Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

tchow_at_lsa.umich.edu
Date: 01/12/05 

Date: 12 Jan 2005 22:31:15 GMT

In article <cs3b0n$kmq$1@ra.nrl.navy.mil>,
Ralph Hartley <hartley@aic.nrl.navy.mil> wrote:
>No. Perhaps "effective" is not exactly the word I intended to use? I
>intended to include *partial* recursive functions. Does "effective" have
>such a precice definition that it excludes them?

O.K., fair enough. I wasn't sure to what extent you were defending the
earlier posters in this thread, who seemed to think that the halting problem
already shows that there are permanently unknowable truths---I took them to
regard partial recursive functions as being not good enough to yield
knowledge. So forget that example.

But in any case, the real issue again is your assumption that
mathematicians use an effective procedure to determine mathematical truth.
As I've argued elsewhere in this thread, CT theses of whatever flavor
don't yield this assumption; CT theses will only say that if mathematicians
are indeed using an effective procedure to find, or are *computing*,
mathematical truths, then the theory of [partial] recursive functions
applies and lets us deduce further conclusions. But it doesn't say
whether mathematicians are indeed using an effective procedure. Even
if the universe is computable in some sense, it doesn't follow that
everything that takes place inside the universe is an effective procedure.
For example, I know of nobody---atheist or believer---who thinks that
theological doctrines are generated by an effective procedure.

You've argued that if something is obtained by a procedure that is not
effective, then it's not *knowledge*. Perhaps that's true, but the
the corpus of what is *generally called* "mathematical knowledge" is
not generated by an effective procedure in any obvious way. "Every
vector space has a basis" is generally considered to be a mathematical
truth. It can be proved using various axioms, including the axiom of
choice. But how did we come to accept the axiom of choice? There were
heated arguments, and through a complex sociological process, the axiom
of choice won out. Was this an effective procedure? Was it even objective?
What's to stop some similar mess from happening again---say, with the axiom
of projective determinacy, which Woodin and others have been advocating?
If these things aren't knowledge, or aren't mathematical truths, then
your assumption comes at the cost of discarding a lot of what most people
consider to be mathematical truth. 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

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