[FOM] What is a proof?
Jesse Alama
alama at stanford.edu
Sun Jan 25 20:46:31 EST 2009
Alasdair Urquhart <urquhart-26n5VD7DAF2Tm46uYYfjYg at public.gmane.org> writes:
> There is an article by the computer scientists
> De Millo, Lipton and Perlis that I think
> is very apposite to this thread. The authors
> argue eloquently against ease of formal verification
> being a criterion of language design. The article
> is called "Social Processes and Proofs of Theorems
> and Programs." It is easily available online --
> just do a search for the title.
>
> The authors sum up their main thesis as follows:
>
> We believe that, in the end, it is
> a social process that determines whether mathematicians
> feel confident about a theorem--and we believe that,
> because no comparable social process can take place
> among program verifiers, program verification is bound
> to fail.
>
> I recommend the article highly.
I also highly recommend that article to get a feel for what was going on
in the debate about program verification in the 70's. For a thorough
discussion of that debate, I also recommend Donald MacKenzie's
_Mechanizing Proof: Computing, Risk, and Trust_ (2004).
I think it's natural to hold that one *feels confident* in a theorem
because the community of mathematicians judges it positively. But that
strikes me as not so interesting from a philosophical perspective; of
course one trusts a mathematical result, especially a complicated one,
if other competent mathematicians also trust it. I suppose this feeling
is based on our recognition of both human fallibility and the degree of
painstaking detail required to get non-trivial mathematics right.
A related, quite different question is whether the positive judgment of
the community of mathematicians *constitutes* the correctness of a
proof. That is a question not about feelings, but about the
epistemology or metaphysics of proof. (This is not a critique of the
Lipton, de Millo, and Perlis article. I am mentioning this only because
I think their article naturally suggests the topic.)
I recommend the entry on social epistemology in the Stanford
Encyclopedia of Philosophy,
http://plato.stanford.edu/entries/epistemology-social/ ,
as an valuable first source for discussion. The entry isn't about the
social character of mathematical knowledge in particular, but it is a
valuable introduction to the wider epistemological issue that underlies
our discussion. One might also consult the entry on the social
dimensions of scientific knowledge
http://plato.stanford.edu/entries/scientific-knowledge-social/
to learn more about these issues in a wider scientific context.
Jesse
--
Jesse Alama (alama at stanford.edu)
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