Major Advances In Knot Theory 230
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."
How many ways are there to tie your shoelaces? (Score:3, Insightful)
42
Unless... (Score:5, 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.
Re:This is so very important... (Score:5, 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.
Re:The hardest math (Score:5, Insightful)
Re:Things like this... (Score:3, Insightful)
Oh really? Would you also say studying topology in general is unimportant? Why or why not? Since you're able to discern which branches of mathematics aren't "important", you're clearly a mathematical authority, so please feel free to enlighten us.
this is great (Score:1, Insightful)
but it will never be able to explain why anyone would want to "tie the knot."
Re:The hardest math (Score:5, Insightful)
Hard problems are only hard because we're using the wrong tools.
Re:This is so very important... (Score:5, Insightful)
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.
Re:This is so very important... (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.
Re:This is so very important... (Score:1, Insightful)
It was whiny, it was irrelevant (how did it relate to knot theory?), it did dismiss pure research -- as if pure research scientists should drop everything and join the war effort, it was scare mongering (as if the only thing we should be thinking about is politics and war)...
Lastly, it is the priorities you're whining about that are irrelevant.
Let politicians work on politics, soldiers work on war, and mathematicians work on math.
If you weren't whining about priorities and politics, then what on earth were you talking about?
Re:The hardest math (Score:2, Insightful)
And I have spent 5 years of my life on the topology of proteins. It is not quite true to refer to "knots" when talking about proteins, as Professor Taylor has shown (http://www.nature.com/nature/journal/v406/n6798/full/406916a0.html) that only a few proteins are actually 'knotted'.
However, mathematical theory of tangled strings is as important as simulations. Estimating the total number of folds, for example. More than just a fancy excursion - but maybe not to your taste?
Re:This is so very important... (Score:3, Insightful)
Yes, this is important.
What do you think where new ideas on saving the world or building a better one will come from? TV studios? Politicians? Hollywood?
Research like this is the foundation of all progress. Note: Not this one specifically, I said "like" this. A lot of the things that you probably wouldn't live very well without started out as ideas with no visible use.
Re:Unless... (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.