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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....)

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3 Answers 3

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.

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One thing which has happened more than once in mathematics is that the calculations which made a breakthrough have been replaced by cleaner concepts after a while. These days Noetherian Rings and Modules are everywhere. Some of the modern Algebra proofs which now look easy because of the Noetherian concept were once fiendish calculations of exciting new results, now they are corollaries and footnotes. Sometimes areas of mathematics resolve in this way - other times, the calculations seem to be an essential part of the process. –  Mark Bennet Apr 3 at 21:01
    
Just a question out of curiosity: are the "calculations" you describe of the kind mentioned in the question: writing down an expression and setting out to determine its value (taken liberally; the value might for instance be a character table)? My impression is that (research) mathematicians almost never mean that when they talk about calculations. Rather I think they usually mean transforming some symbolic expression by some tedious and mostly mechanical procedure into a form that allows making a conclusion. While being low-level and not giving much insight, it doesn't compute a value. –  Marc van Leeuwen Apr 4 at 7:23
    
@MarcvanLeeuwen Yeah the things I computed were simple integrals. It's not pretty but it works. –  Cameron Williams Apr 4 at 12:03

I think the question is a little unclear unless we specify what exactly is meant by calculation.

Using Karatsuba algorithm to calculate the product of two natural numbers is not really a part of mathematics. Proving that it actually works is. Calculating the definite integral of a polynomial function by using the known antiderivatives of monomials is not mathematics, showing that the antiderivatives are what they are and that they can be used to calculate the integral is.

Generally, evaluating an expression once we have an algorithm for evaluating it is not mathematics, but developing an algorithm which does that is (whether it does so in general or just in some particular case). Just because we use computers to prove the four-colouring theorem doesn't mean that the computers are part of the mathematics. That's what I believe.

Of course, sometimes we have to actually apply some algorithm to prove something, but it doesn't mean that the actual calculation is a part of the mathematics. It is that no more than measuring length is a part of physics.

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I think it's much more to do with the definition of mathematics, not calculation. I would say that calculation is mathematics simply because a calculation is the application of the rules of logic to deduce a statement which follows from our axioms and definitions. A calculation is technically a theorem, even if it may not be particularly ground breaking or offer a deeper understanding of mathematics. –  Daniel Rust Apr 3 at 20:46
    
@DanielRust: I don't think it's a good idea to try to find a technical definition of mathematics (except for the purpose of giving money to mathematicians ;) ). Anyhow, I don't believe executing an algorithm is a part of the mathematics as I understand it, but rather applying a tool. Applying tools is not mathematics, knowing which tools to apply is. –  tomasz Apr 3 at 20:52
    
So do you disagree with the statement that calculations are proofs of theorems? –  Daniel Rust Apr 3 at 20:55
    
@DanielRust: not on their own, in general. Whether they are ever really would depend on what you mean by calculation, exactly. –  tomasz Apr 4 at 1:19
    
"Theorem: the first 6 digits of the decimal expansion of $\sqrt{2}$ are $1.41421$. Proof: Using a suitable square root algorithm we calculate the following [insert claculation]. Given that we know the above algorithm is guaranteed to produce the correct output (from a previous proof of this statement), it follows that the result holds. QED." The above constitutes a valid proof and the proof method was direct calculation. Now, normally we wouldn't call the above a proof because it's not the most illuminating proof ever, but the point remains that it is a proof. –  Daniel Rust Apr 4 at 12:28

Calculations are a tool that mathematicians employ to determine what is true.

Let me back up. What is mathematics really? Anything which you can precisely axiomatize and reason about falls within the domain of mathematics. Doing mathematics is reasoning about these axioms and their consequences. Thus you can view chess as mathematics, or any result of physics which proceeds from some set of assumptions (or laws) to a conclusion merely by means of logical argument. (Not everyone agrees with this view of course).

Calculations, as I say, are tools which help with this process. But they are also conclusions themselves since they follow from the axioms. We only prove such things once and then use the result as a shortcut thereafter, but doing so does not remove them from the realm of mathematics.

In some sense, all ends in mathematics are means to other ends. There is no question which mathematicians want to settle exclusively for its own sake. There are always more questions that follow; we always want to know what forthcoming results portend, and in some cases make contingent arguments in anticipation of those results.

A good analogy might be the following: children practice spelling, vocabulary and grammar in school. New words may come into existence, new spelling conventions, and even new grammatical structures (over longer periods of time). All of this is part of language, but is concerned with developing the fundamental tools required to understand language and even make new linguistic innovations. It represents the goals of fluency and comprehension, and is the means to mastery and communication.

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