Every day, startling new accomplishments of Artificial Intelligence (AI) are announced, new records are broken and new problems solved. Things that were lately considered impossible are now effortlessly achieved. The potential benefits are substantial, while the threats to society have been described by some leading authorities as existential. AI technology is greatly improving the health and well-being of many people, but is also used in warfare, promotion of misinformation and disruption of democracy [TM284 or search for “thatsmaths” at irishtimes.com].
Why is AI so good?
Why are LLMs, designed for language processing, so versatile? If two heads are better than one then, by mathematical induction, three are better still. Some maths problems are best solved through large-scale collaboration, where wider experience counts and a broader knowledge-base is available. AI systems benefit from access to enormous databanks and ability to absorb volumes of research far beyond human capacity.
Many people believe that human abilities like judgement, empathy, innovation and creativity will never be replaced by machines. Time will tell but for now the best means of progress is to work in harmony with the new systems. Some professional mathematicians are already doing this. Fields Medallist Terence Tao, an enthusiastic promoter of machine-assisted mathematics, has observed that “Soon we will have systems that automatically check our progress in real time … immediately flagging logical blunders and other inconsistencies.”
Reactions to AI include enthusiasm, anxiety, intimidation and indifference. The Leiden Declaration on AI and Mathematics endeavours to addresses the challenges posed in mathematical research, education, publishing and funding. AI will profoundly influence all these areas. The Declaration calls for responsible reactions to ensure the continued flourishing of the discipline, and provides actionable recommendations for mathematicians, organizations and policymakers.
The Future
In the annual International Mathematical Olympiad (IMO), brilliant secondary-level students attempt six difficult problems over two days. Last year, two AI models reached gold-medal scores. At the Olympiad in Shanghai this year, several AI models scored a perfect 42/42, solving all six competition problems. The models matched the best human performance: only 1% of the students achieved a perfect score.
The IMO problems have known solutions. What about open-ended research problems? LLMs have broken barriers in research mathematics too. In May, a problem originally posed by Paul Erdös, the Unit Distance Problem, was cracked. Thus, an LLM has solved a famous open problem that mathematicians have been grappling with for decades. This is a major breakthrough. Without doubt, there will be many more victories soon.
Will AI systems be able to formulate original and interesting conjectures? What if a machine proof of a theorem is beyond the understanding of human mathematicians? Will humans have a role in the future of mathematics? With AI, experimental mathematics is likely to prevail, where we task computers to examine vast numbers of possibilities and discover results that are interesting and genuinely new. This is an essentially novel way of doing maths.
A Golden Age of mathematics may be at hand, but there are enormous uncertainties and risks. The rapid improvement of LLMs has stunned mathematicians. Some are sceptical about the value of AI in research, some see it as a powerful assistant, and a few fear that the future of the field is in jeopardy. We are living in interesting times.
ThatsMaths 