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A mysterious connection between number theory, algebra and string theory?
- 31reasons 12y ago>>It took several more years before mathematicians succeeded in even constructing the monster group, but they had a good excuse: The monster has more than 10^53 elements, which is more than the number of atoms in a thousand Earths. Did they create a set of numbers the size of 10^53 ? That seems impossible since you need more capacity then all atoms of 1000 Earths!
- AngrySkillzz 12y agoThat's not quite what they mean. By "constructed" they mean "demonstrated that it exists and has the specified properties." They didn't write down every element explicitly.
- wilsynet 12y agoThat's right. For example, you can construct the integers Z without writing all of them down. Which would take a long time ...
- crb002 12y ago53! is how many digits base 10? That's the size of the symmetric group on 53 elements. Groups can be exponentially larger than their underlying set.
- AlexCyrlex 12y agoThe first equation on page 8 of the arXiv paper http://arxiv.org/abs/1503.01472 http://arxiv.org/abs/1503.01472 seems strange to me. It looks like saying "function F(t) given by F(t) + Sum(something)." Shouldn't it use ':=' symbol instead of '+', like "F(t) := Sum(something)"? Or maybe this is the case when math notation is so complex that common symbols have completely different meaning?
- lotsofmangos 12y agoThe naming is fantastic, I mean, Umbral Moonshine Conjecture. By the time we actually find a theory connecting all this stuff, it will probably have aggregated a name so ridiculous that nobody will believe it.
- blrgeek 12y agoAnd one of the connections is Ramanujan's letters. Both the FIRST letter that he wrote to Hardy, and the LAST letter he wrote on his death-bed talking about 'mock-theta functions' that are now called mock-modular functions. > Ken Ono, of Emory University in Atlanta, Ga. “Without either letter, we couldn’t write this story.” Weird, and fascinating!
- ganeumann 12y agoBoth physics and the unreasonable success of math at explaining the world make much more sense if you just assume we are living in a math equation.
- wz1000 12y ago> the unreasonable success of math at explaining the world I don't think that it is unreasonable. Math is the study of formal systems. A formal system can describe pretty much anything. Much study has been concentrated on formal systems that describe some aspect of our universe, as these are the ones that tend to be "useful". However, there are an infinite number of formal systems out there which don't describe any aspect of our physical universe. The math "exists" irregardless of any physical interpretation. It is we as conscious beings that attribute "meaning" to mathematical systems that happen to describe some aspects of our daily life.
- FeepingCreature 12y agoSome other people (most notably Max Tegmark) also agree with this: http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis http://en.wikipedia.org/wiki/Mathematical_universe_hypothesi...
- JonnieCache 12y agoWhat would that even mean though? People throw around these "universe is a computation" ideas without ever making any predictions. Do you think you're living in the matrix? Or do you just think that quantum mechanics can be fully simulated on a classical turing machine? Here is the almighty Scott Aaronson to set us right on the subject: http://www.closertotruth.com/series/the-cosmos-computer#video-1677 http://www.closertotruth.com/series/the-cosmos-computer#vide... http://www.closertotruth.com/series/what-does-quantum-theory-mean#video-1680 http://www.closertotruth.com/series/what-does-quantum-theory... EDIT: lets throw in this full length lecture at IBM for good measure, even though its not 100% related https://www.youtube.com/watch?v=Z7lv4-Bah5c https://www.youtube.com/watch?v=Z7lv4-Bah5c
- selimthegrim 12y ago
- xvilka 12y agoI think this can be connected to the recent work (more like a side product of the main paper) on the ABC Conjecture by S. Mochizuki. [1] http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture http://mathoverflow.net/questions/106560/philosophy-behind-m... [2] http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture http://michaelnielsen.org/polymath1/index.php?title=ABC_conj...
- cschmidt 12y agoInteresting this is a reprint from Quanta magazine, from March 12. https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/ https://www.quantamagazine.org/20150312-mathematicians-chase... The original version has some photographs missing from the Scientific American verison.
- sctb 12y agoThanks, we updated the link from http://www.scientificamerican.com/article/mathematicians-chase-moonshine-s-shadow/ http://www.scientificamerican.com/article/mathematicians-cha....
- littletimmy 12y agoThis is so exciting. Pity one has to devote one's life to study such things to even start to make sense of these discoveries.
- myth_buster 12y agoWell if it excites you then perhaps you can take it as a hobby or side project and get some satisfaction and sense of achievement as you progress on this path.
- littletimmy 12y agoYeah that's what I do, actually. I sneak in a little reading of math from time to time on lunch breaks or after work.
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- rrss1122 12y agoString theorists should stop referring to themselves as physicists and just accept they're pure mathematicians already. String theory makes for an interesting branch of mathematics, but an awful excuse for a physical theory.
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- j2kun 12y agoThe problem is physicists tend to use "intuition" to prove things. I don't know any string theorists, but if they were trained to think like physicists then it's probably not any different.
- yk 12y agoWell, right tool for the job. So 90% of the time math is used in physics to either calculate some quantity, like a scattering amplitude or the position of a planet, or to build models. In both cases you are right. But there is the entire area of mathematical physics, where the goal is formal proofs. ( Admittedly, mathematical physicists are quite often more in the business to get in the position of being able to write down a formal proof, rather than actually proving theorems. )
- privong 12y ago> The problem is physicists tend to use "intuition" to prove things. That is not correct (or I am completely misunderstanding what you intended to say). There's a huge component of physics that is experimental, and those experiments are used to develop and test models and theories. Initution comes into play when developing theories or thinking about new directions to take experiments, but intution is not and cannot be a substitue for experimental evidence.
