Meta-metamathematics

[JOSEPH SHIPMAN]

Are you sure we can’t?
It’s conceivable that an experimental “prediction“ coming from a physical theory could involve a mathematically definable but nonrecursive real number. For example, some quantum gravity theories involve sums over homeomorphism classes of  4-dimensional spacetimes, which can’t be computable classified. In that case, any given mathematical axiom system will only allow the value of such a quantity to be measured to a finite precision, and a more precise measurement could give evidence for a stronger axiom system.

Alternatively: Harvey has shown that certain large cardinals prove the non existence of certain finite patterns quickly that can only be established in ZF by something like exhaustive search, so a prediction that such a search will fail to find such a pattern, confirmed eventually by much computation, provides evidence for the large cardinal axiom.

— JS

Sent from my iPhone

> On Jan 15, 2022, at 5:42 PM, Sam Sanders <sasander at me.com> wrote:
> 
> Dear Joseph,
> 
> you sent us the following proposition, let us call it (R).
> 
>> “There is no physical experiment which could ever provide persuasive evidence for or against statements of set theory that are not absolute. Mathematicians who firmly disagree about such a statement, but who agree about fundamental physics, will never be able to settle the matter by doing computations or any other physical processes.” (R)
>> 
>> If this proposition is correct, then arguing about, say, the continuum hypothesis can be viewed as a “religious dispute”. Even though CH is refuted by ZFC+RVM, CH-believers will not have to regard, say, physical evidence for the consistency of cardinals below a measurable as evidence toward RVM, because direct evidence for CH itself will not be something that correct physical theories can provide. Attention should be focused on axioms with new arithmetical (or at least absolute) consequences.
>> 
>> What is your opinion of this proposition? I especially want to hear from physicists.
> 
> I am not a physicist but close enough to provide a coherent answer, I believe.  
> 
> My first impression of (R) was the following:  it is actually quite weak.  Since we only have access to finite data 
> and (very restricted) computable operations on that, we can never really know what physical reality looks like.  Our best theories
> are just that: theories.  They are the pinnacle of human ingenuity in many ways, but they only approximate reality.
> Our theories do this approximating very well, given how well our experiments are confirmed.  In this way, we cannot
> even hope to have physical experiments that provide persuasive evidence for or against even the most basic infinitary axioms.  
> 
> Along the same lines, one could argue that physical experiments will never provide persuasive evidence for or against e.g. Ed Nelson’s finitism, 
> let alone foundational claims from set theory.  
> 
>> if this proposition (R) is correct, then arguing about, say, the continuum hypothesis can be viewed as a “religious dispute”.
> 
> After some thought, I must say I disagree: one reason to use infinite objects in the exact sciences is because they have nice closure properties.  
> The real numbers are a prime example, as developing calculus over the rationals would be a mess, and over a finite
> set even more so.  This could presumably be done (in some form or other that could encompass enough
> physics), but it would be a drag, and to what end?  
> 
> Hence, if CH (or any axiom) provides nice closure properties that make math go smoother (compared to other axioms), then 
> the axiom at hand has some value beyond merely being the topic of “religious dispute”.  
> 
> The previous paragraphs are non-trivial as there was extensive use of infinitesimals and distributions in physics long before there was a mathematically rigorous 
> theory.  To this day, the path integral only has a full mathematical treatment in special cases.  Hence, it seems that physicists 
> do like their “nice closure properties”, even if they are on shaky mathematical ground.  
> 
> 
> Best,
> 
> Sam
>