Sermon copyright (c) 2025 Dan Harper. As delivered to First Parish in Cohasset. The sermon as delivered contained substantial improvisation. The text below has too many typographical errors, missing words, etc., because I didn’t have time to make the necessary corrections.
Readings
The first reading was an excerpt from the essay “A Mindful Beauty” by the mathematician Joel E. Cohen, from the September, 2009, issue of American Scholar.
“My grade-school education in mathematics included a strict prohibition against mixing apples and oranges. As an adult buying fruit, I often find it convenient to mix the two. If the price of each is the same, the arithmetic works out well. The added thrill of doing something forbidden, like eating dessert first, comes free. In any case, the prohibition against combining apples and oranges falls away as soon as we care about what two subjects, different in some respects, have in common.
“I want to mix apples and oranges by insisting on the important features shared by poetry and applied mathematics. Poetry and applied mathematics both mix apples and oranges by aspiring to combine multiple meanings and beauty using symbols. These symbols point to things outside themselves, and create internal structures that aim for beauty. In addition to meanings conveyed by patterned symbols, poetry and applied mathematics have in common both economy and mystery. A few symbols convey a great deal. The symbols’ full meanings and their effectiveness in creating meanings and beauty remain inexhaustible….
“The differences between poetry and applied mathematics coexist with shared strategies for symbolizing experiences. Understanding those commonalities makes poetry a point of entry into understanding the heart of applied mathematics, and makes applied mathematics a point of entry into understanding the heart of poetry. With this understanding, both poetry and applied mathematics become points of entry into understanding others and ourselves as animals who make and use symbols.”
The second reading was from the poem “Equation” by Caroline Caddy:
…working through difficult equations
was like walking
in a pure and beautiful landscape —
the numbers glowing
like works of art….
The third reading was from a letter written by Albert Einstein, as printed in Albert Einstein, the Human Side: New Glimpses from His Archives (Princeton Univ. Press, 1979):
“If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it.”
Sermon: “Math and Religion”
In honor of Pi Day, which was yesterday, I’d like to talk this morning about the connection of mathematics and religion. The right-wing Christians who make so much noise these days keep trying to tell us that religion has nothing to do with either math or science. But the connection between mathematics and religion in Western culture predates Christianity, and goes back to the ancient Greeks.
The first great mathematician in Western culture was Pythagoras. Pythagoras is best known today for the theorem known as the Pythagorean theorem: in a right triangle, the square of the hypotenuse is equal to the squares of the two other sides. But Pythagoras was not just a mathematician. He also founded a religious community, which was remarkable for combining serious mathematical and scientific inquiry with some fairly strange religious beliefs.
Pythagoras was born in Greece, on the island of Samos. As a young man, he traveled around the Mediterranean Sea seeking learning and wisdom. He supposedly learned arithmetic from the Phoenicians, geometry from the Egyptians, and astronomy from the Chaldeans. He also learned some interesting religious rituals. Tradition tells us that the Egyptians didn’t want to teach him about geometry, so to dissuade him they made him follow strict religious rituals. But Pythagoras wanted to learn the secrets of geometry, and followed all the rituals carefully. So Pythagoras learned his math and science along with religious ritual.(1) Mind you, religion was not the same as it is today.(2) Rather than being focused on personal belief in a transcendent god, religion primarily consisted of ritual, most of promoted social cohesion.
In addition, much of what passed for scientific investigation in that time took place in what we would call religious communities. This actually makes a lot of sense. If you want to gather enough data to be able to predict eclipses — one of the major scientific achievements in Pythagoras’s day, and one which he was directly involved in — then you need a stable community that can support people who spend their time observing the night sky; a community that can collect and safely store data over fairly long periods of time; and a community that brings together people who learn from one another and strive together for the truth. In fact, this kind of community still lies at the root of scientific and mathematical progress. If you’re doing math or science, you have to be in a community of peers that can review your work; that’s how scientific progress happens. Pythagoras not only learned in such communities, he brought the concept back to Greece, and founded his own religious community.
The Pythagorean community was governed by a set of rules, such as the rule prohibiting the consumption of beans.(3) Pythagoras was convinced of the transmigration of souls, and he thought the movement of souls took place through bean plants. There mix of “semi-scientific observation” with superstition sounds alien to us today, but as one scholar puts it, “a network of cleverly designed reasons, with the doctrine of the transmigration of souls at its center, held the whole system together….”(4) Today we would not call this science, but it does represent the beginnings of science.
