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A mysterious connection between number theory, algebra and string theory?

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Re: A mysterious connection between number theory, algebra and string theory?

#81

Post by aikah »

AnimalMuppet wrote: post #76
aikah wrote: post #75

Earlier quoted context omitted.

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 cr…

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.

Re: A mysterious connection between number theory, algebra and string theory?

#83

Post by api »

dimillian wrote: post #34

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.

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.

Re: A mysterious connection between number theory, algebra and string theory?

#84

Post by saraid216 »

eli_gottlieb wrote: post #50
saraid216 wrote: post #32

Earlier quoted context omitted.

> what do I do now ? The basic response to this question is "What should you do?" Western civilization is founded on a particular justification for answering this question.

Really? And what would that be?

"What you want."

Re: A mysterious connection between number theory, algebra and string theory?

#85

Post by littletimmy »

myth_buster wrote: post #28
littletimmy wrote: post #18

This is so exciting. Pity one has to devote one's life to study such things to even start to make sense of these discoveries.

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.

Yeah that's what I do, actually. I sneak in a little reading of math from time to time on lunch breaks or after work.

Re: A mysterious connection between number theory, algebra and string theory?

#86

Post by itsjareds »

saraid216 wrote: post #30
oafitupa wrote: post #25

Math and physics look like a lot of fun. Does anyone know if the real fun only starts after like a decade of studying?

It 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, t…

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.

Re: A mysterious connection between number theory, algebra and string theory?

#87

Post by crb002 »

sweezyjeezy wrote: post #47
crb002 wrote: post #29

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...

That in no way explains why these K3 surfaces have anything to do with mock theta functions...

It explains why iterated functions turn into group like structures and the Mosnster is present.

Re: A mysterious connection between number theory, algebra and string theory?

#88

Post by wz1000 »

ganeumann wrote: post #3

Both 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.

> 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.

Re: A mysterious connection between number theory, algebra and string theory?

#89

Post by crb002 »

31reasons wrote: post #57

>>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!

53! 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.

Re: A mysterious connection between number theory, algebra and string theory?

#90

Post by AnimalMuppet »

aikah wrote: post #81
AnimalMuppet wrote: post #76

Earlier quoted context omitted.

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 h…

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.
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