Grand Strategy Annex

25th February 2012

Post with 1 note

Intuitively Clear Slippery Concepts

Or, into the conceptual wild…

The above phrase, “intuitively clear slippery concepts” just appeared today on the FOM (foundations of mathematics) listserv, as used by Hendrik Boom.

Here’s some context:

“…well-founded is one of these intuitively clear slippery concepts that goes awry when things become too general.”

This appeared in the tread, “iterative conception/cumulative hierarchy,” so this phrase that I ripped out the context of a sentence is also ripped out the context of a long discussion on the foundations of set theory.

Philosophers have used a variety of terms to indicate imprecise concepts and their manipulation: open-textured concepts, loose concepts, fuzzy concepts, and the like. This is the first time I’d seen “slippery concepts” used in this kind of context, but of course we all know what “slippery” means when it is used in colloquial speech. A slippery concept is difficult to pin down.

At the same time, philosophers have used a variety of terms of indicate intuitively clear concepts. The above phrase, “intuitively clear slippery concepts,” has the virtue of combining both the wide recognition that some concepts are intuitively clear, some concepts are slippery, and some concepts are both intuitively clear and slippery at the same time. One might hope or expect intuitively clear concepts to not be slippery, but sad experience has shown us that some of the least slippery concepts are counter-intuitive. 

Moreover, Boom adds the caveat that these concepts, “go awry when things become too general.” In other words, in a suitably restrictive context, we ought not to have much trouble with concepts both intuitively clear and slippery, but once let out into the conceptual wild, as it were, things can go wrong.

This is a constructivist theme. In a couple of posts here, P or not-P and What is the relationship between constructive and non-constructive mathematics? (also inspired by posts on FOM), I have discussed constructivist thought in a general context.

For several years I have been thinking about how to characterize constructivist thought is absolutely general terms, so that all the diverse forms of constructivism can be brought together in a principled unity under a single concept. One obvious approach (which also has obvious problems) is to characterize constructivistic reasoning in terms of the limitations that it places on classical formal reasoning.

For example, “P or not-P” — the law of the excluded middle — doesn’t cause many problems in finite contexts, but once you try to apply it in unlimited or transfinite contexts its seems to generate paradoxes. The more rigorously we limit our reasoning (and this is one of the functions of axiomatization) the more we can be certain about what we say and what it means.

This is especially the point that I tried to make in What is the relationship between constructive and non-constructive mathematics?, as I there suggested that finite mathematics is a common core shared by constructivist and non-constructive thought alike.

That is to say, despite profound philosophical differences, there is a little bit of logic and mathematics upon which all (or almost all) formal thought can agree. It is when we go further afield, “when things become too general” as Boom wrote, or we head out into the “conceptual wild” (as I put it above), that differences emerge. The farther afield we go the more these differences harden into ideological differences informed by a “party spirit.” 

In so far as almost all philosophies of mathematics can recognize the intuitive clarity of finite mathematics, but differences emerge once we pass beyond this restrictive boundary, the whole of mathematics could be said to consist of intuitively clear slippery concepts.

Tagged: constructiveconstructivismconstructivisticphilosophy of mathematicsphilosophyFOMfoundations of mathematicsfinitefinitudeconcepts