David Corfield discusses some speculation originally from Israel Gelfand:
Sporadic simple groups are not groups, they are objects from a still unknown infinite family, some number of which happened to be groups, just by chance.
(In David’s terminology, that means that sporadic finite simple groups are not a natural kind.)
I used to believe this very same thing, so I find it interesting that others have speculated the same thing. A couple of years ago, though, I came across a remark by Michael Aschbacher that made me rethink my view: the classification of finite simple groups is primarily an asymptotic result. Every sufficiently large finite simple group is either cyclic, alternating, or a group of Lie type.
Results that are true only for large enough parameter values are common enough that the existence of small-value counterexamples does not require special explanation. For example, the classification of simple modular Lie algebras looks completely different over small characteristics than it does over large characteristics. The best known results for number theoretic results such as Waring’s problem and Goldbach’s conjecture are asymptotic. Small numbers are just bad news.