From Ben Orlin's "Math with Bad Drawings" blog (but without the charming illustrations) a wonderful summary of the difference between artisan and artist, titled Good Mathematician vs Great Mathematician:
| A good mathematician wants to know how. ("Problem solved. So satisfying!") | A great mathematician wants to know why. ("Problem ... solved? So unsatisfying!") |
| A good mathematician selects the most powerful tool for the job. ("Welding torch it is!") | A great mathematician selects the least powerful tool for the job. ("Nothing a little duct tape can't fix.") |
| A good mathematician answers questions. ("Does it converge? My proof says no!") | A great mathematician questions answers. ("OK, so it doesn't converge ... but ... what if we redefine 'convergence'?") |
| A good mathematician can make arguments intuitive or rigorous. | A great mathematician can make arguments intuitive and rigorous. |
| A good mathematician can get out of a tricky corner. ("Phew! Paradox averted!") | A great mathematician can get into a tricky corner. ("Ooh ... if I change the problem like this, the paradox gets even worse!") |
| A good mathematician solves problems by making them more concrete. ("Before we build the skyscraper, let's make a model.") | A great mathematician solves problems by making them more abstract. ("Before we build the skyscraper, let's study the skyline!") |
| A good mathematician inventories what they know. ("I've got six types of screwdrivers!") | A great mathematician inventories what they don't know. ("I'm still missing so many types of screwdrivers!") |
| A good mathematician loves math. ("How was the math today?" "Good! Even a little better than yesterday, which was already way better than the day before.") | A great mathematician lives math. ("How was the math today?" "f(t) very large! And f'(t)>0, though f"(t)<0.") |
(cf. Good Mistakes (1999-09-06), Millennium Math (2002-12-05), Mystery to Me (2003-05-30), Prime Obsession (2004-01-04), Hardy-Littlewood Rules (2004-06-14), EquationsAndReality (2005-02-21), Staying the Course (2005-07-11), Stokes Theorem (2006-01-27), Mathematics and Poetry (2008-11-09), Roads to Infinity (2010-10-06), One to Nine (2011-01-11), Greatest Inventions (2011-06-09), Music of the Primes (2011-08-11), Probability Problems (2013-02-09), Meta-Analogy (2013-03-20), ...) - ^z - 2016-01-03