来源：ScienceAI
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人工智能设备可能也会对数学产生同样的影响。
In the collection of the Getty Museum in Los Angeles, there is a portrait of the 17th-century ancient Greek mathematician Euclid: ragged, unkempt, hands covered with dirt, holding up his geometry work "Elements".
Portrait of Euclid.
For more than 2,000 years, the writings of Euclid have served as models of mathematical argument and reasoning.
Jeremy Avigad, a logician at Carnegie Mellon University, said: "Euclide, as we all know, started with an almost poetic [definition]. He then built the mathematics of his day on top of that, using fundamental concepts, definitions, and a priori theorems to prove things in such a way that each step [clearly follows] the previous step."
There have been complaints that some of Euclid's "obvious" steps aren't obvious, but the system still works, Dr. Avigad said.
But by the 20th century, mathematicians were no longer willing to base their mathematics on this intuitive geometric foundation. Instead, they developed formal systems—precise symbolic representations, mechanical rules. Ultimately, this formalization allows mathematics to be translated into computer code.
In 1976, the four-color theorem (which states that four colors are sufficient to fill a map such that no two adjacent regions have the same color) became the first major theorem to be proved by computational force.
Now, mathematicians are grappling with the latest transformative force: artificial intelligence.
In 2019, Christian Szegedy, a computer scientist formerly at Google and now at a startup in the San Francisco Bay Area, predicted that within a decade computer systems would match or exceed the problem-solving abilities of the best human mathematicians. Last year he revised the target date to 2026.
Akshay Venkatesh, a mathematician at the Institute for Advanced Study in Princeton and winner of the 2018 Fields Medal, isn’t currently interested in using AI, but he’s keen to talk about it. “I want my students to realize that their field is going to change a lot,” he said in an interview last year. He added recently: "I'm not opposed to the deliberate and deliberate use of technology to support our human understanding. But I firmly believe that it is crucial to be mindful of the way we use it."
In February, Dr. Avigad participated in a workshop on "Machine-Assisted Proofs" at the Institute of Pure and Applied Mathematics at UCLA. The gathering attracted an atypical mix of mathematicians and computer scientists. "It felt important," said Terence Tao, a mathematician at the university, a 2006 Fields Medal winner and the workshop's lead organizer.
Summer school organizers (from left): Dr. Avigad, Patrick Massot of Paris-Saclay University and Heather Macbeth of Fordham.
Dr Tao points out that it is only in recent years that mathematicians have begun to worry about the potential threat of artificial intelligence, either to the aesthetics of mathematics or to themselves. Prominent community members are now asking these questions and exploring the potential to "break taboos," he said. One striking workshop participant sat in the front row: a trapezoidal box called the Hand-Raising Robot, which whispered mechanically and raised its hand whenever an online participant asked a question. "It would be helpful if the robot was cute and not threatening," Dr. Tao said.
Students worked on a group project during the Academy's Mathematical Formalization Summer School.
Bringing in the "proof complainers"
There's no shortage of gadgets to optimize our lives these days -- diet, sleep, exercise. “We like to attach things to ourselves to make it easier to get things done,” said Jordan Ellenberg, a mathematician at the University of Wisconsin-Madison, during a break in the seminar. Artificial intelligence devices may have the same effect on mathematics, he added. "Obviously, the question is what machines can do for us, not what machines will do to us."
One math gadget is called a proof assistant, or interactive theorem prover. (“Automation” was an early incarnation in the 1960s.) Mathematicians turn proofs into code step by step; software programs then check that the reasoning is correct. Validations are accumulated in a repository, which is a dynamic specification reference that others can consult. Dr. Avigad, director of the Hoskinson Center for Formal Mathematics (funded by cryptocurrency entrepreneur Charles Hoskinson), said this formalization laid the foundation for today's mathematics, "just as Euclid tried to codify and organize mathematics. It laid the foundation for the mathematics of his time."
Recently, the open source proof assistant system Lean has attracted much attention. Developed at Microsoft by Leonardo de Moura, a computer scientist now at Amazon, Lean uses automated reasoning powered by what's known as old-fashioned artificial intelligence (GOFAI), symbolic artificial intelligence inspired by logic. So far, the Lean community has verified an interesting theorem about flipping spheres, as well as a key theorem in a scheme to unify the mathematical field, among other strategies.
Emily Riehl, a mathematician at Johns Hopkins University, has been using experimental proof aids.
But the proof assistant has drawbacks: it often complains that it doesn't understand the definitions, axioms, or steps of reasoning entered by the mathematician, so it's called a "proof complainer." All these complaints can make research cumbersome. But the same functionality—providing line-by-line feedback—also makes the system useful for teaching, says Heather Macbeth, a mathematician at Fordham University.
