2004-05-28
When Even Mathematicians Don't Understand the Math
An interesting thought piece on the nature of reality . . .
New York Times, 2004-05-25
By SUSAN KRUGLINSKI
To most of us, smudgy white mathematical scrawls covering a
blackboard epitomize incomprehensibility. The odd symbols
and scattered numerals look like a strange language, and
yet to read them, neurologists tell us, we would have to
use parts of our brains that have nothing to do with what
we normally think of as reading and writing.
Math and physics writers are the interpreters of this
unconventional language. Their books, when written for a
popular audience, may reach thousands of intrepid readers
who barely made it past Algebra II. The abstract concepts
they translate seep into the mainstream through books like
"The Golden Ratio" (its ideas are featured in "The DaVinci
Code") or "The Millennium Problems." Sometimes they even
make the best-seller lists.
And yet much of this subject matter confounds even
mathematicians and physicists, as they use math to
calculate the inconceivable, undetectable, nonexistent and
impossible.
So what does it mean when mainstream explanations of our
physical reality are based on stuff that even scientists
cannot comprehend? When nonscientists read about the
strings and branes of the latest physics theories, or the
Riemann surfaces and Galois fields of higher mathematics,
how close are we to a real understanding? Despite the
writer's best metaphors and analogies, what is lost in
translation?
"It is a bit like trying to explain football to people who
not only have no understanding of the word 'ball,' but are
also rather hazy about the concept of the game, let alone
the prestige attached to winning the Super Bowl," wrote Dr.
Ian Stewart, professor of mathematics at the University of
Warwick in England, in an email message.
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."
Dr. Keith Devlin, a mathematician at Stanford University
and author of "The Millennium Problems," which tries to
describe the most challenging problems in mathematics
today, admits defeat in his last and most impenetrable
chapter, where he is forced to interpret something called
the Hodge conjecture. He suggests to readers, "If you find
the going too hard, then the wise strategy might be to give
up."
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."
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?
In his densely packed, and best-selling, tome on physics,
"Fabric of the Cosmos," Dr. Brian Greene, a Columbia
University physicist, distills the dizzying calculations of
quantum physics and string theory, invoking images of
flowing time and textured space that should not necessarily
be taken literally. "These translations by design suppress
a huge amount of technical machinery that underlies the
everyday English description," he said. "I would say that
there's absolutely always something lost in the
translation."
Dr. Devlin noted that the familiar model of the atom - a
nucleus of protons and neutrons orbited by electrons - was
long obsolete. "Yet physicists can successfully use that
image of the solar system model with its rotating billiard
balls," he said. "It's the same with string theory. I mean,
give me a break - they're not little loops of string! For
one thing, they're in 11 dimensions."
Like Dr. Devlin, Dr. Greene is straightforward about the
impossibility of explaining certain abstractions. But he
thinks there is enough graspable material in the
mathematics of physics to depict just about anything. About
string theory, for example, he said: "The equations that
govern a violin string are pretty close to the equations
that govern the strings we talk about in string theory. So
although the notion of strings is metaphorical, it's pretty
close."
He added, "I suspect that the overarching aim of most every
mathematical study can be described, even if you can't get
to the guts."
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.
In fact, it is difficult to explain what math is, let alone
what it says. Math may be seen as the vigorous structure
supporting the physical world or as a human idea in
development. Some mathematicians say it is not in the same
category as biology, astronomy or geology. While those
fields have empirical systems of experimentation and
discovery, some might say mathematicians rely on something
more intuitive.
"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.
This makes for a frustrating human dilemma. Our brains have
the ability to compute the abstract mathematics they
created to construct theories about reality, and yet they
may never be smart enough to comprehend those theories, let
alone explain them.
Despite his and his colleagues' tireless efforts, Dr.
Greene concedes that this paradox ultimately makes sense.
