FOM: Transfinite Logic

[William Calhoun]

This is ALMOST completely off the topic, but I couldn't resist pointing
out how difficult it is to know where ideas originate.

On Wed, 12 Jun 2002, wiman lucas raymond wrote [selected quotes
only]:

>  In the end, to quote the rock group Guns 'n Roses:
> "what we have here is a failure to communicate."
>

This line appeared in the movie "Cool Hand Luke" years before Guns 'n
Roses.  The warden says it to Luke (Paul Newman) when he refuses to follow
prison rules after repeated punishment.

To bring this back to mathematics, consider the question Heck and Buckner
raised of whether Cantor's ideas represent a "great conceptual leap."
(This question is more interesting to me that all the discussion of
"mysteries" arising from the fact that a thing can have more than one
name.)  Raymond wrote:

> Yes, it's not a great conceptual leap now.  I don't know enough about
> history to posit whether it was then or not.  However, it's a conceptual
> leap forward in the sense that it distills a great deal about what has
> shown to be mathematically important about infinite sets.

I'm not a historian either, but I know that some of Cantor's ideas had
appeared before.  For instance Galileo discussed the idea that there are
as many perfect squares as positive integers since they can be put in a
one-to-one correspondence.  However Galileo concluded that "the attributes
'equal,' 'greater,' and 'less' are not applicable to infinite, but only
finite quantities."

Also, a version of "Cantor's diagonal method" was used by Paul du
Bois-Reymond (circa 1875) to show that any sequence of functions can be
asymptotically dominated by another function.  In 1874, Cantor showed that
the reals are uncountable using a nested intervals arguement.  His more
familiar diagonal argument using decimal representations of reals appeared
in 1891.  I don't know any evidence that Cantor knew du Bois-Reymond's
work, but at least one author (Craig Smorynski) gives du Bois-Reymond
credit for inventing the diagonal argument.

I think it is very difficult to say one person is completely responsible
for an idea.  Cantor was certainly infulenced by previous discussions of
infinity.  But Cantor was the first mathematician to make the bold move of
using one-to-one correspondence to define equal cardinality for infinite
sets and then work out the theory of infinite cardinalities in detail.
By doing so, he changed mathematics forever.  In my book, that is a "great
conceptual leap!"

-Bill Calhoun
-Math, CS, and Stats 		wcalhoun at planetx.bloomu.edu
-Bloomsburg University		Telephone: 570-389-4507
-Bloomsburg, PA 17815		FAX: 570-389-3599

(Research Interest: Computability Theory)