# Conceptual Difficulties in the Roles of Variables - Comments

• Insights
• haruspex
In summary, haruspex submitted a new PF Insights post titled "Conceptual Difficulties in the Roles of Variables" and continued reading the original post. The conversation discussed the opposition between variables and constants, the abuse of notation in using the same function symbol for related but distinct functions, and the use of "temporary variables" in solving math problems. The conversation also touched on the importance of understanding the dependency of derivative calculations on the entire coordinate system.
haruspex
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haruspex submitted a new PF Insights post

Conceptual Difficulties in the Roles of Variables

Continue reading the Original PF Insights Post.

Ssnow, Greg Bernhardt and spaghetti3451
Nice work! Love the first statement!

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.

## What are conceptual difficulties in the roles of variables?

Conceptual difficulties in the roles of variables refer to the challenges that arise when understanding the relationship between variables in a scientific study. This includes understanding the independent and dependent variables, as well as potential confounding variables.

## What is the difference between independent and dependent variables?

The independent variable is the variable that is manipulated or changed by the researcher in an experiment. The dependent variable is the variable that is expected to change as a result of the manipulation of the independent variable.

## What are potential confounding variables?

Potential confounding variables are variables that may influence or affect the relationship between the independent and dependent variables in a study. These variables are not controlled by the researcher and may lead to inaccurate or misleading results.

## How can conceptual difficulties in the roles of variables be addressed?

Conceptual difficulties in the roles of variables can be addressed by clearly defining and identifying the independent and dependent variables in a study, controlling for potential confounding variables, and conducting thorough data analysis to ensure accurate results.

## Why is understanding the roles of variables important in scientific research?

Understanding the roles of variables is important in scientific research because it allows researchers to make accurate conclusions and interpretations about their findings. It also helps to ensure that the results of a study are reliable and can be replicated by other researchers.

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