A Reflection on Math Proofs & Three GSB ’85 Students

Jay Fudemberg

On a Friday afternoon during winter quarter at our weekly LPF beer gathering I was chatting with Reinhard Busch, our esteemed classmate from Munich. As usual, so many things to discuss at the end-of-week, and of course with Reinhard, the list was long.

It was that time of the year to bid on classes for the upcoming spring quarter and our conversation made its way around to what we were both planning to do. I told Reinhard that I was planning on taking an elective in mathematical logic from Stanford’s philosophy department “across the street.” Reinhard gave me his typical enthusiastic grin and chuckle, asking what THAT was about. I said that one of my hobbies was solving math proofs (which continues to this day), and that this course seemed compelling since the sole activity was solving proofs. Not only that, but it was instructor-less; well, it had been created by the famous Stanford mathematician and philosopher, Professor Patrick Suppes, but the course was conducted entirely on computers. It was Pass/Fail. The curriculum consisted solely of solving 50 proofs on the computer. There was no lecture, no instructor and no textbook. Just you and the computer and the proofs. Solve all 50 and you pass. Solve only 49, and you fail.

Much to my amazement, Reinhard was more than enthusiastic; he, like me, found the course description totally exciting. I was quite surprised that anyone else from our class would be equally interested in such a topic and activity. But there it was, at the LPF with beer in hand, we discovered that we shared the same passion for constructing proofs and decided that we would take the course together.

The schedule for the course was very flexible; there was a special computer lab which was available for long stretches within the posted hours. You just showed up, logged-in and worked as long as you liked. Spring quarter came fast, and just a day or two into it Reinhard and I decided to get started. We hiked over to the lab together, which was in some far-off corner of the quad. We weren’t quite sure what to expect, but it turned out to be exactly as billed, a quiet room outfitted with a bunch of computer terminals. After logging in, the computer simply served up a proposition requiring a proof, awaiting the application of insight, creativity, and reasoning. That was it. And, to add to the uniqueness, there was never another person in that room for the entire quarter, regardless of the time of day we showed up. Anyway, on that first day, we selected computers so that we were facing opposite to each other and began. We spent at least a couple of hours working silently and intently and came away feeling thrilled by the activity and challenges. As you might expect, after the session, during our walk back across the quad to the GSB we exploded with conversation. There was so much to discuss and compare. “Did you finish the first one? How many did you get through? What were your approaches; how did you crack each one? Was it hard? Easy? Did you like it? Do you think we’ll have trouble doing all 50? Clearly, they’re going to get progressively harder. What do you think?” On and on. Of course, now the competition was on. And so was the bet. Would we both complete all 50? Who would finish first? We decided that the winner should win a free beer. Reinhard made it clear that he would want German beer if he won. Of course. Laughing, I told him I had no similar constraints.

We had a lot of fun with that course and continued walking over to the lab together throughout much of the quarter. We were silent and intent while working, but on both the walk over and back, there was always a lot of conversation. However, we never divulged key insights about a proof until after we both had completed it. There was victory and a free beer to protect. And yes, the proofs did indeed get progressively harder, so much so that we both started to worry a bit if we would complete all 50 to earn a pass. Failing would be bad. This was an undergraduate course! Not passing would be humiliating. Ugh. Our egos were clearly on the line here. But, in the end, there was good news. With just a few days to spare, we both finished, and as luck would have it, we were in the room together and completed the 50th proof on the same day. It was a draw. No bragging rights. So we celebrated our tie and went for beers. Can’t remember if Reinhard actually got his German beer. I really enjoyed taking that course with Reinhard, our walks, time together, joking about the competition, and all those animated conversations. It was a special time.

But there is more and another GSBer in the story. The experience of that course, which was a most satisfying application of focus, creativity, and reasoning, really stuck with me. Having a computer automatically and accurately evaluate our proofs, our reasoning, was intriguing. Throughout my years at Bain, Enron International, and Pure Markets I often reflected on the experience. Then, finally, it came back into sharp focus after reading an article about the fact that only 6% of college graduates are proficient with critical thinking and problem solving. This obviously was (and still is) a big issue, not only for students, but for employers too. A spark flashed for me.

