TITLE: What's the best way to measure mathematical ability?
QUESTION [41 upvotes]: Very soft question I admit, but it's something that's been bothering me for a while.  
I've been thinking that being self taught has the problem of accreditation.  You can't evaluate a mathematician in a vacuum.  You need an accredited mathematician to decide whether or not someone else is also a mathematician worthy of accreditation.  Well, who evaluated the other mathematician?  Other accredited mathematicians.  It's sort of like becoming a member of an exclusive club.  
We put the job of accreditation on our universities.  But what if some person was discovered, off-the-grid so to speak, who had taught themselves mathematics from library textbooks.  
How could such a person evaluate themselves?  How do you know if you're making progress when you study?  
It's tricky.  It's like language learning.  Do I speak German more fluently now than I did yesterday?  I've no idea.  Who can say?  
It's like playing with Lego.  How do you know if you're getting better with Legos?  You build more complicated things.  But who's to say one person's Lego helicopter is better than another's Lego Enterprise?  What's the goal with Legos?  Is there one?  Should there be one?
I know already that this question will be deleted almost immediately, but I think these are important questions and many people visiting this site are in fact self-taught and I'm sure these questions show up as massive roadblocks.
Thanks for reading.

REPLY [5 votes]: You can do mathematics in a vacuum, though.  It is possible to devise what is necessary without checking whether someone else has done it before.  In this way, it's not like learning German or Old English, where one must ultimately access other speakers of it.
Until the advent of the internet, access to material on a subject was quite hard, and often supposes that one is well versed in some leading subjects to understand it.  I live in the tropics, where you might be lucky to find a single rack of "pop science" books in the larger bookshops.  Getting hold of material was considerably harder then inventing it yourself.
It is interesting now, that while I have discovered many things that have older names, the path to them is different, and the extent of coverage of something is also different.  
One discovers from first principles, the nature of non-euclidean geometry, and then proceeds to do perfectly valid dot-products of vectors on an oblique coordinate system.  Such is something that comes from the "Coxeter-Dynkin" diagram.  Yet in the mathematics one reads on this matter, these diagrams are used of symmetry, and much of Alicia Stott's good work has been for naught.
Mathematics is something that can be discovered, as one might sail ships around the world.  Different nations can sail up the coast of australia, and give different things different names.  But the landscape does not change, just the names and the mental relations.
I came to something like Mobius geometry through a circuitous path.  Because the nature of my geometry is somewhat different, recognising that every circle is a straight line in that geometry is something that I and a famous professor in the USA have chose to disagree on.  We are talking of the same thing, but seeing in a different culture.
It is hard then to assess the competence of a mathematician.  They may understand the mathematics, but have different words for it.  It certainly is true that my endeavours in the geometries of higher dimensions is world-class, and in some parts, well in the lead, but the mathematics is all pre-calculus.