A mysterious connection between number theory, algebra and string theory?
quantamagazine.org
quantamagazine.org
> Ken Ono, of Emory University in Atlanta, Ga. “Without either letter, we couldn’t write this story.”
Weird, and fascinating!
String theory makes for an interesting branch of mathematics, but an awful excuse for a physical theory.
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.
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 "=" :)
"the Maths in Physics isn't very rigorous"
Well, of course it isn't, because of the physical limitations. You can't expect to get Newton's second law and make it work with any kind of mathematical object. (if that's what you mean)
> In Physics, there's just "="
Scalar equality? Vector equality? Magnitude equality?
Also leptons are 'equal' following the Pauli exclusion principle?
Yes. Intuition is a good thing. It just shouldn't be relied on for proofs. To compare, Mathematicians followed their intuitions to bring us all kinds of results (eg. ). It's a good thing. It just shouldn't be used to prone to making mistakes when but shouldn't be relied on for (Mathematical) proof.
> You can't expect to get Newton's second law and make it work with any kind of mathematical object.
That's not rigour, that's generality. For an example of rigour, compare the treatment of (infinitesimal) calculus in Physics and in pure Mathematics (eg. see http://en.wikipedia.org/wiki/Calculus#Foundations ).
>> In Physics, there's just "="
> Scalar equality? Vector equality? Magnitude equality?
I would say the first two are "instances" of the same principle (in the same way that, for example, intensional equality can be "instantiated" for integers, arrays, matrices, etc.). Magnitude equality composes the absolute value operation with "the" equality operation, so I would call it a shorthand rather than a distinct equality principle.
> Also leptons are 'equal' following the Pauli exclusion principle?
What an excellent example of a non-rigorous statement ;) (ie. what is your model of leptons?)
For reference, some of the examples I mentioned are described at http://ncatlab.org/nlab/show/equality
> What an excellent example of a non-rigorous statement
That is completely uncalled for. The difference between leptons (as a kind of fermion, which obey the P.E.P.) and bosons (which do not) is no more and no less than the different definition of identity/equality in the two cases. Rigorously.
http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistic...
https://www.quantamagazine.org/20150312-mathematicians-chase...
The original version has some photographs missing from the Scientific American verison.
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#vide...
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
That all physical systems are governed by equations rather than, say, gnomes pulling levers.
>Or do you just think that quantum mechanics can be fully simulated on a classical turing machine?
This seems quite reasonable. There's no good reason this couldn't be the case, and quantization of fundamental units like Energy-time/Momentum-distance is certainly something that a programmer might reasonably do for a simulation.
If your glass is half full, then of course we are living in a simulation. What is the halting problem for this simulation? That's the real question. IMO that should be true AI.
Not entirely different from the idea of being able to convert from magnetic to electric field and vice versa via a collection of moving bits and bobs as discovered a century or two ago.
It could be useful, in some peculiar way. Or maybe not.
Or blockchains.
> as discovered a century or two ago
Give it up to Michael Faraday!
Humorously, for all we know we're surrounded by entities which can perceive a lot. But we're as aware of them as slugs are of us when confronted by our shoes...
One thing about our math sense is that it seems fairly underdeveloped in comparison to some other abilities; maybe physics is currently dominated by math because we have to work harder at it. Otherwise, the fruits would've come to us earlier.
I doubt we'll ever get the answer to the first one in this life/physical realm. Philosophy can help ponder over the second one though.
From the Stanford Encyclopedia of Philosophy: 'why not?'
http://plato.stanford.edu/entries/nothingness/#WhyTheSomRatT...
If I remember correctly, one of his answers was something like "Because it's the only Universe where we could ask such a question". I think what he means is that all the other possible universes (or most of them) might or might not exist somewhere else, but they certainly don't include intelligent life with the ability to ask "Why is there something instead of nothing?". When you ask that question you are assuming this is the only universe and it was "meant to be". But our sole existence is not evidence at all for a "destination".
"What do I do now" is a complete different question, and I guess people might find their own personal answers in the most varied places. You obviously don't need a physicist for that, although lots of them also think about these things.
The basic response to this question is "What should you do?"
Western civilization is founded on a particular justification for answering this question.
If the effectiveness of math is unreasonable, then the universe is either not logical, unreasonably simple in its logic or math is an unreasonably bad language to investigate facts in.
But of course, the question of why the universe has this particular complexity (it's possible to conceive of life with either simpler or more complex universes) may not be a questions that's "answerable" at all -- there must be some ultimate system in which things to happen (almost by definition), and it must be ultimately arbitrary.
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.
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.
> unless your argument is that mankind's idea of God also corresponds to the reality of what exists outside our heads
But all men do not agree on what God is at first place, or whether he exists or not. It doesn't make the idea of god less powerful(that's my point), however if you define "reality" as what men can experience, no men has ever experienced "God" and created a reproducible experiment of God. Therefore "God" cannot be a reality today. God is a philosophical question.
No, in that God is not (just) a philosophical question. There is a reality which exists. Either there is a God who actually exists in reality, or there is not. It is a question of what is the truth about what actually exists, not just a philosophical question.
