[FOM] The unreasonable soundness of mathematics

[Timothy Y. Chow]

On Mon, 2 May 2016, Mario Carneiro wrote:
> It sounds like what you are objecting to is the very concept of a
> "specification".

Not quite.  As an ultrafinitist, I'll grant you the ability to specify a 
concrete computer with a specific, feasible number of components and a 
specific, feasible amount of memory.  The sticking point arises when 
generalizing/idealizing to an unbounded amount of memory.

The classical mathematician is, on the basis of a *single* formal proof, 
making a prediction about the behavior of an unbounded number of concrete 
machines.  The ultrafinitist, on the other hand, has to separately build a 
megabyte-sized machine, a gigabyte-sized machine, a terabyte-sized 
machine, etc., and only when each of these machines runs to completion can 
the ultrafinitist assert that there are no positive integers A and B *in a 
certain range* such that A^2 = 2*B^2.  For each concrete machine, the 
ultrafinitist can understand what a spec is for that machine, but the 
classical mathematician's "spec" is, in effect, quantifying over specs of 
*arbitrary size* and it is this step that the ultrafinitist cannot 
understand.  Like the memoryless goldfish's little plastic castle in Ani 
DiFranco's poem, the computational result is a surprise every time to the 
ultrafinitist.

Tim