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Major Advances In Knot Theory

Posted by kdawson on from the if-it's-not-theory-then-it-must-be-practice dept.

An anonymous reader sends us to Science News, which is running a survey of recent strides in finding an answer to the age-old question: How many ways are there to tie your shoelaces? "Mathematicians have been puzzling over that question for a century or two, and the main thing they've discovered is that the question is really, really hard. In the last decade, though, they've developed some powerful new tools inspired by physics that have pried a few answers from the universe's clutches. Even more exciting is that the new tools seem to be the tip of a much larger theory that mathematicians are just beginning to uncover. That larger mathematical theory, if it exists, may help crack some of the hardest mathematical questions there are, questions about the mathematical structure of the three- and four-dimensional space where we live. ... Revealing the full ... superstructure may be the work of a generation."

7 of 230 comments (clear)

  1. Unless... by Slur · · Score: 4, Insightful

    Revealing the full... superstructure may be the work of a generation.

    ..assuming computers cease making any new advances.

    Mathematicians do rely on their ability to spot patterns and sense implications that no computer can likely sift for today. But this will not always be the case.

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    -- thinkyhead software and media
    1. Re:Unless... by Anonymous Coward · · Score: 5, Insightful

      No. As a professional computer scientist, I think it is safe to say mathematicians are about the last people in the world to be in danger of losing their job to computers.

      If there's one thing computer science algorithmic theory has told us, it's that computers absolutely do have a limit on what they can do, no matter how fast the microchip gets. Complete searches (and that is what we're talking for computer proofs) are NOT getting any more feasible over time. 2^10000 branches will never be traversable.

      Pretty much the best possible scenario for computer proofs is basic geometry. After all, in US high school, students are taught "2-column" proofs that a computer could actually handle. And even here, computers suck compared to mediocre mathematicians. Why? Because anybody can trace basic implications like a computer does - that's the easy part. The ONLY real hard part is the flash of insight that computers can never do - e.g. why don't we consider this point that is only tangentially related and see how it somehow holds all the structure to solving the problem.

      Once you get into modern math, say knot theory, computers are completely hosed. A math paper might be 100 pages of prose, 80% of which might be insights like that thing above, and 20% of which might be basic implications that a computer can handle. And actually, it couldn't, because 20 pages in prose = 2000 pages in logic statements, and a computer will never be able to traverse that deep.

      There's a reason that every important computer proof up until now has relied on 0 insight from the computer... even something like the 4-color theorem is only using a computer to algorithmically check a finite number of trivial cases that would be impractical to check by hand. This approach does not generalize to making mathematicians obsolete.

  2. Re:This is so very important... by abigor · · Score: 4, Insightful

    The world has been in far worse situations than it's in now. The transient problems of immediate political and social realities shouldn't stop a few people from investigating nature's deep questions via science and mathematics.

  3. Re:The hardest math by tloh · · Score: 4, Insightful

    Once upon a time, I was similarly bored by this area of abstract research. But about a year ago, I attended a seminar where a guest lecturer was a mathematician who applied knot theory to the physical modeling of life processes involving the winding and unwinding of DNA in Chromosomes and the folding and unfolding of peptide strings in protein formation. I didn't understand half of the lecture. But one very important point I got out of it is that no matter how abstract and esoteric a subject might be, there is immense value to be obtained if it can be utilized to model physical processes we seek to understand.

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    Stay sentient. Don't drink bad milk.
  4. Re:The hardest math by oscartheduck · · Score: 4, Insightful

    I think the important thing is that when you're investigating new areas of mathematics and it's _hard_, that's because the tools you're using are not suited for investigating this issue. So you invent a new tool, and that new tool can be applied in many, many places.

    Hard problems are only hard because we're using the wrong tools.

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    How to use coral cache: http://slashdot.org.nyud.net:8090/~oscartheduck
  5. Re:This is so very important... by ScentCone · · Score: 5, Insightful

    I can't believe I got moderated as a troll

    Why? You made a whiny, irrelevent complaint that dismisses the role of pure research in the larger advancement of our knowledge of how the universe works... the very sort of thing that always plays a role in advancing our ability to make more efficient use of energy, more realistic predictions about the behavior of complex systems, and more innovative technological use of things we think we have already fully, or most effectly exploited. This whole "the human race is incapable of doing two things at once" BS never ceases to amaze me. How do you even get out of bed in the morning? Make coffee... take a crap... which to do first? Gaah! I'm paralyzed! Which is the most important fish to fry?

    In other words, you're scare mongering and - if we can assume you have a passable IQ which would suggest you might know better - clearly trolling. And, voila, you were thusly modded.

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    Don't disappoint your bird dog. Go to the range.
  6. Re:This is so very important... by Evanisincontrol · · Score: 5, Insightful

    Suppose you tell us all how solving this knotty problem will help anyone or anything.

    Let's pretend we're in the early 1700s. Leonhard Euler is writing the first ever paper on a field of study called Graph Theory. Simply put, he's figuring out answers to questions about how to arrange circles and lines. Meanwhile, there's fucking WARS going on (Polish succession is going on concurrent to writing this paper; Seven Years' war happens a couple decades later). There are goddamn wars on Euler's front door, and he's writing papers about lines and circles?! What a prick.

    Oh, by the way, without Euler's work we wouldn't have computers, organized roads, efficient data models, efficient sorting algorithms, or countless other instruments that are critical to today's society.

    Don't trivialize work that you don't understand.