Rigor and Intuition in Mathematics
Before I knew much mathematics (and I don't know very much,) things
were obvious to me...
It is a dangerous thing, some say, to have rigor; rigor can
complicate things and clutter one's thinking. Indeed, rigor only walks with baby-steps,
and it is always too close to the heart of matters to capture more than one thing at a time. Yet rigor allows for certainty in one's
conclusions - always there is this balance to be struck between rigor and
intution, but harmony, once acheived, allows one to both have the lofty vision
of Intuition, far more than the near-sighted Rigor could allow one to see, and yet also
the confidence that one's sight is true, and the ability to speak one's sight to others.
So I see first through the eyes of intuition and understanding, and then, if I
feel that something is worth saying, I will formalize it. Yet so long as it is
to myself, I may put it in my own language, I may paint pictures, I may relate it to
my personal knowledge, I may give it fond private names.
Once I will communicate my thoughts to the outside world, though, they must take on
a different form, in an agreed-upon language. Rigor gives us this language that
we need. As a husband and wife put on their social faces to the other world, and yet
are intimate to each other when alone, so I must be with my ideas; when searching
and discovering the ideas, I may be private and speak with the words and the notions,
in the language of Intuition, but in testing and explaining the ideas, only the
language of Rigor may be used.
The intuition is the love that draws us together, and the rigor is the law that
bonds us in marriage. But recall that it is not law that first brings a happy couple
together - the love comes first, the marriage afterwards. Rigor is not, then,
the language of the first meeting, of the first love. That wholly belongs to
Intuition - and just as the love that we know, it will be giddy, unsupported,
insightful, and sometimes wrong. It will seek to know everything, even when it as
yet knows little. It is unrivalled in its enthusiasm and energy.
Yet only in Rigor may it take on its full beauty.
It is only in the marriage that the ideas are understood fully by all, accepted, and bring forth new
and child Ideas, speaking at first only the baby tongue of Intuition, but which
someday will speak in Rigor as fluently as their parents, and which will love
some new form of knowledge, and bring forth surprising new results and thoughts, as broad only
as the collective imagination of humanity.