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I visited this link: http://www.ams.org/journals/mcom/1950-04-030/S0025-5718-50-99474-9/

And I a bit confused by its title "Mathematics of Computation".

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I am not a native English speaker. Could anyone tell me what does this phrase really mean? What's the difference of:

  • Mathematics
  • Calculus
  • Computation

And how could Mathematics be used together with Computation?

I think this could help me get a more deep understanding about What math is. Thanks.

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1  
Back in the day, the journal was called "Mathematical Tables and Other Aids to Computation" (MTAC), since the journal was concerned with methods for hand calculation. Usually this meant tables generated by computer, or descriptions of algorithms that can easily be used by "computers" (in the original sense of the word). After computers became more prominent, AMS changed the title to its current form. As might be surmised, computational methods (usually for numerical analysis or number theory) are the sort of topics this journal deals with. – J. M. Feb 20 at 5:34
    
Mathematics isn't computation. Mathematics is proving deep relations among numbers and other objects, so we can predict the result of computations without actually doing them. – Federico Poloni Feb 20 at 7:53
  • What is $42\times 31$?
  • Why should $\underbrace{42+\cdots+42}_\text{31 terms}\vphantom{\dfrac\int{\displaystyle\int}}$ be the same as $\underbrace{31+\cdots+31}_\text{42 terms}$, and similarly for other pairs of numbers than $42$ and $31$?
  • How efficiently can one calculate things like $42\times31$?

The first item above is a problem of computation.

The second is a problem of mathematics.

The third is a problem in the mathematics of computation.

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So, computation is about the meaning/purpose/rational of some operation. mathematics is about the relation among operations. And mathematics of computation is about techniques to carry out such operations. Am I right? (I deliberately use the word operation 3 times here.) – smwikipedia Feb 20 at 2:08
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Mathematics includes far more than just relationships among operations. Mathematics of computation can ask and answer all sorts of questions about techniques of computation. In mathematics one might ask how many prime numbers there are, and how one knows the answer, what is the limiting probability distribution of the number of times a "head" appears when one tosses a coin $n$ times, under what circumstances the integral of the limit of a sequence of functions equals the limit of the sequence of integrals, and many other things. $\qquad$ – Michael Hardy Feb 20 at 2:12
    
Thanks, I just try to summarize your examples into some kind of formal definition. – smwikipedia Feb 20 at 2:44
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Unfortunately, the journal Mathematics of Computation has articles that are more about "mathematics, using (computer) computation" than the usual English meaning of the phrase "mathematics of computation" you outline here. – Mark S. Feb 20 at 5:30
    
@smwikipedia: Your general intuition about the relative differences between the terms was correct, even if the actual definitions you gave were pretty far off. – Eric Stucky Feb 20 at 10:13

It seems to me that this journal deals with computational mathematics and numerical methods. We're talking about applied mathematics in the realm of computation. Computation in this sense means calculating something. It does not necessarily imply that we are using a modern, transistor based PC, but that is often related.

Mathematics is a rather exact subject, but when it comes to actually computing numerical values, we are faced with challenges. As humans, our solution is to develop tools: stone tablets, the abacus, pencil and paper, the slide rule, the pocket calculator, the computer. These tools have advantages and disadvantages.

Modern PCs are incredibly fast computers, but there are limitations. Take speed for instance. We're always striving for more powerful technology, but part of that research lies in the optimization of preexisting algorithms. For instance, you may be aware that matrices and computer graphics are intimately connected. Image processing and graphics rendering, to my knowledge, boils down to linear algebra. Even a basic operation such as matrix multiplication can turn into a burdensome task for a computer when matrix dimensions grow ever larger. To overcome these limitations, we develop more efficient algorithms.

Another limitation is in the finite nature of computing calculations. How do we perform, say, integration when non-exact values and non-closed form problems are involved? Or forget that, how do we handle continuous real variables? We might, for example, employ finite difference methods. Most non-linear systems of differential equations rely on such methods.

More often than not, we have to sacrifice numerical accuracy for practicality. Anyone who's used $\pi \approx 3.14$ is guilty of this! When employing numerical methods, we often are forced to accept that exact values are pipe dreams. I say this without even broaching the subject of floating point error...

floating point error

... which would bridge us towards computer science. Anyway, the subject of computational mathematics can also deal with addressing these deficiencies.

These are the sorts of maths that journal likely explores, both in a theoretical and applied sense.

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