Re: Cantorian pseudomathematics

david petry said:

briggs@xxxxxxxxxxxxxxxxx wrote:

Whenever scientists make a measurement, they take into
consideration the uncertainty associated with the measurement. What I'm
suggesting is that we could start with that idea to build up a notion
of "finite precision real numbers" (i.e. the kind of real numbers that
scientists actually use when they make measurements), and then take the
mathematicians' infinite precision reals to be the limiting case of the
finite precision reals as the uncertainty goes to zero.

The number 3.14 has no uncertainty specified. One way to specify an
uncertainty would be to write 3.14(3) which means that there is an
uncertainty of 3 in the last decimal place. As a start, you could
think of that '3' as the standard deviation of a normal distribution
with mean value 3.14.

But that's just a crutch for thinking. No real measurement is going
to have an error distribution that is either normal or exactly
quantified.

But logic itself is "just a crutch for thinking". I do accept the
Bayesian view of probability, and I am especially fond of the approach
taken by Jaynes.


If one goes to the limiting case it turns out that the shape of the
error distribution or its exact size are equally irrelevant. And
one can approach the same limiting case by considering Cauchy
sequences of rationals, so there is little value in considering
finite precision measurements as being foundational.

I really think that it would be more intellectually honest of you to
say "I don't see the value ..." rather than to say "there is little
value ...".

The claim I made that started this thread is that mathematics may be
viewed as the science that studies the phenomena observable in the
world of computation, and my claim was that there is great value in
looking at things this way. From this point of view, it makes a great
deal of sense to develop a theory finite precision real numbers first,
as they are things that can be observed, and then consider the limit as
the uncertainty goes to zero. At the same time, it makes sense to
consider probabilistic logic as foundational, and then Boolean logic is
the limiting case as the uncertainty goes to zero. I argue that the
"foundations" of mathematics should include a study of the connections
between human intuitions about mathematics and mathematical formalisms.



I think that's very good point. Considering exact quantities without error is a
bit like considering 2-valued Boolean logic values, which some consider to be
just conveniently labeled 0 and 1 for false and true, but which really cover
the entire range from 0 to 1, as a range of probabilities. I can see David's
point that, when dealing with symbolic representations of quantity rather than
raw quantities, it makes sense to build up the symbolic strings in the language
from null and finite strings, and consider the possibilities for representation
as the limit of the string length approaches oo. Given this approach, any
number which is not finitely representable in the symbolic language is
successively approximated as additional digits are added, and the difference
between the actual real value and that represented by the finite string can be
viewed as a margin of error due to the finiteness of representation. If one is
considering actual infinite languages, the number of infinitely long strings
will far outnumber the number of finite strings, and so this idea of a
monotonically decreasing margin of error as string length increases makes sense
for almost all real numbers that one could speak of in any given language with
a finite alphabet. So, I don't consider Petry's point at all vacuous. It's the
kind of scientific/experimental approach to mathematics that has been gaining
popularity over the last couple decades, since computers became commonplace.
--
Smiles,

Tony
.

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