[FOM] methodology

Harvey Friedman 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  
little prose?

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  
interpretability, etc.?

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.

Harvey Friedman

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