The one thing I hate about how mathematics is taught at the higher level in the US and in the UK (and by extension I assume likewise for Austria and Canada), is the lack of opportunity to ruminate on the content we learn. Let me illustrate what I mean.

In a lecture the other day, the professor cites a theorem. The form of the theorem was a straight forward conditional, if p, then q. When such claims are made, the immediate questions that come to mind are is the converse true?* During lecture, of course, there is no time to speculate on side questions. Unfortunately, the attention is limited even outside class because focus is on absorbing the given content sufficient enough to tackle the immediate problem-set due or prepping for the forthcoming midterm or final exam.

Where is the time to ponder such things?

Unless the answers happen to be trivial or obvious, mathematician lingo for ‘no’ and easy to prove, respectively, there should be allowances for digressive related thoughts.

Part the joy in doing mathematics, for me at least, is the idle exploration and discovery involving these abstract objects.

*A quick lesson in logic. A conditional statement is of the form if . . . , then . . ., for example, if there is snow, then school is cancelled. The clause ‘if there is snow’ is called the hypothesis while ‘then school is cancelled’ in the example is the called the conclusion of the statement. What follows the ‘if’ is the hypothesis and what follows ‘then’ is the conclusion.

The converse of a conditional is switching the hypothesis and conclusion. Continuing using the example, the converse is if school is cancelled, then there is snow.

The inverse introduces a negation to both hypothesis and conclusion, so inverse of the example conditional is if there is no snow, then school is not cancelled.

The contrapositive of a conditional is combination of a converse and inverse, negating and switching hypothesis and conclusion, thus if school is not cancelled, then there is no snow.