The following is the prepared text of the address delivered by Professor of Mathematics Judy Holdener to the Class of 2019 at the Baccalaureate service on May 17, 2019.
Thank you, President Decatur, for those kind words, and thank you, Class of 2019, for inviting me to be your Baccalaureate speaker today. I am thrilled to have this opportunity to share in the celebration of your success, because you are truly an inspiring group of people, and I imagine I can speak for everyone on this stage when I say it has been a privilege to work with you.
As grateful as I am to be here, however, I have to admit that your invitation to speak has given me flashbacks — flashbacks to my high school graduation, where I delivered a commencement speech to parents and community members in my hometown of Ravenna, Ohio, just two hours northeast of here.
The year was 1983, and I had finished high school ranked fourth in my class. At the time, I was very happy with this ranking, because I figured I had achieved the highest possible GPA I could get while remaining safe from having to deliver a big scary commencement speech. In previous years it had always been the valedictorian and the salutatorian who delivered the speeches. But as luck would have it, two of my classmates tied for the number two spot, and the teachers decided to break away from the well-established school protocol by voting for the second speaker. So like Kenyon’s Class of 2019, they elected me.
But it isn’t so much the invitation to speak here that gives me flashbacks. More significant is the tumultuous political backdrop of our time … because much like the situation in our nation and the world today, my hometown of Ravenna was deeply divided in the early ’80s. During the spring of my sophomore year, my teachers went on strike for 85 school days, breaking the record for the longest teacher’s strike in U.S. history. Ravenna High School still holds that record in the state of Ohio. Substitutes replaced nearly all of my teachers in the classroom, and my school devolved into chaos. I witnessed acts of vandalism, hatred and revenge that truly shocked me.
I, myself, was no angel throughout the ordeal. With teachers having no knowledge of who we were or where we belonged, my classmates and I took full advantage of our newfound freedom. We enjoyed two-hour lunches and skipped class to play cards. We swapped instruments in band, and I played the clarinet without any knowledge of the fingerings. The conductor couldn’t understand why we sounded so bad. In my English class I invented a fake student named Jack, and I submitted two papers for each and every assignment — one for him and one for me. I did this without telling anyone in the class and without any hope of reading the teacher’s feedback on Jack’s satirical responses to her prompts. What was I thinking?
The strike finally ended after the Ravenna Board of Education raised teachers’ wages, but they refused to renew the contracts of 51 non-tenured teachers who had participated in the strike, and there were a lot of hard feelings. So when I delivered my graduation speech 36 years ago to a divided community, I spoke about healing and unification. I don’t recall the specifics of my speech; I just recall wanting the conflict to end.
And so my life comes full circle as I stand here today, except the stage is bigger now (at least figuratively; I actually think my high school stage might have been bigger than this one). My country is divided, and so is much of the world. Emotions run amok, and we live in a world of misinformation and alternative facts. I believe truth is necessary for a stable society, and truth is central to my field of mathematics, so I have decided to speak with you today about how it is that mathematicians are able to establish truth.
This subject is a bit on the heavy side for this celebratory occasion, so I thought I would keep it light by presenting the main ideas in the context of a fun true/false exam. I’m thinking the Class of 2019 might enjoy this chance to have one last exam before leaving this hill.
Don’t worry … it won’t be long. The exam consists of just three questions, and they are designed to highlight important components of the mathematician’s pursuit of truth. But before we get started, let me first say what I mean when I refer to a truth in mathematics. Briefly, a truth in mathematics is a statement that can be logically deduced from other established or accepted truths in mathematics. For example, a very famous mathematical truth provided by Archimedes over 2000 years ago is the statement that the circle of radius r has area equal to πr^2. This truth is probably familiar to many of you, because it is a useful result; it has been used in both large and small ways to establish other mathematical truths.
So are you ready for your first question?
Question 1. True or False: “Kenyon College has the most beautiful campus.”
Despite what Admissions might tell you, this sentence has no well-defined truth value, because it is a subjective claim expressing an opinion. So while you may well hold the belief that Kenyon has the most beautiful campus, others who are probably graduating from other schools may disagree. As such, mathematicians would not permit this sort of sentence to be used within a logical argument. We are very careful to separate beliefs from truths, and within logical arguments we are only permitted to use sentences that can be identified as either true or false; there can be no ambiguity.
Now if I were to modify the sentence to read “Kenyon was ranked first on the 2019 Best Colleges list of the ‘50 Most Amazing College Campuses,’” then there would be a well-defined truth value, and we could identify that truth value by looking up this list. It turns out that the statement is false … we were actually ranked second.
