Mathematicians of Veeky Forums:

Mathematicians of Veeky Forums:
How do mathematicians know what to go after? In physics I can see it: you see a phenomenom and you get an insight like that stuff about the apple falling on newton's head or you see a phenomenon and make some basic questions to yourself.
But I can't see how Mathematicians get their inspiration I don't even understand how they even start finding out the things they do. It's like it comes out of nowhere and it's just a bunch of disconnected rules. It's like: How do you even start thinking about roots or those other properties?

Your image gives away your troll.

How is my post trolling? I'm asking a genuine question. I also like Wildberger.

user, in mathematics a believing heart is your magic.

Someone said it was something about defining variables and equating certain reactions with those variables. a = hit b = movement
a + b = speed

something like that I believe. Although the hard part is working out the kinks and the logic behind your equations most likely.

What does that even mean my man? Are Mathematicians connected to other realities than us? Do you guys just dream about this stuff? Fuck why don't mathematicians answer this shit. I want to see a mathematician recording what he does to arrive at a proof or some new concept.

Mathematicians attempt to discover patterns in nature while finding correlations between different phenomena. For example, the famous E=mc2, which tells us that energy is equal to mass.

>What does that even mean my man?
If you believe it in your heart and can imagine it, it is real.

>Are Mathematicians connected to other realities than us?
Yes

>Do you guys just dream about this stuff?
Not always but mathematics dreams are always the best dreams.

>I want to see a mathematician recording what he does to arrive at a proof or some new concept.
It's called a book lol come the fuck on man.

physicist*

No man. I'm talking about the guy who comes up with the stuff that makes writing a book possible in the first place. I'm talking about how did Galois and other mathematicians find the stuff he did?

What I'm asking is what made he even start exploring in the direction he was going. What were the questions and motivations. For someone like me it looks like it's something out of nowhere.

But nevermind. It's not like I'd understand it anyways. Life sucks.

Everyones brain is better at certain things than others. Some are just great at math, others spatial reasoning, deduction, etc. What do ypu think yours is good at?

When you have a hard problem and a lot of time, crazy stuff starts happening.

Galois, as everyone, was trying to solve a really hard problem and he had a couple of years to work on it.

>How do mathematicians know what to go after?

Mathematicians, like physicists, want to find answers to interesting questions posed by themselves or other people. Like a physicist would maybe ask, how to planets orbit about the sun?
The mathematician would ask, what is the general form of the solution to a polynomial equation?

Then physicists would look to find general patterns in the orbits of planets a find a fitting equation. The mathematicians would start playing around with polynomials to start finding the roots expressed in terms of the coefficients.

But then the physicists find that there are unexplained phenomenae that occur that cannot be explained by the proposed equations of gravity. Similarly, the mathematicians find that there simply are no general solutions for degree 5 or more polynomials.

Then physicists start thinking, hey, maybe there is something more fundamental than a simple square law to gravity? It would help if instead of going forward, we check to see if the fundamentals of our theory were right to begin with. Analogously, the mathematician may ask, where do rules for polynomials come from? What determines its roots? If we make our theory more abstract and generalising, we could find patterns and rules that we couldn't think of before.

Then physicists discover special and general relativity, which makes more sense and fits the data much better, and mathematicians invent field theory and Galois theory which explains and finds a way of finding solutions to general polynomials.

But then physicists discover that their rules might be wrong again. Is there something that their equations don't account for? What is this 'dark matter'? Similarly, a mathematician may ask, ok, we can solve these equations over the complex numbers. But can we solve them to find exclusively integer solutions? How does one come about this? Well, you generalise the integers to rings, and develop ring theory, which appears to solve some equations over the integers.

Alternatively, someone may be working in a different field for a change, and notices that one of the completely unrelated objects he's working with is extremely similar to something he's used before. Is there a link between these? Why not find out? pic related shows how two unrelated fields linking can be of importance.

Notice that everything you study in mathematics stems from a question somebody at some point asked, and the theory is developed in an attempt to answer it.

They are LITERALLY are devils born from the left hand of God. There comes a point when you see something...someone, with such a vicious aptitude for STEM fields that you can only conclude that it comes from a higher power.

go for another mans wife.

Rings were formalized after field and Galois theory? Odd, elementary number theory made it seem as if rings were a thing since forever, since it was what we started with (albeit the abelian requirement should probably key one in on its recent development, I just assumed the various requirements were implicit, which I suppose goes against my surprise at the lack of formalization...)

Fields were conceptualized by Gauss in the late 1700s, and used by Abel and Galois, while Galois conceptualized groups in the 1830s, but only formalized in the 1870s, and rings were only formalized in the 1920s.

