There is a preconceived notion that mathematics is a subject for the gifted or very intelligent, that the whole idea is having hundreds of equations on a blackboard with symbols flying here and there. I’ll tell you here, that’s it not. When I think of mathematics, there are two parts for me: the idea and the execution.
The idea is the principle behind what I’m trying to accomplish and the execution is bashing the problem with compositions of known methods and tools.
When my study of mathematics began, I used to make a lot of mistakes in my calculations ( well I still do), but over time I started to realize that was mostly because I didn’t really segregate out the two steps mentioned before.
As in, I kept doubting what I was doing while executing the algorithms of the standard techniques. After much trial and error, I picked up that we need to understand the tools we are using, figure out a game plan and only then start the mindless bashing with algorithms of known techniques.
Critical points of change
The second big thing I’ve learned is the optimization principle involving the rates of change. The idea goes as follows : If we have a quantity-A dependent on another quantity-B, then the quantity-A is maximized or minimized under quantity-B right at the small interval where quantity-A becomes independent of quantity-B.
Simply put, when a maximum is occurring, it almost seems like ‘incrementing inputs’ into the system doesn’t exactly disturb equilibrium but just make your increment large enough and BOOM! Everything collapses.
Breaking da rulez
The third important life lesson is that the possibilities are endless if we simply expand our horizons. If we keep our view limited/ stick to the well-known rules, we may be in the way of discovering novel ideas. Take for example operational calculus discovered by Oliver Heaviside, the methods were totally unrigorous but it worked and gave him the desired results. When he was questioned about the methods, he gave the response:
“Shall I refuse my dinner because I do not fully understand the process of digestion?”
Being too rigorous and trusting of the rules can limit our thinking. The best way to think is to understand the core ideas behind the rules / the motivation behind them and work from those principles, rather than the conception of them in the form of rules.
Notation is something I used to care about a lot, that is displaying the symbols in the standard set of ideas which were taught. However, I realized that, unless you want to speak to someone about what you are doing, the way you present something is completely irrelevant!
You can write any symbol to denote an idea you wish, no need it for be the standard set. For example, instead of putting the elongated S for denoting integration, I could denote it as a symbol like Z before the quantity integrated.
The point being that anyone can illustrate any idea in any way of presentation as long as they have the correct conception of what the object actually is! It’s fun to see at times, there are actually newly proposed notations for certain mathematical operations such as exponentiation (*)
As long as you know the kind of object you are dealing with, the way you represent the thought is completely irrelevant!