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In the twenty-seventh episode of the second season of the “Saturdays at Seven” conversation series, Todd Ream talks with Russell W. Howell, the Kathleen Smith Chair of Natural and Behavioral Sciences at Westmont College. Howell opens by discussing how mathematicians often find themselves confronted by results they find mysterious and for which no immediate explanation exists. He provides a couple of examples of such mysteries within mathematics that have attracted the interest of mathematicians for generations. Howell explains how his initial sense of calling to the ministry was refocused toward mathematics and the ways he believes his platform as a mathematician has given him unique opportunities to share the love of Jesus Christ. For Howell, his focus on mathematics is most profoundly expressed through his appreciation for the subdiscipline of complex analysis and the ways he has introduced students to the joy that emerges from such a focus. As a leader in conversations concerning the relationship mathematics and faith share, Howell describes what prompted him to co-edit two book projects on that topic and the impact of those texts within and beyond the classroom. Howell then closes by detailing the virtues mathematicians need to cultivate in order to pursue work in their discipline as well as the virtues that will allow mathematicians to share their work with scholars in other disciplines.
- Russell W. Howell and W. James Bradley (eds.), Mathematics: Through the Eyes of Faith (HarperOne, 2011)
- John H. Matthews and Russell W. Howell, Complex Analysis for Mathematics and Engineering (Sixth Edition) (Jones and Bartlett, 2011)
- Russell W. Howell and W. James Bradley (eds.), Mathematics in a Postmodern Age: A Christian Perspective (William B. Eerdmans, 2001)
Todd Ream: Welcome to Saturdays at Seven, Christian Scholar’s Review’s conversation series with thought leaders about the academic vocation and the relationship that vocation shares with the Church. My name is Todd Ream. I have the privilege of serving as the publisher for Christian Scholar’s Review and as the host for Saturdays at Seven. I also have the privilege of serving on the faculty and the administration at Indiana Wesleyan University.
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Our guest is Russell W. Howell, the Kathleen Smith Chair of Natural and Behavioral Sciences at Westmont College. Thank you for joining us.
Russell Howell: It’s a pleasure. Thank you for inviting me.
Todd Ream: As with other guests who contributed to this series concerning mathematics as a vocation, I’d like to once again start our conversation by asking for your impressions of a 1960 article by Eugene Wigner.
For individuals listening today who are unfamiliar with Wigner, he served most of his career on the faculty at Princeton University, served briefly as the Director of Research and Development at what became the Oak Ridge National Laboratory, and won the Nobel Prize in 1963 for his contributions to the understanding of the atomic nucleus. In February 1960, he published “The Unreasonable Effectiveness of Mathematics and the Natural Sciences” in communications in pure and applied mathematics.
When reading this article, I found Wigner resorting to curious and inviting forms of language. For example, in his first two points, Wigner claims that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious, for which there is no rational explanation. In his second point, he then contends that it’s just uncanny that this usefulness of mathematical concepts rises to the level it does in terms of physical theories. To begin, how prone are mathematicians and physicists to using terms such as mysterious or uncanny in summations of their efforts?
Russell Howell: It’s a little hard to quantify how frequently that occurs, but it certainly does occur. I would say they would not add there’s no rational explanation for it, because the uncanny feature has come out of some sort of a rational proof. I can give you a couple of examples of things that strike me as just bizarre, that I think I could explain without too much effort.
If you take any odd number like 15 and ask what are its proper divisors, 1, 3 and 5 are the numbers that go into it. If you add those up, it’s less than 15. 1 plus 3 plus 5 is 9. And there was a conjecture that every odd number, if you add up its proper divisors, will be less than the number, and that works for 3, 5, 7, 9, 11, all the way up to, let’s see, I think it’s something like 943.
But 945, if you add up all the proper divisors, it comes out to 975, which is just bizarre that all of a sudden the pattern holds and then bang, it breaks. Or the fact that the number pi comes up in so many unexpected ways. I mean, everyone knows the formula for the area of a circle is pi times the square of the radius, pi r squared. But if you can imagine taking a graph let’s see, it would look this way, y equals 1 over x, and just rotating that around, and extending it out to infinity, the volume is actually finite, and it’s pi, if you go from 1 to infinity. Why would that be?
If you add up the sum of the reciprocals of all the square numbers, so, 1 plus 1/4 plus 1/9 plus 1/16 plus 1/25 and so on, that comes out to pi squared over 6. Why in the world pi would get involved in that is maybe you could use the word mysterious. Uncanny is another, another word.
