"In the expectation that there may perhaps be conceptions which relate a priori to objects, not as pure or sensuous intuitions, but merely as acts of pure thought (which are therefore conceptions, but neither of empirical nor aesthetical origin) in this expectation, I say, we form to ourselves, by anticipation, the idea of a science of pure understanding and rational cognition, by means of which we may cogitate objects entirely a priori."

Thus does Immanuel Kant provide the basis for the intuitionist school of thought, in his classic work Critique of Pure Reason. A most constructivist philosophy in nature, intuitionism is based upon the idea that one can work with mathematical conceptions which could possibly exist in the future, and will work with these ideas in a sort of intermediate state until they are either proved or countered sufficiently. Rather than formal rigor, or symbolic posturing, this branch of mathematical philosophy looks upon the intellectual endeavor as mainly an act of the individual mind. As Arend Heyting describes, mathematical objects are by their very nature dependent on human thought. Their existence is guaranteed only insofar as they can be determined by thought.

Where foundationalists strive to discover the inevitable underlying truth, intuitionists work to imagine and predict these truths. Ironically enough, the weight of empiricism also weighs in favor of the intuitionists, as many of the most profound mathematical breakthroughs, (such as those given by the likes of Fermat), were not proved until after sometimes even the deaths of the original theorizer. Often times intuitive creativity is beyond the scope of the current formal style. Intuitionism is unique in that it recognizes that the current mathematical process is fallible, and will inevitably be fixed so one need not waste time attempting to compromise ideas to fit into the mold of the status quo.

Of all of the schools of philosophical thought on mathematics, intuitionism is the most closely tied to physicalism, and the intial realm of Euclidean geometry. This is evident in the maintained reliance on possible visualization of mathematical objects, much in the same manner that a student of geometry evaluates a problem based on perceived physical traits of a figure even if the figure is yet nothing but an equation for points on a Cartesian graph.

Through the recognition of the corrigibility of mathematics, intuitionism is more of an epistemological process (one having to do with the transmission of knowledge; teaching) as opposed to the foundationalist ontological (having to do with being of some sort) perspective.

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