A mysterious connection between number theory, algebra and string theory? • Hacker News Board

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

#51

Post by j2kun » Wed, Apr 08, 2015, 4:54 PM UTC

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…

Nobody is criticizing the results of physics. But you can't call them mathematicians unless they're giving mathematical proofs.

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

#52

Post by ruggeri » Wed, Apr 08, 2015, 5:05 PM UTC

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.

Kinda both, right? Mathematics is a human-created formalism inspired by, and intended to explain, the natural world.

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...

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

#53

Post by pjonesdotca » Wed, Apr 08, 2015, 5:09 PM UTC

tormeh wrote: post #13

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.

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 no…

I wish I could upvote this response more than once.

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

#54

Post by hliyan » Wed, Apr 08, 2015, 5:15 PM UTC

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.

   As humans, we created mathematics

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!

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

#55

Post by yk » Wed, Apr 08, 2015, 5:23 PM UTC

j2kun wrote: post #27

rrss1122 wrote: post #21

String theorists should stop referring to themselves as physicists and just accept they're pure mathematicians already. String theory makes for an interesting branch of mathematics, but an awful excuse for a physical theory.

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.

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. )

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

#56

Post by AnimalMuppet » Wed, Apr 08, 2015, 5:56 PM UTC

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.

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.

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

#57

Post by 31reasons » Wed, Apr 08, 2015, 6:06 PM UTC

>>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!

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

#58

Post by AngrySkillzz » Wed, Apr 08, 2015, 6:11 PM UTC

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.

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

#59

Post by AngrySkillzz » Wed, Apr 08, 2015, 6:15 PM UTC

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 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.

[1] https://en.wikipedia.org/wiki/Philosophy_of_mathematics

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

#60

Post by phkahler » Wed, Apr 08, 2015, 6:26 PM UTC

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?

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.