If you make the graph of a function, the sine of X divided by X, and ask yourself, what’s the algebraic area of that graph, it turns out to be pi over 2. If you take instead of the numerator, just the sine of X and the denominator X, if you have the numerator, sine of x times x over 3, and the denominator x times x over 3, watch that algebraic area, pi over 2.
Now take sine of x times x over 3 times x over 5 in the numerator, and then the denominator x times x over 3 times x over 5. Watch the area under that, pi over 2, and so on, up to let’s see, something like, sine of x over 15 over x over 15, so you have all those products in the numerator and all in the denominator, that’s not pi over 2. It’s some bizarre fraction, and they know the exact answer, but it’s not pi over 2. It’s something close to pi over 2, but not that. So why all of a sudden the pattern would change? That’s kind of a surprise, if not a mystery.
I don’t know if that’s an example that’s relevant for the people listening, but there are dozens and dozens of things like that where mathematics is all about studying patterns, and all of a sudden the pattern doesn’t work. Why would that be?
Todd Ream: At these moments when the patterns don’t work, such as what Wigner encountered, such as what you just described through a variety of examples, in what ways would the lessons from other academic disciplines perhaps be helpful in creating and offering explanations? I think here in particular, perhaps philosophy or theology that may grapple with some of these questions just in different ways or through different means.
Russell Howell: Yeah, I don’t know if mathematics or physics as a discipline itself would be able to provide answers other than that’s just the way it came out. But the question, why did it come out that way, it would be helpful for philosophers and mathematics and even theologians to be able to be conversation partners in those discussions. So I think there’s fruitful room for that kind of thing.
The discussions would have to begin with, what is your opinion of the nature of a mathematical object? Are they just human constructions? Or are they representative of God’s thoughts, for instance? The latter is an Augustinian view. The former would be sort of a view of a contemporary philosopher, William Craig or, or George Barclay also had that view, a nominalist kind of perspective of mathematics. So, I mean, I think there’s really an opportunity to get into some good conversations. I’m not sure you would be able to settle the issue.
Todd Ream: Yeah. In what ways, if any, then is mathematics as an expression of the academic vocation defined by persistence and persistence in the pursuit of truth through that which otherwise may seem mysterious or uncanny, at least for a season, if not, you know, perhaps even after persisting through as far as rational explanations may offer, a sense of peace with the what we’ve found or what we’ve uncovered?
Russell Howell: Well, I think this is not unique to mathematics. I think this would be true for any discipline, the notion of persistence. Thinking deeply about something that’s simple. By the way, that’s a phrase that was coined by the physicist Alfred Landé, who would have students at the Ohio State University over to his home for discussion about various philosophical issues.
And one student asked, what is it that distinguishes great thinkers from the mainstream? And the response was to be a great thinker, you have to learn to think deeply about simple things. And I tell my students often you have to be able to live with the problem or issue that you’re working on. And by living with it, you just, it’s just become part of your being.
One of the best results that I got in my doctoral dissertation came as I was sitting in the back of a car, riding from Wheaton, Illinois, back to Columbus, Ohio, in a snowstorm. I was not the driver, and I was just sitting back there, half daydreaming, thinking, why don’t you try this? And it sort of made sense in the car, I couldn’t really do it then. When I got back to campus, I tried it, and it worked. Why do you get that thought? You only get it because, prior to that, you’ve invested a lot of hard thinking about looking at other issues.
Todd Ream: And perhaps we’d like to think, at least in this part of the country, aided by the mundane nature of driving along the Indiana tollway there across the northern part of the state between Wheaton and Columbus?
Russell Howell: Yes, I have no doubt that that was part of it. I can’t remember the route we took it. All I remember is there was snow all over the place.
Todd Ream: A common feature of the Indiana tollway four or five months of the year, yeah.
I want to ask you then actually about your commitment and calling to mathematics and the time that you spent at Wheaton and Ohio State. You earned an undergraduate degree in math from Wheaton and then did doctorate in mathematics from The Ohio State University.
I also want to pause at this point in time to mention that you’re also a graduate of Sunny Hills High School.
Russell Howell: We share that, we share that distinction.
Todd Ream: Fullerton, California, which is the same hometown for both of us there.
But then you also earned a Master’s in Computer Systems Engineering from the University of Edinburgh. At what point did you begin to sense that mathematics would prove central to your vocation?
