Paradoxes of free will and the limits of human reason.

MMANUEL KANT, the eighteenth-century German philosopher, and Niels Bohr, the twentieth-century Danish physicist, both noted that driving human reason too far in the analysis of deep problems often leads to irresolvable contradictions. Kant (1934) epitomized his insight into this fundamental limitation of human reason with his aphorism “Out of timber so crooked as that from which man is made nothing entirely straight can be built.” And Bohr (1949) drew attention to the limits of human reason by citing what he referred to as an “old saying.” According to that old saying, there are two kinds of truths: one kind is an ordinary truth , which is so simple and clear that its opposite is obviously false, while the other kind is a deep truth , whose opposite is also a deep truth. Kant’s and Bohr’s ruminations on the ultimate limits of human reason arose from their encounter of paradoxes inhering in our rational faculty. One kind of paradox presents a proposition that seems selfcontradictory or incompatible with generally accepted opinions. An example of that kind of paradox is the claim made by Aristarchus of Samos in the third century b.c.e. that our obviously stationary Earth rotates about our obviously moving Sun. Another kind of paradox presents two paired propositions, either of which, when considered alone, is supported by apparently sound arguments. But when the paired propositions are considered together, they turn out to be mutually contradictory. An example of the paired proposition kind of paradox is the religious quandary to which the seventeenth-century German philosopher and mathematician Gottfried Wilhelm Leibniz (1710) gave the name