There is a question that came to my mind that I'd like to discuss here. I hope it is clear what I want to express since English is not my mother tongue.
Since I have started studying mathematics and physics there is one thing I have noticed especially in comparison to the math-courses I've attended in Gymnasium (High School equivalent). It is the lack of calculations. And by calculation I mean things like $2+2$, $\int_a^b dx (f(x)) $
Whereas in physics I have moments where I need a calculator, in maths the closest thing I've come to that is by looking for a derivative. And since I have been attending mathematics I have encountered a rather interesting view on the relation between calculation and maths by asking other mathematicians what they think of that.
It is a view that sees calculation as an offspring or even just a byproduct of mathematical work. A view that claims that mathematics just examines, defines and checks. It is somehow like a searching for new things within its boundaries. In case of simple arithmetic operations in $\mathbb{R}$ that would mean that the real mathematics lies in the pure construction of the logic behind. Same for those math text problems everyone knows from High-School.
But is this perception really reasonable or just too extreme?
I would argue that indeed it is too extreme. One may even need simple calculations for examining new theories, proving or disproving them the most evident example being to find a counterexample.
But what if one adds the assumption that mathematics only uses calculations as a means to an end? That would not exclude all practical calculation from mathematics but most done for example in High-School or financial mathematics, physics etc. Solving those math text problems in High School would not really be mathematics then. It somehow would be like the relationship between physics and engineering.
So what do you think?
Thank you very much and every answer is appreciated,
FunkyPeanut
(P.S.: I kind of don't know what tags to include here….)
Best Answer
Calculations can serve as part of a whole of a mathematical work and in rare instances are the mathematical work itself (some very complicated counterexamples in Banach space and $C^*$-algebra theory have been like this). For instance, a paper I just put up on arXiv had some pretty grueling calculations. I was defining a family of integral operators on a dense subset of $L^2(\mathbb{R})$ and wanted to establish that they are isometries. This required quite a bit of really unfortunate calculation but there was no reasonable way around it. Then by extension theorems, the operators could be lifted to all of $L^2(\mathbb{R})$. In general this is a pretty hard thing to do and some degree of calculation is needed. However many computations do not serve as a true mathematical work. Like most things in life, there isn't much black and white; it's all shades of gray.