And the Pythagorean community managed to come up with some pretty interesting discoveries in math and science. The Pythagorean community discovered the connection between numbers and music; predicted eclipses; developed the idea of numbers as shapes, as in the square of a number or the cube of a number; and with the Pythagorean theorem helped lay the foundations for geometrical proof later perfected by Euclid. Finally, Pythagoras is supposed to have said that “all things are numbers,” which in a generous interpretation resembles the way science today uses mathematics to model reality.(5)
Also noteworthy is that the Pythagorean community admitted women as members.(6) By today’s standards we would doubtless consider the Pythagorean community to be hopelessly sexist, but by the standards of their day they were unbelievably progressive. Women in the Pythagorean community contributed to the theoretical work of the community, and wrote their own treatises. This may be the earliest recognition that women have just as much to contribute to math and science as men; a fact that certain elements in today’s scientific and mathematical communities are still trying to accommodate themselves to.(7)
We also get our word “theory” from the ancient Greek word “theoria.” For the Pythagorean community, “theoria” meant a kind of “passionate sympathetic contemplation” that came out of mathematical knowledge; it represented a kind of “ecstatic revelation.”(8) While I am not especially adept at mathematics, this does describe the feeling I’ve gotten at times when I’ve finally managed to follow a proof of a challenging theory — a very satisfying feeling that comes upon perceiving something that’s really true and good and beautiful and unchanging. Right-wing Christians would be horrified to hear me say this, but this is indeed a kind of religious experience.
This brings me to Kurt Godel, the next mathematician I’d like to talk about. You may have heard of Godel from the bestselling book Godel, Escher and Bach: An Eternal Golden Braid, written in 1979 by Douglas Hofstader. However, I first encountered Godel in 1981 when I took an introductory course in mathematical logic. This class was designed to give us enough background so that we could follow the proof of Kurt Godel’s famous incompleteness theorems.
I remember being blown away by the implications of Godel’s incompleteness theorems. The first incompleteness theorem can be summarized like this: “Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.”(9) What I took from Godel was this: that within a logically consistent system like arithmetic, you have to accept some statements that cannot be proven within that system. And even though you might be able to construct another logically consistent theorem that would allow you to prove those unproved axioms, there would be other axioms that you couldn’t prove within that second system.
Godel’s theorems obviously have implications for mathematics, but Godel himself believed that they had also implications for all human thought. John W. Dawson, a mathematician and biographer of Godel, put it this way, quoting in part from one of Godel’s lectures:
“[Godel] believed [there was] a disjunction of philosophical alternatives. Either ‘the working of the human mind cannot be reduced to the working of the brain, which to all appearances is a finite machine,’ or else ‘mathematical objects and facts … exist objectively and independently of our mental acts and decisions.’ Those alternatives were not … mutually exclusive. Indeed, Godel was firmly convinced of the truth of both.”(10)
If Godel was correct, this becomes very interesting. First, if the human mind is indeed something more than the workings of the brain, what is that something more? Perhaps this is no unlike what the ancient Pythagoreans called “soul.” We Unitarian Universalists affirm the inherent worthiness and dignity of every personality. In this sense, we agree with Godel that human beings, and other sentient beings, are something more than mere machines.
Second, if mathematical objects exist objectively and independently of our mental acts, what does that mean for science? Most of us these days believe that mathematics is useful because it creates models to help us understand the physical world. We typically believe that the greatness of the mathematics in Einstein’s theory of relativity, for example, is that the mathematics helps us understand observations made in real world scientific experiments. But Godel understood mathematical objects to have an independent existence. Since they are not bound to things in the real world, these pure mathematical objects are not perceived through the usual senses. We intuit them directly, through our minds. Compare this to Ralph Waldo Emerson. Emerson, a Unitarian who remains the biggest single influence on , was a Transcendentalist who said that we could directly apprehend truth and beauty. Thus, we Unitarian Universalists are like Godel in that we have a tendency to think that we can apprehend truth directly with our minds.
This brings me to the third and last mathematician I’d like to talk about: Karen Uhlenbeck, a Unitarian Universalist who also happens to be one of the greatest of living mathematicians. Uhlenbeck received a MacArthur “genius grant” in 1982, and in 2019 became the first woman to win the Abel Prize, the most prestigious prize in mathematics. The Abel award cited Uhlenbeck for “her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.”(12)
Sadly, I don’t have the background to understand Uhlenbeck’s mathematical achievements.(13) I did discover, however, that she has spoken about the connection between mathematics and introspection, and between mathematics and community. Both introspection and community are characteristic of Unitarian Universalist notions of religion, and I wondered if this might represent a connection between mathematics and religion.
