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For centuries, computers (machines) have been the mathematician's best friend, using them to calculate, formulate conjectures, and perform mathematical proofs. With the advent of more advanced tools such as interactive theorem provers, machine learning algorithms, and generative AI, machines are being used in more innovative and in-depth ways.
Recently, at the 40th anniversary celebration of MSRI (American Institute of Mathematical Sciences, now renamed SLMath), Tao Zhexuan (Terence Tao) gave an overview of the history and latest development of machine-assisted mathematical proofs, and the future of machine-assisted mathematics in mathematics role to predict.
He said: Early computer-assisted mathematics research was dominated by women, and a unit of computing power was derived: "thousand female hours", how many calculations can be completed by a thousand women using adding machines. The emergence of "Thousand Girls' Hours" also marks the origin of large-scale computing.
And now the computer has gone beyond "brute force exhaustion", it is gradually able to provide creative suggestions to help people find new directions and inspiration in mathematical research. Especially GPT-4: "Conversing with these advanced language models can stimulate its mathematical thinking in an indirect way. It is like a very patient colleague who is good at listening without judging."
Regarding the role of computers in mathematics in the future, Tao Terence said: "The contribution ratio of humans and computers in the field of mathematics creation is 95% and 5%, so in ten or twenty years, this ratio will become 50-50 ."
Reason for recommendation: I hope to use this as an opportunity to call on researchers to pay attention to the amazing role of AI in assisting mathematical proofs. Machine learning provides a powerful framework and opens up new ideas for mathematicians. In the future, AI has great potential to help solve mathematical problems that are extremely challenging for humans, simulate and perform advanced mathematical operations, and even propose new mathematical conjectures and theories. The following is the full text of the speech, edited and compiled without changing the original meaning, please enjoy:
Video address: https://vimeo.com/817520060
When I was asked about the Mathematical Sciences Research Institute (MSRI) and the direction of mathematics development, I thought that the application and role of computers in the field of mathematics will become more and more important.
Computers already play an important role in mathematical communication, such as writing in LaTeX and communicating with collaborators by email and video conferencing. In pure mathematics, the role of computers has not been appreciated, but that is changing.
In the past, the contribution ratio of humans and computers in the field of mathematical creation was 95% and 5%, so in ten or twenty years, this ratio will become 50-50. It may take a long time to get to 5(person):95(computer).
Early Machines and Mathematics: Artificial, Mechanical, Feminist
The keywords of early computers were artificial and mechanical. In the 18th and 19th centuries, the main function of computers was to perform digital calculations and establish logarithmic tables and trigonometric function tables.
In the 1920s, physicist Hendrik Lorenz used a team of human computers to simulate the flow of water around a dam (Avsruyterdijk). It was an important engineering project that required a lot of complex calculations, and Lorenz and his team used floating-point arithmetic, a method of calculation that was not common at the time.
Before the war, computers had been able to partially solve partial differential equations. In World War I and World War II, computers helped humans do a lot of math in ballistics, navigation, the Manhattan Project, and more. In fact, most of the work is done by human computers, mostly operated by women. Because men at the time were going to war. This also derived a unit of computing power: "a thousand female hours", how many calculations can be completed by a thousand women using adding machines. The emergence of "Thousand Girls' Hours" also marks the origin of large-scale computing.
Legend: Gauss and Legendre used the prime number table (up to about 10^6) to study a basic question: how many prime numbers (denoted as π(x)) before a given threshold x
How do research mathematicians use computers? One of the earliest uses, aside from building tables and such, was to make conjectures. In the 18th century, when Gauss and Legendre studied prime numbers, they used large-scale artificial computers to construct a large number of prime number tables, which were used to "guess" the law of prime number distribution.
Legendre's prime number distribution conjecture formula
Despite some problems with the accuracy of Legendre's prime number conjecture, it was still very influential. Later, this conjecture was revised by Dirichlet into a more accurate form, which is now widely accepted and included in textbooks.
Legend: Dirichlet's revised formula
Numerical calculations powered by computers have now become a routine tool in analytic number theory, used to help form and verify conjectures. A good example is the Birch and Swinnerton-Dell conjecture, an unsolved mathematical problem posed through extensive numerical calculations and data analysis.