Well, which/what "God"? the definition of God itself is purely a philosophical question. You can't ask yourself whether "God" exists or not, until you define precisely what you mean by "God". And men do not agree at all on what "God" is, even those that claim they follow the same religion. That's why you can't ask "science" to answer a question on a matter that has no formal definition to begin with. God is an idea first and foremost. a vague idea at best that bear no exact definition, thus a philosophy.
Mathematics is not the real world; it's a map, not the territory. But of course it's not surprising if a map resembles the real world...
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...
You might also want to read about Euclidean and Non-Euclidean geometries and the history of the parallel postulate for how humans have grappled with this question in the past, and how our thinking has evolved in the last 150ish years. This is a fantastic book with a mix of math, philosophy, and history:
http://www.amazon.com/Euclidean-Non-Euclidean-Geometries-Dev...
(I'm sure you can find a used copy for a lot cheaper than that link!)
It's sort of an open question whether mathematics is invented or discovered.
"The opposition between the analytic and the synthetic approach to mathematics in the first half of the nineteenth century is well-known.. Less well-known is that both distinctions were rooted in a cultural clash, in the period of the first French Republic, between the established analytical tradition, guided by Lagrange and Laplace, and a new, geometrically-oriented approach, swayed by the revolutionary upstart Monge. These mathematicians had been assigned by the government to normalize and rectify mathematics to a perfectly transparent and hence universally learnable 'language'"
Gaspard Monge was the Director of Ecole Polytechnique, the pioneering French military engineering school, educational predecessor of West Point, MIT and others (http://www.uh.edu/engines/asmedall.htm).
"Monge... expressly rejected the reduction of mathematical reasoning to a formalism. He insisted on the indivisibility of form and content, and denied that any rules, mechanical or otherwise, could be given for the conduct of mathematical investigations. For him, analysis was not a language, closed in itself, but merely the 'script' for the notation of reasonings about quasi-empirical, especially geometric contents."
All we can do is observe that the behavior of an object in the physical world behaves as the model predicted. With enough observations and enough predictions, we can say with some certainty that the model is accurate. We invented the model, then we tested it and confirmed that it approximates reality, usually within a known range of uncertainty.
The universe is still logical in that causality is relatively logical. We're just describing the universe as we see it.
As humans, we created mathematics
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!
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.
I'd argue a number is just an idea, and all ideas exist whether you've thought of them or not.
Therefore the practice of mathematics can only take place in a universe where universal, repeatable, rule-following procedures are possible. It has effective regularity in the cosmos as a dependency.
Looked at this way, the thing to be surprised at is not so much the effectiveness of mathematics at describing the universe, but the universe's admission of regularity - once that is granted, both mathematics and its effectiveness seem to follow unproblematically, imo.
Perhaps phrased in a more rigorous way, the universe is a spectacularly huge and complex system, and a single human brain is a positively tiny piece of that system, which is very much a part of the system and not external to it. So the question really becomes something like this: to what extent can a tiny little piece of a huge system contain within it a fully functional model of the entire system, which inevitably means it contains a model of itself?
Mathematics is not unreasonably effective any more than English is.
Some of these patterns appear so fundamental that it's virtually impossible to imagine them (in a detailed way) as being different. Maybe this is because our brains, being physically embodied, are constrained by these same properties of nature in terms of how they process information. We can't picture 1+1=5 because we literally can't think it.
Once you have a language with a grammar, you can also start exploring the "pure" properties of the language. Writers do this with written languages, constructing oddities like Ulysses and "a rose is a rose is a rose" and buffalo^8: https://en.wikipedia.org/wiki/Buffalo_buffalo_Buffalo_buffal...
Since the language was induced from nature, exploring its structure can sometimes let you deduce very strong and powerful hypotheses about nature. But these hypotheses are not true (a.k.a. theories) until they are confirmed by experiment or observation. The combinatorial rule space of a language is so large it will contain an effectively infinite number of meaningless coincidental patterns, and Turing proved that the halting problem is undecidable so you can never be "done." The inverse is also true: there will always be facts of nature that cannot be hypothesized by studying any language or logic system. This is the primary consequence of Godel's incompleteness theorem.
TL;DR: Languages are induced to describe reality, therefore you can make conjectures about reality by playing with them. But not all such conjectures are true, and some things must be true that cannot be thus conjectured.
Edit:
Come to think of it, I am not certain that "The combinatorial rule space of a language is so large it will contain an effectively infinite number of meaningless coincidental patterns" is true, or at least I'm not aware of a proof of this akin to Godel's theorem (wouldn't this be the inverse of Godel's theorem?). It could be that all theorems and patterns in mathematics (and other languages for that matter) either directly reflect something in nature or are isomorphic with something that reflects something in nature.
If this were true it would be impossible to make a meaningless statement that is syntactically correct in any language. Tolkien's endless discussions about orcs and elves are talking about something, just maybe not literal orcs and elves.
[1] http://mathoverflow.net/questions/106560/philosophy-behind-m...
[2] http://michaelnielsen.org/polymath1/index.php?title=ABC_conj...
It's notable that the Mathieu Groups (like M24) were discovered in the 19th century, long before any other sporadic group was known.
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!
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.
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.
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.