• Choose a category
• All
• Classic-Fiction

By Root 799 0

lead to unacceptable conclusions. In the example above, the village barber both shaves and doesn’t shave himself. Can this particular paradox be resolved? One possible resolution to the paradox, strictly as stated above, is simple: The barber is a woman! On the other hand, were we told in advance that the barber had to be a man, then the absurd conclusion would have been the result of accepting the premise in the first place. In other words, such a barber simply cannot exist. But what does any of this have to do with mathematics? As it turns out, mathematics and logic are intimately related. Here is how Russell himself described the linkage:

Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now [in 1919] become wholly impossible to draw a line between the two; in fact the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic.

Russell holds here that, largely, mathematics can be reduced to logic. In other words, that the basic concepts of mathematics, even objects such as numbers, can in fact be defined in terms of the fundamental laws of reasoning. Furthermore, Russell would later argue that one can use those definitions in conjunction with logical principles to give birth to the theorems of mathematics.

Originally, this view of the nature of mathematics (known as logicism) had received the blessing of both those who regarded mathematics as nothing but a human-invented, elaborate game (the formalists), and the troubled Platonists. The former were initially happy to see a collection of seemingly unrelated “games” coalesce into one “mother of all games.” The latter saw a ray of hope in the idea that the whole of mathematics could have stemmed from one indubitable source. In the Platonists’ eyes, this enhanced the probability of a single metaphysical origin. Needless to say, a single root of mathematics could have also helped, in principle at least, to identify the cause for its powers.

For completeness, I should note that there was one school of thought—intuitionism—that was vehemently opposed to both logicism and formalism. The torch-bearer of this school was the rather fanatical Dutch mathematician Luitzen E. J. Brouwer (1881–1966). Brouwer believed that the natural numbers derive from a human intuition of time and of discrete moments in our experience. To him, there was no question that mathematics was a result of human thought, and he therefore saw no need for universal logical laws of the type that Russell envisioned. Brouwer did go much further, however, and declared that the only meaningful mathematical entities were those that could be explicitly constructed on the basis of the natural numbers, using a finite number of steps. Consequently, he rejected large parts of mathematics for which constructive proofs were not possible. Another logical concept denied by Brouwer was the principle of the excluded middle—the stipulation that any statement is either true or false. Instead, he allowed for statements to linger in a third limbo state in which they were “undecided.” These, and a few other intuitionist limiting constraints, somewhat marginalized this school of thought. Nevertheless, intuitionist ideas did anticipate some of the findings of cognitive scientists concerning the question of how humans actually acquire mathematical knowledge (a topic to be discussed in chapter 9), and they also informed the discussions of some modern philosophers of mathematics (such as Michael Dummett). Dummett’s approach is basically linguistic, stating forcefully that “the meaning of a mathematical statement determines and is exhaustively determined by its use.”

But how did such a close partnership between mathematics and logic develop? And was the logicist program at all viable? Let me briefly review a few of the milestones of the