On Sun, 22 May 2005, Hal Finney wrote:
Regarding the nature of Tegmark's mathematical objects, I found some
old discussion on the list, a debate between me and Russell Standish,
in which Russell argued that Tegmark's objects should be understood as
formal systems, while I claimed that they should be seen more as pure
Platonic objects which can only be approximated via axiomatization.
I see that Russell is right, and that Tegmark does identify mathematical
structures with formal systems. His chart at the first link above shows
"Formal Systems" as the foundation for all mathematical structures.
And the discussion in his paper is entirely in terms of formal systems
and their properties. He does not seem to consider the implications if
any of Godel's theorem.
Actually he does both. Most of the time he implies that universes are
formal systems, e.g. formal systems can be self-consistent but objects
just exist: only statements about objects (e.g. theorems) have to be
consistent. But he also says that *our* universe corresponds to the
solution of a set of equations, i.e. a mathematical object in Platonia.
This is definitely muddled thinking.
Specifying a universe by a set of statements about it seems to be a highly
redundant way to go about things, whether you go for all theorems or the
much larger class of true statements. (Of course, many statements,
including nearly all the ones interesting to mathematicians, apply to
whole classes of objects). Hence despite all the guff about formal
systems, I think any attempt to make sense of Tegmark's thesis has to
go with the object = universe.