friedman at math.ohio-state.edu
Mon May 5 21:30:32 EDT 2008
I call your attention to a very well thought out and interesting
posting of Tennant http://www.cs.nyu.edu/pipermail/fom/2008-May/012872.html
which fairly reflects what I am trying to accomplish in this discussion.
In particular, Neil writes
"Indeed, much of what is judged as constituting intellectual progress
in contemporary analytical philosophy can fairly be described as
Q-ifying erstwhile influential Ps."
For easy reference:
THESIS. Suppose that a philosophical paper P, in any part of
philosophy, consisting of informal prose, without rigorous
systemization, represents intellectual progress. Then there exists a
paper Q with the following properties.
1. Q focuses on rigorous systemization.
2. Q has a relatively small amount of informal prose.
3. Q can be written using the current level of practice in rigorous
systemization and foundational thinking.
4. P is fully subsumed by Q.
Do subscribers view the general passage from Natural Philosophy to
Physical Science as providing plenty of instances of this THESIS?
I want to respond later to some of the recent postings on methodology.
But first I want to make some further remarks.
1. Preparing Q's is generally very difficult, but within the range of
expectation over a period of time by suitable scholars with suitable
ambition, who know the kind of thing that is required, know the severe
limitations of P's, and value Q's properly compared to P's.
2. A large book may be equivalent to roughly a dozen or more P's. So
for a large book, there may be something like a dozen or so Q's
needed, each with a little bit of prose. So the prose does add up,
somewhat. But of course, nothing like the amount of prose in the
original large book.
3. Consequently I view any statement that Q's don't exist as dubious
on the face of it, given the great many examples of Q's and how these
Q's have rendered entirely obsolete massive amounts of earlier
discussions and papers.
4. I have personally thought most deeply about issues surrounding
mathematical thought. In this realm, it would be interesting to see if
I can meet the following challenge. Subscribers can offer or point to
a philosophical discussion surrounding mathematical thought, for which
there is intellectual progress, and for which they know no Q. The
challenge is for me to create a Q, or a good stab at a Q, in response.
5. I do not think that the rest of philosophy is intrinsically any
different than philosophy surrounding mathematical thought. So perhaps
somebody else can be presented with a challenge analogous to 4 for
other philosophical contexts.
6. It is definitely illuminating to work backwards, and see just how
much philosophical discussion is subsumed by various Q's. Consider the
epsilon-delta development by Cauchy. Now just how many reams and reams
of philosophical discussions are subsumed by this - together with a
7. Sure, you can say that perhaps there are other aspects not
subsumed, in particular if you take infinitesmals seriously. OK, but
there are some other Q's for that. E.g., nonstandard arithmetic/
analysis, ultrapowers, etcetera. And when I was a student (say about
1966), just for this purpose, I invented and proved principal theorems
about formal theories of infinitesmals. This supports some reasoning
with infinitesmals without requiring that one use a particular model
of them in ordinary terms. If yet more subtle informal uses of
infinitesmals are discussed, then this would be subsumed again by more
delicate systemizations - always with some explanatory prose.
8. Now let's move on to Frege, and work backwards. Frege is generally
credited with the predicate calculus. However, what he wrote there is
of course unreadable in comparison to the best setups and
presentations that we have now - we have lots of Q's for this.
9. But instead of going forward from Frege, let's go backwards, and
give him credit for predicate calculus, as usual. Now just how much
philosophical discussion concerning logical matters, before Frege, is
subsumed by predicate calculus, with a limited amount of additional
10. Incidentally, moving forward a little from Frege, Frege emphasized
that quantifiers should range over everything. That is not reflected
at all in the usual work on predicate calculus. I set up, several
decades ago, the basics of: Logic in the Universal Domain: a complete
theory of everything, a version of that being on my website. This
should subsume a lot. Of course, there are other logical adventures of
Frege that are not part of this - but they too are quite amenable to
Q's. In fact, if my memory serves me right, I posted some new
alternative axiomatizations of arithmetic apparently based on writings
of his concerning the notion of natural number - and proved key
properties of the axiomatization.
11. Now let's come to Goedel. Just how much philosophical discussion
is subsumed, obsoleted, or greatly clarified to the point of
obsolesence, by his Completeness Theorem, together with some prose?
And by his Incompleteness Theorems, together with some prose?
12. What about Goedel's theorem that every "logically simple"
statement provable in formal arithmetic is provable in formal
intuitionistic arithmetic, together with some prose?
13. And what about that great absolutely fundamental definition from
Tarski, of Interpretability? How much does that subsume, together with
some prose? And related offshoot notions such as synonymy, faithful
14. For those of you who are still quite skeptical about this Thesis,
it would be quite interesting to see the closest Thesis to mine that
you are willing to defend. Some of these substitutes I have seen on
the FOM seem very weak - too weak to be interesting.
15. Note that 15 is not in tune with the most common methodology in
academic philosophy: namely, criticize a Thesis without working
comparatively hard to put up a good replacement. Since the main goal
is intellectual progress, we should try to further intellectual
16. It should be clear that I have serious objections to various
aspects of the usual methodology of academic philosophers. To avoid
any misunderstanding, I have equally serious objections, of an
entirely different nature, to the usual methodology of academic
mathematicians, and to the usual methodology of academic computer
scientists. More about that later.
More information about the FOM