The “Horgan Surface” and “The Death of Proof”
Behold the Horgan surface! Actually, it is only a pseudo-surface, because it has gaps, as the image below reveals. But the image below that is a genuine surface, the “triply periodic Horgan surface.” I found these images on the website of Matthias Weber.
HOBOKEN, JULY 12, 2024. Decades ago, mathematicians named a geometric structure, or pseudo-structure, or something after me: the Horgan surface. They meant to insult me, not honor me. Here’s the backstory, which touches on disturbing trends in mathematics (and yeah, this column is an updated, free version of one on ScientificAmerican.com).
In 1993 my boss at Scientific American, excited by the news that a Princeton mathematician had proved Fermat’s last theorem, ordered me to write an in-depth report on mathematics. I resisted. My degrees were in literature and journalism, I whined; I could handle a quickie news story on Fermat's last theorem, but a major article would be too hard. My boss insisted.
I began interviewing prominent mathematicians, and eventually I came up with one of the biggest stories of my career, which goes as follows: For millennia the logical, step-by-step arguments known as proofs served as the gold standard for truth. What could be truer than the Pythagorean theorem? But mathematics was evolving in ways that undermined the status of conventional proofs.
First, mathematics kept growing more complex and specialized, making confirmation of some alleged proofs difficult. A case in point was the claim of Andrew Wiles that he had proved Fermat’s last theorem. Only a handful of experts were qualified to evaluate Wiles’s long, dense proof; they found an error, which Wiles corrected.
Meanwhile, mathematicians were becoming more reliant on computers to explore mathematical structures and to construct and confirm proofs. Plus, funding agencies were pressuring mathematicians to work on applications like cryptography and pattern recognition, where the motivating question shifts from Is it true? to Does it work?
I reported on all this in “The Death of Proof,” the cover story of the October 1993 Scientific American. “Computers are transforming the way mathematicians discover, prove and communicate ideas,” I wrote, “but is there a place for absolute certainty in this brave new world?”
Nothing I’ve written has provoked a more ferocious backlash. One critic was one of my sources, David Hoffman, whose research involved computer models. In a letter printed by Scientific American, Hoffman contended that my article “defies logic and accuracy in favor of controversy and sensationalism.”
Hoffman was still irate in 1998 when he reviewed my book The End of Science in Notices of the American Mathematical Society. Hoffman (not unreasonably) saw "Death of Proof" as a prelude to The End of Science, which he loathed. He accused me of “a clear antipathy toward mathematical thinking and a fundamental misunderstanding of the uses of mathematics in science.”
Hoffman pushed back against my claim that string theory is untestable “ironic science,” more akin to science fiction than science. He suggested that CERN’s Large Hadron Collider might soon yield evidence for string theory (which needless to say hasn’t happened).
Hoffman compared me to the apocryphal 19th-century patent commissioner who quit his job because everything had been invented. Hoffman wrote: “People who write about, think about, and do science and mathematics at the end of the twenty-first century will not have to dredge up half-true stories of patent commissioners to point out the shortsightedness of their post-postmodern critics. They will have Horgan’s End to kick around.”
I certainly hope so.
Meanwhile, Hoffman had been using computers to explore “minimal surfaces,” which are the smallest possible continuous surfaces conforming to certain rules. In the mid-1990s, Hoffman and Hermann Karcher found an object that initially seemed like a minimal surface but turned out to be a pseudo-surface, which has gaps (see below).
On closer inspection, the Horgan surface turns out to have gaps. Oops.
Hoffman and Karcher called the object the “Horgan surface” because it supposedly contradicts my claim that computer modeling can supplant traditional proofs. They described the surface in their 1995 paper “Complete Embedded Minimal Surfaces of Finite Total Curvature.”
I pieced this history together in 2019 when I discovered the writings of mathematician Matthias Weber (yes, I was self-googling). In a 1998 paper, “On the Horgan minimal non-surface,” Weber notes that Hoffman and Karcher coined the phrase “Horgan surface” as “a small revenge” for “Death of Proof.” Weber adds that other mathematicians “were amused but asked for a nonexistence proof,” which is a proof that something does not exist.
Irony alert! Hoffman and Karcher, annoyed by my disrespect toward proof, presented no proof that the Horgan surface is actually not a surface! Weber supplied the proof. In his 1998 paper, he presents “a very simple nonexistence proof of the most symmetric Horgan surface”—or non-surface, as he prefers to call it.
In a 2019 blog post titled “To be or not to be,” Weber says the Horgan surface blurs the line between existence and non-existence, which is, like, totally appropriate. He also describes the Horgan non-surface here and here. Weber calls the non-surface “non-trivial,” which makes me irrationally proud. So does Weber’s proof of the existence of a “triply periodic Horgan surface” (below) that is a genuine minimal surface. Yeah, baby.
Behold the triply periodic Horgan surface!
Mentions of the Horgan surface can also be found in "An Embedded Minimal Surface with no Symmetries,” Journal of Differential Geometry; and the Wolfram Demonstrations Project. Moreover, someone, presumably a hostile mathematician, has stuck a reference to the “sarcastically” named Horgan surface into my Wikipedia page.
Do I regret “Death of Proof”? Hell no, no more than I regret The End of Science, which still provokes controversy. I’m thrilled that an object in the realm of eternal Platonic forms has been named after me, if only “sarcastically.”
I’m even more gratified that mathematicians continue to cite “Death of Proof,” and not only to bash it. Steven Krantz, who like David Hoffman was both a source for and critic of “Death of Proof,” notes in his 2011 book The Proof Is in the Pudding that my main claims “are well worth considering.” Nice to know. This website lists 133 publications that cite “Death of Proof.”
Meanwhile, the trends on which I reported in 1993 have accelerated. As I noted last year, some mathematicians fear that advances in artificial intelligence are leading to the “mechanization” of their field. Artificial mathematicians might construct proofs beyond the comprehension of mere humans, which is depressing for those who cling to the idea that the primary purpose of math is illumination.
But let me end this column on a positive note. Although mechanization looms, humans are still discovering new mathematical objects, like the phony and real Horgan surfaces. That makes me happy.
Further Reading:
See quantum-computing theorist Scott Aaronson’s 2019 critique of the “Death of Proof” here.
And here are relevant columns on this website:
Should Machines Replace Mathematicians?
Is Ultimate Truth an Equation? Nah
Is the Schrödinger Equation True?
Sabine Hossenfelder, The End of Science and My Quantum Experiment
Confessions of a Namedropping Humblebragger
My free, online book My Quantum Experiment also touches on the mysteries of mathematics.