Russell Howell: Well, that’s a good question. I think some people just are born knowing what they want to do. I didn’t. My career path was number 1, I’m going to be a cowboy. I had a Roy Rogers bedspread and the whole thing. And then I got a chemistry set. Now I’m going to be a chemist. In high school, I wanted to be a pilot. And by the time I headed off to college, I was going to be a pastor. So I was gonna go through college, major in something, and then go to seminary and be a pastor.
But for sure—in my generation, you didn’t take calculus in high school, for sure I was going to take a calculus course because of an idle comment made by a geometry teacher after class one day. I was standing, talking with him, and I was very interested in pi, as I mentioned earlier, how that shows up. The area of a circle is pi r squared. No matter what the radius, pi always shows up. That’s kind of weird.
And he drew a picture of a curve that looks like this. It’s called a parabola. And he said, what do you think the area under that curve is between the origin and one? And he said, now, look, it’s a curved line. You can’t use the area of a triangle to get it because it’s not a straight line. And the curve has the equation y equals x squared. So I said to him, well, the area has got to be the square root of something. He said, nope, it’s exactly equal to 1/3. I said, it can’t be that simple a number. It just can’t be. And he said, well, if you take calculus, you’ll see why the area is one third.
So this little bell goes off. Okay, no matter what I got to find out why that’s the case. And so I took calculus. And when we got to that part, it was so satisfying, the elegant theory. It was so beautiful. You’ll see mathematicians using that phrase a lot to describe results. It’s just so elegant, so beautiful, so symmetric that I had to take another course.
And then sophomore year. I thought I’m going to major in mathematics, but I was on a plane ride back to California from Wheaton and got into a discussion with a person about my faith. And towards the end, he said, what’s your major? And I said, mathematics. And that really jolted him. It sort of gave what I was saying more credence in his judgment because he was scientifically oriented.
And I imagined what would have happened if I had that conversation 20 years from now, and the person says, what do you do? And I say, I’m a pastor. Will it have the same effect? Now, of course, pastors have a tremendous role, but I thought there’s a role for someone who loves mathematics, which by that time I did to be a witness for Jesus Christ to people who are maybe drawn to that field that think science disproves God or something like that. So I don’t know if that’s what you’re getting at, but that was sort of my career path.
Todd Ream: Yeah, you mentioned a geometry teacher who challenged you in the way that you noted, are there any other mentors who are helpful along the way? Any other comparable experiences, perhaps with elegant explanations details that really hooked your interest?
Russell Howell: I guess I would have to say Arthur Holmes, who is a philosophy teacher at Wheaton. So I was toying with, do I do a double major in mathematics or philosophy? And I had several courses from Arthur. And I was so impressed with how thinking deeply about philosophical issues can not only enrich your life, but also give you a bigger perspective on important issues of the day.
Todd Ream: Thank you. Broadly speaking, your area of expertise is complex analysis, with one of your signature efforts being Complex Analysis for Mathematics and Engineering. A book you co-authored with John Matthews, was published by Jones and Bartlett and has gone through numerous editions. For individuals unfamiliar with complex analysis, how would you define it? And then of all the sub disciplines in mathematics that could have garnered your attention, why complex analysis?
Russell Howell: First of all, it has such an incredible history. So for those that may not be familiar, complex analysis is a discipline that studies what we call sometimes imaginary numbers, things like the square root of minus one. And back in the day, in the 1400s, those kinds of things were scoffed at.
Negative numbers, by the way, didn’t hit the mathematical scene until later, even in the 1400s. There was no need for negative numbers. So they were around, but people didn’t pay attention to them. So if negative numbers were suspicious, taking square roots of negative numbers were ridiculous. If you multiply any number by itself, you get a positive number. So you can’t have the square root of a negative number.
But mathematicians kept using their imagination, and there’s a very colorful history of how complex numbers develop. What was needed was number one, some sort of a geometric representation. What would they be? And then how do you multiply and divide them? Things like that.
And a regular number, you can think of as just a point on a straight line with maybe zero arbitrarily designated, and then if you go to the right, that’s plus one. If you go to the left, that’s minus one. I don’t know if that’s right and left as you’re looking on the screen, but a complex number is simply now a point on the plane. So if down here is the straight line, you take a point on the plane and multiply it by itself, you can get to that number minus 1.
And now you mentioned Eugene Wigner earlier, he used complex numbers all the time as ways to describe the quantum world. The complex numbers play a big role. I’d even say they’re indispensable in doing that. So that’s just amazing by itself. That’s something that would have no practical application.
