Many of my classmates have
their first exposure to proofs in CSC165 so this class caters towards them. I think this is fair but I am sure that at least some of my classmates did proofs in
other classes. I become used to the essay-like
proofs of my previous proof-oriented math classes: MAT138 (Intro to Proofs) and MAT240 (Algebra I). So when the first proofs in
CSC165 presented themselves in mainly logical sentences instead of mainly English sentences, I
was pleasantly surprised. Their aesthetic appearance reminds me of
programming code. Why are the proofs in CSC165 presented this way?
The Pros
After thinking about it for a few days,
I arrive at two justifications. Proofs can seem
overwhelming to those of us who are new it; this certainly was the case for
me. Presenting the proofs in the style of programming code can make it
less intimidating to those us who are comfortable with programming.
In addition, the use of mainly logical sentences instead of mainly
English sentences can make it easier to catch logic errors.
The Pitfalls
However, there is a downside to the CSC165 way of writing the proofs. Where is the intuition? There is emphasis on annotating the logical sentences correctly but not much focus on the how to prove a claim. How do we decide between a direct proof or proof by contradiction if we are looking for a sound and efficient way to prove? Do we memorize the cases for when one proof strategy works better than another? Experience tells me that this is a bad idea because we will be at a loss if we see a new, unfamiliar claim to prove.
So what is the best way to gain
intuition? My opinion is to try things out. We should think way
about different ways to prove a claim. We should think how the
mathematical structures in a claim make a specific proof strategy efficient or
inefficient. Unfortunately, the focus on annotating the logical sentences
correctly and making the aesthetics of a proof comparable to programming code is contrary to the goal of gaining intuition. If CSC165
can focus more on the strategies for approaching proofs, then we might benefit
more from the logical rigour through which the course presents its proofs.
Zane, I share your sentiment towards "gaining proof intuition" in CSC165. There is quite a bit of emphasis on writing proper proof structures and for readabilities sake, I can see why that's important. But as I work on a proof, I sometimes feel like I'm stuck in rote mode, where I'm simply following a memorized proof pattern as established from previous examples. What about the intuition? I think following a proof pattern is constricting and leaves less room for experimentation. While constructing a proof, if we're able to intuitively establish a logical flow for the reader, where each new line of a proof is implied by earlier lines in the sequence, then why all the focus on "proper proof structures"? Let us do as we will!
ReplyDeleteYou're right Dragan. Rote mode is dangerous in math. I think they relax the structure in CSC236 but the logic remains the same.
ReplyDelete