Definition:Abuse of Notation

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Definition

Mathematical notation can be considered in two ways:

  • As an aid to mathematical understanding, no more and no less than a useful convention to encapsulate more or less complicated ideas in a completely unambiguous format;
  • As the reason for mathematical effort, so as to encapsulate a truth as a documented piece of aesthetic beauty in its own right.


Abuse of notation is a technique of using a system of symbology in a way different from that for which it was originally defined.

Such abuse may make a train of thought more streamlined, as it is often possible to save considerable redefinition of one's terms.

However, such abuse is frequently considered to be incorrect, improper and (in the eyes of many mathematicians) illegal.

Philosophers of the various schools practising pragmatism tend to consider that if a notation has been adequately explained, then one should be allowed to use it in whatever way is most useful to communicate one's ideas.

On the other hand, such an attitude causes indignation, rage and fury among philosophers whose attraction to mathematics is purely aesthetic.


Examples

Correct Use of Logarithms

Using this mnemonically useful identity:

$\log_a b \ \log_b c = \log_a c$

rather than the less easy to remember:

$\displaystyle \log_b c = \frac {\log_a c} {\log_a b}$

is likely to get you banned from the teaching profession in certain of the less liberal realms of the Western hemisphere.

You are supposed to remember (and teach, and cause to be remembered) the second of these identities, as it indicates directly what it is for. That is, its purpose is to allow you to convert from logarithms to the base $a$ to those of base $b$.


Non-perpendicular Coordinate Axes

When one is constructing a pair of coordinate axes, it is possible to set them up at an angle different from 90 degrees from each other. In particular, when investigating symmetries of patterns and friezes based on the equilateral triangle, it can be convenient to set up a pair of axes at 60 degrees to each other. If you are tempted to do so, then according to the aesthetic school, you may not call them $x$ and $y$. Those terms are reserved for a pair of axes which are strictly perpendicular.


Differentials

It is convenient, in many situations, to treat derivatives as fractions even though they, strictly speaking, are not.

For example:

\(\displaystyle \frac {\d y} {\d x}\) \(=\) \(\displaystyle \frac {\d z} {\d x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \d y\) \(=\) \(\displaystyle \frac {\d z} {\d x} \rd x\)


Thinking about why this sort of manipulation may or may not be justified is usually discouraged.


Algebraic Structures and Underlying Sets

In Abstract Algebra, it is very common to refer to the underlying set as the algebraic structure itself.


For example, let's say we have a group $\left({G, \circ}\right)$.

Many people refer to $\left({G, \circ}\right)$ as $G$. But when viewed from a notational perspective, $G$ is not a group, but the underlying set of the group $\left({G, \circ}\right)$.


Be warned

Ignoring these strictures can ensure that you spend the rest of your mathematical career editing websites under a carefully guarded pseudonym. Be careful your alias is not uncovered, or a lengthy prison sentence awaits.


Also see

Compare: