Constructivism Is Difficult

In a recent issue of this MONTHLY, Fred Richman [8] discussed existence proofs. Richman’s conclusion, as I understood it, was that once a mathematician sees the distinction between constructive and nonconstructive mathematics, he or she will choose the former. That conclusion, if extrapolated further than Professor Richman intended, suggests that any mathematician can learn constructivism easily if he or she so desires. But in fact constructivism is unusually difficult to learn. Learning most mathematical subjects merely involves adding to one’s knowledge, but learning constructivism involves modifying all aspects of one’s knowledge: theorems, methods of reasoning, technical vocabulary, and even the use of everyday words that do not seem technical, such as “or”. I discuss, in the language of mainstream mathematicians, some of those modifications; perhaps newcomers to constructivism will not be so overwhelmed by it if they know what kinds of difficulties to expect.