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I am currently taking a course involving the use of numerical methods to solve partial differential equations. I have not yet been exposed to such a technique and as an aspiring computer scientist, am particularly intrigued as to its implications. Specifically, can anyone speak to how pervasive the discretization of continuous mathematics is? Can anyone recommend further reading on the subject? I am particularly interested in whether or not it would be possible to redefine fundamental physical laws, previously described using continuous mathematics, in terms of discrete mathematics.

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Depending on the direction you take within computer science, this question may be fundamental; floating-point arithmetic is actually very subtle. For instance, floating-point addition is not associative.

The paper at http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html seems to give a good overview, but a quick Google search will give you more resources than you could ever hope to read. The main point to take away from all of them is that if you hope to do scientific work with non-integers, you must understand exactly how floating point operations work in the language you're using. In many cases, this is the IEEE 754 standard -- not light reading.

This can bite you, especially early in your career, in very surprising ways. For instance, I once wrote a short graphical program in which the camera would abruptly point straight up, with no remedy except restarting the program. It turned out that all of my camera rotations were very slightly reducing the length of the camera's "look" vector, and it eventually became $\vec{0}$.

To address your final question about physics, there is some philosophical debate about whether in fact the real world is discrete, including time. Even if that were so, I think many people would continue to use continuous approximations, because it is simply cleaner. It is also not directly falsifiable, because any measurement we take has associated error.

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Can you elaborate on what you mean by "it is also not directly falsifiable"? What do you mean by "it" and why is it not falsifiable? –  bam54 Apr 23 '13 at 22:10

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