[FOM] More Clarifications on Platonism and Formalism
Dmytro Taranovsky
dmytro at mit.edu
Tue Oct 21 23:44:18 EDT 2003
Some Clarifications on Platonism and Formalism.
>>If for all n, PA proves phi(n), then for all n phi(n).
>I do not understand these "all n".
This is the reflection principle for PA, and it suffices to prove
results like the Goodstein's theorem and the Paris-Harrington theorem.
It is a recursive set of axioms.
>How much "almost" [complete is PA]? Who can measure?
One can measure almost completeness by taking the ratio of mathematical
importance of theorems to the mathematical (excluding metamathematical)
importance of provably independent conjectures. For PA, the ratio is
large; I believe, it is over 100.
>But I also do not understand your argument [that mathematical theories
>are not like works of fiction] as a whole.
In writing fiction, one can liberally choose the story, characters, etc.
even once the main idea is specified. In developing the theory of
integers (and of real numbers), once one implicitly defines the notion
of an integer, further development is highly constrained, and
incompletenesses in formalizations appear to have unique natural
solutions. Such uniqueness is a major cause of the general acceptance
of absolute truth and falsity of arithmetical statements.
>I would advise you to read my paper "On feasible numbers"
>available via my homepage. This is an attempt to formalize
>what are (feasible) numbers which can be "physically realized
>as the number of "cobble-stones" in a particular region of
>space" taking into account some (astro)physical
>***real fact*** that our universe is bounded.
The notion of a feasible number is inherently vague because it means
different things in different contexts. For every particular context,
there exists the greatest feasible integer N; N is feasible but N+1 is
not. As such, feasible numbers are not subject to ordinary
formalization. Instead, one should deal with feasible numbers is
probabilistically: For every feasible number, there is a probability
that it will be the one to appear in a given practical application.
This way, one can measure the probability that a feasible number has a
feasible successor, and the probability that adding two feasible numbers
is feasible.
Vladimir Sazonov appears to object to the iterated modus ponen rule.
His beliefs seem to be along the line that one penny cannot make you
rich, but 10000000 pennies can make you rich, so the iterated modus
ponen rule is not valid. However, in ordinary discourse, one often says
things like "one penny cannot make you rich" when one means that with
probability 99.99999% or so it will not make you rich. Similarly, when
one says that every feasible integer has a feasible successor, one means
that with a very high probability the relevant feasible integer has a
feasible successor. Technically, one penny can make you rich; not every
feasible integer has a feasible successor, and the iterated modus ponen
rule is valid.
---------------------------
Both Professor Sazonov and I agree that we can reference mathematical
objects. However, there appears to be a fundamental metaphysical
disagreement between us. I continue to believe that, logically
speaking, "the mere reference to sets implies that they are something,
and hence exist". I explained the reasoning with
>However,"object exists" means "object is" which means "object is
>something", so "object does not exist" means "object is not" which
>means "object is not something" which means "object is nothing". If
>you refer to an object, you do not refer to nothing, so you claim or
>assume that the object exists.
Sazonov replied, "Sorry, I consider this rather as some linguistic
exercises." However, being a linguistic exercise, the reasoning is a
logical truth. The derivation of the Fermat's theorem from the axioms
of the Peano Arithmetic can also be considered as a linguistic exercise,
and as such is a logical implication.
Sazonov, however, believes (or gives the impression that he believes)
that the world is not logical; specifically, "The world
is not black-and-white as, it seems, you see it and should not obey the
logical laws you like. In particular, the application of the low [sic]
of excluded middle in the way as you do and in such a context is not
acceptable," "our real world even does not follow the classical logic".
I view logic as an absolute truth. For every two sets x and y, either x
is a member of y or it is not. The truth value of a disjunction is
determined by the truth value of the disjuncts, and the truth value of a
negation is determined by the truth value of the negatum. For a given
property, either a set that satisfies the property exists or it does
not. By induction, the first order language of set theory is not
vague. I hope that future mathematical results will back up the
conclusion with natural solutions to the outstanding set theoretical
independences.
-----------------------------
I fully understand Sazonov's views and I deeply appreciate his postings,
especially his lively counter-arguments to my claims. However, due to
space and time constraints, I cannot address at this time some of his
concerns, such as whether our a priori intuitions of absolute time are
disproved by the theory of relativity (Sazonov claimed in the
affirmative, and used the example to doubt our intuitions on natural
numbers).
Best Wishes,
Dmytro Taranovsky
P.S. I was surprised to see that no one disputed my claim that
projective determinacy is generally accepted as true and is
of fundamental importance in mathematics (see FOM Projective
Determinacy).
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