Re: Why does Cantor a target for cranks?
- From: "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx>
- Date: 10 Dec 2006 01:57:13 -0800
Proginoskes wrote:
Larry Hammick wrote:
On Dec 9, 9:32 pm, "Andrew Usher" <k_over_hb...@xxxxxxxxx> wrote:
Why are there so many on this groups whose mathematical goal is toWell, when Cantor himself was around, it wasn't just the cranks who had
disprove uncountability? I can't imagine, really, why it would be a
crank target.
their doubts about his stuff. Remember Kronecker and Poincaré, who
were far from cranks. Set theory, axioms of choice, cardinals, and all
that stuff, were hotly disputed for quite a few years.
A survey of this branch of crankology:
http://www.crank.net/cantor.html
Wilfrid Hodges's paper "An Editor Recalls Some Hopeless Papers" at this
website (
http://www.crank.net/cgi-bin/redirect.cgi?http://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps
) is especially telling; on page 3, he states: "It's nothing more than
a guess, but I do guess that the problem with Can-tor's argument is as
follows. This argument is often the first mathematical argument that
people meet in which the conclusion bears no relation to any-thing in
their practical experience or their visual imagination."
I'll add a minor detail: Nowadays many people work on computers and
software and they acquire a variety of bad habits of thought in the
process. One example (putting it informally) is the notion that if no
amount of RAM will ever be enough to distinguish one infinite ordinal
from another, then there must be something objectionable about
comparing infinite ordinals. That's just an example.
N. J. Wildberger's rant about Set Theory (I think it's "Set Theory:
Should You Believe?") also springs to mind. He states that we should
never consider big numbers, because we have no way to represent them in
the Universe. My reaction was the question of why we should abandon
something just because it's beyond the grasp of one particular
individual (Wildberger). The fact that that particular individual
should know better just reinforces my "beliefs".
--- Christopher Heckman
Cantor had a universe in his naive set theory.
Cantor's cardinals get a lot of examination, and for some,
disagreement, because they're part of a framework advanced as "the
foundation of mathematics."
For people with deep personal interests in mathematics, interest in the
foundations is often a given. That there are fundamental truths of a
sort, generally expressed as axiomatics, from which very rich
structures of symmetry and complement for in- and deduction logically
ensue, is a cognitive pleasure. The knowledge of how and why things
work in the mathematical realm (pure mathematics) and their use for
solving real world problems (applied mathematics) is for some a
practice in pursuit justified in its own right.
So, the questions (and answers) about what are the "foundations" of
mathematics, those things that are to be accepted as truths, are dear
issues to those with a mathematical conscience of a sort, and there are
people with deeply embedded, and not on the surface unreasonable,
beliefs on both, or of many, sides of the arguments about whether or
not transfinite cardinals are meaningful or whether the or not the
"real" real numbers biject to the real "natural" integers and so on.
Not all those who argue against transfinite cardinals and thus
diagonalization to incompleteness are necessarily cranks nor
mathematically illiterate nor haven't found errors in Hodges' paper
there.
As consequences of incompleteness imply no truth, I don't find the
collected rationalizations to that effect compelling. Consider the
notion of a theory of everything that is every true statement, where
many of those "axioms" would be redundant. That there is any existence
at all is a counterexample to incompleteness. Everything at once is.
Where that may only be so for some "non-standard" model of arithmetic,
ie, with some infinite in the natural integers, and it is so because of
existence, then the "non-standard" is the "true".
There is a universe, it would be its own powerset, reality shows that
the powerset result is seen to not hold.
That there are no known applications of transfinite cardinals
specifically and in a required way to explain anything in the real
world, would not by itself diminish them as purely mathematical
objects, but it doesn't validate them. Pretty much anything else in
mathematics has definite analogs in the real.
About the reals as "uncountable", with no least positive real for
example, consider the particle/wave duality in physics, and consider it
with regards to real numbers as somehow, because they are, dually
complete ordered field and contiguous sequence of points. In fact, if
infinite sets are equivalent as the universe and your and my existence
in it illustrates, one of the few ways to consider the arguments said
to imply uncountability of the reals to not apply is that there are
those points in and on the real line.
Does Zeno's arrow ever arrive? Zero is the additive identity.
Obviously, there is no universe in ZF. Where ZF in toto, the domain of
objects in the theory (besides the quantifier(s), element, and
identity) is the collection of all well-founded sets: that's very
similar to the Russell set, one of the stated reasons for regularity in
the first place. ZF never got away from Russell's "paradox", it's in
it, it is it. The existence of a universal quantifier seems to
contradict with the explicit invalidity of one, or in stronger terms it
does. Quantify, or don't. There is a universe, in Not ZF.
You shouldn't abandon everything because it's beyond the grasp of
Goedel.
Ross
.
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