Legend: Expression of energy equipartition theorem
Computer Era: Click to open a number of mathematics auxiliary skill trees
After World War II, physicists Enrico Fermi, John Pasta, Stanislaw Ulam, and programmer Mary Tsingou worked together to give The electronic computer "MANIAC" finds new applications and decides to simulate the evolution of crystals. They viewed the crystal as a chain of one-dimensional particles that evolve through Hamiltonian equations. According to the equipartition theorem, all energy should be evenly distributed among all modes (ie, individual particles in the crystal).
Legend: The experimental results show that the energy is not evenly distributed, and what is found is a stable local solution (solitary wave)
But when they actually ran the numerical simulations, the results did not meet their expected equipartition theorem. At first, they set an initial state in which only one particle's energy was excited. However, the energy is not distributed uniformly over time, but rather a long-term stabilization between one or two particles occurs. That is to say, the phenomenon of solitary waves (or solitons) appears.
However, with long-running experiments on a computer, the phenomenon disappears and the energy ends up evenly distributed.
This phenomenon occurs because the system they study is actually an approximation of a fully integrable system—Toda lattice equations, also known as Toda chains. In this fully integrable system, solitons will exist forever. And their actual system wasn't perfectly integrable, but it was close enough that they could see soliton phenomena in simulations.
This hugely influential numerical experiment started the study of integrable systems and also the study of the stability of solitary waves, a research field that is still active today.
Legend: Appel and Wolfgang Haken proved the four-color theorem with computer assistance in 1976.
In the early days of modern electronic computers, the power of computers lay in their ability to reveal "unexpected" phenomena. The four-color theorem is probably the most well-known theorem proved by a computer. In their proof, Kenneth Appel and Wolfgang Haken write specific programs that process specific graphs and examine certain aspects of these graphs with a computer. nature. If a particular graph satisfies these properties, it can be colored with four colors. This is the four-color theorem proved by computers.
The proof of the four-color theorem is divided into two main parts: naturalization and inevitability. The naturalizable part shows that any graph containing any of the 1834 configurations can be transformed into a smaller graph, and if the smaller graph can be four-colored, so can the original graph. Part of this proof is done by entering these configurations into the computer one by one.
The inevitability part shows that if there exists a graph that cannot be four-coloured, then it must contain at least one of the 1834 configurations, which is verified in part by manual inspection of more than 400 pages of microfilm.
Judging from today's standards, Appel-Haken's original proof contains some repairable errors and does not meet the standards of modern computer verifiable proofs. In 1996, Robertson, Sanders, Seymour, and Thomas presented for the first time a truly computer-verifiable proof using 633 configurations whose naturalizable and non- Avoidance can be checked by computer. Then in 2005, Werner and Gonthier fully formalized the proof using the auxiliary tool Coq, making it a truly computer-verifiable proof.
The Kepler conjecture is another example of a computer-aided mathematical proof. In the 17th century, Kepler studied the most dense packing method of unit spheres in three-dimensional space, and found that two packing methods, namely hexagonal close packing and cubic close packing, both have a density of about 74%. Kepler guessed that this was the maximum possible packing density that no other packing method could exceed. This conjecture is still an important mathematical problem.
Kepler's conjecture is essentially an optimization problem involving infinitely many variables, which makes it not easy to verify by computer. Although in 1951 Toth proposed a possible solution to finding an upper bound on the packing density of spheres by some weighted inequalities on the volume of a finite number of Voronoi cells, however, despite many attempts, But no convincing proofs have been found that lead to precise results.
Thomas Hales and Samuel Ferguson adopted a flexible strategy for solving minimization problems: tweaking the scoring function, but this in turn creates more problems. However, their task became somewhat easier despite the increasing complexity of the scoring functions. After publishing several papers, the two scholars finally successfully proved Kepler's conjecture in 1998 by designing a complex scoring function using linear programming.
It is said that Hales initially did not intend to use a computer to help him prove this conjecture, but as the project became more and more complex, he found that he had to rely on a computer. In the end their proof contained 250 pages of mathematical notes, and 3GB of computer code, data and results.
In order to "dispel" doubts about the proof of Kepler's conjecture, Thomas Hales launched the "Flyspeck" project in 2003. They use the language of a proof assistant to formalize the proof process so that computers can automatically verify it. Although originally estimated to take two decades, the project was completed in 2014 through the combined efforts of Hales and his 21 collaborators.