The theory of complex numbers was not motivated. by any desire for practical application. It was purely abstraction. Let’s see, you know, let’s see if we can somehow create the square root of minus one, and then maybe 200 years later, applications start showing up.
And there was a philosopher at University of Notre Dame who unfortunately died in about five years ago, named Mick Detlefsen, philosopher in mathematics. He sent me an email one day saying, you know, I am puzzled by how complex analysis, those theories, can simplify calculations so much for regular mathematics.
And that’s true also. I tell my classes when I teach the course, if you want to make things simple, complex analysis does it. Complex analysis makes real analysis simple. It’s sort of the standard line. Maybe I’m just sort of rambling here, but all those things combined are what kind of drew me to the subject.
Todd Ream: Perhaps the answer to the question I’m about to ask then, you know, points back to the examples you were offering early on in our conversation, but in what ways, if any, might the study of complex analysis lead at least initially to conclusions, which can be described as nothing other than mysterious or uncanny?
Russell Howell: In what ways can the study of complex analysis lead to conclusions that are uncanny? I’ll go with that. So this gets back to Mike Detlefsen’s question. Why is it that complex analysis is able to simplify calculations that would otherwise be very laborious? That’s just what it does. I mean, so in order to evaluate real integrals, certain types of real integrals, the process is greatly simplified by using complex analysis. I guess I don’t want to get more technical than that.
By the way, about a year later, I emailed Mick and I said, in my initial response, he wanted to have examples of things, and I gave him a few. And about a year later, I emailed him back and I said, I’m curious about your philosophical take on this. Why is it that complex analysis makes real analysis calculations simpler? And he said, I got involved in some other projects, and I’m going to turn my attention to this pretty soon, but I don’t have anything to say yet. And then, unfortunately, he passed away.
Todd Ream: Would you please describe one of your recent research efforts to us?
Russell Howell: Oh sure. I had three great research students in the summer of 2023. We worked for 10 weeks in the summer, and came up with results that are going to be published next month in the American Mathematical Monthly. And one of those results led to a question that’s not solved yet, so I can talk about that.
So take the number line and just take the numbers between zero and one and write the number in a decimal expansion. So one third would be 0.33333. 3/4 would be 0.75. Take those decimal numbers and let’s just say for simplicity, if the numbers are between 0 and 5, associate that with a plus sign, 6 through 9 associated with a minus sign. And then depending on the number that you pick, you can get an infinite sequence of plus and minus signs. So if the number is 0.33333, then you get all pluses. If the number was 0.3821, you might get a plus, a minus, a plus, a minus, and so on.
With those sequence of pluses or minuses, then you can form a polynomial. X minus X squared plus X cubed and so forth. And the question we asked is, what’s the probability if you pick a number between 0 or 1, and you form the polynomial with pluses or minuses, that it will have a certain property? And we were able to prove that there are infinitely many numbers that give polynomials with that property, and also infinitely many numbers that give polynomials that don’t have that property.
And we were also able to prove that the probability is either 0 or 1, that you pick a number that will have that property. So there’s infinitely many that do, infinitely many that don’t, the probability is either zero or one. Watch the probability. That’s the unsolved question. I don’t know if that’s been helpful or if that flew by you too fast, but.
Todd Ream: No. What about that pursuit and that effort that you and your students engaged in, in the summer of 2023, did you all find appealing and or gratifying intrinsically so perhaps it had some of the elegant qualities to it in terms of what began to emerge that you sort of reference back that mathematicians also find so appealing?
Russell Howell: Well, first of all, it was a great pleasure to be working with such highly gifted students. I mean, one is just about to graduate this year. One graduated, two others graduated in 2024. And we’re still in touch. So the fellowship that comes from working on, I think C.S. Lewis talked about friendship develops when you’re pursuing a common area, that’s where real friendship comes from, where you’re both engaged in a common effort. And so I just think those three guys are just tremendous people, and so that was a real joy.
But in terms of the results itself, Andrew Wiles, in a very interesting NOVA video, a Princeton mathematician was asked about the process in which he proved an amazing result, he proved Fermat’s Last Theorem, something that had been unproved for over 300 years. And he says something like this: sometimes it involves reading something in a book over here and seeing how that calculation was done. Sometimes it involves looking at some other effort and trying to expand the context. And then sometimes you just have to come up with something completely new. It’s a mystery where that comes from.
