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Why did I choose to do CM? Because I found it interesting, and because the idea of actually finding objects, instead of merely showing that they could not possibly fail to exist, was one that appealed to me. When I first came across Errett Bishop�s book, in 1968, I had been working on von Neumann algebra theory as a graduate student, and had been vaguely�and certainly inarticulately�dissatisfied with the prevailing style of existence proofs in my reading in that subject. Such proofs typically either proceeded by assuming the non�existence of the desired object and deducing a contradiction, or else applying Zorn�s lemma to �construct� a maximal family of projections with some property or other. Somehow, beautiful though those proofs were, they left me with a feeling that I had been cheated. What did those objects whose existence was proved really look like? How could they be described explicitly? It was only on reading Errett�s book that I understood what was bothering me and that it was possible to give satisfactory answers to those questions.
Now, don't get me wrong. Just because I find CM particularly appealing it is not the case that I dislike classical (that is, traditional) mathematics, let alone that I am advocating that classical mathematics is somehow not a proper activity for mathematicians to be engaged in. I find most classical mathematics (at least, what I can understand of it) very interesting and a worthy scientific/cultural pursuit. However, if, as I am, you are interested in computability/constructivity within pure mathematics (as distinct from, say, numerical analysis), then you should seriously investigate constructive mathematics. By working constructively�that is, with intuitionistic logic�you will learn to appreciate the distinction between idealistic existence (the impossibility of non�existence) and constructive existence. This distinction is one that, in my view, should be heeded and appreciated far more than it is. As Bishop wrote,
"Meaningful distinctions deserve to be maintained" [6]
If, however, you are not interested in questions of computability, then you should stick to classical logic. There are even areas of mathematics where the content is so highly nonconstructive that it would make little sense to give up classical logic; the higher reaches of modern set theory would seem to be just such an area.
To summarise: CM should interest people who would like to understand better the distinction between classical and constructive existence, and who are interested in pushing beyond the former to a real construction of the object whose existence is asserted.
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