Insights Conceptual Difficulties in the Roles of Variables - Comments

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The discussion focuses on the conceptual challenges surrounding the roles of variables and constants in mathematics, highlighting that these are not merely mathematical objects but notation tools. It emphasizes the distinction between mathematical objects and notation, particularly in the context of Leibnitz notations, which can lead to confusion when mixing function symbols with variable representations. A significant point raised is the abuse of notation when using the same function symbol for different functions that depend on varying coordinate systems, which can obscure their distinct nature. Additionally, the conversation touches on the informal use of "temporary variables" by learners to simplify calculations, illustrating a personal approach to problem-solving. Overall, the thread underscores the need for clarity in understanding the roles and implications of variables in mathematical expressions.
haruspex
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Conceptual Difficulties in the Roles of Variables

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haruspex said:

I think there are still important pitfalls that might deserve deeper explanations and details:
  1. The opposition variable/constant is not a mathematical one but a meta-mathematical/logical one. Variables and constants are not mathematical objects but notation tools to describes mathematical objects (that's why for example you can say at the same time that ##x## is a variable and ##x## is a real number). A big pitfall with the ubiquitous Leibnitz notations (##\frac{d f}{d x}##, ##\frac{\partial f}{\partial x}##, ...) is that it mixes mathematical objects (like function ##f##) with notation devices (like variable ##x## which represents one of the function "slots", i.e. argument position and absolutely not a number here).
  2. As mentioned it is a common abuse of notation to use the same function symbol for related but distinct functions when their output has the same semantical meaning but the arguments (= the coordinate system) are different. One of the big issue with this abuse of notation is this one:
    Consider ## (x, y) ## and ## (x, u) ## two coordinate systems with the common coordinate ##x## and a smooth function ##f##. I can then compute ## (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) ## and ## (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial u}) ##. But the two objects ## \frac{\partial f}{\partial x} ## are in fact entirely different functions : ## \left. \frac{\partial f}{\partial x}\right|_y ## and ## \left. \frac{\partial f}{\partial x}\right|_u ##. The fact that derivation along a single coordinate depends on the entire coordinate system would really deserve to be insisted upon.
 
Nice article, though I can't follow the section on derivatives as I haven't reached calculus yet. At a much simpler level (which is where I'm at as an older adult/amateur learner revisiting high school math), I sometimes invent "temporary variables" just for purposes of calculating; e.g. there was a homework question recently about a word problem in basic algebra that stated, among other things, "##d## is ##k## percent less than ##c##". Solving it myself for fun, rather than work directly with ##d = (1-k)c##, I made a temporary variable ##k' = 1-k## so I could get rid of the parens and just say ##d = k'c##, postponing the subtraction until after everything else was done. But that's just me working on my own; I have no idea if "temporary variables" are used by other people.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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