And that, that bit about coming up with something new reminds me of a quote by the logician Augustus De Morgan, which I have my classes memorize and recite, which is simply this, the moving power of mathematical invention is not reasoning, but imagination.
I just think that’s really a great quote, because it’s true. It’s the creative part, the imagination part, and there was one case where one of my students read a paper that someone else had written and said, hey, we can use this. It just bang. Look, this is a great thing we can use. And when we did, we got a result because of that.
Todd Ream: Yeah. That’s great. What a great example.
I want to ask you about some of your other work then now, because addition to complex analysis and your efforts in that subfield, you’ve led some prominent efforts pursuing the relationship shared by mathematics, philosophy, and theology. When pursuing those efforts, what historical conversation partners have been part of your thinking? And then what contemporary conversation partners have been part of your thinking? To whom have you turned for inspiration and perhaps fire your imagination?
Russell Howell: Well, I guess the first one I’d have to mention from the past would be Augustine, Bishop of Hippo. He has a view of mathematics that I’m very drawn to, and it’s most thoroughly articulated in his book on the free choice of the will. It’s he, I guess, you could say, he Christianized Plato.
Plato had this view as those who study philosophy know there’s this ideal world of forms. It’s a little hard to say where they are, how they exist or how humans interact with them. But for instance, the reason why I can recognize something as a chair is because it, according to Plato, participates in the universal form of sureness.
Well, Augustine, at least for the logical side of mathematics, like 2 plus 2 equals 4, 5 plus 7 equals 12, those logical truths have always been true because they are representative of God’s thoughts. And humans being created in God’s image can think God’s thoughts after Him. And that’s why we’re able to do mathematics. And that’s partly why mathematics is so successful in explaining how the universe works.
I think it was C. S. Lewis who said something like this, the disorderly or chaotic world that we refuse to believe in is the same disorderly or chaotic world that he refused to create. So God’s mind is orderly, it seems. And He created a world that can be described by means of these orderly logical thoughts. Now that’s, that was Augustine’s view. It has, there’s a lot of philosophical problems with that position that I don’t want to go into, but nevertheless, I’m still drawn to them.
As far as the opposite view for a conversation partner, someone from again, a while ago, Bishop George Berkeley in the 1700s, he was a nominalist. And so you have the Christian Augustine, who is, let’s say, a Platonist in mathematics, he Christianized Platonism. But the Christian George Barclay was a nominalist. And so I’m really attracted to some of the thinking that George Barclay has to say about the nature of mathematics.
A more contemporary person with that view, with that nominalistic view that I mentioned earlier is William Craig. He has written extensively on that and he is a nominalist. A more contemporary person who would be a Platonist is Chris Menzel at Texas A& M. He wrote a very interesting article called, ” Theism, Platonism, and the Metaphysics of Mathematics.: So I’ve learned a whole bunch from those.
I said I’m drawn to the Augustinian view, but that’s sort of when I, on Tuesdays and Thursdays. I am also, wow, you know, there are problems with it, and I understand the other view too. So I just enjoy reading about that stuff.
Todd Ream: Thank you. I want to ask you about these efforts and begin by asking you about one of the articles that you published, this one in particular in June 2015 in Perspectives on Science and Christian Faith was entitled “The Matter of Mathematics.” After working through some of the questions and challenges that emerged in terms of the relationship shared by mathematics and faith, you offer an elaborate overview of opportunities for fostering relationships between them.
In the 10 years now that have just about passed since the publication of that article, have any of those opportunities perhaps proved more promising than others? And if you were to write that article again today, in what ways, if any, might you rephrase it or reconstruct some of those?
Russell Howell: Well, let’s see, I let’s go back to Berkeley again, because there’s something that he wrote in 1734. The essay is called “The Analyst or a Discourse Addressed to an Infidel Mathematician.” And he is arguing against the freethinking movement in the UK at the time and saying essentially, you freethinkers criticize religious people for relying on authority, tradition, and revelation. You say we’re superstitious. You’re worse in mathematics.
This is when Newton’s calculus was first getting launched, and his criticisms of Newton’s calculus have stood the test of time, at least as it was developed then. There are lots of stuff mathematicians didn’t know what they were doing. Now, since then, the foundations have been firmed up.
But Berkeley concludes the essay with a series of pithy rhetorical questions, that relate in part to mystery. And I think there are mysterious conclusions that mathematicians draw that can help us deal with paradoxes that arise in the Christian faith.
