The previous article discussed the importance of definitions to mathematical thought. We looked at a definition (of an end-vertex in a graph), and picked it apart by finding multiple ways to look at it. We also directly used the definition in a practical manner to find “weak links” in a network. This time, we’ll look at the structure of mathematical theorems and proofs, and how we can read the proofs to understand the theorems. A mathematical theorem (informally) is a statement that takes the following structure:

IF {we have this stuff}, THEN {we get to say this ../about other stuff}.

Let’s pick this apart a bit even at the abstract level, because even this simple structure is important when we consider using theorems:

*If a real-valued function $f $ is differentiable, then $f $ is continuous.
*

$f(x) = x+5 $. Fits nicely. Try some nonpolynomials now.

$f(x) = \sin(x) $. Still good.

$f(x) = e^{x} $. Really nice.

As you’re playing with these, try to move between formally showing these functions fit the definitions and developing a visual picture of what it means for these functions to fit the definitions. (My high school calculus teacher, Andy Kohler, gave a nice intuitive picture of continuity—you shouldn’t have to pick up your pen to draw the function.) This step helps you visualize your starting point and your destination. Every single time I develop a theorem, this is the process I go through. This isn’t always short either. I’ve spent weeks on this part—exploring connections between my start and my target to develop an intuition. Often, this leads me to a way to prove the desired theorem. Occasionally I manage to create an example that renders my statement untrue. This isn’t a trivial or unimportant part, so I implore you to temper any budding frustration with the speed of this stage. However, since here we’re focused on reading proofs rather than writing them, we’ll likely assume we’re working with a true statement.“Because $f $ is differentiable, [STATEMENT]”

Study this part of the proof. Flag it. Do this for each time an IF condition is invoked. Make sure you can use the definition or condition invoked to make the leap the author does. Here I implore you to not just “pass over”. Really take the time to convince yourself this is true. Go back to the definition. Return to the proof. Again, this study isn’t necessarily quick.(Particularly if the proof's author likes to skip steps. Again, here I've spent hours or days on one line of a proof someone else wrote.) Doing this for each line in the proof will help you see what’s going on between IF and THEN.