Rise of the chatbots
Machine learning tools are already helping mathematicians to formulate new theories and solve tough problems. However, it is predicted that they are to shake up the field even more in the near future.
With global interest in chatbots spreading like wildfire, mathematicians are starting to explore how Artificial Intelligence (AI) could assist in their work. AI is already used to verify human-written work and suggesting new ways to solve difficult problems. According to researchers, automation is starting to change the known maths fields, venturing far beyond mere calculation.
According to Andrew Granville, a number theorist at the University of Montreal in Canada, the possibility of building bridges between mathematicians and computer scientists, are on the cards. However, he said that most mathematicians are completely unaware of these opportunities.
Akshay Venkatesh, one of the international 2018 winners of the prestigious Fields Medal, one of the highest honours a mathematician can receive, at the Institute for Advanced Study in Princeton, New Jersey, kick-started a conversation on how computers will change maths at a symposium in his honour in October. Two other recipients of the medal, Timothy Gowers at the Collège de France in Paris and Terence Tao at UCLA, have also taken leading roles in the debate.
The Fields Medal is awarded to only two, three, or at the most, four mathematicians under 40 years of age by the International Congress of the International Mathematical Union (IMU). The name of the award honours Canadian mathematician, John Charles Fields.
Scientists are discussing concerns which type of automation tools would be most useful. AI comes in two major flavours; Symbolic wherein programmers embed rules of logic or calculation into their code. The other approach, which has become extremely successful in the past decade, is based on artificial neural networks. In this type of AI, the computer starts more or less from a clean slate and learns patterns by digesting large amounts of data. This is called machine-learning and forms the basis of large language models, including chatbots such as ChatGPT. These systems can beat human players at complex games or even predict how proteins fold. Whereas symbolic AI is inherently rigorous, neural networks can only make statistical guesses and their operations are often mysterious.
Scientists helped symbolic AI to score certain early mathematical successes by creating a system called Lean. This interactive software tool forces researchers to write out each logical step of a problem down to the most basic details, ensuring the maths is correct. Two years ago, a team of mathematicians succeeded at translating an important but impenetrable proof – one so complicated even its author was unsure of it — into Lean, thereby confirming that it was correct. Researchers say the process helped them to understand the proof and even to find ways to simplify it and admit that “even in our wildest dreams, we didn’t imagine that.”
As well as making solitary work easier, this sort of proof assistant could change how mathematicians work together by eliminating what is called a trust bottleneck. A proof assistant shows your collaborators that the work is trustworthy.
At the other extreme are chatbots, neural-network-based large language models. At Google in Mountain View, California, former physicist, Ethan Dyer and his team have developed a chatbot called Minerva, which specialises in solving maths problems. At heart, Minerva is a very sophisticated version of the autocomplete function on messaging apps. It has learnt to write down step-by-step solutions to problems in the same way that certain apps can predict words and phrases. Unlike Lean, which communicates using something similar to computer code, Minerva takes questions and writes answers in conversational English, solving some problems automatically.
Minerva shows both the power and possible limitations. It can accurately factor integer numbers into primes (numbers that can’t be divided evenly into smaller ones), but starts making mistakes once the numbers exceed a certain size, showing that it has not understood the general procedure.
Still, Minerva’s neural network seems to be able to acquire some general technique as opposed to just statistical patterns. The Google team is trying to understand how it does that and hoping that eventually they would they would be able to brainstorm with it. It could also be useful for non-mathematicians needing to extract information from specialised literature. Further extensions will expand Minerva’s skills by studying textbooks and interfacing with dedicated maths software. The motivation behind the Minerva project was to see how far the machine-learning approach could be pushed, making it possible that such a a powerful automated tool could end up combining symbolic AI techniques with neural networks.
Maths v. machines
In the longer term, will programmes remain part of the supporting cast, or will they be able to conduct mathematical research independently? AI might get better at producing correct mathematical statements and proofs, but some researchers worry that most of those would be uninteresting or impossible to understand. To do this, computers will have to judge what is interesting and worth proving. If they can do that, the future of humans in the field looks uncertain.
AI aids mathematicians to unsolvable problem
A team of researchers has stumbled on a question which is mathematically unanswerable because it is linked to logical paradoxes discovered by Austrian mathematician Kurt Gödel in the 1930’s that can’t be solved using standard mathematics.
The mathematicians, who were working on a machine-learning problem, showed that the question of learnability (whether an algorithm can extract a pattern from limited data) is linked to a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. The latest results appeared in Nature Machine Intelligence.
“For us, it was a surprise,” says Amir Yehudayoff at the Technicon–Israel Institute of Technology in Haifa, who is a co-author on the paper. He says that although there are a number of technical maths questions that are known to be similarly undecidable, he did not expect this phenomenon to show up in a relatively simple problem in machine learning.