Pure mathematics fascinates me precisely because it is so inaccessible. I picture it as a remote, cold, dangerous realm, similar to Antarctica’s Sentinel Ridge. Hardy souls who rise to the heights of mathematics seem superhuman.
I once asked Andre Weil, a legendary climber of mathematical peaks, if it bothered him that few people knew about his achievements in number theory and algebraic geometry, and even fewer understood them. He seemed confused by the question. No, he replied, “it makes it more exciting.” In autobiographyWeil says that his work transports him to “a state of conscious exaltation in which one thought replaces another, as if miraculously.”
Perhaps because I romanticize mathematicians, the idea that machines might replace them bothers me. I touched on this possibility in “The death of evidence“, published in October 1993 Scientific American. In response to the increasing complexity of mathematics, as I have reported, mathematicians are becoming increasingly dependent on computers. I asked, “Will the great mathematicians of the next century be made of silicon?”
Mathematics is still given to me sorry about that articleeven as the trends I have described continued. Anthony Borge, a mathematician at the University of Cambridge, worries that his field could face a “replication crisis” similar to the one afflicting scientific research. Mathematicians, Bordge notes Art A mathematical geniussometimes they accept a proof not because they have checked it step by step, but because they trust the methods of the proof and the author.
Given the “increasing difficulty in verifying the correctness of mathematical arguments,” Borge says, old-fashioned peer review may no longer be sufficient. Famous mathematicians have published “proofs” so new and complex that even experts in the relevant mathematics cannot verify them. Take the 2012 proof in which Shinichi Machizuki claims to have proved the ABC conjecture, a problem in number theory. For the past decade, mathematicians have organized conferences to determine whether Machizuki’s proof is true, but to no avail. Some accept it, others don’t.
Borge suggests that computerized “evidence assistants” will help verify the evidence. Microsoft researchers have already invented “interactive proof of theorems” called Lean, which can check proofs and even suggest improvements—just as word processors check our prose for errors and finish our sentences for us. Lean is linked to a database of established results. New mathematical work must be carefully translated into a language that Lean recognizes. But augmented by artificial intelligence, programs like Lean could eventually “discover new math and find new solutions to old problems,” according to report to Quanta Magazine.
Some mathematicians welcome the “digitization” of mathematics, which will facilitate computer verification and make mathematics more reliable. Others, like Michael Harris, a mathematician at Columbia University, are ambivalent. Advances in computer mathematics, says Harris, raise a serious question: What is destination mathematics? Harris sees mathematics as “free, creative activity”, which, like art, is practiced for its own sake, for the simple joy of discovery and understanding.
Harris is not opposed to the mechanization of mathematics as such. In a recent article in Pour La ScienceFrench edition Scientific American (see its partial translation here), Harris notes that mathematicians have used mechanical devices such as the abacus for millennia. And mathematicians, after all, invented the computer.
But Harris worries that tools like Lean will encourage “vision delay” of mathematics as an economic good or product, not “way of being human.” After all, math research funders like Google and the National Security Agency value math primarily for its applications. As Harris saysmathematics is “indispensable for engineering, technology, record keeping, and any activity that involves predicting the future.”
We also value science for its application. Sentimental scientific writings, including mine, imply that the purpose of science is to understand nature. However, in the modern era, the main goal of science is power. Science helps us manipulate nature for various purposes: to prolong our lives, to enrich and entertain us, to boost our economy, to defeat our enemies. Modern physics is incomprehensible to most of us, but who cares if physics gives us smartphones and hydrogen bombs?
Physicists often follow a utilitarian mindset, exemplified by the slogan “Shut up and count!” This is what professors allegedly tell students confused by quantum mechanics. The information is that students must apply quantum formulas, such as by the creation of quantum computers— without caring about their meaning. Stephen Hawking and Martin Rees have predicted that artificial intelligence will play an increasingly important role in physics. Wouldn’t it be funny if quantum artificial intelligence found have been looking for a unified theory of physics for a long timebut not even the brilliant string theorist Edward Witten understands this?
The mechanization of knowledge is suggestive Chinese room experiment. In this famous philosophical argument, questions written in Chinese are asked of a person in a room. Although the man does not understand Chinese, he has a manual that tells him how to answer one line of Chinese characters with another line that represents the appropriate answer to the question. So, the person in the room facial expressions understanding of the Chinese language.
Philosopher John Searle conceived the Chinese room experiment as a critique of the claim that machines can think. Searle compares a computer to a person in a room mindlessly processing symbols without knowing what they mean. The more mathematicians and scientists rely on machines to do their work, the more they resemble the man in the China room.
When I raised the specter of artificial mathematicians a few years ago, Scott Aaronsonwhose work spans computer science, mathematics and physics, scolded me. “It’s possible that someday,” Aaronson said, “computers will replace humans in all aspects of mathematical research, but it’s also possible that when they do, they’ll be able to replace humans in music and science journalism and everything else!” Wait, science journalism? never!
By the way, the question in my title “Should machines replace mathematicians?” perhaps doesn’t matter because it means mathematicians have a choice. A better question to ask is whether machines can replace mathematicians. I am skeptical about some statements about artificial intelligence. But given the powerful forces behind automation, if machines it is possible they will replace mathematicians, probably just as they replace drivers, bank tellers, travel agents, cashiers and other workers. Mathematicians’ desires, such as their desire to seek truth solely for its own sake, can be controversial.
In the future, mathematics may not resemble a remote mountain range, but a factory where robots assemble cars. A few technicians wander around the factory making sure the robots are working properly, but the robots do all the hard work. Meanwhile, the human overlords who own the factories—and perhaps the future of mathematics—keep getting richer and stronger.
This is an opinion and analytical article and the views expressed by the author or authors do not necessarily reflect the Scientific American.