[FOM] Hamkins's multiverse and ultrafinitism

[Timothy Y. Chow]

On Wed, 29 Nov 2017, John Baldwin wrote:
> I am indebted to Roman Kossak for the following insight. It is a big 
> leap from thinking we have a clear grasp of the set of natural numbers 
> with the successor operation tothinking we have a clear grasp of the 
> natural numbers with successor, addition and multiplication. It seems to 
> me that any illusion that we have such a clear grasp stems (contra Tim's 
> point 3) precisely categoricity of 2nd order Peano arithmetic.

I don't dispute that addition and multiplication introduces extra 
conceptual complexity.  What I don't see is how this provides any support 
for the multiverse view.  If we claim that our grasp of the natural 
numbers equipped with addition and multiplication is unclear, then surely 
a fortiori our grasp of a *nonstandard model of arithmetic* equipped with 
addition and multiplication is even more unclear.  Somehow we have managed 
to talk ourselves into thinking that this multiverse of nonstandard models 
is clear whereas the standard model is not.  This makes no sense to me. 
It seems to me to be a much larger self-delusion than any self-delusion 
that might be involved in convincing ourselves that we know what the 
standard model is.

Here's another remark that may help drive home the point.  If we are going 
to adopt the "sophisticated" viewpoint that "nonstandard integers are 
normal" then nonstandardly long proofs must be normal, as must algorithms 
that terminate after a nonstandard number of steps.  I cannot see the 
mathematical community coming to accept that, say, a ZFC-proof of 0=1 
whose length is a nonstandard integer is just as "normal" as a ZFC-proof 
of 0=1 whose length is a standard integer.  Nor can I see people being 
convinced that an algorithm that always terminates after N steps where N 
is a nonstandard integer should be viewed on a equal footing with an 
algorithm that always terminates after N steps where N is a standard 
natural number.  The assumption that we have a clear understanding of 
finitude is so pervasive in mathematics that it is easy to underestimate. 
I think the multiverse view can seem plausible only if we conveniently 
neglect to trace out all its logical consequences for how we think about 
and do mathematics.

Tim

P.S. I don't think this is directly relevant to the main issue of 
multiverses, but I really don't see that the categoricity of second-order 
Peano arithmetic somehow gives us "clarity" about what the standard 
natural numbers are.  It is a very general fact that uniqueness proofs 
proceed as follows: I start with some object X; I identify some properties 
of X; I then prove that X is the unique object satisfying those 
properties.  If I am unclear what X is initially, this process will not 
suddenly clarify for me what X is.  Indeed, if I am unclear what X is, how 
am I supposed to go about proving anything about it?  What distinguishes 
math from any other subject is that we don't even consider the possibility 
of proving anything mathematically until we have a precise understanding 
of what we're trying to prove.  (Well, maybe I'm a mindless robot 
programmed to generate proofs without "understanding" anything.  But in 
that case, I will remain mindless after producing the proof and will not 
attain "clarity" as a result.)  The thing that is unclear before I 
generate a uniqueness proof is *whether there's some other object Y* that 
I have overlooked; it's not *what X is* that is unclear.