Is God a Mathematician_ - Mario Livio [91]
This development had dramatic philosophical consequences. As in the case of the non-Euclidean geometries in the nineteenth century, there wasn’t just one definitive set theory, but rather at least four! One could make different assumptions about infinite sets and end up with mutually exclusive set theories. For instance, one could assume that both the axiom of choice and the continuum hypothesis hold true and obtain one version, or that both do not hold, and obtain an entirely different theory. Similarly, assuming the validity of one of the two axioms and the negation of the other would have led to yet two other set theories.
This was the non-Euclidean crisis revisited, only worse. The fundamental role of set theory as the potential basis for the whole of mathematics made the problem for the Platonists much more acute. If indeed one could formulate many set theories simply by choosing a different collection of axioms, didn’t this argue for mathematics being nothing but a human invention? The formalists’ victory looked virtually assured.
An Incomplete Truth
While Frege was very much concerned with the meaning of axioms, the main proponent of formalism, the great German mathematician David Hilbert (figure 52), advocated complete avoidance of any interpretation of mathematical formulae. Hilbert was not interested in questions such as whether mathematics could be derived from logical notions. Rather, to him, mathematics proper consisted simply of a collection of meaningless formulae—structured patterns composed of arbitrary symbols. The job of guaranteeing the foundations of mathematics was assigned by Hilbert to a new discipline, one he referred to as “metamathematics.” That is, metamathematics was concerned with using the very methods of mathematical analysis to prove that the entire process invoked by the formal system, of deriving theorems from axioms by following strict rules of inference, was consistent. Put differently, Hilbert thought that he could prove mathematically that mathematics works. In his words:
Figure 52
My investigations in the new grounding of mathematics have as their goal nothing less than this: to eliminate, once and for all, the general doubt about the reliability of mathematical inference…Everything that previously made up mathematics is to be rigorously formalized, so that mathematics proper or mathematics in the strict sense becomes a stock of formulae…In addition to this formalized mathematics proper, we have a mathematics that is to some extent new: a metamathematics that is necessary for securing mathematics, and in which—in contrast to the purely formal modes of inference in mathematics proper—one applies contextual inference, but only to prove the consistency of the axioms…Thus the development of mathematical science as a whole takes place in two ways that constantly alternate: on the one hand we derive provable formulae from the axioms by formal inference; on the other, we adjoin new axioms and prove their consistency by contextual inference.
Hilbert’s program sacrificed meaning to secure the foundations. Consequently, to his formalist followers, mathematics was indeed just a game, but their aim was to rigorously prove it to be a fully consistent game. With all the developments in axiomatization, the realization of this formalist “proof-theoretic” dream appeared to be just around the corner.
Not all were convinced, however, that the path taken by Hilbert was the right one. Ludwig Wittgenstein (1889–1951), considered by some to be the greatest philosopher of the twentieth century, regarded Hilbert’s efforts with metamathematics as a waste of time. “We cannot lay down a rule for the application of another rule,” he argued. In other words, Wittgenstein did not believe that the understanding of one “game” could depend on the construction