[FOM] Hamkins's multiverse and ultrafinitism

[Timothy Y. Chow]

After reading a bit of the relevant sections of the paper by Fletcher et 
al. that Mikhail Katz cited, I have a few additional comments.

If I understand correctly, the frame of mind that we are being invited to 
adopt is that we don't have, and perhaps never have had, a particularly 
clear notion of what *the* natural numbers are.  Instead, because of 
independence phenomena (we are supposed to imagine that we have powerful 
tools available to construct nonstandard models of arithmetic almost at 
will, satisfying almost any properties that we want), we come to realize 
that "natural number system" is a vague term referring to any member of an 
infinite family of nonstandard models of arithmetic.  Unenlightened people 
of the past, when they thought they were talking about a specific member 
of this family, were really not singling out any particular member at all; 
rather, they were just narrowing down the field a little by picking a few 
properties that excluded some candidates but left many others in the 
running.  In particular, a "generic" choice of viable candidate would 
*not* be the minimal model.

I don't dispute that it's possible to adopt this frame of mind.  What I 
dispute is that the mathematical community at large will ever be tempted 
into adopting it, no matter what technical tools for constructing 
nonstandard models are developed.  There are a couple of reasons I say 
this.

1. We don't have to wait for those technical tools to materialize to 
imagine what it would be like to have them.  It would just mean that a 
book entitled "Models of Peano Arithmetic" would no longer be a specialist 
monograph; rather, it would be a standard graduate text like "Ring Theory" 
where some axioms are laid down and lots of examples and tools are 
studied.  On page 3, Example 1 would be the standard natural numbers. 
Everyone is still going to know exactly what Example 1 is and it will 
remain the the most important example.  For comparison, algebraic number 
theorists routinely study number fields and are interested in theorems 
that hold for arbitrary number fields as well as for various classes of 
number fields, but none of this causes anyone to think that "the 
rationals" is a vague term that might refer to any number field.  For 
another comparison, computational complexity theorists routinely construct 
oracles to separate or collapse various complexity classes, but this does 
not lead anybody to think that "P = NP" is a vague statement that doesn't 
have a privileged interpretation.  The empty oracle is privileged.

2. I said this before but it bears repeating: I cannot see any way of 
forming a clear grasp of a multiverse of nonstandard models of arithmetic 
without first forming a clear grasp of the standard model.  This is a key 
point where the analogy with set theory breaks down.  It is plausible to 
argue that we can form a clear concept of a countable model of ZFC without 
first forming a completely clear concept of the entirety of V.

3. I maintain that our confidence that we know what the standard natural 
numbers is does *not* stem from examining a set-theoretic proof that the 
second-order Peano axioms are categorical.  If it did, then I could see a 
devil's advocate trying to argue that the words in that set-theoretic 
proof could be interpreted as an argument in first-order set theory and 
hence not necessarily mean what we think it means.  A clear conception of 
the standard natural numbers must come first before we can even form a 
clear concept of, say, "well-formed formula."  No human being has ever 
arrived at a clear concept of "well-formed formula" at an earlier age than 
arriving at a clear concept of "natural number."  Theorems about the weak 
expressive power of first-order logic cannot lead us to confidently 
declare that "natural number" is vague, because if we take that conclusion 
seriously, then it automatically creates doubts about whether we really 
know what those theorems are asserting.  It doesn't make sense to saw off 
the branch one is sitting on.

4. Doubting that we have a clear concept of "natural number" can be done, 
but that takes us down the ultrafinitist route, not the multiverse route.

To summarize, I still see no reason to believe in the multiverse view at 
the level of the natural numbers.  It can of course be studied as a 
mathematical curiosity, but the analogy with set theory fails and I have 
trouble believing that it will ever catch on with the mathematical 
community at large.

Tim