Translated from french by Grok

The starting point for this note is a question from Thomas Riepe (a member of the GLIA society) that appeared in THOTH (page 6, issue 14, April 99):
“In math, sometimes a distinction is made between an ‘abstract’ and an ‘algorithmic’ style. Do you make that distinction too, and if so, which one do you prefer?”

Let’s consider the Rubik’s Cube paradigm. When we try to solve this particular problem, we begin with an experimental approach, then we notice the stability of certain patterns, and using those patterns we can develop an algorithmic method that treats them as steps toward the solution. Once the problem is solved, if we’re truly curious and interested, we try to find other solving methods. Having several different methods, we can then compare them and identify what they have in common. To do that, though, we need to know—at least for this specific problem—some basics of finite group theory. In fact, if we already have that prerequisite knowledge, we can solve the problem using an abstract method. Still, it feels difficult to apply a purely abstract method directly as a heuristic approach to a brand-new problem (i.e., one with no similarities to previously known problems). So for us, the abstract style seems to serve as a medium for achieving a better and deeper understanding of an already-solved problem.

For example, if someone wants to understand the classification of the sporadic groups (whose proof spans several thousand pages), they are practically forced to adopt an algorithmic style. Only after going through that phase can they use an abstract style to organize and classify their complete knowledge of the theorem.

To conclude, let’s consider the maze model. We can understand it in two ways: the algorithmic style (like Ariadne’s) and the abstract style (like Daedalus’s). Clearly, the first requires only local knowledge, whereas the second demands global knowledge.

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