Not long after she was announced as the winner of the Abel Prize, Uhlenbeck was asked if she though success in mathematics is partly due to concentration. She replied, “I think you can’t do mathematics without the ability to concentrate. But also, that’s where the fun is, the rest of the world fades away and it’s you and the mathematics.” In that same interview, Uhlenbeck said: “You struggle with a problem, it can be over a period of years, and you suddenly get some insight. You’re suddenly seeing it from a different point of view and you say: ‘My goodness, it has to be like that.’ You may think all along that it has to be like that, but you don’t see why, and then suddenly at some moment you see why it is true….”(14)
To me, the way Uhlenbeck describes what it feels like to solve mathematical problems sounds similar to how people who have meditation or mindfulness practices describe their epxeriences. The process goes something like this: you concentrate, and the world fades away, and it’s just you and something beyond yourself. Then, if you concentrate long enough, you may have an “aha” experience that really feels out of the ordinary, where you feel like you’ve seen something new and (dare I say it) beautiful. So I emailed Uhlenbeck to ask if she thought there was a similarity between doing math and doing meditation. She replied, in part: “When I try to meditate, I usually end up thinking about math. They are very similar.”(15)
Indeed, this experience occurs in many different pursuits. In the first two readings, mathematician Joel E. Cohen and poet Caroline Caddy both find a deep connection between poetry and mathematics, because both “create internal structures that aim for beauty.”(16) In these kinds of experiences, we use symbols to help us perceive the beauty and order of the universe. The poets and mathematicians have the original insights, and then we ordinary folks can experience some of the same wonder by following the mathematical proof, or reading the poem, or reading one of Ralph Waldo Emerson’s essays. Although right-wing Christians would disagree, I would call these religious experiences.
Mathematics and religion are also connected in that human community is central to both. Most obviously for mathematics, when a mathematician thinks they have done some original work in mathematics, they have to write it up and publish it so that their work can be reviewed by other mathematicians. Individual mathematicians may work alone, but overall mathematical progress happens in community, as mathematicians check each other’s work, and then build upon the work of others.
Religion also requires human community, for much the same reasons. Take Ralph Waldo Emerson as an example. Emerson had one or two insights on religious matters, and wrote them up in an essay he titled “Nature.” When he first published the essay, some people thought it was brilliant and others thought it was garbage. Over time and after much discussion, a consensus arose that Emerson really had come up with some genuine insights into religion. Still others came along and extended Emerson’s insights, including people like Henry David Thoreau.(17) Emerson’s new ideas first had to be carefully considered by a human community, and then extended by other people.
Karen Uhlenbeck refers aspect of human community in an interview. When Uhlenbeck was doing postdoctoral work at the University of California in Berkeley in the 1960s, she found herself in the midst of tumultuous political activity concerning the Vietnam War, women’s rights, and so on. Uhlenbeck had always thought of mathematics as somehow separate from politics. But, she told an interviewer, “I was startled to see the politics appear in the math department. It was eye-opening to me… up until that time I had seen mathematics as a very bookish thing and that what went on in the mathematical community had nothing to do with the life out there on the streets, and this is not true.” In other words, Uhlenbeck realized that mathematics is a human activity that’s done by humans. This means that “all of what goes on between humans appears in the mathematics community, perhaps toned down quite a bit, but it’s not a world of pure brains, people behaving rationally and unemotionally.” (18)
One of the very human problems in the mathematics community that Uhlenbeck became aware of was that nearly all mathematicians were men. She told one interviewer, “if I had been five years older, I could not have become a mathematician because disapproval would be so strong.”(19) Thus while human community is necessary, human community also has problems that must be addressed. If you’re a mathematician, you can’t just take the human community for granted, you have to be willing to confront the faults and problems of that human community. Obviously, the same is true for any human community, including religious communities.
