Re: Cantorian pseudomathematics

david petry wrote:
> MoeBlee wrote:
> > david petry wrote:
> > > A statement has observable content if it makes predictions about the
> > > results of a computational experiment.
> >
> > Same as last post:
> >
> > Suppose I give you two natural numbers such that the second natural
> > number is the Godel number of the string of symbols that mathmaticians
> > point to as the theorem that the set of real numbers is uncountable.
> > You put both natural numbers into your computer and let your computer
> > compute whether the first natural number is the Godel number of a proof
> > of the formula whose Godel number is the second natural number.
> >
> > If a mathematician claims to have proven a theorem from the axioms of
> > set theory, then the claim can be decided (either confirmed or
> > falsified) by sticking the numbers into your computer and letting it
> > compute.
> >
> > If a mathematician makes the statement, "It is a theorem of set theory
> > that the set of real numbers is uncountable, and the following is a
> > proof: [fill in purported proof]" then just run the Godel numbers and
> > you have your computational experiment. You'll always get an answer,
> > yes or no.
>
> I'm claiming that an assertion is meaningful if it makes a prediction
> about the results of a computational experiment. So it's entirely
> possible that the assertion "A is a theorem of ZFC" is meaningful,
> while 'A' itself is not. That is, 'A' could be just a syntactically
> correct string of meaningless symbols.

You give me a formula that you consider to be meaningful, say "1+1=2".
I ask, what makes this formula meaningful? You say the formula is
meaningful because it predicts the results of a computational
experiment. The computer confirms that 1+1=2.

I give you a mathematical formula that I consider to be meaningful, say
"The Godel number of sequence S is the Godel number of a proof of the
uncountability of reals". You ask, what makes this formula meaningful?
I say the formula is meaningful because it predicts the results of a
computational experiment. The computer confirms that the Godel number
of sequence S is the Godel number of a proof of the uncountability of
the reals.

Then you say that the statement 'The reals are not countable' is not
meaningful on its own.

And I say I have more than one available response: (1) Even IF the
statement 'The reals are not countable' is not meaningful on its own, I
did give you a statement about its Godel number that is meaningful by
your own definition of 'meaningful', and I don't need to assert any
meaning beyond that, since my computational experiement is just as
"computational" as yours; (2) Meaning is given by structures for a
language. The statement 'The reals are not countable' can be examined
for truth in some structures and falsehood in others. That the inquiry
is not computational does not make it a meaningless inquiry, especially
since we CAN, again, COMPUTE proofs of truth or falsehood in
structures, and especially since truth and falsehood in arithmetic is
not computable anyway. (3) If all formulas must predict the results of
a computational experiment, then, for example, what computational
experiment and what results are predicted by the formula 'pi is
irrational'?

MoeBlee

.

Relevant Pages

  • Re: Cantorian pseudomathematics
    ... >> A statement has observable content if it makes predictions about the ... >> results of a computational experiment. ... > compute whether the first natural number is the Godel number of a proof ... > If a mathematician makes the statement, "It is a theorem of set theory ...
    (sci.math)
  • Re: this has no proof, therefore formal compilers dont work
    ... TOTAL BULLSHIT mathematics today ... an extra number to the list of reals, ... any quotes about Godel or Cantor is not understood by I-Jerc, since his vocabulary is barely bigger than 'BS'. ...
    (sci.math)
  • Re: Computable functions/reals.
    ... knowing that the natural numbers are recursive. ... computable subset of N in this sense. ... subset of the reals - there is no machine ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)
  • Re: Computable functions/reasls: followup.
    ... equivalent to what various "idiots" suggested the definition ... (we need to assume that the uniform continuity ... with reals - in any case it's curious you don't ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)
  • Re: Simple vector space question-
    ... spaces is built ON TOP of the reals. ... David is following the standard definitions. ... of what the facts are - maybe I have the facts wrong. ... "Understanding Godel isn't about following his formal proof. ...
    (sci.math)
Loading