(For the first part of this article, see here.)
“Learning takes wings when it’s related to what is passing in the student’s inner life.” So says Thornton Wilder in a wonderful book, Theophilus North, which will teach you why its hero could have been called Amadeus or Gottlieb, and what is creative kindness.
How can locally integrable functions be related to anyone’s inner life?
Let us reformulate the question in what seems to me the spirit in which it was asked: Why are we going through material which is hard and demanding, not particularly interesting, and irrelevant to anything of importance in my future career? Is this not the way to regurgitation for a qualification?
Here is what I think. University is a very peculiar institution. It differs from any other institution in that people who work there like ideas. It is a joy to work in a University precisely because, and only to the extent that, the University allows you to think. There is much to perturb a soul when you work in a University, but usually not enough to hound a person into industry, civil service or, God forbid, merchant banking. An added beauty is that one can teach, because what is teaching in a University but constantly extending an invitation to think? I agree that this picture is idealised, but from time to time this invitation is accepted and acted upon, and one is then privileged to see the ability to think develop in a student.
I think you should see by now the outlines of my answer to the sad student. Every bit of material we teach, in mathematics and English, in physics and philosophy (as opposed to engineering which is partly vocational training, and certainly as opposed to various non-disciplines, such as Hospitality Science, Sport Science, Media Studies — to name just the least litigious — in which no thinking ever happens even by chance) is an invitation to think. OK, is never zero, but is there any other complex number that this function cannot hit for some ? What are the implications of the fact that for some function from the unit interval into itself, , , ?
This is what we do: we invite you to think, first about simple matters (is there a function that is differentiable 5 times but not 6?), then about harder ones. Someone once said that saintliness is offered to every person anew every single day, and if one rejects it, the rejection has to be repeated again and again, every day. The same is true about an invitation to think: every time you come to a lecture, a tutorial, or sit down to do homework, the choice is yours, accept the invitation or refuse it. If time after time you accept the invitation to think, thinking can become a habit, which you might indulge in areas beyond mathematics — and this is the one habit that distinguishes an educated person from an uneducated one: all the rest is frippery.