STOP THE PRESSES !
I feel the urge to fisk violently upon reading an article in today's NYT titled 'When Even Mathematicians Don't Understand the Math':
<.. deep breaths ...>
Here are some choice excerpts:
Asked if there exist mathematical concepts that defy explanation to a popular audience, Dr. Stewart, author of "Flatterland: Like Flatland, Only More So" replied: "Oh, yes - possibly most of them. I have never even dared to try to explain noncommutative geometry or the cohomology of sheaves, even though both are at least as important as, say, chaos theory or fractals."
I see. so this is news to the NYT that there are (perish the thought)mathematical ideas that cannot be explained to layman...
The Hodge conjecture deals not only with cohomology classes, a complicated group construct, but involves algebraic varieties, which Dr. Devlin describes as generalizations of geometric figures that really do not have any shape at all. "Those equations represent things that not only can we not visualize, we can't even imagine being able to visualize them," he said. "They are beyond visualization." This difficulty points to a math truism that ultimately framed his entire project.
"What the book was really saying was, 'You're not going to understand what this problem is about as a layperson, but neither will the experts,' '' he said, adding, "The story is that mathematics has reached a stage of such abstraction that many of its frontier problems cannot be understood even by the experts."
Well sure - an expert in sub area X may not always understand the terms in sub-area Y. Is this (a) surprising or (b) a BAD BAD thing ? Isn't this just the consequence of living in a rich and incredibly complex field ?
At the same time, higher math is used to decipher the existence and composition of the world. But how can it make sense that a nearly unintelligible language can explain the physical world?
There seems to be the unstated implication that a language unintelligible to the layman cannot be used to explain the world.. see below...
But if science writers described breakthroughs in genetics or zoology in terms of overarching aims and not concrete facts, readers would question the foundations of that field. That lay readers and scientists alike accept that they will never wrap their heads around much of higher math is evidence that it is a world unto itself.
But surely one can say the same thing about some of the denser theories of post-modern literary criticism, that it is incredibly hard for the layman, or even many literary theorists, to understand them. Is there some kind of subtle anti-intellectualism at work here - knowledge that is not accessible to the masses is not real knowledge ?
In fact, it is difficult to explain what math is, let alone what it says.
Sigh...two pages into the article, the writer figures it out...
"It isn't science," said Dr. John L. Casti, the author of "Five Golden Rules: Great Theories of 20th-Century Mathematics and Why They Matter." "Mathematics is an intellectual activity - at a linguistic level, you might say - whose output is very useful in the natural sciences. I think the criteria that mathematicians use for what constitutes good versus bad mathematics is much more close to that of a poet or a sculptor or a musician than it is to a chemist."
And just as one cannot define what it is that makes a moving phrase played on a violin moving, the essence of the superb equation may also be ineffable.
Finally something I can live with ;)...