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.  This translation
was created by opening Laurent Siebenmann's web page containing
the original French version of Galois's essay using the Alta 
Vista translation service, and then "manually" adjusting some
of the words to make it flow a little better.  Be warned: Laurent 
advises me that this translation is full of misinterpretations,
so it shouldn't be taken too seriously.

I'd be interested if someone could send me a proper translation 
of the essay.  By the way, if you produce one, you could enter
it in the translation contest described on Laurent's web page.


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 memories 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

For some reason this reminds me of the wonderfully wry comment
with which Descartes concluded his Geometrie:

  "I hope that posterity will judge me kindly, not only as 
   to the things which I have explained, but also as to those 
   which I have intentionally omitted so as to leave to others
   the pleasure of discovery.
                                       Rene Descartes, 1637