[FOM] Tangential Epistemological Comment

[A.P. Hazen]

    Harvey Friedman, writing about a hypothetical alien artifact that 
purports to be a hypercomputer, says:
    "...experiments are done on the device as a "black box" with absolutely
	 no understanding whatsoever of its internal mechanism.
    "I claim that we would then obtain a "proof" that Goldbach conjecture
	 is false, that is not as convincing as one would hope, but
   	 substantially more convincing than any knowledge that we presently
	 have in the sciences and engineering outside mathematics."
I would like to applaud the scare-quotes he has put around "proof" 
here.  If I were writing very carefully, and not in an informal 
symposium like FoM, I would prefer to put his thought by saying that 
the experiments would provide EVIDENCE, rather than proof (and go on 
to say that this evidence was more convincing than blah blah, and 
even that it justified a higher degree of confidence than most 
published proofs!).
    I think we have (as part of the conceptual resources provided by 
what philosophers call "ordinary language") a variety of ways of 
classifying and evaluating kinds of evidence (in the broadest sense 
of the word: grounds for rationally changing our degree of belief in 
various propositions) and that it is ***NOT*** the case that they fit 
into a single hierarchy of more-to-less convincing (or more-to-less 
conclusive). PROOF is a special kind of evidence that is available in 
mathematics.  Being presented with a proof doesn't automatically 
justify complete, 100%, confidence in the truth of the claimed 
theorem: authors, and editors and referees, make mistakes.  Sometimes 
a non-proof may be more convincing (and RATIONALLY more convincing) 
than a proof.
    Compare legal proceedings.  I can readily imagine cases in which 
hearsay evidence would raise my degree of subjective confidence in 
the guilt of the defendant more than eyewitness testimony, and I 
think rationally.  (I have good grounds to believe in the reliability 
and probity of the person reporting the overheard conversation, I 
know the eyewitness's vision is no better than mine ....) 
Nonetheless, the eyewitness testimony is, and the hearsay is not, 
"admissible in evidence."
     (((Apologies for going off on a philosophical tangent not 
directly f.o.m.-relevant.  I'll try to be good in the future.)))
---
Allen Hazen
Philosophy Department
University of Melbourne