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Insights Conceptual Difficulties in the Roles of Variables - Comments

  1. May 17, 2015 #1

    haruspex

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  3. May 18, 2015 #2
    Nice work! Love the first statement!
     
  4. May 11, 2017 #3
    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.
     
  5. May 11, 2017 #4
    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.
     
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