[FOM] CH and mathematics
Alex Blum
blumal at mail.biu.ac.il
Thu Jan 24 10:46:18 EST 2008
The expression "accepted by a consensus of mathematicians" or its
cognates have lately been stated a number of times. Joe Shipman stated
it in his recent discussion of 'definiteness'. I understand that it is
a state of art expression. What is unclear to me is the problem it is
addressed to. Is " accepted..." one with mathematical truth? It can't
be, for an inconsistency may lurk at the very foundations; a
possiblitity that crossed Keisler's mind. We then revise: the
mathematical proposition p is true iff p is both consistent and is
accepted by a consensus of the mathematical community. But It is
difficult to think that the truth of '2>1' has anything to do with what
anyone thinks. True, it may be difficult to sustain the belief if
objected to, but that may be true of any belief. Consesus, or common
intuition may of course indicate that there is a truth lurking which is
being intuited.
Regarding 'definite', if it means in effect 'provable from
immaculate assumptions' then being indefinite is a problem only if one
identifies mathematical truth with provability,etc. But why do that? If
it is a bad account, no account may be better.
Alex Blum
joeshipman at aol.com wrote:
>I should probably add to my remarks below that, according to my
>"necessary condition", the Continuum Hypothesis seems unlikely to be
>"definite". I am sympathetic to Woodin's view that "if CH is definite,
>then CH is false", but I haven't seen good arguments for CH's
>definiteness.
>
>My "sufficient condition" says that pi^0 statements are definite, and
>also that any provable statement is definite. Pi^1 statements have an
>asymmetry, in that false ones are refutable and hence definite. So, to
>find a number-theoretic candidate for indefiniteness, we should look at
>Pi^2. The twin primes conjecture and "P does not equal NP" are much
>more generally believed to be true than false. Can anyone suggest a
>natural example of a pi_2 statement of number theory for which there is
>no strong general opinion about its truth or falsity?
>
>-- JS
>
>
>*********************
>The following is sufficient for "A is definite":
>
>***
>One can effectively find a Turing machine T and prove that
>1) Either T halts with output "True" or T halts with output "False"
>2) If T halts with output "True", then A
>3) If T halts with output "False, then not-A
>***
>
>The following is necessary for "A is definite":
>
>***
>Mathematicians cannot permanently disagree on the truth-value of A in
>the sense that some will insist "A is true" and others will insist "A
>is false" -- they may disagree on whether it HAS a truth-value, and
>they may disagree on whether a particular truth-value has been
>established, but at most one of the two truth values {True, False} has
>the possibility of becoming permanently accepted by a consensus of
>mathematicians.
>***
>
>
>
>
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