In today’s world, we have a strong tendency to separate religion from mathematics and science. Yet by so doing, I think we place unwarranted restrictions on religion. The right-wing Christians are wrong — religion, religious experience and activity, can not be restricted to the very narrow sphere of personal belief in a transcendent god. Religion includes the introspection that occurs not only in meditation and centering prayer and mindfulness practice, but also introspection of doing math and science. Religion and mathematics can both result in ecstatic experiences that come when you gain insight into truth. Both religion and mathematics are rooted in human community. And while you, personally, may not have ecstatic experiences or pursue introspective practices, yet as a part of a human community we accept the differences between us, and try to lrean from those differences.
Notes
(1) Christopher Riedwig, Pythagoras: His Life, Teaching, and Influence, p. 8.
(2) See, e.g., Brent Nongbri, Before Religion: A History of Modern Concept, Yale Univ. Press, 2013.
(3) Bertrand Russell, The History of Western Philosopy, p. 32.
(4) Riedwig, p. 71.
(5) Russell, p. 35.
(6) Much of what follows is taken from Sarah B. Pomeroy, Pythagorean Women: Their History and Writings, Johns Hopkins Univ. Press, 2013.
(7) See, for example, the 2005 remarks of Lawrence Summers, then president of Harvard University. According to the Harvard Crimson, “Summers’ Comments of Women and Science Draw Ire” (14 Jan. 2005, article by Daniel J. Hemel), Summer said “the under-representation of female scientists at elite universities may stem in part from ‘innate’ differences between men and women….” Admittedly, this is not precisely what Summers said, but for a good discussion of the implications of Summers’s remarks, see “What Larry Summers Said — and Didn’t Say,” Swarthmore College Bulletin, Jan. 2009, article signed “D.M.,” available online: https://www.swarthmore.edu/bulletin/archive/wp/january-2009_what-larry-summers-said-and-didnt-say.html
(8) Russell, p. 33. The OED says that “theoria” refers to contemplation, including the contemplation of beauty.
(9) Panu Raatikainen, “Godel’s Incompleteness Theorems,” Stanford Encyclopedia of Philosophy (Spring 2022 ed.), ed. Edward N. Zalta, https://plato.stanford.edu/archives/spr2022/entries/goedel-incompleteness/ accessed 15 March 2025. For an explanation of both Godel’s proof, and its implications, designed for the intelligent layperson, see: Ernest Nagel and James R. Newman, Godel’s Proof, New York Univ. Press, 1958; this book is available to read online at the Internet Archive: https://archive.org/details/gdelsproof00nage/page/n5/mode/2up
(10) John W. Dawson Jr., Logical Dilemmas: The Life and Work of Kurt Godel, p. 198.
(12) Isaac Chotiner, “A Groundbreaking Mathematician on the Gender Politics of Her Field,” New Yorker, 28 March 2019.
(13) For those who do have the background to understand Uhlenbeck’s work, a discussion of her achievements in variational problems in differential geometry is freely available online in Simon Donaldson, “Karen Uhlenbeck and the Calculus of Variations,” Notices of the American Mathematical Society, March 2019, pp. 303-313 DOI: https://doi.org/10.1090/noti1806. There may well be other such technical summaries available online and not hidden behind paywalls.
(14) Bjørn Ian Dundas and Christian Skau, “Interview with Abel Laureate Karen Uhlenbeck,” Notices of the American Mathematical Society, March 2020 [reprint of an interview originally published in Newsletter of the European Mathematical Society, September 2019], p. 400.
(15) Karen Uhlenbeck, personal communication, 11 Feb. 2025.
(16) Joel E. Cohen, “A Mindful Beauty,” American Scholar, September 2009.
(17) As an aside on Emerson: In his book The American Evasion of Philosophy (Univ. of Wisconsin, 1989), Cornell West argues that Emerson also lies at the root of the American philosophical tradition: “The fundamental argument of this book is that the evasion of epistemology-centered philosophy — from Emerson to Rorty — results in a conception of philosophy as a form of cultural criticism in which the meaning of America is put forward by intellectuals in response to distinct social and cultural crises.” (p. 5) Our Unitarian Universalist religious tradition is directly influenced by this philosophical tradition.
(18) Isaac Chotiner, “A Groundbreaking Mathematician on the Gender Politics of Her Field,” New Yorker, 28 March 2019.
(19) Ibid. A side note: To help inspire more young women to go into mathematics, Uhlenbeck wrote an essay for the book Journeys of Women in Science and Engineering: No Universal Constants, ed. Susan A. Ambrose, Temple Univ. Press, 1997, pp. 395 ff.; this essay is freely available on her website here: https://web.ma.utexas.edu/users/uhlen/vita/pers.html.
