Mathematics
from the "The unity of mathematics" in Connes' A View of Mathematics:
The scientific life of mathematicians can be pictured as a trip inside the geography of the "mathematical reality" which they unveil gradually in their own private mental frame.
It often begins
by an act of rebellion with respect to the existing dogmatic description of
that reality that one will find in existing books. The young "to be
mathematician" realize in their own mind that their perception of the
mathematical world captures some features which do not quite fit with the
existing dogma. This first act is often due in most cases to ignorance but it
allows one to free oneself from the reverence to authority by relying on one's
intuition provided it is backed up by actual proofs. Once mathematicians get to
really know, in an original and "personal" manner, a small part of the
mathematical world, as esoteric as it can look at first, their trip can really
start. It is of course vital all along not to break the "fil d'Arianne" which allows
to constantly keep a fresh eye on whatever one will encounter along the way,
and also to go back to the source if one feels lost at times...
It is also vital to always keep moving. The risk otherwise is to confine
oneself in a relatively small area of extreme technical specialization, thus
shrinking one's perception of the mathematical world and of its bewildering
diversity.
The really fundamental point in that
respect is that while so many mathematicians have been spending their entire
scientific life exploring that world they all agree on its contours and on its connexity: whatever the origin of one's itinerary, one day
or another if one walks long enough, one is bound to reach a well known town
i.e. for instance to meet elliptic functions, modular forms, zeta functions.
"All roads lead to Rome" and the mathematical world is "connected".
In other words there is just "one" mathematical world, whose
exploration is the task of all mathematicians and they are all in the same boat
somehow.
...
Most mathematicians adopt a pragmatic attitude and see themselves as the explorers of this "mathematical world" whose existence they don't have any wish to question, and whose structure they uncover by a mixture of intuition, not so foreign from "poetical desire", and of a great deal of rationality requiring intense periods of concentration.
Each generation
builds a "mental picture" of their own understanding of this world
and constructs more and more penetrating mental tools to explore previously
hidden aspects of that reality.
Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were previously believed to be very far remote from each other in the natural mental picture that a generation had elaborated. At that point one gets the feeling that a sudden wind has blown out the fog that was hiding parts of a beautiful landscape.
some thoughts of Gromov on being questioned by Remi Langevin regarding the reality of mathematics:
[The axiomatic approach] was taken as the credo (of formalism) by Hilbert, where the body of mathematics is visualized as an immense exponentially expanding "tree", "the tree of all possible mathematical formulas" growing from axioms.
Now questions come easily. What is the overall geometry and topology of this "tree"? What is the "human scale", the size of the branches accessible to human mathematicians? What patterns of this "tree" represent the entities we call "theorems"?
Godel pointed out that this "tree" was disconnected and consequently may have branches of infinite complexity where no mathematical creature is given a chance for survival. ... besides the scare of "Godel branches", the sheer exponential size of Hilbert's "tree" makes it inadequate for representing "earthly mathematics". We need a better model.
...
[climbing back to the Hilbert tree where] we find ourselves confined to a tiny region, that is our "human mathematics", some kind of a little cloud drifting in the immensity of exponentially stretching branches.
What guides us through the Hilbert tree and keeps away from the damnation of meaningless complexity and undecidability? ...This needs a scrupulous study and we are tempted to follow our gut feelings, shut the eyes and embrace the Platonic viewpoint. Here, there exists "the true world of mathematics where our brain serves as a relay between ourselves and this world, acting via some mysterious force of "semantic field". Sounds romantic (and can be turned into a great piece of science fiction) but not acceptable unless we are ready for a complete intellectual surrender!
(Yet the religious fervour of the Platonic faith into the absolute truth and internal harmony of mathematics is the prerequisite for success in discovering and proving great theorems. The process of creation is fueled as much by irrational as much as procreation.)
from van der Waerdens recollections about how the proof of Baudets conjecture was found
Once in 1926,
while lunching with Emil Artin and Otto Schreier, I told them about the conjecture of the Dutch
mathematician Baudet: If a sequence of integers of 1,2,3, etc. is divided into two classes, at least one of the
classes contains an arithmetic progression of l terms - a,a+b,a+2b, . . . ,
a+(l-1)b - no matter how large the length l is. After lunch we went into Artins office . . . and tried to find a proof.
. . . One of the
main difficulties in the psychology of invention is that most mathematicians
publish their results with condensed proofs, but do not tell us how they found
them. In many cases they do not even remember their original ideas. Moreover,
it is difficult to explain our vague ideas and tentative attempts in such a way
that others can understand them.
. . . All ideas
we formed in our minds were at once put into words and explained by little
drawings on the blackboard. We represented the integers 1,2,3,etc.
in two classes by means of vertical strokes on two parallel lines. Whatever one
makes explicit and draws is much easier to remember and to reproduce than mere
thoughts.
THEOREM (Perron). The greatest natural integer is 1.
Proof. Let N be the greatest natural integer. Assume, by contradiction, that N>1. By multiplying both sides by N, we obtain NN>N. So N is not the greatest. Contradiction. So N=1.