In recent years, Peter Scholze's liquid tensor experiment is arguably the most well-known formal proof experimental project involving Scholze's theory of "condensed mathematics" with the goal of fixing topology Some problems of space classes, such as topological Abelian groups, are solved by using "condensed" new mathematical objects instead of traditional topological space classes.
A key result in the theory of "condensed mathematics" is the "vanishing theorem". It is found that in a specific mathematical structure, certain elements or attributes will "disappear" under certain conditions. The disappearance theorem is the core of condensed mathematics, and it is the basis of functional analysis research using condensed mathematics. Proving the theorem was so difficult that Schultz spent a year working on it, but progress was slow.
Therefore, Schultz initiated the "Liquid Tensor Experiment", hoping to use the mathematical software Lean to formalize the results. During the implementation process, they added several basic mathematical theories to Lean's mathlib library. Finally, with the efforts of Schultz and his team, a key sub-theorem was formalized in May 2021, and the complete theorem was formalized in July 2022. The main achievement of this project is the substantial expansion of Lean's math library. In fact, other Lean formalization projects have become more convenient and efficient due to the rich library routines built by this project.
Legend: liquid tensor experiment blueprint
The process of formalization is accompanied by the creation of a human-readable proof "blueprint". This "blueprint" synthesized the formal and informal proofs in an interactive fashion that allowed Schultz and others to gain a deeper understanding of the proof.
Legend: Use Lean to prove the topology theorem. Tao envisions that future topology textbooks might use this interactive step-by-step explanation of proofs to make the learning process clearer and easier.
Inspired by this work, a growing number of researchers are developing software tools to automatically convert formal proofs into human-readable interactive proofs.
As an auxiliary tool, computers usually function in two ways: 1. Gradually increase the use of formal proofs; 2. Use computers to perform violent calculations. For example, the solution Helfgott used to solve the odd-number Goldbach conjecture in 2013 was violent calculation + ingenious optimization.
The Era of Artificial Intelligence: Neural Networks and Large Models Enter Mathematics
An example I particularly like is Ramsey's theorem from 2016. Let's first look at the "Boolean Pythagoras triplet problem", which is roughly like this: If you divide the numbers from 1 to 7825 into two groups (that is, mark them with two colors), then no matter what Grouping, you can find at least one set of Pythagorean triples (three numbers satisfying a² + b² = c²) in one of the groups, and these three numbers are the same color (in the same group).
This is actually a 3-SAT problem with a very large amount of calculation (3-satisfiability problem, an example of NP complete problem). To solve it, the standard 3-SAT solver cannot be applied, and the 3-SAT solver is required. Make special customizations or modifications.
Finally, Huele, Kullman and Marek solved the problem using computers. They used a tool called a "Boolean Satisfaction Solver" and ran it on the Stampede supercomputer for four years to finally find all possible groupings and show that no matter how you group them, you can always find at least one of them in one of them. A Pythagorean triplet of the same color. The amount of data in this proof is very large, the original data size reached 200TB, but it can be reduced to 68GB through compression.
Among the fluid equations, there are two famous unsolved problems: the Navier-Stokes equations and the finite-time explosion of the Euler equations. A decade ago, numerical simulations gave solutions that seemed to develop singularities: They started out smooth, but became more and more turbulent over time, with energy starting to concentrate at certain points. However, these numerical results become unreliable when simulations approach singularity occurrences due to computational precision limitations.
It was recognized that if a particular type of solution, known as self-similar solutions, could be found, then it would be possible to demonstrate that the fluid equations develop singularities in finite time. This is similar to finding closed geodesics or orbits in general relativity.
An "approximate" self-similar solution can be found using a computer. The solution is not perfect, but it is close to a perfect self-similar solution with only a small error. However, if combined with rigorous stability analysis, the existence of practical self-similar solutions can be proved. This is an idealized approach, which can be regarded as a "dream", which may be difficult to realize in practice.
