[FOM] Harvey on invariant maximality

[Andreas Weiermann]

Dear all,

I do not want to define what natural means
but my experience is that senior people in the field
know what mathematical taste is and what a beautiful theorem is.

As Hardy wrote
"Beauty is the first test: there is no permanent place in this world  
for ugly mathematics."
For significance of a theorem Hardy further mentioned:
generality, depth, unexpectedness, inevitability, economy.

I think that Harvey's results meet all these criteria.

What naturality concerns: There is the famous question of
what makes an ordinal notation system natural. People working
in the area know whether something is natural when they see
it. But attempts to define the notion precisely didn't succeed
yet. But one can nevertheless work with these notations and
sometimes even beautiful results can emerge.

I would like to pose a question which might be out of scope:

Given a proof of a significant theorem. Why is the most
beautiful one typically the shortest and vice versa?
(Counterexamples will be appreciated.)


Best,
Andreas Weiermann




Citeren "Sam Sanders" <sasander at cage.ugent.be>:

> 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
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>