[FOM] Are proofs in mathematics based on sufficient evidence?

[Vaughan Pratt]

On 7/21/2010 8:12 AM, Charles Silver wrote:
>   Tangentially, Richard L. Epstein has a critical thinking book,
> workbook, teacher's edition, CD (I think), in which, after establishing
> examples of deductively valid arguments in English, then presents a
> myriad of examples of arguments that are not deductively valid (like
> those we encounter daily), but are evaluated in terms of their
> strengths. Some may be (from my weak memory) extremely strong: very,
> very strong; very strong; strong; not very strong; etc., etc. One can
> consider the arguments at the top of the strength list to be
> "inductively valid," or close to it--whatever that means. Since these
> are everyday arguments, the non-deductively valid ones that are
> super-strong are very valuable outside of the realm of pure logic. He
> provides answers to exercises, which he invites the reader to disagree
> with. I regard some of the examples given on this thread to appear among
> his close-to-deductively-valid arguments. (This is not an ordinary
> "critical thinking" book. For one thing, Dick is a recursion theorist
> and has also written a book on computability.)

Wikipedia has a separate article on the topic of Argument.  The topic of 
"Proof (truth)" is not quite the same thing.

The critical difference as I see it is that even a very strong argument 
may nevertheless not be sufficient to establish the truth of a proposition.

Statistical considerations provide an extremely strong argument for 
Goldbach's conjecture.  This fact earned me a free lunch in 1968 in 
return for explaining to an amateur mathematician friend of our 
department chair (over that lunch) why that tremendously compelling 
argument, for which he was claiming credit, was regrettably still not 
enough to prove the conjecture.

My favorite way of making the distinction is as follows.  Consider these 
two scenarios.

1.  Alice has an argument for P and an argument for not-P.

2.  Paul has a proof of P and a proof of not-P.

Do you find anything inconsistent about either scenario?  Are they both 
equally consistent?

I did not at any time point this out on Wikipedia because it would have 
created more problems than it solves.  The subscribers to FOM are in a 
better position to appreciate it than the Wikipedia editing community, 
to which every political persuasion under the sun belongs.  Never argue 
back on Wikipedia, you'll get back every conceivable interpretation of 
your argument, just get your ducks (sources) in a row.

Incidentally "Proof (truth)" could become "Proof" without the "(truth)" 
bit if enough people would weigh in on that question at the discussion 
page of "Proof," which is currently a mere dab or disambiguation page.

Vaughan Pratt