FOM: surreal numbers; Conway's foundational ideas

[Stephen G Simpson]

Randall Holmes 26 May 1999 11:13:04

 > On p. 43, Conway says "as an abstract Field, No is the unique
 > universally embedding totally ordered Field",

Ah, thanks.  I had overlooked that.  That ties in nicely with the last
corollary in my posting of 21 May 1999 19:59:44.

 > which strongly suggests that he knows the model theory ...

I don't glean from this that Conway knew the model theory.  After all,
he didn't use the model-theoretic terminology or cite the
model-theoretic literature.  It seems more likely that he simply
rediscovered this special case of an argument which was well known in
the model-theoretic literature.

 > Conway is well aware that the surreal numbers are not a new
 > structure -- in the sense of not being a new field.

Why do you assume Conway knew that the field of surreal numbers was
not new?  He didn't cite any of the earlier literature, in which this
field was constructed.

 > However, the surreal numbers have a good deal more structure than
 > that.  The construction {A | B} of a surreal number from sets A and B  
 > of surreal numbers is not definable from field operations.  ...

I agree.  My only point was that the surreal numbers qua ordered field
were already known from the model-theoretic literature.

 > Simpson has been very firm himself about the importance of the
 > details of the definitions of mathematical structures (dare we
 > recall the discussion of Boolean rings?).

Well, if you want to bring that up, I have to point out that Boolean
rings are *not* isomorphic to Boolean algebras, but the surreal field
No(kappa) *is* isomorphic to the saturated real closed field of
cardinality 2^kappa, assuming GCH at kappa.

 > On p. 44, we find ...  "So we can say in fact the Field No is
 > really irrelevant to nonstandard analysis".

Ah, thanks, I missed that.  So Conway *was* aware of nonstandard
analysis, even if he didn't know the model-theoretic literature on
saturated models, etc.

In any case, this whole question of who knew what when is not very
important.  What's interesting is the tie-in between Conway's
construction and the model-theoretic concept.

 > I find the whole tone of Simpson's remarks about this book simply
 > astonishing. ...

What bothers me about Conway's book is the foundational remarks.  I
explained my concerns in my posting of 24 May 1999 19:25:58.

Randall, what do *you* think of Conway's foundational ideas?

Specifically, what do you think of Conway's idea that formalization is
irrelevant, ``even for foundational studies'' (page 66)?  Do you take
this idea seriously?  Some people apparently don't take it seriously,
e.g. Martin Davis 24 May 1999 20:53:44.  I *do* take it seriously, for
reasons explained in my posting.

-- Steve