the path that, in my opinion, alone can lead to greater heights in analysis. Note that I do not speak here about simplistic analyses; my assertions applied to the more basic operations of mathematics would be paradoxical.


Long calculations initially were not very necessary to the progress of mathematics, as extremely simple theorems will hardly benefit from being represented in the language of analysis. It is only since Euler that a more compact language became essential to making further progress beyond what that eminent mathematician bequeathed to science.  Since Euler, calculations have become increasingly necessary, but also increasingly difficult, as they applied to more advanced objects of science. By the beginning of this century mathematics had reached such a level of complexity that any further progress had become impossible without the elegance by which modern mathematicians express their research, allowing the mind to grasp, all at once, a great number of operations.


It's obvious that elegance, graced with so fine a title, can have no other goal. Noting well that the most advanced mathematicians strive for elegance, we can thus conclude with certainty that it becomes increasingly necessary to embrace several operations at the same time, because the mind has not the time to dwell on each detail.


I believe that the elegance and simplification (conceptual simplifications, not material ones) achievable by means of calculations have their limits; I believe that the time will come when the algebraic transformations envisaged in the speculations of mathematicians will no longer find the time or the place to occur; at that point it will be necessary to be satisfied to have envisaged them. I do not wish to say that nothing more can be accomplished in analysis without this change in outlook, but I believe that one day, without this change, mathematics will be exhausted.


To leap to the common foundations joining our calculations; to group the operations, to classify them according to their difficulties and not according to their forms; such is, in my view, the mission of future mathematicians; such is the manner in which I have approached this work.


One should not confuse the opinion I've expressed here with the affectations of certain people who avoid any kind of calculation, people who translate into long sentences what is very briefly expressed by algebra, and adding thus to the length of these operations the lengths of a language which is not made to express them. Such people are a hundred years behind the times.


That of which I speak is entirely different; I speak of the analysis of analysis. In this context the highest calculations that have been so far performed will be regarded simply as particular cases, the treatment of which was useful, even essential, but which it would be disastrous not to go beyond and embark on a broader search. There will be time to carry out the detailed calculations envisaged by this higher analysis, in which concepts are classified according to their difficulties, but not specified in their form, when special questions call for it.


The general thesis that I've advanced here may be fully appreciated only when one attentively reads my work, which is an application of this thesis. This is not to say that the theoretical point of view preceded the application; rather, in considering the finished work - which I knew would seem so strange to the majority of readers - and reflecting on the process by which I had progressed, I became aware of my tendency to avoid calculations in the subjects which I covered and, moreover, I recognized that carrying out such calculations in these matters would have been an insurmountable difficulty.


One must admit that, covering such new subjects, treated in such a novel way, very often difficulties arose that I could not overcome. Therefore in these two memoirs and especially in the second, which is more recent, there will often be found the phrase "I do not know". The class of readers about whom I spoke at the beginning will undoubtedly laugh at this.  Unfortunately one does not expect that the most valuable books of learning are those that acknowledge what they do not know, and that an author never harms his readers more than when he obfuscates a difficulty. When competition, i.e. selfishness, no longer reigns in the sciences, when one cooperates in his studies instead of sending sealed packages to the academies, then one will hasten to share his least observations that contain something new, and one will add: "I do not know the rest".


                              Evariste Galois                 De Ste Pelagie, December 1831



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