FOM: Wolfram and logic/foundations of mathematics

[Henry Cohn]

>    A friend of mine has just informed me that some persons believe Stephen
>Wolfram's new book contains "exciting new material on logic and
>mathematics."   On the basis of a rudimentary internet search, I have not

It does contain some discussion of logic and the foundations of mathematics.
However, it didn't seem to me to contain many original ideas in these areas.
There was a lot of rehashing of standard ideas (I wasn't sure to what extent
Wolfram was aware they were standard), together with some wild speculation.
For example, he suggests that there may be a universal Diophatine equation
of degree 4 in two variables.

I wrote a review, which you can read at
http://research.microsoft.com/~cohn/wolfram.html.
It mentions one of the really thought-provoking things Wolfram brings
up.  He conjectures that (roughly) almost all systems are either
computationally universal or obviously not computationally universal
(because they do something trivial).  This seems like something that
could be investigated mathematically: is it true that, say, in a random
cellular automaton, if we are look at the problem of predicting whether
a certain pattern will occur, and condition on automata for which this isn't
computable, will the probability of universal computation go to 1 as the
size of the automaton rule goes to infinity?

Henry Cohn