And 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.”
The 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.
That's a bit of an oversimplification that I'd like to point out. Math proofs are certainly more rigorous and physics does rely on a bit more of an intuitive property in some ways, but that doesn't mean that intuition is the main source of proving things in physics. A bit of empirical evidence falls in there too. (Something mathematicians don't rely on as heavily :P)
> 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.
There'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 "=" :)
That "intuition" is what brought us Special and General Relativity
"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)
> That "intuition" is what brought us Special and General Relativity
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 ).
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?)
>> Also leptons are 'equal' following the Pauli exclusion principle?
> 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.
Well, 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. )
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.
Have some fun with mixing automata theory and the universe as a machine, where there's whole classes of machine that can or cannot be simulated by other machines or classes of machines and algorithms to convert from one machine to another or run one class of machine on others, and limitations and all kinds of big O notation results.
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.
Math, fundamentally, is just a standardized formal language in which certain facts can be encoded. This is in contrast to someone's random rigorous thoughts, which are not standardized. Math being standardized allows anyone doing math to apply a large set of ready made transforms to their math-encoded facts to derive new math-encoded facts.
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.
I disagree. It's not a scientific discussion to begin with, but given that life and consciousness exists at all it's to be expected that things are not too random, they must have some order. Mathematics just lets us sift through the simplest explanations of our physical world until we find one that fits.
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.
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 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.
Humans should have enough humility to accept that our limited cognitive abilities probably don't define the universe. If you had a certain mutation, you might in principle discover things out of reach of my mere math perceptions. Just like my number sense apparently lets me notice things out of the cognitive reach of cats. (Or so they let me think.)
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.
The first equation on page 8 of the arXiv paper 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?
You are quite correct. (Professional mathematician myself, and indeed a former grad student of Ken Ono, so I have at least a passing familiarity with the subject matter.)
So 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.
This 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.
Think of math as a more precise version of spoken/written language. We use math to describe the world around us in very precise terms. Ultimately though, it's impossible to completely abstract mathematics from language - something that should be taken into account by a discerning reader.
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.
Sometimes this is called "the unreasonable effectiveness of mathematics"---if you want something to Google for. I feel like the order of the world is vaguely miraculous.
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:
"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."
Yes, I think this is a philosophical question. The way that I've always come to terms with this question is that humans are a natural outcome of pattern in the universe. We are, after all, just atoms observing other atoms. Fortunately, we're rather complex patterns, so the emergent patterns that form from us are also complex. Math is just a language for describing pattern. The further you go into math, the more patterns on patterns you discover (that's the point of abstraction). To answer your question, I think it's some amount of both. There are basic patterns in the universe that allow for emergent patterns/behavior. So the mathematical equations that we create are just eddies in abstract, emergent pattern.
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!
If 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.
Did the Mandelbrot set exist prior to someone creating an image of it? I think so, so it was discovered. We're so used to fitting curves to data and things like that, that we think of mathematical objects as constructs rather than discoveries. By looking at something more complex like a fractal, it should be easier to see it the other way round.
If mankind could come up with Math and Physics, it's more likely than mankind made up God too. The modern monotheist God isn't that old ,roughly 3000 years. "History" itself is between 10,000 and 20,000 year old. Gods come and go as men do and undo them.
But the only value in "coming up" with physics is that it corresponds to the external world - that it's not just a game inside our heads. That is, I don't think you can use mankind coming up with physics to support your argument - unless your argument is that mankind's idea of God also corresponds to the reality of what exists outside our heads.
My point is mankind can come up with powerful ideas.
> 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.
Yes and no. Yes, you can't run an experiment to prove God to five sigma certainty.
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.
> Either there is a God who actually exists in reality
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.
I come at this the other way around. If there is a God that exists in reality, then that God is the relevant one. If there is no God that exists in reality, then none of them are relevant. Again, the driving question is what exists in reality, not philosophy.
we created mathematics as a logical progression from axioms. We invented the axioms (they're about whatever seems "reasonable" to us) and they're the structure on which we build everything in mathematics.
Mathematics, in one sense, is an exploration of universal, repeatable, rule-following procedures - if you can't get the same result as me, today, yesterday, and next year, by following my procedure, then whatever my procedure is, it can't be a mathematical one.
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.
You couldn't ever teach general relativity to your dog. It seems absurd to me to think that there is no theory about the world which is unteachable to humans. We're not that different from dogs in the long run.
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?
We didn't create mathematics from nothing. It's a language that we created to describe certain patterns that we saw in the universe, beginning with elementary patterns such as counting and working up to things like calculus.
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.
The 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.
There 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...
Well 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.
>>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!
That'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.
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 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.
> Ken Ono, of Emory University in Atlanta, Ga. “Without either letter, we couldn’t write this story.”
Weird, and fascinating!