Take into account that elementary number theory meant just that, until only recently. One of Gauss's many breakthroughs was the quadratic reciprocity theorem, and generalizing that up to general powers of n was the leading direction of number theory, until the conception of rings.

For example

You have a problem, in this case it is square roots of negative numbers. They occur when finding roots of polynomials, but that shit doesn't exist. So you invent i, with the definition that i^2 = -1. That brings us complex numbers, which bring us the complex plane. Then you find out a shitload of stuff about the complex plane. Then you can use all of the theory about complex numbers and apply it to, say, pure geometry problems, and it works. Bam, you just invented new mathematics

Basically inventions are made in order to deal with problems that can't be dealt with using the tools we have currently. Then you start to develop the new theory, new questions arise, you start linking it to other (seemingly unrelated fields of math), that brings even more questions with it, etc etc. I'm not a researcher but this is how I see it and I'm pretty sure it's correct

This is very interesting to me, user. I'd love for you to describe more about the importance of rings and the development of number theory, I'm only a freshman (haven't even had Algebra 1, which was a prereq for elementary number theory, but I took it despite being in an engineering-type major because I'm a madman/retard)

Any information is new to me, almost. Just learning that rings are a thing is something that only happened when I started number theory, so I'd love to learn as much about the history and development that you'd be willing to talk about

John Derbyshire has this magnificent anecdote about working construction as a math student. He's supposed to collect (or set out) ceiling tiles, for someone else to actually install. Instead he decides to balance the stacked tiles so that each tile is halfway on the last, seeing how many can mass-cantilever like that. This replicates one of the patterns associated with Euler's Number.

Rings are motivated from solving diophantine equations in the following way (as an example):

Suppose you want to find the solution to [math]x^2+y^2=M[/math], where M is any (known) integer. Solving this is very easy using elementary number theory methods, given that you can check the equation mod some small integers and see what happens, and using quadratic residues you can check which numbers can solve it or not, etc. But this method is very hard for exponents bigger than 2.

So to keep going with the same equation, we want to consider an alternate approach. In elementary number theory, we can solve the equation ax=b for x by checking the prime decomposition of a and b, and then we can see what should x be by comparing them. We want to use a similar approach to this problem. What if we extend the numbers that we're working with so that such a factorisation is possible? Luckily, there is a well-known factorisation [math](x+iy)(x−iy)=M=z\bar z[/math] over the complex numbers, so we only had to extend our integers by some object [math]i[/math].

Now we can do some work: first, we want to find all the prime factors of M over the extended integers; ie: we want to find all the factors of M in the form of [math]a+bi[/math] such that [math]a,b\in\mathbb{Z}[/math]. Next, we want to find all the factors that divide both [math]z[/math] and [math]\bar z[/math]. For example, if both [math]x[/math] and [math]y[/math] are even, then [math]2[/math] would divide [math]z[/math] and [math]\bar z[/math]. Then we can write [eqn]z\bar z=M= \pi_1^{r_1}...\pi_n^{r_n} p_1^{s_1} ... p_k^{s_k}[/eqn], where the [math]\pi_j[/math] are factors of both and [math]p_k[/math] are other factors. So we can find what z is by comparing the factors, and hence we get a solution for x and y.

This process is quite easy, albeit slightly computationally exhausting, and a very powerful method of solving diophantine equations algebraically.

Cont

So using this method, we can solve Fermat's Last Theorem: [eqn]x^n+y^n=z^n[/eqn]
We can factorise the equation into roots of unity, check the factors on both sides, and we get a contradiction on the possible powers of the decomposition (very lax here, cba typing too much more). Easy!

But of course not, there's a caveat: the extended integers generated by the 22nd root of unity has a slight problem; there are two different decompositions of numbers into primes! We can see this in an even simpler extended integer ring generated by [math]\sqrt{-5}[/math]: [math]2\cdot 3=6=(1+\sqrt{-5})(1-\sqrt{-5})[/math], and one can check that they are all "prime" in their ring. But here we must make a distinction; until we arrived here, prime and irreducible meant the same thing, so intuitively we think of primes as irreducibles. Here, the 4 numbers are irreducible, and not prime, which has a different definition.

So now think: we want to create an algebraic object that even when extended, it behaves like the integers: voilà, rings. Now, we want to check which of these rings has the property that every element has a unique decomposition, call these unique factorisation domains (UFDs). We find that not many rings are UFDs; in fact, it turns out most of them aren't, so this process fails badly.

This calls for something à la imaginary numbers; what if we did have some "imaginary object" that did always factorise uniquely in every ring, uhh, yeah I know what I'll call them: ideal numbers, or for short, "ideals". After a lot of work, one eventually finds that these ideals do have a lot of rich properties. And yeah, that's algebraic number theory's beginnings explained.

Damn, Satan, thanks for the thorough explanation! I truly appreciate it