I came across this quote by Eric Weinstein that "when things got supposedly more advanced, they actually got simpler because mathematicians started revealing what was powering all the things that you previously learn." I was wondering if anyone came to the same conclusion, and if so, care to give an example of such a revelation.
[Math] Does advanced math “power” more rudimentary math
educationphilosophysoft-question
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In many ways, I am atypical in the way that I approach a problem, but it works for me. Specifically, I try to understand an example in as much detail as I possibly can. If the example, is too complicated, then I make a simpler example. As much of the intricate detail that I can bring to bear on the example is brought.
For example, instead of trying to understand Lie groups and Lie algebras in general, start with the circle and the line that is tangent at the point (1,0). What is the exponential map? Oh, OK. Now how about $SU(2)$ and $su(2)$? Can you understand that the Lie group is the $3$-dimensional sphere? Can you understand the coordinates? Can you understand the equators? How do $i,j$ and $k$ really work? What is the difference between the multiplication rule $i\times i =0$ and $i^2=-1$?
I spend time pondering. And often my notebooks will contain tangential problems or specific computations. I will keep doing the computation until I get it right! If necessary, I will write a program to complete the computation. When I understand the example completely, it is usually easy to abstract.
Then I follow up, usually writing in a notebook or several notebooks before I begin writing on the computer. I have an advantage in that I have long-distance collaborators, so it becomes necessary to explain the idea to the collaborator(s). That is the first writing stage: write for someone who knows your short-hand and your metaphors. the second stage is to write for someone who does not. Then I write with a set of colleagues in mind, but I assume the colleagues do not remember anything from the previous work. I also try to explicate the notation writing for example "the function $f$, the knot $k$, or the tubular neighborhood $N$.
A complex analytical colleague only uses $z$ for a complex number, $x$ for a real variable, and $n$ for an integer. These variable choices are culturally determined, and so one keeps with the culture of the discipline unless there is good reason to deviate. As a final example of this, the variable $A$ in the bracket polynomial is known to everyone in the field. The variables $q$, $t$, $X$ etc. are less known and involve different normalizations. So it is the burden of the author to relate these to the more well known choices.
I'm a student of mathematics at the third year undergraduate level.
Lets begin with what's unsatisfying. Example are when:
- you perceive that a branch of math "isn't set up right", or
- a theorem is clumsily worded or clumsily proved, possibly because the writer hasn't had enough exposure to adjacent fields, or
- an author's definition is suboptimal for ease of expression, or
- terminology/notation in the literature is inconsistent and there seems to no way to fix the problem - its like you can't win,
etc.
On the other hand, the existence of a huge mathematical universe is not unsatisfying. It's wondrous. I love the fact that there's always more to explore, always harder problems to face... in fact, at any level of difficulty, there's always progress to be made - its like a video game set in an enormously massive world that is just full of secrets, and it just stretches forever.
It can become pretty addictive. I'll leave you with the following quote.
Yeah, I used to think it was just recreational... then I started doin' it during the week... you know, simple stuff: differentiation, kinematics. Then I got into integration by parts... I started doin' it every night: path integrals, holomorphic functions. Now I'm on diophantine equations and sinking deeper into transfinite analysis. Don't let them tell you it's just recreational.
Fortunately, I can quit any time I want.
Best Answer
This is true whenever you learn a "powerful tool" in high school maths. The two main examples that come to mind are trigonometry and calculus.
In trigonometry, we had to memorise a large number of trig identities. We would learn that $\tan$ is basically "defined by" $\tan(x) = \sin(x)/\cos(x)$, but apart from this it was all just memorising the double angle formulae and so on. Early on in university maths, I learned that $e^{iz} = \cos(z) + i\sin(z)$, and all of the trig identities become obvious. But even then, I did not really understand what $e^{iz} = \cos(z) + i\sin(z)$ meant. Eventually, I studied complex analysis, and the notion of complex exponentiation made sense. At this point, everything that flowed on from it became clearer. (As an added bonus, understanding the proof behind the residue theorem meant I was also finally able to see why various integrals I memorised for my physics courses worked.)
A second related concept was calculus. When I first "learned" that $\int \frac{\mathrm{d}x}{x} = \ln(x)$, I had no idea why. Most of integration was just memorisation in high school. In my first maths course at university, I was finally properly taught what functions were, and what some of their properties are - and about inverse functions. Now, together with the chain rule, and knowing that $x\mapsto \ln(x)$ is the inverse of $x\mapsto e^x$, the lecturer showed us a simple reason why the derivative of $\ln(x)$ is $1/x$ (assuming it is in fact differentiable).
$$1 = x' = [e^{\ln(x)}]' = e^{\ln(x)}\cdot \ln'(x) = x\ln'(x)$$
Thus $\ln'(x) = 1/x$, and so an antiderivative of $1/x$ is $\ln(x)$. Seeing the proofs of the fundamental theorem of calculus was fantastic as well. In high school I asked why integration was antidifferentiation, and I was told "of course it is", but it was never obvious at all. They seemed like entirely different concepts so I couldn't understand how everyone was so okay with the close way they're tied together. Studying real analysis really helped to understand why calculus works.
As for the maths I've learned at university, all of my courses were well designed so they didn't use any tools that weren't proven in the course/earlier courses. I can't give any higher-level examples for your question.