On abstract and algorithmic styles in mathematics

N. Lygeros

Let us consider the paradigm of the Rubik's cube. When you try to solve this particular problem you start with an experimental approach. After that you remark the stability of some patterns and with those you can find an algorithmic method which uses them as steps to the solution. When the problem is solved if you are realy curious and interested you try to find other solving methods. Then getting few different methods you can compare them and find their common point. But to do this, you have to know, for this particular problem, some notions of the theory of finite groups. In fact, if you have already this background you can solve the problem with an abstract method. But it seems for us difficult to directly apply an abstract method, as an heuristic approach, on a new problem (i.e. with no similarities with other known problems). So, for us, an abstract style seems to be a medium for a better and deeper understanding of a solved problem.
For example, if someone wants to understand the classification of sporadic groups (which proof represents many thousands pages) then he's forced to adopt an algorithmic style. And only after this phase, he can use an abstract style to classify his complete knowledge of the theorem.

As a conclusion, let us consider the model of the labyrinth. We can take up two styles to understand it : the algorithmic style (these of Ariadne) and the abstract style (these of Daedalus). Nevertheless it's clear that for the first one you only need a local knowlegde whereas for the second you must have a global knowlegde.

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