Last week (last Sunday), I tried to explain to a friend who does not have a math background (she is going to recognise herself ^.^) the following equality
... and I un-fortunately (even though this was the right thing to do), showed her the classical geometric argument...
...given that she would not have understood the analytic argument, which among other things would have required me to correctly define limit of series as limit of the sequence of their partial sums. It was not so much that the move from series to sequences would have lost her, but the epsilon-delta definition of limits of sequences of real numbers (or elements of any normed spaces , or more generally elements of any metric space) would have been too much for her to digest (despite its triviality).
Interestingly, my friend mentioned Zeno's paradoxe as a justification of the fact that the equality could not be true. I have a problem with people bringing this paradox up, since their every day experience (being able to move from one point to another in a finite amount of time) contradicts it. It would never occur to me to try and make a point in a conversation using an argument that I have being violating from birth... anyway...
I spent one week trying not to think about it, but today I decided to try and fully understand why it was difficult for her to understand. Of course, I remember what I told her last week about the need not to confuse a process' individual steps and the process itself. I then went on trying to give her a definition of limit using a geometric interpretation of espilon-delta using circles and the idea of attending a show (that one was funny...), but still she could not make the last mental step towards understanding mathematical limits.
Moments ago, it occurred to me that I have observed a strange phenomenon among undergraduate students, that they understand some constructions "from above" better than they understand construction "from below". For instance, they understand easier the "above" definition of the linear sub space of a vector space generated by a subset (as the intersection of all subspaces that contain the subset), than the "below" definition of the collection as all finite linear combination of elements of the subspace (union the null vector if we want to allow the subset to be empty).
I thought I had reached a some sort of a conclusion as was happy about it, but nothing would then explain to me what exactly in the definition involving intersections made it easier to understand that the one involving unions, when it hit me: the thing is that we never really try to understand, we try to prove it, to justify it mentally (which is obviously a reflex of the mind), and the human mind has less problem building internal representations of sets defined by statements of the form (for all set)[property of object] than those of the form (exists set)[property of object], (where object is a free variable in both templates).
Now, if you ever wrote a single math proof in your life, the above should strike you as obvious, isn't it :-) Now my problem is how can I express the intuition behind the original formula as a universal statement with a free variable (whose corresponding set is singleton 1) ? The answer was actually in my "attending a show analogy", so in the end I simply need to become better at giving that analogy :-)
[ps] ... and expecting the "Yes but that's all mathematics, nothing to do with the real life" argument, my answer to that is "Step away from the iPhone m'am! It is dangerous and will corrupt your mind since it was made using mathematics!" (^_^)
[epilogue] ... and an additional question is: what exactly happens in students' minds as they mature so that they don't have this problem anymore ? I think the answer is simple, they stop thinking of infinite processes as "taking an infinite amount of time". Fully understanding that an infinite process, a process with an infinite number of steps (and not only physical processes but computational processes as well), can happen, can terminate, in a finite amount of time (or better: in no time at all), is probably what makes mathematicians who they are...