Alexander Grothendieck, Récoltes et Semailles (Reapings and Sowings), from Promenade 9, 1986 (English translation by Roy Lisker)
That is to say that, if there is one thing in Mathematics
which (no doubt this has always been so) fascinates me more than
anything else, it is neither number
, nor magnitude
but above
all form
. And. among the thousand and one faces that form
chooses in presenting itself to our attention, the one that has
fascinated me more than any other, and continues to fascinate
me, is the structure buried within mathematical objects.
One cannot invent the structure of an object. The most we can
do is to patiently bring it to the light of day, with humility
- in making it known it is discovered
. If there is some
sort of inventiveness in this work, and if it happens that we
find ourselves the maker or indefatigable builder, we aren't
in any sense making
or building
these
structures. They hardly waited for us to find them in order to
exist, exactly as they are! But it is in order to express, as
faithfully as possible, the things that we've been detecting
or discovering, to deliver up that reticent structure, which
we can only grasp at, perhaps with a language no better than
babbling. Thereby are we constantly driven to invent the
language most appropriate to express, with increasing
refinement, the intimate structure of the mathematical object,
and to construct
with the help of this language, bit by
bit, those theories
which claim to give a fair account
of what has been apprehended and seen. There is a continual
coming and going, uninterrupted, between the apprehension of
things, and the means of expressing them, by a language in a
constant state [of] improvement, and constantly in a process of
recreation, under the pressure of immediate necessity.
As the reader must have realized by now,
these theories
, constructed out of whole cloth
,
are nothing less than the stately mansions
treated in
previous sections: those which we inherit from our
predecessors, and those which we are led to build with our own
hands, in response to the way things develop. When I refer
to inventiveness
(or imagination) of the maker and the
builder, I am obliged to adjoin to that what really
constitures it soul or secret nerve. It does not refer in any
way to the arrogance of someone who says This is the way I
want things to be!
and ask that they attend him at his
leisure, the kind of lousy architect who has all of his plans
ready made in his head without having scouted the terrain,
investigated the possibilities and all that is required.
The sole thing that constitutes the true inventiveness
and imagination of the researcher is the quality of his
attention as he listens to the voices of things. For nothing
in the Universe speaks on its own or reveals itself just
because someone is listening to it. And the most beautiful
mansion, the one that best reflects the love of the true
workman, is not the one that is bigger or higher than all the
others. The most beautiful mansion is that which is a faithful
reflection of the structure and beauty concealed within
things.