At this point, you might be losing trust in my exam because it was not possible to answer the first question correctly. Perhaps you will like Question 2 better.
Question 2. True or False: “President Decatur is currently seated on a wooden throne.”
President Decatur is certainly seated on a chair, and the chair is made of wood, so it seems that the truth value of this statement rests on whether or not his wooden chair would be considered a throne. According to Dictionary.com, a throne is defined to be “the chair or seat occupied by a sovereign, bishop or other exalted personage on ceremonial occasions.” So if we accept that Baccalaureate is a ceremonial occasion, then it seems that the truth value of this sentence depends on what is meant by “exalted personage” and whether or not President Decatur satisfies that criteria. He has a scepter, so I would be inclined to say that he does, but we would have to agree on the specific criteria, and that criteria would have to be clear and unambiguous. The matter comes down to definitions. Let us agree that exalted personage includes any president of Kenyon College, so the statement is true.
Axioms and definitions play fundamental roles in mathematics — serving as the foundation of proof and problem solving. I want to say a few words about this, but I will warn you that my husband described this discussion as the most boring paragraph of my speech. I have retained it anyway, because 1) it is important, and 2) I want to allay a common misconception that many people seem to have about mathematics — that everything in math is either black or white.
In fact, when defining objects in a new theory, things can get very messy. There are often competing viewpoints over which definition best captures the essence of the relevant concepts, and the mathematical community works deliberately to reach agreement. Deliberations come in the form of presentations, publication and individual conversations, and the process can be a lengthy one — sometimes extending across decades or even centuries. Achieving uniformity is well worth the effort, however, because it allows us to communicate mathematical ideas more effectively, and it is through this common ground that we are able to identify objective truths.
Okay, are you ready for the final question?
Question 3. True or False: “1 + 1 = 2.”
Certainly, this statement is true in the context of standard arithmetic, but there are other algebraic systems in which the statement is false. For example, binary is a way of representing numbers as a sequence of 0s and 1s. If I had intended to compute 1 + 1 using the binary number system, then 1 + 1 = 2 would be false, because in binary form, 1 + 1 = 10 (“one zero,” not to be confused with 1o, “ten”!). Indeed, in binary the number 2 will never appear in the representation of any number.
More generally, we cannot really say any given equation is true without specifying the context, and, in fact, by changing the context it is possible to turn almost any true statement into a false one. Mathematicians are very careful to pin down the necessary context when presenting mathematical truths, and doing so can take some work; social networking services with 280-character limits would not be our platform of choice.
So what was the purpose of this exercise? I can assure you it was not an attempt to haze you one last time. Rather, my hope was to illustrate that the notion of truth is a complicated one, and if we want to identify objective truths we have to establish some ground rules. The process relies on a widespread agreement on the fundamental starting points, and achieving such agreement takes time and patience, but the benefits are worth the effort. It is amazing for me to think that mathematicians from all over the world and from all religions and political backgrounds have been able to reach this agreement. Given what we have seen in politics over the past several years, I am starting to realize that despite the stereotype, mathematicians have incredibly good social skills!
Of course, I recognize that truth in the real world is not the same as truth in the mathematical world — with one big difference being the direction of reasoning. For example, in the scientific world we often assume there is some set of rules defining “truth,” and we observe outcomes in order to figure out what those rules are. In mathematics, we know what the rules are because we define them, and then we examine the possible outcomes following from those rules.
I believe the world would gain much by adopting the mathematicians’ deliberate approach in the establishment of common ground for truth and problem solving. Today’s society faces immense challenges: poverty, inequality, violence and climate change. Solutions to these difficult problems will require collaboration and innovative thinking, and decisions should be guided by tolerant and informed deliberation. The polarization in today’s political climate does not provide the necessary framework for such deliberation.
Class of 2019, tomorrow you will graduate and start the next chapters of your lives. Some of you are heading off to jobs or internships, perhaps a Fulbright or a graduate program. Others of you will take some time to reflect on the next steps of your lives. These are all fine options. Wherever you go, whatever you do, I hope you can work to create a culture of deliberative dialogue. Listen carefully to your peers and co-workers and take the time to understand their viewpoints. Consider the contexts of their previous life experiences. Work to find common ground and seek truth using data and facts from reputable sources. I know that you have the ability to make this world a better, more stable place.
Seniors, for all your hard work and accomplishments, I congratulate you and wish you the best of luck moving forward.
And parents, I want to take a quick moment to recognize your efforts as well. You have worked through many challenges to get to this point, and you have good reason to be proud. I hope you have an amazing weekend.