FOM: epochal f.o.m. advances; philosophical exercises
Stephen G Simpson
simpson at math.psu.edu
Mon Aug 3 13:52:26 EDT 1998
In his brilliant FOM posting "explicating the epochal" of 2 Aug 1998
09:57:29, Neil Tennant established an appropriate context and thereby
paved the way for an explicit discussion of the following key f.o.m.
question:
Which f.o.m. advances deserve to be called epochal, and why?
Obviously such a discussion is of the highest importance for the FOM
list, in several ways. First, it explicitly raises the issue of
standards for evaluating f.o.m. advances. Second, it draws the
attention of f.o.m. professionals to the best, the greatest, the most
outstanding achievements in our field. Third, it raises the ante by
urging us to consider f.o.m. in the context of outstanding human
achievement generally. Thank you, Neil.
I urge all FOM subscribers to ponder the above question and try to
answer it, at least in their own minds. What do you think are the
greatest f.o.m. advances? Are they of epochal significance, as
explicated by Neil? Why are they epochal? What is their general
intellectual or cultural significance? I would be very grateful to
any FOM subscribers who choose to post their answers.
Neil's remarks tie in very well with my proposed series of
"philosophical exercises" dating from March (my postings of 17 Mar
1998 13:13:02 and 19 Mar 1998 09:00:18). After presenting a concise,
high-level formulation of the philosophical aspect or general
intellectual interest (g.i.i.) of Friedman's recent results, I said:
As a useful series of philosophical exercises, let's try to give
similarly concise g.i.i. formulations of other well known high
points of f.o.m. research. I refer to f.o.m. advances such as:
1. Frege's invention of the predicate calculus
2. G"odel's completeness theorem
3. G"odel's incompleteness theorems
4. Turing's work on computability
5. consistency and independence of the continuum hypothesis
6. the large cardinal hierarchy
7. the MRDP theorem
8. determinacy and large cardinals
etc.
Would FOM subscribers care to suggest additions to this list? If so,
please be prepared to justify them. In any case, it seems to me that
this series of "philosophical exercises" would be extremely valuable
for all students of f.o.m.
As an example, here is a concise g.i.i. formulation of items 1 and 2,
predicate calculus and the G"odel completeness theorem:
Frege discovered a precise formal language and rules of inference
which appeared to embody all correct reasoning of a certain kind,
including all rigorous mathematical reasoning. In its later
equivalent formulation, this is known as the predicate calculus.
G"odel shored it up by showing that syntactic deducibility in the
predicate calculus corresponds precisely to a clear, semantic
notion of logical consequence defined in terms of models.
This is only a first stab, and I would welcome corrections and
improvements. In addition, let's try to compile similar
g.i.i. formulations of items 3-8 and other epochal f.o.m. advances.
-- Steve
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