While at Stanford, I (sadly) had never met Professor Suppes, the creator of the mathematical logic course that both Reinhard and I so thoroughly enjoyed. But now, I had this idea of a web-based platform that could develop, exercise, and assess critical thinking and problem solving skills by using real world problem scenarios instead of abstract logical expressions. Each custom-created problem scenario on the platform would be unconstrained – it could encompass any issue in any context and be created by anyone. Further, the platform would have the capability to evaluate the analysis and reasoning that any user/learner provided to support their resolution to the scenario problem, and to automatically assess the quality of that reasoning and provide personalized instructive feedback about it. Might Professor Suppes still be around to discuss this? If so, would he even be interested? Creating a software engine that could assess the quality of the reasoning behind proposed solutions to unconstrained real-world problems was quite a leap from evaluating highly constrained mathematical logic proofs. But his course was an inspiration and it was possible that he might have a perspective on my thoughts and plan. However, since it had been almost three decades from the time Reinhard and I had taken his course I thought it unlikely that he would still be around. But I was in luck. Professor Suppes was indeed still on campus and academically active. I emailed his assistant, and learned that yes, not only would he see me, but was quite interested in what I was suggesting. So, down to Stanford (from San Francisco) I went. We had a very warm, engaging first meeting in his office, which I might add was more like a large comfortable living room than an office. That was followed up with additional conversations which were also inspirational and affirming. A remarkable man, and I was lucky to have had the opportunity to engage with him.

So, after a bit more thinking and design work, I hired a team of programmers to build the web-based platform that has become findingQED. Enter our dear classmate, Kathy Dewenter, who as many of you may know passed away in 2018, far too young and too soon. I spoke with Kathy shortly after the beta version of findingQED was completed in 2015, and she was totally excited about using it to teach cases in her international business economics classes. As a professor at the University of Washington Foster School of Business she agreed that students were woefully weak in critical thinking and that the findingQED platform would be a powerful and engaging way to develop and assess these skills. We agreed to create a case scenario for her classroom, and promptly set about doing it. She flew down from Seattle to San Francisco to spend some time with my family (my wife Judy and daughter Rin), to sketch out what she wanted to accomplish, and to learn about how the findingQED platform could enable her to easily create web-based case scenarios. It was delightful quality time with the wonderful human being that Kathy was. The case involved PitchBook, a financial information services firm, and Kathy arranged to have the CEO and several of the senior executives be interviewed for various video scenes in the problem scenario. Together we worked on molding the case and its issues and had a lot of fun. Just a few months later she let me know the week the students would be engaging with the scenario on the platform, and the day she would be discussing it. I flew up for the class and thoroughly enjoyed watching her facilitate the debate and discussion among the students, which she said were the most animated she’d experienced all semester. That was partly due to the platform’s x-ray of each student’s thinking, which Kathy utilized to great effect. We both enjoyed it, and she made sure to leave some time for me to share a bit about findingQED with the class, discuss the underlying technology, and engage in some Q&A, all of which was a pleasure. I am especially grateful that we were able to spend some time working together, however briefly. A very dear person.

I still own findingQED and we’re currently working with the U.S. Air Force Intelligence Community. The platform can help them to be more effective in their mission and activities. Another totally unpredicted, but exciting opportunity.

When I think about the interesting, fulfilling, and winding path since the GSB, I am struck by its unpredictability and variety, but simultaneously, by its seeming coherence. I could not have come anywhere close to predicting back in 1984, while chatting about proofs with Reinhard in the GSB courtyard, that taking a mathematical logic course with him would ultimately lead decades later to creating technology that sharpens higher-order thinking skills for students and professionals alike. But somehow, looking back, all the connections and pathways seem to make sense. A lot of fulfilling adventures and great stories along the way. It certainly has been a joy. I am lucky for it, and for knowing many of you.