Re: Cantor Confusion


Andy Smith wrote:
You are comfortable in the formalism and paradigm that you have been
taught,

I surmise you mean 'formalism' not necessarily to refer to the
philosophy of formalism but, I suspect, rather to refer to such formal
theories (and the notation that goes with them) as Z set theory.

And it is true that I am pretty comfortable with first order logic, set
theory, and mathematical logic, though I have a staggering amount to
learn just to catch up to a certain basic level. But it's not so much a
matter of my having been taught in any sense of having been
intellectually raised in a certain context from which I had no choice
but to receive it. Instead, it's more like I got intererested in the
subject and did, of course, first come across the dominant approach of
classical predicate logic and I did, very quickly, find it to make
great sense and to be, conceptually, extremely useful. Yet, in that
process I have always QUESTIONED and put to SCRUTINTY everything I've
learned, as well as I have done a lot of work myself, devising many of
my OWN formulations (within first order logic and Z set theory), of
many details that are usually not subjected to such scrutiny in the
textbooks. And meanwhile, I always have my ears open for ideas not in
the mainstream, including intuitionism, finitism, ultra-finitism,
non-standard logics, paraconsistent logic, as well as the many
philosophical approaches such as constructivism, realism,
structuralism, fictionalism, and even as farflung as certain mystical
views of logic and mathematics. (But that doesn't entail that I don't
also exercise my prerogative to skewer postings by cranks.) And though
I am not even close now to being well versed in all these, I do hope to
become much better informed as I continue my amateur studies. And
meanwhile I am perplexed by certain philosophical problems that I find
to come from classical mathematical logic and set theory.

but everything that you know rests on the shoulders of giants.
Given a clean slate, could you create infinite set theory and a systemic
formalism from the ground up? If not, you can cut me some slack.

Oh, but I am not AT ALL expecting you to devise your own formal
systems. I am just telling you some of what mathematics has to offer
and I would suggest also that you avail yourself of the well written
textbooks.

Here's a thought for you. Possibly you are so locked into your paradigm
that you cannot think of any thought not expressed in its terms.

Possibly, but I doubt it highly. Meanwhile, plain old mathematics does
offer an approach that cuts right through so much confusion and
vagueness such as is found in the homespun mathematical ruminations of
people who have not availed themselves of the basics of a mathematical
education. Thus, I believe it is very constructive to offer formal
mathematical definitions as an alternative to floundering in
picture-word mathematics.

So
maybe the only way in which you get something new is when some
neanderthal like me blunders in and asks some stupid questions.

I haven't found your questions to be stupid. But I do think your
questions would benefit by being put in the light of mathematical
definitions and some familiarity with certain basics of the subject.

Of
course intelligence comes into it and I cheerfully accept that the
probability of me making any useful contribution has a lot of 0's
following the decimal point.

I hold even less hope for myself in that regard. Mathematics is among
the very least of my talents and I am really not good at it when it
comes to coming up with clever proofs and things like that (just check
it out when I post requests for help with proofs that are not, for a
talented mind, difficult at all).

I only discovered usenet a week or so ago, but the sci.math forum is
something of a revelation. Snakes on a plane! You have sects, schisms
and heretics, massive egos ... no quiet, polite, thoughtful exchanges
coming to a common consensus, but ad hominem the norm. All very amusing
to an idealist.

It is indeed a real trip! And, you're right that it is not at all an
oasis of intellectual colloquy or bon homie. On the other hand, there
are some very knowledgable and generous people from whom one can learn
a lot. It is a beautiful thing that, just by typing, I can get help
with problems and questions, for no charge, from experts around the
world. And it's my impression that if one has questions, then one will
be treated with an indifferent (usually not even what could be
described as 'cordiality') but still a fair working respect if one has
at least first done the prior leg work of thoughtfully reading a basic
textbook. On the other hand, what often meets, as is to the good,
sarcasm, ridicule, and derision are the postings of cranks - especially
the most strident, self-righteous, and self-satisfied ones - who are
virtually ignorant of the subject of discussion and who won't give even
a hint as to the norms of whatever special personal logic they use to
reach their errant conclusions.

MoeBlee

.

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