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 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. 

Too Many Words, Not Enough Numbers

I suspect the reviewer, like the author, has no concept of testing data, and prefers sweet little anecdotes.

There is a concept that is the corrupter and destroyer of all others. I speak not of Evil, whose limited empire is that of ethics; I speak of the infinite.
Jorge Luis Borges, author
Hey… infinity isn’t that bad!

David Brooks can’t add


Fact-checking David Brooks could be a full-time job. Just yesterday, he wrote this about the federal budget problem:

Raising taxes on the rich will not do it. There aren’t enough rich people to generate the tens of trillions of dollars required to pay for Medicare, let alone all the other programs.

Almost every word of this is wrong.

Medicare doesn’t require “tens of trillions,” unless your budget horizon is something like twenty years. This year, Medicare will cost $572 billion. In 2020, according to the CBO, it will cost $949 billion. Over the next ten years, it will cost $7.6 trillion, which isn’t even a ten of trillion, much less “tens of trillions.” (And that doesn’t include “offsetting receipts”—$80 billion this year, and $1.2 trillion over the next ten, which reduce those outlays significantly. Supporting spreadsheet is here.) Right now, the top 1% of the U.S. pop has something like $1.4 trillion in income. The next 4%, $1.3 trillion. The next 5% has almost a trillion. (Computed from Piketty and Saez data here.) In other words, you could entirely fund Medicare by hitting up the top 1% for about a third of its income. Yeah, I know that’s politically impossible, but they’ve got the money—we just can’t have any of it.

As for the rest of it, Social Security is a trivial budget problem, if it’s one at all. Medicaid is a problem, but, as with Medicare, the best way to solve that budgetary problem is with a single-payer system, which wouldn’t require any new tax revenue at all, but would actually save money by eliminating administrative costs.

As the saying goes, it’s a fool that goes looking for meaning in the columns of David Brooks. 

The bigger issue is that journalists are completely innumerate. I can count on one hand the number of journalists who have any understanding of mathematics. 

Fire all journalists who are innumerate. Fire them now.

(Source: azspot)