"Our brains evolved so that we could survive out there in
the jungle," he said. "Why in the world should a brain
develop for the purpose of being at all good at grasping
the true underlying nature of reality?"
http://www.nytimes.com/2004/05/25/science/25math.html
New York Times, 2004-05-25
By SUSAN KRUGLINSKI
To most of us, smudgy white mathematical scrawls covering a
blackboard epitomize incomprehensibility. The odd symbols
and scattered numerals look like a strange language, and
yet to read them, neurologists tell us, we would have to
use parts of our brains that have nothing to do with what
we normally think of as reading and writing.
Math and physics writers are the interpreters of this
unconventional language. Their books, when written for a
popular audience, may reach thousands of intrepid readers
who barely made it past Algebra II. The abstract concepts
they translate seep into the mainstream through books like
"The Golden Ratio" (its ideas are featured in "The DaVinci
Code") or "The Millennium Problems." Sometimes they even
make the best-seller lists.
And yet much of this subject matter confounds even
mathematicians and physicists, as they use math to
calculate the inconceivable, undetectable, nonexistent and
impossible.
So what does it mean when mainstream explanations of our
physical reality are based on stuff that even scientists
cannot comprehend? When nonscientists read about the
strings and branes of the latest physics theories, or the
Riemann surfaces and Galois fields of higher mathematics,
how close are we to a real understanding? Despite the
writer's best metaphors and analogies, what is lost in
translation?
"It is a bit like trying to explain football to people who
not only have no understanding of the word 'ball,' but are
also rather hazy about the concept of the game, let alone
the prestige attached to winning the Super Bowl," wrote Dr.
Ian Stewart, professor of mathematics at the University of
Warwick in England, in an email message.
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."
Dr. Keith Devlin, a mathematician at Stanford University
and author of "The Millennium Problems," which tries to
describe the most challenging problems in mathematics
today, admits defeat in his last and most impenetrable
chapter, where he is forced to interpret something called
the Hodge conjecture. He suggests to readers, "If you find
the going too hard, then the wise strategy might be to give
up."
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."
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?
In his densely packed, and best-selling, tome on physics,
"Fabric of the Cosmos," Dr. Brian Greene, a Columbia
University physicist, distills the dizzying calculations of
quantum physics and string theory, invoking images of
flowing time and textured space that should not necessarily
be taken literally. "These translations by design suppress
a huge amount of technical machinery that underlies the
everyday English description," he said. "I would say that
there's absolutely always something lost in the
translation."
Dr. Devlin noted that the familiar model of the atom - a
nucleus of protons and neutrons orbited by electrons - was
long obsolete. "Yet physicists can successfully use that
image of the solar system model with its rotating billiard
balls," he said. "It's the same with string theory. I mean,
give me a break - they're not little loops of string! For
one thing, they're in 11 dimensions."
Like Dr. Devlin, Dr. Greene is straightforward about the
impossibility of explaining certain abstractions. But he
thinks there is enough graspable material in the
mathematics of physics to depict just about anything. About
string theory, for example, he said: "The equations that
govern a violin string are pretty close to the equations
that govern the strings we talk about in string theory. So
although the notion of strings is metaphorical, it's pretty
close."
He added, "I suspect that the overarching aim of most every
mathematical study can be described, even if you can't get
to the guts."
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.
In fact, it is difficult to explain what math is, let alone
what it says. Math may be seen as the vigorous structure
supporting the physical world or as a human idea in
development. Some mathematicians say it is not in the same
category as biology, astronomy or geology. While those
fields have empirical systems of experimentation and
discovery, some might say mathematicians rely on something
more intuitive.
"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.
This makes for a frustrating human dilemma. Our brains have
the ability to compute the abstract mathematics they
created to construct theories about reality, and yet they
may never be smart enough to comprehend those theories, let
alone explain them.
Despite his and his colleagues' tireless efforts, Dr.
Greene concedes that this paradox ultimately makes sense.
"Our brains evolved so that we could survive out there in
the jungle," he said. "Why in the world should a brain
develop for the purpose of being at all good at grasping
the true underlying nature of reality?"
http://www.nytimes.com/2004/05/25/science/25math.html
Subscribe to Posts [Atom]