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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?

#61

Post by chriswarbo »

raverbashing wrote: post #49
chriswarbo wrote: post #45

Earlier quoted context omitted.

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 th…

> 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 ).

>> 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

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

#62

Post by wilsynet »

AngrySkillzz wrote: post #58
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!

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.

That's right. For example, you can construct the integers Z without writing all of them down. Which would take a long time ...

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

#63

Post by snarfy »

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.

Did we create mathematics, or discover it?

I'd argue a number is just an idea, and all ideas exist whether you've thought of them or not.

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

#64

Post by BrainInAJar »

heed wrote: post #36
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, we discovered it.

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.

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

#65

Post by AlexCyrlex »

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?

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

#66

Post by rwissmann »

AlexCyrlex wrote: post #65

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?

Fairly sure that is indeed a typo, in particular given how F+ is introduced a few lines above.

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

#67

Post by kordless »

VLM wrote: post #48
kordless wrote: post #40

Earlier quoted context omitted.

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 electr…

> limitations and all kinds of big O notation results

Or blockchains.

> as discovered a century or two ago

Give it up to Michael Faraday!

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

#68

Post by DougMerritt »

chriswarbo wrote: post #61
raverbashing wrote: post #49

Earlier quoted context omitted.

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 th…

> 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 m…

>> 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.

http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistic...

http://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics

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

#69

Post by taeric »

VLM wrote: post #46
dimillian wrote: post #38

Earlier quoted context omitted.

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

But would you instead argue that you discovered how to draw the map?

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

#70

Post by aikah »

amelius wrote: post #37
heed wrote: post #36

Earlier quoted context omitted.

We didn't create mathematics, we discovered it.

"God made the integers, all else is the work of man." -- Leopold Kronecker

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.
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