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

quantamagazine.org

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

#41

Post by john_b »

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.

This may be interesting reading for you if you find this question compelling.

https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...

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

#42

Post by pjungwir »

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.

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:

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.

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

#43

Post by walterbell »

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.

From Eduard Glas, Between Form and Function, Social Issues in Mathematical Change, http://logica.ugent.be/philosophica/fulltexts/42-3.pdf

"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."

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

#44

Post by exelius »

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.

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.

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

#45

Post by chriswarbo »

privong wrote: post #31
j2kun wrote: post #27

Earlier quoted context omitted.

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.

> 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 b…

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 "=" :)

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

#46

Post by VLM »

dimillian wrote: post #38
heed wrote: post #36

Earlier quoted context omitted.

We didn't create mathematics, we discovered it.

Is this backed? I mean, this is still and will forever be a debate, do we discovered or invented mathematics?

Geography analogy, some people can discover a continent for some bunch of people, but the only thing you can invent or create is a map.

http://en.wikipedia.org/wiki/Map%E2%80%93territory_relation

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

#47

Post by sweezyjeezy »

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...

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

#48

Post by VLM »

kordless wrote: post #40
wyager wrote: post #22

Earlier quoted context omitted.

>What would that even mean though? 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…

Don't forget worrying about hackers.

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.

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

#49

Post by raverbashing »

chriswarbo wrote: post #45
privong wrote: post #31

Earlier quoted context omitted.

> 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 b…

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 ex…

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)

> In Physics, there's just "="

Scalar equality? Vector equality? Magnitude equality?

Also leptons are 'equal' following the Pauli exclusion principle?

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

#50

Post by eli_gottlieb »

saraid216 wrote: post #32
johnchristopher wrote: post #10

Earlier quoted context omitted.

It may make much more sens but it doesn't answer two fundamental questions of mine: why is there something instead of nothing and what do I do now ? 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.

> 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?
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