In 2022, two methods for finding self-similar solutions have been developed: 1. The Chen-Hou method, which uses traditional numerical partial differential equations and conducts rigorous stability analysis with computer assistance. This is the first time for the incompressible Euler equation The smooth solution of the rigorous proof of the finite-time blasting results outside the cylinder boundary; 2. Wang-Lai-Gomez-Serrano-Buckmaster method: using the Physical Information Neural Network (PINN, Physics Informed Neural Network ), a technique that combines knowledge of physics with neural networks.
There are also many researchers who are actively using machine-assisted technology to explore the "explosive solution" of fluid equations. However, a very complex problem in fluid dynamics: the global regularity of the Navier-Stokes equations, still seems to be difficult to solve.
In the field of knot theory, there is an interesting combination of computer and human mathematical methods. Conclusion Theory studies the properties of knots formed by winding wires in space. Efforts are currently being made to relate the hyperbolic invariants of the conclusion (a set of real and complex numbers) to the signature of the conclusion (an integer) to better understand the properties and structure of knots. Representative work was done by the team of Alex Davies et al.
Their most recent advances have been made possible by artificial neural networks. They used a neural network, trained on a database containing more than two million "knots." The trained neural network was very successful, accurately predicting the transition from hyperbolic invariants to signatures. This also shows that there is indeed a connection between hyperbolic invariants and signatures.
However, the neural network works like a "black box" that can see its input and output, but what is actually going on inside is not known. Therefore, the specific relationship between hyperbolic invariants and signatures cannot be known through artificial neural networks at present.
Legend: Main hyperbolic invariants for predicting signatures
They then performed a significance analysis to determine which hyperbolic invariants were most important for predicting the signature relationship. In other words, they want to know which input parameters to change will have the greatest impact on the output. Ultimately, they found that only 3 of the 24 hyperbolic invariants had a significant effect on the predicted signature relationship. This means that while they may initially think that all invariants are important, only a small number of them are actually critical. Therefore, neural network + saliency analysis helps to gain a deeper understanding of the relationship between this hyperbolic invariant and the signature.
Hidden connections between different areas of mathematics can be unearthed using special databases like the Online Encyclopedia of Integer Sequences. Such a connection could manifest itself as two completely different fields producing the same sequence of numbers, as in the famous "Monstrous moonshine" phenomenon in mathematics.
In the future, it is hoped that more automated tools can discover the connection between different fields at a deeper level. For example, two papers from different fields may prove similar theorems, so these tools can clearly "identify". While such "semantic search" technology is likely at least a year away, it represents an exciting future.
Legend: GPT-4 solves math problems
Large models have also shown some potential in the process of mathematical research. For example, GPT-4 can even solve problems in the 2022 International Mathematical Olympiad, demonstrating its application capabilities in the field of mathematics.
Legend: The original answer given by the large model is 120
But this system is not stable when dealing with mathematical problems. While it is sometimes unexpected, it can also produce errors in explaining or calculating mathematical concepts, and even in basic arithmetic problems (such as the error of 7*4+8*8=120). Therefore, the current state of the art can be seen as a "confused undergrad" who has memorized everything but doesn't really understand it.
Some scholars have tried to use some new methods to make those large language models understand mathematics better, including combining with traditional computing tools such as Wolfram Alpha and integrating traditional mathematical problem-solving strategies. In addition, dialogue with these advanced language models can also stimulate its mathematical thinking in an indirect way. It is like a very patient colleague who is good at listening without judging. This feature exhibited by large models can also promote human exploration of new ideas and creative thinking.
Summary: Encourage experimentation with AI tools
So, where are computer-aided proofs now? First of all, it seems unlikely that it can solve the main mathematical problem on its own; leaving aside the previous Olympic problem, if you simply enter "Please solve the Riemann Hypothesis for me" into ChatGPT, you may get specious gibberish.
Secondly, it is increasingly being used to support the work of human mathematicians. Unlike brute force or calculation, computers are now gradually able to provide creative suggestions and help people find new directions and inspirations in mathematical research.
To sum up, artificial intelligence technology, in the short term, may have more impact on the auxiliary aspects of mathematical research, such as automatically summarizing a large amount of literature or suggesting related work, but it has not been able to touch the core aspects. I encourage everyone to try AI tools, which, unlike the nitpicking tools we usually use (like LaTeX, where one parenthesis can cause a whole lot of mistakes), are tolerant of very ambiguous input, and get structured and interesting answers from it.
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