by Gordon Honerkamp-Smith

One of the most interesting things about mathematics is how it warps and exploits language for its own purposes.  This is an aspect of math that even the most non-mathematical of minds can appreciate.  Instances of the mathematical sabotage of language abound; for example, consider the word "integral."  Once confined to the lowly class of adjective (which is, if one thinks about it, the proverbial "linguistic parasite"--the way they latch on to other words), mathematics has liberated the integral, it has become a noun! 
Another great example of such lingual contortions is the phrase "in general."  When we use this term in our day-to-day communications, we aim to confer a certain amount of "looseness"; it's a phrase that signifies truth only to within some margin of error.  In fact, we can often take this expression as a cue that something will most certainly not be true at least some of the time. When someone tells you something is true in general, they usually have some specific instance in mind of when it can be or has been false ("The parachutes?  Yeah, uh, they've worked pretty well in general…").  In short, the conventional definition of generality refers to a kind of average truth. 
Now contrast this with the term as it is used by mathematicians.  In the mathematical context, we use generality in a specific way.  We may make reference to a "general solution" of a differential equation, or perhaps--whilst proving some lemma or theorem--make an assumption "without loss of generality."  In these mathematical environments, the use of the concept of generality is a carefully meditated choice, meant to convey specific information about a statement or body of information.  According to thefreedictionary.com, an online encyclopedia, the term without loss of generality "…is used to purport that one might as well assume whatever follows, as this suffices to prove all situations."  In other words, the term is a blanket-statement that encompasses all possible situations.  When we give a general solution to a differential equation, we are asserting that any solution to the

equation one can think of will fit the given format.  When we make assumptions without loss of generality, we are proclaiming that all circumstances derived under an old set of assumptions are still plausible under the new ones.

To summarize, mathematicians use the term generality only when all possibilities have been accounted for; mathematical generality is absolute rather than average.
As a final example of how mathematics so often has its way with language (and doesn't even call the next day), let's consider the body of terms which occur in mathematics seemingly to lull one into a false sense of security before subsequently biting one in the ass.  You know what I'm talking about.  These are words like "ordinary" and "simple;" lackadaisical words that normally exude ease and quiescence.  However, when used by mathematicians, these words take on a much more sinister edge, and they should always be met with a mistrustful, foreboding wariness.  Naïveté in the implications of these words  won't  leave a bitter taste in one's mouth; consider:  "Oh good, partial derivatives; these should be easier to understand…" "Don't worry, we just have to solve ordinary differential equations," "Elementary Analysis?  How hard can it be?"
To be sure, mathematics could not exist without the aid of language.  The effective mathematician must be eloquent.  She or he must be knowledgeable not only in mathematics but also in the ways in which mathematical ideas can be expressed and communicated.  Beyond this, however, mathematicians can do what we want, and if this means wrestling language, kicking and screaming, to the ground for a good pummeling, then so be it.  I, for one, am looking forward to the fray.

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