Richard Courant said it nicely in "What is mathematics?", my all time favourite book:<p><i>There seems to be a great danger in the prevailing overemphasis on the deductive-postulational character of mathematics. True, the element of constructive invention, of directing and motivating intuition, is apt to elude a simple philosophical formulation; but it remains the core of any mathematical achievement, even in the most abstract fields. If the crystallized deductive form is the goal, intuition and construction are at least the driving forces. A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal. The notion that the intellect can create meaningful postulational systems at its whim is a deceptive half-truth. Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value…To establish once again an organic union between pure and applied science and a sound balance between abstract generality and colourful individuality may well be the paramount task of mathematics in the immediate future.</i>
Another mathematician basis function from Freeman Dyson, <i>Birds & Frogs</i>:<p><a href="http://www.ams.org/notices/200902/rtx090200212p.pdf" rel="nofollow">http://www.ams.org/notices/200902/rtx090200212p.pdf</a><p>Dyson contrasts the ADHD of the multi-disciplinary cross-fertilizing polymath, with the meticulous obsessive immersion of the specialist.<p>There is also the Kuhnian distinction between <i>revolutionary</i> and <i>ordinary</i> progress: the leap of the system creator compared to the incremental steps of the system extender.<p>I think it is very relevant that the intuitionist geometric thinkers are often the mathematical physicists. They have a vivid kinaesthetic imagination, where mathematical entities have a visual appearance, complete with spatial properties and forceful interaction. Sometimes these take the form of practical <i>thought experiments</i>, which open a problem to precise mental tests. But they can also involve more abstract explorations, like the slow precession of a mobile sculpture, such that alignments and symmetries become manifest: not yet understood, but actually there for examination.<p>A list of the greats would include Newton, Gauss, Riemann, Poincaré himself, Einstein and Feynman. Many of these used spatial representations to inspire algebraic progress. A more recent example (although necessarily on a lower tier) would be Penrose.
It seems to me that the absolute intuitionist and absolute logician can not exist as real persons. I see them rather as two extremes that are always mixed in some proportion (and always in fight with each other for more space, so the proportion changes over time). For instance, Euclid is mentioned as logician, but at least one of his axioms (about parrallel lines that could not intersect) is based on intuition that, turns out, is not always "true". Therefore we have got other Non-Euclidian geometries.
Here's a more readable version: <a href="http://www.scribd.com/doc/150574528/Intuition-and-Logic-in-Mathematics" rel="nofollow">http://www.scribd.com/doc/150574528/Intuition-and-Logic-in-M...</a>