So for instance, using the axioms of Zermelo and Frankel, ZFC is the accepted set of axioms that mathematicians now use. You can take a sphere, of let’s say the volume is one and decompose it into just five sections without distorting or bending them or twisting them, reassemble it into two spheres with the same volume. How can that be? Well, there are ways to resolve it. The two things that you get are not measurable sets, it turns out, and that’s more technical than I want to get into.
But I had a New Testament teacher at Wheaton, Morris Inch, who said one day, you know, I have a problem set, we’re talking about synoptic problems, something like that. And, and he paused and he said, I have a problem set of issues with the Scripture that just, you know, they’re, they’re hard to, hard to sort out. And then he said, I take comfort that this problem set is different than it was 20 years ago. But it’s always there.
So if things can get convoluted, so to speak, in a logically defined, very precise language framework, such as mathematics, it shouldn’t surprise us that paradoxes and antimonies or things like that show up in our Christian faith. And the fact that there are these surprising things in mathematics, some of which you go, how could that be?
Let me give you another example. I mentioned these two sets of numbers. There’s infinitely many numbers that have the property, there’s infinitely many numbers that don’t. Well, the fractions are one set of infinite numbers. Things that can’t be written as fractions, we call irrational numbers, are another set. And you can actually show that you can’t match those up. There’s actually more of the irrationals than rationals. How can that be, if they’re both infinitely large? And then you just shrug your shoulders and say, that’s the way things are with infinity.
And sometimes looking at those aspects of mathematics has helped me to accept the possibility of there being mysteries in my faith, things that I won’t necessarily be able to sort out, even though the context of this scriptural passage is slightly different. As far as the heavens are above the earth, so are My thoughts above Your thoughts. So it creates maybe a humility about the limits of human knowledge, I suppose.
Todd Ream: Thank you. I want to ask you now about two of your co-edited works. Perhaps one for which you are most well-known was published in 2011 by HarperOne, a volume you co-edited with James Bradley entitled Mathematics Through the Eyes of Faith. 10 years prior to that, also with James Bradley, you co-edited Mathematics in a Postmodern Age that was published by Eerdmans.
What were your motivations for co-leading those efforts? What did you learn, through those efforts and the conversations that you had with your fellow contributors?
Russell Howell: There’s an organization that was founded by Robert Brabenec of Wheaton College called the Association of Christians in Mathematical Sciences. It has about 400 members. And I had read a book, I don’t want to give the title because I was very critical of the book. It was dealing with mathematics and faith. And I thought it was very, very confrontational, maybe naive philosophically.
For instance, there’s a line early in the book, and it says, secular thinking teaches us that man began counting by grunting and groaning in the cave. A biblical view of man presents him as a fully functioning automaton. And when I read that, I got embarrassed at the thought of a non-Christian reading that, regardless of what one’s view of evolution is, I thought that was not the right approach.
So we had a conference of this organization in 1997, and I gave a little talk at the conference criticizing that book. And saying, look, I think collectively we can write a better one. I don’t think I’m capable of writing a book by myself that deals with mathematics and faith, but collectively, come on, we can do this. And I outlined, here’s what I think such a book would look like. So that was part of the talk.
That night there was a knock on my door and it was Jim Bradley and he said boy, you know I’ve been thinking about this too and I think we can together. And so we wound up getting a grant from the Calvin Center of Christian Scholarship, brought in a bunch of scholars to work and that’s where Mathematics in a Postmodern Age came from. The opportunity to work with other Christians who are pursuing a common interest, I think I referred to C. S. Lewis, that brings great joy, and I’m still in touch with those people.
10 years later, Jim left Calvin, and he was at the Templeton Foundation. I had an appointment at Oxford at the time, and I got an email out of the blue, and he said, hey, I just found out that HarperOne is revising their Through the Eyes of Faith series. This would be a good time to propose that we have a mathematics volume. I can’t do it because I’m at Templeton. I’m not allowed to do this, but why don’t you contact Harper One?
So I did, and we sent out surveys, and we had another workshop with people. A year or two later, Jim had wound up leaving Templeton, and so he came on, you know, he was allowed to join the project. So that’s how that got going.
Todd Ream: In what ways do you think those volumes and their widespread use in undergraduate classes have impacted how conversations are taking place concerning the relationship shared by theology, math, and faith?
