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Hilbert famously said

The art of doing mathematics consists in finding that special case which contains all the germs of generality.

Can you give an example of a situation in mathematics where this quote applies? I don't remember many situations like this where finding the special case was really the essential core of the argument. I wonder why Hilbert gives such a significance to finding the special case -- in my opinion the creativity mostly lies somewhere else. But since it is Hilbert, there must me something to this quote, and I want to understand it.

Question can possibly be community wiki.

I just saw Examples for Hilbert's Quote but the answers there do not seem satisfactory: they (at least to me) don't indicate why finding special cases containing all generality is that important that Hilbert puts so much significance to it.

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  • $\begingroup$ I think the point is not that the special case is forming the crux of some mathematical argument, but that rather it exemplifies all the diverse phenomena that can occur. There’s a lot of value of seeing the general concept “in action”. $\endgroup$ – Sir Jective May 24 at 21:25
  • $\begingroup$ Ah, thanks, nice interpretation. I mainly thought about the situation if one has, as an assumption, given a for all statement, which one then may use by plugging in a specific value. $\endgroup$ – user7280899 May 24 at 21:27
  • $\begingroup$ You will find examples in the literature if you search for "representative special case" (with the quotes, so you get that exact phrase). $\endgroup$ – Ethan Bolker May 24 at 21:28
  • $\begingroup$ @EthanBolker: Thanks! Maybe we can use this thread to collect some of these examples and make them more accesible? $\endgroup$ – user7280899 May 24 at 21:30
  • $\begingroup$ That would be a good thing to do but I don't have time to do it. This from Polya is a favorite of mine but posting it calls for a screen capture. books.google.com/… $\endgroup$ – Ethan Bolker May 24 at 21:34
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I don't know what Hilbert actually intended with that quote as I haven't read the context let alone exhaustively gone through Hilbert's writings with this quote in mind. I strongly suspect Hilbert meant this in a rather informal way. Nevertheless, let's brazenly charge forward with trying to take this fairly literally. As a nice touch, what I discuss is quite friendly with Formalism.

One interpretation is the mathematicians are (or should be) focused on discovering and analyzing things like free objects, e.g. the free monoid. For a class of relevant properties, any of those properties that hold of the free object hold for every other similar sort of object. These free objects are often rather syntactical. (A significant generalization of this is the notion of syntactic categories and internal languages. A different object that has similar properties is classifying spaces/classifying toposes.)

Continuing the spirit of being overly literal, a good example of this is the Lindenbaum-Tarksi algebra for concreteness say for classical propositional logic. This is essentially a free Boolean algebra. It has the property that a formula of classical propositional logic is valid if and only if it is true in this particular model. This is a very literal interpretation of a "special case" that contains the "germs of generality".

Being less literal but maybe closer to the spirit of Hilbert's statement. Syntax is the handle we have on mathematical objects. To the extent that we want to actually calculate things (often even for the very amorphous notion of "calculate" that mathematicians often use), we need a (more) syntactic/combinatorial description of an object. Often that isn't quite possible, but what is possible is finding an equivalent object that is describable in a more combinatorial manner. Probably one of the best examples of this is the notion of a simplicial complex from algebraic topology1. I'm fairly confident that Hilbert would view simplicial complexes as an example of a special case containing the germ of generality.

1 To connect to earlier, the closely related notion of a simplicial set gives rise to the category of simplicial sets and is a classifying topos.

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A good example for the above that I noticed is the proof that any matrix can be written as the product of upper triangular by G Strang in his video lectures in linear algebra at Open Course Ware at MIT.

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