Is God a Mathematician_ - Mario Livio [87]
Figure 48
Central to Frege’s philosophy was the assertion that truth is independent of human judgment. In his Basic Laws of Arithmetic he writes: “Being true is different from being taken to be true, whether by one or many or everybody, and in no case is it to be reduced to it. There is no contradiction in something’s being true which everybody takes to be false. I understand by ‘laws of logic’ not psychological laws of takings-to-be-true, but laws of truth…they [the laws of truth] are boundary stones set in an eternal foundation, which our thought can overflow, but never displace.”
Frege’s logical axioms generally take the form “for all…if…then…” For instance, one of the axioms reads: “for all p, if not-(not-p) then p.” This axiom basically states that if a proposition that is contradictory to the one under discussion is false, then the proposition is true. For instance, if it is not true that you do not have to stop your car at a stop sign, then you definitely do have to stop at a stop sign. To actually develop a logical “language,” Frege supplemented the set of axioms with an important new feature. He replaced the traditional subject/predicate style of classical logic by concepts borrowed from the mathematical theory of functions. Let me briefly explain. When one writes in mathematics expressions such as: f(x) 3x 1, this means that f is a function of the variable x and that the value of the function can be obtained by multiplying the value of the variable by three and then adding one. Frege defined what he called concepts as functions. For example, suppose you want to discuss the concept “eats meat.” This concept would be denoted symbolically by a function “F(x),” and the value of this function would be “true” if x lion, and “false” if x deer. Similarly, with respect to numbers, the concept (function) “being smaller than 7” would map every number equal to or larger than 7 to “false” and all numbers smaller than 7 to “true.” Frege referred to objects for which a certain concept gave the value of “true” as “falling under” that concept.
As I noted above, Frege firmly believed that every proposition concerning the natural numbers was knowable and derivable solely from logical definitions and laws. Accordingly, he started his exposition of the subject of natural numbers without requiring any prior understanding of the notion of “number.” For instance, in Frege’s logical language, two concepts are equinumerous (that is, they have the same number associated with them) if there is a one-to-one correspondence between the objects “falling under” one concept and the objects “falling under” the other. That is, garbage can lids are equinumerous with the garbage cans themselves (if every can has a lid), and this definition does not require any mention of numbers. Frege then introduced an ingenious logical definition of the number 0. Imagine a concept F defined by “not identical to itself.” Since every object has to be identical to itself, no objects fall under F. In other words, for any object x, F(x) false. Frege defined the common number zero as being the “number of the concept F.” He then went on to define all the natural numbers in terms of entities he called extensions. The extension of a concept was the class of all the objects that fall under that concept. While this definition may not be the easiest to digest for the nonlogician, it is really quite simple. The extension of the concept “woman,” for instance, was the class of all women. Note that the extension of “woman” is not in itself a woman.
You may wonder how this abstract logical definition helped to define, say, the number 4. According to Frege, the number 4 was the extension (or class) of all the concepts that have four objects falling under them. So, the concept “being a leg of a particular dog named Snoopy” belongs to that class (and therefore to the number 4), as does the concept “being a grandparent of Gottlob Frege.”
Frege’s program was extraordinarily impressive, but it also suffered from some serious drawbacks.