Dr. Calhoun came into class the first day and set his mug of coffee on the desk. His white hair and gray beard gave him an air of wisdom and authority.
He turned to the sixteen of us there and smiled. “Welcome to the graduate course, Algebra Theory 602. We are going to learn the beauty of mathematical proofs. I hope you are all here for one of the greatest, artistically fulfilling adventures of your life.”
I admired his way with words. I had already been through many mathematical theory classes, and I had yet to find one I thought was artistically fulfilling, though adventure was probably a good word.
“Good proofs should be like a beautiful painting,” Dr. Calhoun continued. “The words should flow like vibrant colors across the page, leading from one logical concept into the next in an unbroken chain. These should start with the opening statement or thesis of the proof, interweaving every logical validation until, like a master painting, an undeniable and emphatic conclusion can be drawn.
“During this year that you are in the class, I will assign proofs for you to read so that you can gain more understanding of this logical flow. We will also have theorems that you will prove. By the end, you will have become like a great master painter, yourself having overcome the doldrums of minimal thought to be able to think in a deeper, richer plane.”
He went on for quite a while, expanding on the idea that nothing was more beautiful than a well-written proof, with logic and understanding at its core. By the time he finished, his words hung over me like a great symphony, and I was excited about the course ahead of me.
Dr. Calhoun then assigned us some proofs to read and gave us a list of theorems to prove. He told us we didn’t need to finish all of them immediately. We would start with the first one on that list, and he would call someone to the board. That person would do their best to prove the problem, and if they failed, Dr. Calhoun would call on the next person. As soon as he was satisfied with the proof of that theorem, we would go to the next one on the list. He suggested we start with the first five and try to keep five ahead of where we ended in class.
Dr. Calhoun was good at building confidence, and I left, almost excited to work through the first five problems. Over the next couple of days, I read the assigned proofs and carefully worked on proving the first five theorems.
When we met for the next class, he called someone to the board. He informed us we would not be chosen in any order, alphabetically or otherwise, but would be randomly selected at his discretion. The first person was able to get about halfway through the proof before getting stuck. Dr. Calhoun went through most of the class, and others made more headway, but it was not logically finalized to his specifications.
Eventually, I was chosen. I had some challenging classes and felt confident in my work. I went up to the board and started the proof from the beginning. My logic flowed well through it, but my wording was nothing like Dr. Calhoun’s. Mine was much more straightforward and more pragmatic.
When I finished, he frowned. “There is no doubt you have proven the theorem,” he said. “However, I must take back my words that a proof is logical and beautiful. I have just realized it can also be logical and unsightly.”
From then on, Dr. Calhoun never called on me to go to the board unless everyone else in the class had failed. I was always able to prove what everyone else had not, but his words were only somewhat complimentary. He would always say something like, “There is no doubt you have proven it, but you basically beat it into submission.” Other times, he would say, “Well, you really killed that one, and by that, I mean you pretty much kicked it to death.”
When the course ended, I applied for a job and asked Dr. Calhoun for a reference. After I got hired, I got to see my folder, which included his letter. He said, “His logic is flawless. If there is something that can be proven, he can prove it. However, it’s never pretty.”
