FOM: Maddy on method

[Patrick Peccatte]

Traditional fom member digest:
Name: Patrick Peccatte
Position: Software developper and philosopher
Institution: Independant private company and Paris 7 University
Research interest: Philosophy of sciences and of maths (specially
quasi-empiricism, experimental maths, ontology)
More information: http://peccatte.rever.fr
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Penelope Maddy wrote:
>One can think that all mathematical objects are ultimately (modelled as)
>sets without thinking that all effective mathematical methods are set
>theoretic methods. Compare: one can think that everything studied in the
>sciences is ultimately physical without thinking that the only effective
>scientific methods are those of physics. Who would expect botanists or
>psychologists to use the same methods as physicists?

Another analogy:
mathematics/biology or psychology/end-user computer applications [resp.]
are modelled (or based upon) set theory/physical objects/computer
assembly languages (or more precisely binary machine code) [resp.].
It is possible in principle for humans to write end-user complex
programs directly in binary machine code - exactly as specific theorems
in algebra are provable in principle from the axioms of set theory. But
*in the real world*, most of the more useful programs - from a human
point of view, of course - are now written using high level languages.
Their reductibility to machine code is not performed by humans; even if
this task is possible in principle, this is almost impossible in fact.
Programmers are using high level languages implementing a lot of
features that don't exist in machine code (from 'loops' to
'heritability'). The *supervenience* mentionned by Neil Tennant is in
this case the intelligence embedded in high level languages and
compilers, and this is very important to note that it is a human
product.
I understand the Maddy's sentence: "That's what I mean when I deny that
the methods of algebra are the methods of set theory" (Fri, 20 Feb 1998
16:07:52) as "I deny we can write any sufficiently complex program
directly in machine code and from scratch, even if it is possible
theoretically". To pursuie the paraphrase, we found a very similar human
impossibility to prove an algebraic theorem without thinking
'algebraically', even if it is possible theoretically thinking
exclusively 'sets'. Large and complex programs *supervene* upon low
level machine code as all mathematical objects are modelled in the
low-level universe of sets.
We can even put the comparison forward in that sense:
we can write the same end-user program (fonctionnaly) on different
machines or operating systems. Maybe it's possible to compare set theory
with a standard and largely accepted {hardware + operating system +
primitive libraries} as a whole, and the search for foundations of
mathematics could be compared with the research in computer
architectures, electronics, language theory, compilers technology, etc ?
And a step forward, the largely accepted {hardware + operating system +
primitive libraries} (Unix, Mac, Windows, for ex), are similar to the
varieties of logics (aristotelian, intuitionnistic, etc.) or ontologies
(sets, categories, others?) currently in focus in fom debates.
I prefer this analogy to the biology/psychology's analogy because the
*supervenience* evoked is a human and well-known product and is more
understandable than the evolution product in biology/psychology.
Regards
-- 
Patrick Peccatte
peccatte at club-internet.fr