Chris and Chris, (01)
CM:
> Barry's criticisms of the use of a non-well-founded set theory like Aczel's
> AFA are on the money. He notes that it is a greatly overpowered for the needs
> of the document; it entails, among other things, the entire massively infinite
> hierarchy of transfinite numbers... (02)
CP:
> Not every logician agrees with Chris (which I am sure is no surprise to
> anyone - including Chris). (03)
On this point, I sympathize with CM. (04)
There are many mathematical systems that I find interesting in the same
way that I enjoy chess, go, and other games. But I tend to be skeptical
about their value until I see practical applications. (05)
I'm even agnostic about Cantor's hierarchy of infinities. When I first
came across them, I shared Hilbert's amazement about the "paradise"
that Cantor discovered. But I'm am now inclined toward Wittgenstein's
view of them as a "swamp". (06)
The primary reason for my skepticism about the infinities is that
the only entry point to that huge body of theory is just a single,
non-constructive method of proof: the diagonalization method. (07)
In most branches of mathematics, the central theorems can be proved
in many different ways. Many different mathematicians starting from
very different axioms with very different methods often develop
isomorphic systems. (08)
As another point of view, Marvin Minsky is fond of saying that
you don't really understand a problem if you only know one way
of solving it. I have come to agree with that point. (09)
But the diagonalization method is a non-constructive proof by
contradiction, which many mathematicians treat with caution.
When you find a contradiction, the only thing it proves is that
somewhere among all the steps and axioms, there is a contradiction.
It doesn't directly identify which step or axiom is at fault. (010)
Finally, I am constantly amazed by the creativity of children
from the age of 3 to 5. They put all of our most sophisticated
AI systems of learning, data mining, etc., to shame. (011)
But note that those misguided mathematicians who tried to teach
set theory to kindergarten kids failed miserably. Instead of
promoting more creativity and faster learning, it just confused
them. It was definitely misguided. (012)
John (013)
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