[FOM] methodological thesis

Vaughan Pratt pratt at cs.stanford.edu
Wed Apr 30 12:37:47 EDT 2008

Claims of the form "paper Q in area B has subsumed paper P in area A" 
should be taken with a large grain of salt.

First, good papers are rarely written in a vacuum, and even when they 
are they cannot be readily communicated when the audience is listening 
in a vacuum.  Bill Thurston's wonderful Bull. AMS essay "Proof and 
progress in mathematics" at 
http://arxiv.org/PS_cache/math/pdf/9404/9404236v1.pdf (1994) , 
republished in Reuben Hersh's recent anthology of essays, underlines the 
point that even mathematics is very much a social process in several 
respects including the following.

> We held an AMS summer workshop at Bowdoin in 1980, where many mathematicans
> in the subfields of low-dimensional topology, dynamical systems and Kleinian
> groups came.
> It was an interesting experience exchanging cultures. It became dramatically
> clear how much proofs depend on the audience. We prove things in a social context
> and address them to a certain audience. Parts of this proof I could communicate in
> two minutes to the topologists, but the analysts would need an hour lecture before
> they would begin to understand it. Similarly, there were some things that could be
> said in two minutes to the analysts that would take an hour before the topologists
> would begin to get it. And there were many other parts of the proof which should
> take two minutes in the abstract, but that none of the audience at the time had
> the mental infrastructure to get in less than an hour.

If such a large gap can exist between analysis and topology, imagine how 
much larger it must be between philosophy and mathematics!

Particularly impressive is that Thurston was able to communicate certain 
material in two minutes to topologists, and certain other material in 
two minutes to analysts.  I take this as an object lesson, that anyone 
claiming paper Q in area B subsumes paper P in area A should first check 
that he can at least convey the gist of P to experts in area A in two 
minutes and obtain their agreement that he has understood P.  When you 
rely solely on the experts in area B to judge the quality of paper Q, 
the bit about Q subsuming P suggests that "what we have here is a 
failure to communicate."

Second, the presumption of subsumption is always optimistic when it 
spans such disparate cultures.  A Frenchman might take offense at "Tale 
of Two Cities" as expressing too one-sided a view and rewrite it from 
the French perspective.  The two versions may be factually consistent, 
but only the French will consider their version to have not only 
subsumed but improved on the original.  The English may well complain 
that this translation misrepresents the English perception of the two 
cities.  The French retort that the English have no taste and worse 
judgment in such matters, and now a slinging match ensues.

What the two parties should be able to agree to is that the respective 
works represent the best efforts of both sides to express the situation 
from their respective vantage points.  To call this process 
"subsumption" introduces a disrespectful and uncalled-for asymmetry.

Set theory and category theory as competing foundations for mathematics 
have seen this sort of warfare, waged more strenuously on FOM in the 
late 1990's than perhaps at any other place and time, with at least one 
person denying for a long time the right of categorical foundations to 
even exist, much as Hamas and Fatah have denied until now the right of 
Israel to exist.

An area can afford to place a premium on precision of thought within 
that area.  Between areas however precision is less important than 
respect for the differences in outlook.

Third, there is the chicken-and-egg question of which comes first, area 
A or area B.  I have seen this argued both ways in computer science. 
Stanford president John Hennessy took the position when he was chair of 
the CS department that computer architects innovate and the theorists 
comes along afterwards to organize systems thinking more coherently 
("clean up after them" as he put it).  The opposite view is that the 
architects cannot innovate in a conceptual vacuum and therefore depend 
on theory to provide the concepts that they then turn into working products.

The truth of the matter is that computer science works like a hen house: 
every hen came from an egg and vice versa.  This may not hold as 
strongly for other pairings of areas, but it is at least worth being 
reasonably calibrated on the extent to which it does.

Vaughan Pratt

Harvey Friedman wrote:
> I would like to discuss a methodological issue related to philosophy.
> THESIS. Suppose that a philosophical paper P, in any part of  
> philosophy, consisting of informal prose, without new formalisms or  
> new theorems or new formal conjectures, represents intellectual  
> progress. Then there exists a paper Q with the following properties.
> 1. Q focuses on associated new formal definitions, new formalisms, new  
> formal conjectures, and new theorems.
> 2. Q has a relatively small amount of informal prose.
> 3. Q can be written using the current level of practice in formal  
> methods and foundational thinking.
> 4. P is fully subsumed by Q.
> [...] philosophical progress of any real kind is  
> always followed, or is realistically follow-able, by formal, or  
> formally systemized, progress.
> Most philosophers believe (or would believe if they looked into it)  
> that what I do is not philosophy. On the contrary, what I do is  
> appropriately viewed as philosophy of type Q, that subsumes any type P  
> philosophy that does or could have preceded it.
> In other words, I omit writing the P papers, keeping their essence in  
> my head, to be used to create Q papers. Only the Q papers are then  
> published.
> Proposed COUNTEREXAMPLES to this thesis would be greatly appreciated.  
> The challenge to me would be to subsume the proposed P paper into the  
> subsuming Q paper.
> Harvey Friedman
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