- chriswarbo 12y agoThere's a distinction between "mathematical proof" and "empirical proof". Physicists are certainly good at empirical proof, but I think the 'appeal to intuition' complaint was aimed at the mathematical side (ie. the Maths in Physics isn't very rigorous). As a former Physicist and current Computer Scientist, I would agree with that complaint (although our use of empirical evidence is FAR below that in Physics). For example, in Programming Language Theory it's amazing how many different notions of "equal" there are (isomorphism, definitional, judgemental, propositional, extensional, etc.). In Physics, there's just "=" :)
- oafitupa 12y agoMath and physics look like a lot of fun. Does anyone know if the real fun only starts after like a decade of studying?
- saraid216 12y agoIt depends on what you find fun. I get a lot of enjoyment out of simply solving problems that I find in the wild. (I spent some time proving that my particular walking pattern was, in fact, more efficient than an alternative because the actual distance traveled was shorter.) If that's sufficient for you, then all it requires is a little studying and a little imagination. If you want to contemplate the cutting edge, then yeah, a decade of studying is necessary mostly because it's hard to actually comprehend all the implications on the cutting edge until you've done so. On the upside, there are plenty of MOOCs on physics and math these days; if you find a set you like, you can absorb plenty that way.
- itsjareds 12y agoI guess we find the same problems in the wild enjoyable. I spent a few nights procrastinating on homework to brainstorm how I could try to find the most efficient walking path between points on campus. I figured that I would need to be able to represent the campus on some sort of plane where each point has a value referring to its elevation, and then finding the geodesic. I figured I could refer to some physiology literature and find if anyone has tabulated average energy expenditures for walking at different grades (downhill, flat, or uphill). Now, my math skills are really sub-par (haven't gotten past single variable calculus), so this question had me asking various physicist friends how to solve the problem. I just found myself learning along the way. I'm curious how you determined efficiency for each of the candidate walking patterns. Did you compare only the traveled distance, or did you also take into account the energy spent per unit distance? I think that could make a difference if, for example, you had these walking strategies: Walking strategy S => A "normal" human gait, except you travel in an squiggle path (i.e. not a straight line) to your destination. Walking strategy T => Do continuous jumping jacks while walking, but continue in a straight path. S may travel a longer distance, but will exert less energy overall and therefore be more efficient than T. Now, that's a pathological scenario, but I wonder if your walking pattern could have the same issue where you are actually exerting more energy despite walking a shorter distance. It'd be interesting to do more research on the biophysics of how your body moves.
- alricb 12y agoThe M24 and Monster groups mentioned in the article are examples of the "sporadic" groups, which are 26 exceptional finite simple groups. John Conway (mentioned in the article, inventor of his game of life) discovered four of these groups, Co0, Co1, Co2 and Co3. It's notable that the Mathieu Groups (like M24) were discovered in the 19th century, long before any other sporadic group was known. http://en.wikipedia.org/wiki/Sporadic_group http://en.wikipedia.org/wiki/Sporadic_group
- crb002 12y agoThere is nothing mysterious. Endofunctions on sets are transformations. Think permutations but you are allowed to have repeats. If you iterate them f(x), f(f(x)), f(f(f(x))) ... the function forgets elements until you have a stable partition which cycles. http://chadbrewbaker.github.io/combinatorics/transformations/permutations/2014/08/07/endo.html http://chadbrewbaker.github.io/combinatorics/transformations...
- sweezyjeezy 12y agoThat in no way explains why these K3 surfaces have anything to do with mock theta functions...
- crb002 12y agoIt explains why iterated functions turn into group like structures and the Mosnster is present.
- dimillian 12y agoSo I think this is more a philosophical question but ... As humans, we created mathematics, this is something we created from "nothing"(?) in order to explain, rationalise things we observe. Now, how can we assume that the universe is logical, and that it can be explained by mathematical equations that WE create? This is bugging my mind every time I think about it.
- hliyan 12y agoAs humans, we created mathematics That depends on whether we are talking about mathematics, the method of rigorous notation or, mathematics, the inquiry into quantities without reference to quantities of what (i.e. numbers). We created the former but discovered the latter. I'm neither mathematician nor philosopher, but there seem to be some a priori truths in mathematics (I hope I'm using that term correctly) which are not defined or created by us: the ratio between a circle's circumference and diameter (Pi), the distribution of prime numbers, etc. If you are bugged by where those come from, then I don't think you're alone!
- AnimalMuppet 12y agoIf our brains are kludges hacked together by evolution, it is very difficult to understand. If God is a mathematician, and he created humans in his own image, though, then it makes perfect sense.
- AngrySkillzz 12y agoThis question is usually called the philosophy of mathematics[1] which is distinct from mathematical logic and has essentially no bearing on the actual practice of mathematics. Some people are Platonists and think that mathematics exists as a sort of idealized mental realm (similar to Plato's theory of forms, but more reasonable in this context). If this is the case, we would say that mathematics is mind-independent and trans-universal; mathematical truths are true regardless of whether we exist to observe them, and they are true in all possible worlds. Others think mathematics is a linguistic "game" played by man, that it is mind-dependent and completely of our creation. Still others think mathematics is a set of derivations within a "formal system" (or formal language); this is sort of a middle ground, as it implies that mathematics is a construct of man but that it has a weaker mind-independence/trans-universal property, namely that any entity in any universe examining the same formal system comes to the same conclusions. But as I said, how you feel about this question has no effect on the real-world practice of mathematics. Most mathematicians you ask will have some opinion on the matter, but it's not like you can say "I'm a Formalist so I don't believe in that theorem." The human process of doing mathematics and the validity of theorems with respect to their assumptions has no relation to the questions of the ontological or metaphysical status of mathematics as a whole. [1] https://en.wikipedia.org/wiki/Philosophy_of_mathematics https://en.wikipedia.org/wiki/Philosophy_of_mathematics