andrew danks

Gödel’s first incompleteness theorem was the first to suggest that Hilbert’s programme to create a complete and consistent system of axioms of all mathematical truths is impossible. It says, in particular, that any consistent system of “strong enough” axioms about arithmetic on the natural numbers that can be listen by any effective procedure is incomplete. A theory (P) is complete if either (P) can be proven or (\neg P) can be proven. (P) is consistent if not both (P) and (\neg P) can be proven. The theorem also says that if such a consistent set of axioms is infinite and can be enumerated by a computer program, it is still incomplete. As a result, for any axiom added to the system which maintains its consistency, there will always be other truths that cannot be proven with the system’s own language. If an axiom is added to the system such that it makes it complete, it does so at the cost of making it inconsistent. If a system is inconsistent (i.e., contains (P) and (\neg P)), then one may be able to prove that anything is true.

These results are rather intriguing. It means that there are mathematical truths of arithmetic that we cannot prove with our current system, which may implicitly promote the Platonist account for mathematics. Perhaps this is because we are unable to capture certain mathematical facts, perhaps due to their independence. The implications of the theorem was also devastating for Hilbert’s programme.

Hilbert’s programme was inspired by the fundamental crisis of mathematics which encompassed many paradoxes including Russell’s paradox. He wanted to ground all existing mathematical theories to a finite, consistent, and complete set of axioms. He proposed that this could be reduced to just doing so for arithmetic as it appeared easier and more reasonable.

Gödel’s first incompleteness theorem showed, however, that even for arithmetic, it is impossible to completely axiomatize all truths of arithmetic. In fact, Smorynski (1977) argued that the first theorem alone is sufficient for a fatal blow to Hilbert’s programme because Hilbert required that for any real formula (P), it must be able to be finitistically proven, which was shown to be impossible.

It is quite convincing that if there is no consistent and complete axiomatization of arithmetic, then there is none for mathematics in general. However, I do believe there are weaker statements of Hilbert’s programme that doesn’t require completeness so we could axiomatize only some existing results. However, this is far from his original goal and such a small axiomatization would be meaningless.

Constructive mathematics can be defined and distinguished from classical mathematics by interpreting “there exists” as “we can construct.” With this constraint, all logical connectives and quantifiers must also be re-interpreted in addition to the existential quantifier. For example, consider the connective OR (“\(\lor\)“). In classical mathematics, the expression \(P\lor Q\) can be interpreted as \(P\) is true or \(Q\) is true. In the constructive account, however, it would be \(P\) is proven or \(Q\) is proven. This approach is embodied by constructive logic. Constructive mathematicians believe that we make or construct mathematical facts, in contrast with the Platonist account that says we discover mathematical truths which exist independently from us. The constructivist thesis says that only those mathematical truths inferred by constructive rules of inference are sound.

The constructivist approach to mathematical proofs has some advantages. The Platonist approach allows for a large variety of investigative techniques to be used as empirical evidence for theorems, but this flexibility has a cost. It opens the door to many disagreements in what kind of evidence is legitimate. For example, it is still debated on whether pictures can ever be used as evidence (“picture proofs”). The merit of computer proofs have also been disputed; some argue that computers should only be used for verification. Constructive mathematics, on the other hand, provides a more consistent yet strict procedure for how mathematical facts can be derived. This is perhaps useful for early mathematics education. There are even further non-ideological applications in foundational mathematics and computer science. Physicist Smolin stated this account provides a good basis for the logic of cosmology:

The statements an observer can make about the universe are divided into at least three groups: those that we can judge to be true, those that we can judge to be false and those whose truth we cannot decide upon at the present time

There are many downsides to constructivism, however, and I do not think these non-ideological applications warrant conducting ourselves in a constructivist manner when trying to derive new truths in the realm of mathematics.

If we adopt the constructivist thesis, we would lose many important results. For example, the Bolzano-Weierstrass theorem, which has applications in complex analysis, topology, economics, and other fields would need to be discarded. This is because the theorem guarantees the existence of a cluster of points, but not how to find it, and constructive logic does not allow this. Moreover, it also does not allow for the law of excluded middle which says that any statement of the form \(P\lor\neg P\) is true. For example, consider the theorem that says there exists irrational numbers \(a\) and \(b\) such that \(a^b\) is rational. The proof goes like this: if \(a^b\) is rational, then let \(a=b=\sqrt{2}\).  Otherwise, let \(a=\sqrt{2}^{\sqrt{2}}\) and \(b=\sqrt{2}\). Then \(a^b=2\) which is rational. The constructive point of view would forbid this because we make no claim regarding the rationality of  \(\sqrt{2}^{\sqrt{2}}\).

Hilbert said that taking away this law would be like taking away a telescope from an astronomer. It could take centuries to reproduce all results in a constructivist manner and may not even be possible with today’s mathematics. With the flexibility of the Platonist point of view, there does not exist counter-proofs or inconsistencies in our results (there are only refinements like in the case of the definition of a function or the axiom of comprehension). Thus there is no reason to adopt constructivism as an exclusive procedure to discovering mathematical truths.