My name has the perfect number of characters.
The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.
I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.
In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining —- they depend very heavily on the community of mathematicians.”
— The mathematicial community just had a grave loss. - Bill Thurston
The Mystery of 3-Manifolds - William Thurston (by PoincareDuality)
Very sad to say, but Twitter informs me that Bill Thurston recently died. He was an amazing teacher and mathematician. I never was able to take courses with him at Cornell, but one of my professors was very influenced by his thinking and pedagogy.
“Complex systems are ones with a large effective number of strongly-interdependent variables.
This excludes both low-dimensional systems, and high-dimensional ones where the variables are either independent, or so strongly coupled that only a few variables effectively determine all the rest.”
“Grothendieck came to consider the Institut des Hautes Etudes Scientifiques at Bures a gilded cage that kept him away from real life.
The solidarity of outcasts had created in him a strong feeling of compassion. Grothendieck had always been uncomfortable frequenting the “better” places and felt more at ease among the poor, even the impoverished.
The son of a militant anarchist who had devoted his life to revolution, Alexander lived as an outcast throughout his entire childhood. His home was always wide open to “stray cats”.”
Pierre Cartier, A Mad Day’s Work. as quoted in Wer ist Alexander Grothendieck?
(I liberally rearranged the text.)
Grothendieck is a hero.