Immanuel Kant had proposed that certain mathematical intuitions are part of “synthetic a priori knowledge.” For Kant (1929: 68) “[s]pace is not an empirical concept which has been derived from outer experiences,” rather, it is “necessarily a priori representation, which underlies all outer intuitions.” For Kant our intuitive knowledge of space is systematized in geometry:
The apodeictic certainty of all geometrical propositions and their possibility of their a priori construction, is grounded in this a priori necessity of space.
Kant (1929: 86) provides the following example from elementary geometry:
Take for instance, the proposition, Two straight lines cannot enclose a space, and with them alone no figure is possible,” and try to derive it from the concept of straight lines and of the number two. Or take the proposition, “Given three straight lines a figure is possible,” and try, in like manner, to derive it from the concepts involved. All your labour is in vain; and you find that you are constrained to have recourse to intuition, as is always done in geometry.
Davis and Hersh (1981: 150) recognize that there are certain pivotal junctures in routine geometrical proofs that depend on intuitive leaps. They point to the:
extraneous lines which in high school are often called “construction lines” complicate the figure, but form an essential part of the deductive process… Now, how does one know where to draw these lines so as to reason with them? It would seem that these lines are accidental or fortuitous. In a sense this is true and constitutes the genius or trick of the thing. Finding the lines is part of finding a proof, and this may be no easy matter. With experience come insight and skill at finding proper construction lines.
The element of intuition in proof to some extent unsettles notions of consistency and certainty in mathematics.
Davis, Philip J. and Hersh, Reuben (1981) The Mathematical Experience. Houghton Mifflin, Boston and New York.
Kant, Immanuel (1929) Critique of Pure Reason. Translated by Norman Kemp Smith. St Martin’s Press, New York. First published as Kritik der reinen Vernunft: Riga 1787.
We use the term ‘intuition’ as a catch-all label for a variety of effortless, inescapable, self-evident perceptions or insights that seem to arrive all at once and fully-formed. The origins of most of our intuitions are hidden below the threshold of conscious awareness, but remain a vivid aspect of our subjective experience. Davis and Hersh caution (1981: 391) that intuition in mathematics can be viewed as “a dangerous and illegitimate substitute for rigorous proof.” They also recognize a different context where:
it seems to denote an inexplicable flash of insight by which the happy few gain mathematical knowledge which others can attain only by long efforts.
They (1981: xviii) are aware that not all mathematicians are created equal. Some:
are more endowed with the talent of drawing pictures, others with that of juggling symbols and yet others with the ability to pick a flaw in an argument.
Davis and Hersh (1981: 319) cite mathematical iconoclast Kurt Gödel; who seems to have thought deeply about the implications of “the fact that the axioms force themselves upon us as being true.” Gödel sees no reason why:
we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception…
WHEN INTUITION ERRS ― THE MONTY HALL PROBLEM
“Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to take the switch?”
Posed a reader of Marilyn Vos Savant's column in Sunday Parade, September 1991.