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Sunday, December 11, 2005

Why is math so hard?

I am near the end of the semester and hope to make a few more entries in this blog over break. One of the things I hope to do is to talk a bit about the Nash embedding theorem.

I am teaching a numerical methods class and gave a take-home exam. One student included this photo in his work, though he crossed out the word "you" and wrote "I". This reminded me of a quote from the mathematician Ronald Graham:, "Our brains have evolved to get us out of the rain, find where the berries are and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." (source: )
This link goes to an interesting discussion which I've reproduced here:
Will There Be Anything Left To Discover?
Is the great era of scientific inquiry over? Have all the big theories been formulated and important discoveries made—leaving future scientists nothing but fine tuning? Or is the real fun about to begin?By JOHN HORGAN and PAUL HOFFMAN

A spirited debate, conducted via e-mail, between two acclaimed science journalists: John Horgan, author of the controversial book The End of Science, and Paul Hoffman, former editor of Discover magazine and past president of the Encyclopaedia Britannica.
HOFFMAN: The past decade has brought a spate of books sounding the death knell for a host of subjects. Francis Fukuyama served up The End of History and David Lindley The End of Physics. But your more sweeping work The End of Science (1997) attracted a lot more attention and controversy—and with good reason. The idea that science may have had its run—that we've discovered all we can realistically expect to discover and that anything we come up with in the future will be pretty much small-bore stuff—left people either intrigued or outraged. With today's seemingly frenetic pace of scientific discovery, however, how can you say that the whole enterprise is coming to an end? The scientists I know, far from preparing for the undertaker, are ebullient about the future of their field.
HORGAN: Sure, scientists are keeping busy, but what are they actually accomplishing? My argument is that science in its grandest sense—the attempt to comprehend the universe and our place in it—has entered an era of diminishing returns. Scientists will continue making incremental advances, but they will never achieve their most ambitious goals, such as understanding the origin of the universe, of life and of human consciousness. Most people find this prediction hard to believe, because scientists and journalists breathlessly hype each new breakthrough, whether genuine or spurious, and ignore all the areas in which science makes little or no progress. The human mind, in particular, remains as mysterious as ever. Some prominent mind scientists, including [Time Visions contributor] Steven Pinker, have reluctantly conceded that consciousness might be scientifically intractable. Paul, you should jump on the end-of-science bandwagon before it gets too crowded.
HOFFMAN: Don't save a seat for me quite yet, John. Take the human mind. I agree that we are not close to an understanding of consciousness, despite the efforts of some of the best minds in science. And perhaps you're even right that we may never understand it. But what is the evidence for your position? You've criticized scientists for having faith—a dirty word in the scientific lexicon—that our era of explosive progress will continue unabated. Isn't it at least as much a leap to think that the progress will abruptly end—particularly since the trajectory of discoveries so far suggests just the opposite, that supposedly unanswerable questions eventually do get answered?
HORGAN: My faith is based on common sense, Paul, and on science itself. As science advances, it imposes limits on its own power. Relativity theory prohibits faster-than-light travel or communication. Quantum mechanics and chaos theory constrain our predictive abilities. Science's limits are glaringly obvious in particle physics, which, as Steven Weinberg describes [in the Visions issue], seeks a "theory of everything" that will explain the origin of matter, energy and even space and time. The leading theory postulates that reality arises from infinitesimal "strings" wriggling in a hyperspace of 10 (or more) dimensions. Unfortunately, these hypothetical strings are so small that it would take a particle accelerator the size of the Milky Way to detect them! I am not alone in fearing that string theorists are not really practicing science anymore; one leading physicist has derided string theory as "medieval theology." Paul, here is persuasive evidence of science's plight.
HOFFMAN: Yes, but who is to say that all these scientific theories won't ultimately be replaced by ones with greater explanatory power? Galileo and Newton thought their laws of motion were the cat's pajamas, explaining everything under the sun and many things beyond, but 2 1/2 centuries later a Swiss patent clerk toppled their notions of space and time. Obviously, Galileo and Newton did not foresee what Einstein found. I think it's ahistorical to assert that in the future there will never be an Einstein of, say, the mind who will be able to pull together a theory of consciousness. And even if it's true that some of the big unanswered questions of science may never be answered, a lot of new and exciting science could still come from overturning truths that we now take for granted. Robert Gallo, the AIDS researcher, once told me that at the end of the 1970s, he was at a conference where a prominent scientist confidently summed up the truths of biomedicine— such truths as: epidemic diseases are things of the past, at least in so-called developed nations; a widespread outbreak of infectious disease is impossible unless the microbe is casually transmitted; the kind of virus found in animals known as the retrovirus doesn't exist in man; and no virus causes cancer in humans. By the end of the 1980s, these four truisms had hit the dustbin. Or take a more recent example: the newfound plasticity of the human brain. Until a year and half ago, it was a dogma taught in every medical school in the country that the adult human brain is rigid, that its nerve cells can never regenerate. Now we know our brains do have the ability to generate new cells—a discovery that may not only open up a new understanding of the brain but also lead to novel treatments for a host of brain disorders. HORGAN: Here's the big question we're dancing around: Can we keep discovering profound new truths about reality forever, or is the process finite? You seem to assume that because science has advanced so rapidly over the past few centuries, it will continue to do so, possibly forever. But this view is, to use your word, ahistorical, based on faulty inductive logic. In fact, inductive logic suggests that the modern era of explosive scientific progress might be an anomaly, a product of a singular convergence of social, intellectual and political factors. If you accept this, then the only question is when, not if, science will reach its limits. The American historian Henry Adams observed almost a century ago that science accelerates through a positive-feedback effect. Knowledge begets more knowledge; power begets more power. This so-called acceleration principle has an intriguing corollary: If science has limits, then it might be moving at maximum speed just before it hits the wall.
HOFFMAN: Of course, I accept that science has limits—and may even be up against them in some fields. But I believe there is still room for science, even on its grandest scale, that awe-inspiring discoveries will continue to be made over this millennium. The mathematician Ronald Graham once said, "Our brains have evolved to get us out of the rain, find where the berries are and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." Sounds reasonable, except when you consider that it could be similarly said that our brains didn't evolve to invent computers, design spaceships, play chess and compose symphonies. John, I think we'll continue to be surprised by what the brains of scientists turn up.
HORGAN: I hope you're right, Paul. I became a science writer because I believe science is humanity's most meaningful creation. We are here to figure out why we are here. The thought that this grand adventure of discovery might end haunts me. What would it be like to live in a world without the possibility of further revelations as profound as evolution or quantum mechanics? Not everyone finds this prospect disturbing. The science editor of the Economist once pointed out to me that if science does end, we will still have sex and beer. Maybe that's the right attitude, but there aren't any Nobels in it. No matter how far science does or doesn't advance, however, there's one wild card in even the most pessimistic scenario. If we encounter extraterrestrial life—and especially life intelligent enough to have developed its own science—then all bets are off.


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