🚀 go-pugleaf

zach@csdec1.tuwien.ac.at wrote:
> Many, if not most, "great" logicians did not "solve problems", as one
> usually does in mathematics, in that someone states a problem, and a
> the you go ahead and solve it (positively or negatively); furthermore,
> their solutions are more instructive in themselves than the original
> problems where and they not only develop new concepts to solve an existing
> problem (this is commonplace in mathematics), but rather invent a new
> concept which they think is interesting IN ITSELF, prove something about it,
> and incidentially some open problem follows as a corollary. In fact, more
> often than not, they do not solve problems in the usual sense, but they
> say something about the problem itself, eg, that it was mis-posed, mis-
> conceived, or is meaningless.

I think this gets the practice of both ordinary mathematics and logic wrong.
In both cases many of the major achievements are kinds of classification of
informal ideas: 19th-century analysis only got results by seeing that the
general idea of a function split into many, depending on continuity,
uniform continuity, degree of differentiability or integrability, and so on;
what Cantor did by breaking the pre-existing idea of "infinity" up into
cardinal and ordinal hierarchies is not all that methodologically different.
(And Cantor's work *did* start from a conventional mathematical problem, the
convergence behaviour of Fourier series).

Move on a bit: later work in logic tries to pin down the idea of "proof",
originally as vague as "function" was in 1800, and ends up with a vast
range of classifications by proof-theoretic complexity, degree of
constructivity, ontological assumptions about the infinite, strength of
choice principles employed, and so on.  Meanwhile in "normal" mathematics
you have a whole industry going on inventing new kinds of "geometric"
entity and classifying them: Banach and Hilbert spaces, topological spaces,
lattices,...  yes, these *do* come up in the context of solving existing
problems like stability of dynamical systems, but so does the definability
hierarchy of set theory arise out of conventional mathematics.  And the
notion of "algorithmic problem" was no more invented by the logicians of
the 30s than the idea of "curve" was by 19th-century differential
geometers; both are thousands of years old.  What recursion theory did was
to find *structure* inside that notion, the degree hierarchy.

Any takers for renaming mathematical logic as "hierarchy theory"?  It would
be a lot less misleading in some ways.  And a "Frege-free" history of the
subject might make it clearer what most logicians have actually been doing
with their time over the last few decades.

-- 
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