"For someone whose only tool is a hammer, every problem looks like a nail" (EWD 690)

This note is about paradigms. In short, I would define a paradigm as being the belief that a particular abstraction or theory fits a certain context. We can see, for example, in programming, the paradigm of object oriented programming as being the belief that focusing one's analysis and design efforts on the relations between different pieces of data is an efficient way to cope with the variation in requirements. Also, we can see that applying game theory to economic behavior gives rise to the paradigm that we might call "the rational choice". I think a paradigm is a very useful belief to have because it allows one to deal in simple terms with problems that might get complicated. In my opinion, a healthy use of a paradigm is like controlled schizophrenia: when you deal with a problem within the realm of applicability of the paradigm, it is useful and often necessary to assume its postulate without questions. This is very efficient when dealing with related problems. However, one should be able to defend the use of the paradigm and more specifically, should consciously stop believing in the postulates while doing so. I find it frustrating when discussing with somebody when I realize that, not only can I identify the paradigm he adopts (which is not bad) but I realize that he is unable to step out of it even after several direct attacks at his assumptions. This happens both in technical discussions about computing and in humanities. On the other hand, it is also fruitless to discuss hard problems with somebody who can't assume the postulates once they have been defended. In technical terms, this might be seen as somebody that keeps reasoning about a method call in terms of the instructions that gets executed as a substitute instead of reasoning in terms of an abstract specification. In the first case, the person is hardly schizophrenic at all (what a fraud!) and in the second case the schizophrenia is uncontrolled so the person's reasoning is all over the place: he can't focus on one aspect of his problems at a time because he can't see a clear separation between the two. As a second example of uncontrolled schizophrenia, I see some people rely strongly on set theory and have to understand everything in terms of set. Some of them for example, can't bear to use sequences for their reasoning because the proofs with sequences are so complicated and messy. This is a recent conviction of mine but I keep seeing examples where having a clear and simple abstraction would be much more effective. In the case of sequences, we need a set of postulates that describe sequences as objects of their own right and don't drag sets or anything else into the picture unless a useful relationship can be defined. And even then, the relationship is not central to understanding the abstraction. For soundness purposes, it might be useful to translate postulates of sequence theory in set theory. This becomes metamathematics though. It does not affect the use of sequences.