This spring, Dr. Macbeth designed a "bilingual" course: She translated every problem on the blackboard into Lean code on the lecture notes, and students submitted solutions to homework problems in both Lean and prose. “It gave them confidence,” Dr. Macbeth said, because they got instant feedback on when the proof was complete and whether each step in the process was right or wrong.
After attending the workshop, Johns Hopkins mathematician Emily Riehl used an experimental proof assistant program to formalize a proof she had previously published with a coauthor. At the end of one proof, she said, "I understood the proof really, really deeply, much more deeply than I had ever understood before. I thought it out so well that I could explain it to a really stupid computer."
Brute Force Reasoning—But Is It Math?
Another automated reasoning tool used by Carnegie Mellon University computer scientist and Amazon scholar Marijn Heule is what he colloquially calls "brute reasoning." Just say which "strange object" you want to find in a carefully crafted code, and a network of supercomputers will churn through the search space and determine whether the entity exists, he said.
Just before the seminar, Dr. Heule and one of his Ph.D. Student Bernardo Subercaseaux finally came up with a solution to a long-standing 50 TB file problem. However, the document bears little comparison with the results of Dr. Heule and his collaborators in 2016: "200 terabytes mathematically proven largest ever", proclaims a headline in the journal Nature. The article goes on to ask whether solving problems with these tools really counts as mathematics. In Dr. Heule's view, this approach is necessary to "solve problems that humans cannot solve."
Marijn Heule and a student recently used automated inference tools to solve a "package coloring" problem, which is a bit like the four-color map problem, but much more complicated.
Another set of tools uses machine learning, which can synthesize large amounts of data and detect patterns, but is not good at logical, step-by-step reasoning. Google's DeepMind has designed machine learning algorithms to solve problems such as protein folding (AlphaFold) and chess winning (AlphaZero). In a 2021 paper in the journal Nature, a team described their work as "advancing mathematics through artificial intelligence guiding human intuition."
Yuhuai "Tony" Wu, a former Google computer scientist who is now a Bay Area startup, outlined a more ambitious goal for machine learning: "Solving math problems." At Google, Dr. Wu explored how large language models underpinning chatbots could help mathematics. The model the team used was trained on internet data and then fine-tuned using large math-rich datasets such as online archives of math and scientific papers. The specialized chatbot, called Minerva, is "very good at imitating humans" when asked to solve math problems in everyday English, Dr. Wu told the workshop. The model outperformed the average score of 16-year-old students on a high school math test.
Ultimately, Dr. Wu says, he envisions an "automated mathematician" with "the ability to solve mathematical theorems by itself."
math as touchstone
Mathematicians have paid varying degrees of attention to these disturbances.
Columbia University's Michael Harris expressed doubts in his "Silicon Reckoner" substack. He is troubled by the potential conflicting goals and values between research mathematics and technology and the defense industry.
Dr Harris lamented the lack of discussion about the larger impact of AI. Mathematics research, in particular, is "almost ubiquitous outside of mathematics" "compared to a very lively conversation going on".
DeepMind collaborator Geordie Williamson of the University of Sydney spoke at NAS. Bring together and encourage more participation by mathematicians and computer scientists in such conversations. At the Los Angeles seminar, he began his talk with a line adapted from George Orwell's 1945 essay "You and the Atom Bomb." "Deep learning hasn't generated as much discussion as might have been expected, given how deeply we're all likely to be affected over the next five years," said Dr Williamson.
San Francisco Bay Area computer scientist Yuhuai "Tony" Wu envisions an "automated mathematician" — a general-purpose research assistant "capable of solving mathematical theorems by itself."
Dr. Williamson sees mathematics as the litmus test of what machine learning can or cannot do. Inference is the essence of mathematical processes and a key unsolved problem in machine learning.
Early in the collaboration with DeepMind, the team discovered a simple neural network that could predict “mathematical quantities that I really care about,” and it was “ridiculously accurate,” Dr. Williamson said in an interview. Dr. Williamson struggled to understand why - which would form the basis of a theorem - but could not. No one at DeepMind could do it. Like the ancient geometer Euclid, neural networks somehow intuitively discern mathematical truths, but the logical "why" is far from obvious.
A prominent theme at the Los Angeles workshop was how to combine intuition and logic. If artificial intelligence can do both of these things at the same time, everything will be solved.
However, Dr. Williamson observed, there is little incentive to understand the black box that machine learning presents. “It’s the hacker culture in tech, and it would be nice if it worked most of the time,” he says. But the situation makes mathematicians unhappy.
Trying to understand what's going on inside neural networks raises "fascinating mathematical questions," and finding answers offers mathematicians the opportunity to "make meaningful contributions to the world," he added.
Reference content:
https://www.nytimes.com/2023/07/02/science/ai-mathematics-machine-learning.html
Editor: Wen Jing