Russell Howell: Oh, that’s a great question. I can give you a personal example. I got an email out of the blue from an atheist mathematician. And he said hey, I read somewhere that you wrote about a book about math and your faith. Having no faith in me, I don’t even know what that means, except perhaps in my ability to reason, I can’t see how a rational person can subscribe to the Christian dogma. And I just wonder if you have anything to say about that. That was essentially his email.
And so I wrote back and I said, let’s get together for breakfast. He concluded his email by saying, by the way, I’m a happy member of the Skeptic Society of Altadena. And so about a year later, maybe less than a year, we got together and had a very good conversation, but it was just the fact of that book, rather than the content that produced some good conversation. He remained a skeptic as far as I know, but I think more appreciative of people in studying mathematics, who also have a faith.
Todd Ream: Before we close our conversation I want to ask you a couple of questions about the academic vocation, your understanding of it, and the practice of mathematics. As a mathematician and a scholar of complex analysis, what qualities and or characteristics define how you understand and come to terms with your calling to the academic vocation?
Russell Howell: There’s a great quote by Frederick, Frederick Buechner, who talks about calling. I hope I can get it right. The place God calls you is where your deep gladness and the world’s deep hunger meet. Frequently, I counsel students that come into my office and say, I want to be such and such, and I’m not sure what major to do because I want to do such and such. And I say, you’re asking the wrong question.
You should ask not what you’re going to do with your major. You should ask what your major is going to do with you. How is it going to shape you? How is it going to develop your character? How is it going to increase your imagination? Your creativity, any major can do that. And the way you serve God is going to be by using your passion and contributing to God’s kingdom with the gifts that He’s given you.
I’ll give you an example of something that shows up in missions work, that would have been maybe would have been surprising. This is back maybe in the 1950s, I’m not sure of the year, but Ivan Lowe, a British mathematician, and Kevin Pike were called to consult on a language translation problem in Bari, in the Sudan Interior Mission. And the linguist had spent time learning the language, it was a monolingual language at the time.
And finally thought they had it, and they preached. Jesus said, I am the Light of the world. And the response of the natives was, be wilderness. What do you mean? You want us to worship you if you’re the light of the world? So it would be as if I said Jesus said I, meaning the speaker and the Light of the world, as opposed to Jesus Himself is the Light of the world.
So Ivan Lowe solved that problem, believe it or not, using permutation groups, which is part of a branch of mathematics known as group theory, that you usually study about your junior or senior year as a mathematics major in college. And he solved that problem and wrote an article in and said the name of the article was “Christian Mathematicians, Where Are You?”
And he said, we need people who are trained in formal skills to be able to help with language translation efforts. And I frequently counsel students, you major in what you like and what challenges you, and then you will find a way to serve God with it, in ways that might surprise you.
Todd Ream: Thank you. You may have mentioned a couple of these already. We talked, for example, about humility but as a mathematician, what virtues, whether they be intellectual, moral, or theological, do you believe are most important to cultivate?
Russell Howell: Boy, that just, what you’re saying that reminds me of a comment that Stanley Hauerwas made once, and I think it’s written down also somewhere, but he was visiting campus. And he was talking about moral virtue. And he said, look, the best moral training you can give someone is to make them take a good course in mathematics. He actually said that.
And at the time he was reacting to postmodernism, where all truth is relative, and this is my truth and that’s your truth. And he said, in mathematics you’re confronted with an objective reality. And you might get the wrong answer. You might be wrong, and you have to realize it’s just obvious that you’re wrong.
That might be more true in mathematics than some other disciplines, where you hear students say, well, just because I didn’t say things the way the teacher wanted it, I got marked down. You don’t hear that too much in mathematics. You have the students say, no, I agree that I’m wrong. So there’s a moral quality there about submitting yourself to some sort of higher authority. In this case, it’s objectivity and reason and so forth.
I think also the quality of being able to stick with the task and persist in it. So this research group that I mentioned for five weeks, we were, we didn’t get anything. So we were just fiddling around. But as I said, living with the problem, then you say, wait a minute. Things start coming and I think that’s relating to a moral component also. Persistence, humility before higher authority.
Todd Ream: If I may ask sort of the reverse side of that for mathematicians, are there any vices which you think are important to guard against and to be vigilant?
Russell Howell: Sure. The opposite would be arrogance, and think that your discipline is the only thing that’s important and to sort of scoff. You know, you hear this a lot of people, you hear the word, the soft sciences, you also hear sometimes in secular settings, I’ll go to a math conference and people in the humanities are sometimes ridiculed, oh, sloppy thinking, soft thinking.
