Editing Type theory (section)

==== Constructive mathematics ====
Per Martin-Löf proposed his intuitionistic type theory as a foundation for [[constructive mathematics]].<ref name=":6" /> Constructive mathematics requires when proving "there exists an <math>x</math> with property <math>P(x)</math>", one must construct a particular <math>x</math> and a proof that it has property <math>P</math>. In type theory, existence is accomplished using the dependent product type, and its proof requires a term of that type.

An example of a non-constructive proof is [[proof by contradiction]]. The first step is assuming that <math>x</math> does not exist and refuting it by contradiction. The conclusion from that step is "it is not the case that <math>x</math> does not exist". The last step is, by double negation, concluding that <math>x</math> exists. Constructive mathematics does not allow the last step of removing the double negation to conclude that <math>x</math> exists.<ref>{{cite web|title=proof by contradiction|url=https://ncatlab.org/nlab/show/proof+by+contradiction|website=nlab|access-date=29 December 2021|archive-date=13 August 2023|archive-url=https://web.archive.org/web/20230813170132/https://ncatlab.org/nlab/show/proof+by+contradiction|url-status=live}}</ref>

Most of the type theories proposed as foundations are constructive, and this includes most of the ones used by proof assistants.{{Cn|date=January 2024}} It is possible to add non-constructive features to a type theory, by rule or assumption. These include operators on continuations such as [[Call/cc#Relation to non-constructive logic|call with current continuation]]. However, these operators tend to break desirable properties such as [[canonicity (type theory)|canonicity]] and [[parametricity]].