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I've come to understand "calculus" as the mathematical study of continuous changes in a mathematical function or physical system. Differential and integral calculus are broad examples of this.

R.C. Jones studied transformations of an optical polarization state, and called the vector/matrix operations a "calculus" [1]. To me, matrix multiplication is a discontinuous operation, for example, as it is in quantum mechanics. In the conventional Jones method, a polarization state is discontinuously transformed into another after passing through a waveplate. So in my mind, this is more like algebra. Why then is this method termed a "calculus"? Was Jones being a bit pretentious?

[1] R. C. Jones, “A New Calculus for the Treatment of Optical Systems: I Description and Discussion of the Calculus,” Journal of the Optical Society of America, vol. 31, no. 7, p. 488, Jul. 1941, doi: https://doi.org/10.1364/josa.31.000488.

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  • $\begingroup$ This really does sound like more of a mathematics question than it does physics, but a note on the matrix multiplication part, considering that calculus works with matrix operations I'm not sure where you consider that matrix multiplication is discontinuous in that sense. $\endgroup$
    – Triatticus
    Jan 20 at 18:42
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    $\begingroup$ This is a vocabulary question. In mathematics, the word "calculus" usually refers to any set of rules for calculating something. See, for example, lambda calculus or Schubert calculus. $\endgroup$
    – WillO
    Jan 20 at 18:49
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    $\begingroup$ I’m voting to close this question because it's not about Physics. $\endgroup$ Jan 20 at 19:15
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    $\begingroup$ As the top answer says, it's a vocabulary question and a mathematically-based vocabulary question on top. It has basically no bearing on physics whatsoever. $\endgroup$
    – Kyle Kanos
    Jan 20 at 21:33
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    $\begingroup$ The problem is not that your usage of "continuous" is imprecise or distracting. The problem is that your usage of "continuous" is irrelevant. In standard usage, the word "calculus" has nothing to do with continuity. Calculus of fractions, intersection calculus, lambda-calculus, etc, occur most naturally in contexts where continuity is irrelevant. $\endgroup$
    – WillO
    Jan 21 at 4:22

3 Answers 3

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This is a vocabulary question. In mathematics, the word "calculus" usually refers to any set of rules for calculating something. See, for example, lambda calculus or Schubert calculus. Or browse the list here.

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There isn't really a proper mathematical definition of a calculus. There is a branch of mathematics which is called calculus, but one might also speak of lambda calculus or a functional calculus.

Distinct from the branch of mathematics, a calculus is a method of reasoning or analysis. For example, one could speak of the calculus of deciding which restaurant to go to for dinner, in which case the calculus might be e.g. some balance between price, preference, and location. It's not the most common word, but it's not deprecated and still enjoys a reasonable degree of use.

As a side note, there's nothing discontinuous about matrix multiplication, in quantum mechanics or anywhere else - it is about as well-behaved as a mathematical operation can be, unless continuity is being defined in an exotic way. Indeed, the continuity (and in fact, smoothness) of the map $\mu(A,B) = AB$ is central to a large part of e.g. Lie theory.

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First off, the thing that you are calling calculus, which is roughly “trying to see if $f(x+\delta x)$ is simpler to analyze than $f(x)$ is, for small $\delta x$, then rebuilding/analyzing $f$ out of this understanding,” is much bigger than continuous functions. One example is that it was used as part of “differential cryptanalysis” in the 80s and 90s (I believe?) to break a bunch of cryptographic primitives. A physicist even proposed using it in business calling it the “theory of constraints,” basically just “figure out what tweak actually causes your factory to produce less as opposed to just being absorbed by the feedback loops of the system, trace the tweak into the system to find out why it does that, call what you find your ‘bottleneck,’ try these strategies to improve flow at the bottleneck or add capacity or allow backpressure from the bottleneck to reach the inputs of the system.”

Indeed I prefer to teach that form of calculus when I can by teaching the discrete calculus on infinite sequences $X = (x_1, x_2,x_3,\dots),~~\Delta X=(x_1, x_2-x_1,x_3-x_2,\dots)$ before introducing integrals and derivatives as the continuous generalizations of these operations.

Second, matrices do have ways of being considered close to each other such that we can have topologies on them and consider them continuously.

But yes, “the” calculus these days is the Leibniz-Newton differential-integral calculus and various derivatives like vector calculus. But people use the word for other ways of calculating things, for instance “umbral calculus” being some noted similarities between various polynomials $P_n$ and some operator $D$ such that $D[P_n] = n~P_{n-1}$ (which is a very discrete notion!) and maybe this allows some mnemonics for the forms of the $P_n$ or so. I'm sure that I’ve heard of Young tableaux as having been referred to as a “calculus of Young tableaux” at least once in my life, etc. ... It's just a way of performing calculations in it's original meaning (at least the original meaning that does not refer to a calcified rock found in your body). It just happens to be the case that one of these ways of performing calculations is one that has become part of our scientific literacy, almost every scientist gets taught this calculus at some point.

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