oopsilon-deltoids

Proving, or disproving, the existence of limits is a great target for the tools of logic. The bare-bones limit concept involves a cascade of three quantifiers, mixing universal and existential quantification:

For every positive real number epsilon, there exists a positive real number delta, for every real number x, if |x-c| is less than delta, then |f(x)-f(c)| is less than epsilon.

Limit notation conceals some of the quantification of epsilon and delta:

As x approaches c, f(x) approaches L.

"Approaching" means getting arbitrarily close, and the limit forms says you can get within epsilon of L by getting within delta of c.

Continuity adds an extra feature to the limit concept --- the limit L that is approached by f(x) is exactly f(c). Now pile on another feature: f is continuous at every real number. As a limit this says that for every real number c, the limit of f(x) as x approaches c is f(c). Expressed with quantifiers, the whole bundle becomes:

For every real number c, for every positive real number delta, there exists a positive real number delta, for every real number x, if |x-c| is less than delta, then |f(x)-f(c)| is less than epsilon.

Okay so far? This is the sort of statement we learn how to structure proofs (and disproofs) of in CSC165. We become adept at juggling long strings of symbols and realizing the importance of having one sort of quantifier precede another. We become used to freely changing the symbolic names of variables as suits our purposes. We (and here I think I mean me) can lose sight of conventions that attach particular meaning to particular symbols.

The generic statement above talks about the function f(x). To make it more concrete, let's suppose this is the square function, so f(x) is x². Suppose, in addition, that I use letters near the end of the Latin alphabet to stand for real numbers, so y seems as good a choice as c for the point in the domain we approach for the limit. Now the above statement about continuity becomes:

For every real number y, for every positive real number epsilon, there exists a positive real number delta, for every real number x, if |x-y| is less than delta, then |x² - y²| is less than epsilon.

From a logical point-of-view, this is extremely similar to the first statement about continuity at every real point, except instead of a generic function f(x), there is a definite function x². However, from a psychological and cognitive point-of-view, many people are in the habit of thinking of y as the dependent variable that (graphically) expresses position in the vertical dimension of a graph. Looking at the statement above, they picture y² shooting out sideways or something.

The lesson is that logical expression is very sensitive to context and connotations that it is afloat in. Although logic can't completely concede to the surrounding context, since it tries to be precise and self-sufficient, logic has to be aware of strange, distracting associations that the use of a particular symbol or word may cause.