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.
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