[FOM] Harvey on invariant maximality

Sam Sanders sasander at cage.ugent.be
Mon Apr 2 22:30:02 EDT 2012

The short answer: 

No, we cannot provide a full and precise formalization 
of naturalness. 

Yes, there are partial steps towards understanding naturalness
we will take in the near future.  These are intimately tied with progress in the exact sciences.
Ultimately, this understanding is important for Mathematics and her related notion of naturalness.

The long answer:

For the sake of clarity/neutrality, I will use Biology as an example.
This discipline is a body of knowledge that came about by countless
small additions, revisions, many iterations and small steps forward.
No superman philosopher could ever deduce this body of knowledge
via pure reason:  Biology came about via a relatively *long* evolutionary
process which cannot be replaced by *short* ad hoc reasoning.  
Similar arguments can be made for languages, medicine, chemistry,
physics, mathematics, logic, intelligence, living matter, health, brains,
conscienceness, knowledge we have acquired, learning process, …  

Correct me if I am wrong, but currently, we have few techniques to 
adequately model such evolutionary phenomena.  These are all 
*finite* entities, but came about by a *large* number of iterations.
They continue to evolve, but have a certain stability/robustness 
which makes them indestructible /essentially unchanged by any 
*small* numbers of steps.  Understanding these finite phenomena
where the notion of large vs small steps is central, is one of the next great 
challenges for Mathematics, and science in general: a next leap
in science will only come about if we start properly modeling/under-
standing such complex evolutionary systems like intelligence,   
health, brain, consciousness, etc.  The challenge to Mathematics
here is not to look at infinite objects, which usually have nice
closure/stability properties) but at finite objects which have such 
properties, due to the presence of large vs small parameters.

The link with naturalness here is the following: 

A question/result in discipline X is natural or g.i.i (general intellectual interest) if we do 
not have to undergo such evolutionary learning process wrt X to understand/appreciate it.
Returning to biology, everyone appreciates that understanding living matter is of g.i.i.
On the other hand, a topic like "The role of Killer-T cells in the amino-hypersystase
of the metacyste formation in patients of mixed backgrounds in 1900 Industrial England"
would require an involved (evolutionary) learning process for most people, hence this topic
is not g.i.i., although it might be an interesting niche discipline.  

Concretely, a result like Harvey Friedman's unprovable statement may be deemed natural/g.i.i.
(which I believe it is) if a mathematician can understand it in a small number of steps.  Arguably
many results in logic require a large number of steps to understand, perhaps even an "evolutionary
large" number of steps, after which one has (partially) evolved into a logician.  Thus, these are not g.i.i.  

To sum it up:

The next big leap in science will be the better (general) understanding of such finite phenomena
where large vs small parameters play a role.  This is directly linked to naturnalness and will probably
be developed in tandem.  

> I would like to ask the chief participants in this debate to
> predict the future, as suggested by the above remarks;
> with or without giving their reasons.
> Most mathematical ideas start off informal, and become more precise
> until they are finally given a "frozen" formal description.
> This has happened with continuity, smoothness, computability,
> and so forth.  It seems unlikely to happen with something so vague
> as (say) elegance.
> My question is: Do you think this will happen with "naturalness"?
> Within a meaningful time, say, the next century.
> -- Bill Taylor

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