Now, someone that studied a fair amount of philosophy in undergrad school, I go, you have no idea what you’re talking about. You get in a conversation with a philosopher, and they will lose you faster than you can say, you know, snap your fingers. So, I think, a vice would be intellectual arrogance. And again, that’s not just limited to mathematics.
I will say something that grieves me, though, in our culture. This is maybe going the other way. It’s sort of cool to brag about your ignorance of mathematics. Oh, I’ve never been any good in mathematics, ha ha ha. It doesn’t seem as equally acceptable to say, oh, I’ve never heard about Shakespeare, ha ha ha. If you had that attitude, you’d be, wait, what’s wrong with you? But it’s somehow cool to say, oh, I’m never good in mathematics and I can ignore it.
You’re going to be interviewing Francis Su in the following week. And he has a very good book called Mathematics for Human Flourishing, in which he argues that everyone does mathematics at some level, whether they realize it or not.
The famous British historian Arnold Toynbee laments that he was not required to take a course in calculus because it is the basis of all modern science. He said, I was too immature and I just decided to go on and do classics.
So I wish there would be some sort of in a liberal arts college, I would like there to be, I talk my students into taking humanities classes. And I hope, and I think they do, my humanities college colleagues do likewise. And say, look, you need to take a mathematics course to be a truly liberally, liberally educated person.
Todd Ream: I think we could have a whole nother conversation just about the role that mathematics has historically played and should continue to play, but is perhaps declining right now in what we call general education or core curriculum and in terms how we seek to shape the sensibilities of one’s mind and the orientation of their thinking. And yeah work needs to be done there.
Before we close though, if I may ,because we’re talking about, you know, of these details right now, what contributions, if any then, do you think are mathematicians uniquely positioned to make to colleagues who are called other disciplines? In what ways can those contributions become more part of our thinking and more publicly aware in terms of our sense of understanding?
Russell Howell: Yeah. Well, let me begin by criticizing mathematicians and then maybe offering a remedy. And the criticism is that we have developed, as a discipline, a vocabulary that’s so specialized that if it would be very, very difficult for me to have a conversation, say, with my wife even, hey, what are you doing in research? Three words out of my mouth, and I have no idea what you’re talking about.
And that’s different from, I mean, if I talk to a chemist, they can, they have ways to describe what they do. Because words like molecule and so on are in the general vocabulary. And I think that’s true for humanities people. They can describe what they do because they have the vocabulary for it.
I think a person in philosophy or theology could lose someone instantly if they wanted to talk about what are they doing in their research with technicalities and so on, but they don’t do it because they’re not arrogant and they have a vocabulary for doing. So mathematicians need to do a better job at communicating in vocabulary that’s part of the culture, communicating what their discipline is about.
There are two arguments often that are given for the value of mathematics, of pure mathematical research. One is the future value argument. Maybe what you’re doing in these abstract things don’t have any application now, but the history of mathematics reveals that maybe even a hundred years later, application is found. Complex analysis being one example that we talked about. So there’s that and there’s also the aesthetic argument, just like a painter paints objects that are beautiful so are mathematical theories are beautiful. And in that sense, they’re intrinsically worthwhile.
Well, you know, the future value argument is lessened to the extent that mathematics is becoming so specialized that only maybe 20 people in the world can read and understand an article that’s been written. There needs to be a way to somehow communicate the results to a larger audience. And that also impacts on the aesthetic argument. If there’s only 20 people that can understand what you’re doing, what’s the aesthetic value for that? If it’s only limited to 20 people.
So just circling back to your question, I think that we, just as a community, have work cut out for us to be able to engage with other people. And part of that has to do with finding better ways to communicate.
Todd Ream: Yeah, thank you very much. Our guest has been Russell W. Howell, the Kathleen Smith Chair of Natural and Behavioral Sciences at Westmont College. Thank you for taking the time to share your insights and wisdom with us.
Russell Howell: You’re welcome, Todd. And as I said earlier, it was a great pleasure to have been asked.
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Todd Ream: Thank you for joining us for Saturdays at Seven, Christian Scholar’s Review’s conversation series with thought leaders about the academic vocation and the relationship that vocation shares with the Church. We invite you to join us again next week for Saturdays at Seven.
It seems that Westmont is blessed to have Howell, and he is enthused with them! I love his borrowed mantra — “the moving power of mathematical invention is not reasoning, but imagination.”