There are ongoing rapid advances in the power and versatility of AI or artificial intelligence. Computers are now producing results in several fields that are far beyond human capability. The trend is unstoppable, and is having profound effects in many areas of our lives. Will mathematicians be replaced by computers?
Let us reflect on what has happened in other areas. Early computer-generated music was crude; the composer’s job was safe. Now computers produce compositions with the intricacy of a Bach fugue, difficult to differentiate from “the real thing”.
The first attempts to translate language using computers were pathetic; now machine translation is invaluable. Years ago, we were confident that a computer would never beat a chess grand master; now machines are unbeatable. It’s the same story for the subtle boardgame Go.
A leading mathematician Paul Halmos opined decades ago that computers had no role in pure mathematics. Around the same time, the renowned logician Paul Cohen expressed the view that at some future time research mathematicians would be replaced by computers.
Nobody knows for sure, but the truth is likely to lie somewhere between these extremes: the computer will generate candidate conjectures, and produce theorems or refutations, while human intuition will determine which candidates are interesting and should be explored more deeply and which appear sterile and can be disregarded.
What is this mathematical intuition? Intuition is insight that does not depend on rigorous reasoning. It guides us towards plausible results in the absence of proof. It is holistic rather than analytical. It is essential in the creative process of mathematics.
But in the end, a result arising through intuition remains a conjecture until a rigorous proof is found. There are many sparkling conjectures crying out for some brilliant mathematician to prove them.
Axiom, theorem, proof
Mathematics is often viewed as a deductive science: assumptions (axioms) are made and the laws of logic are applied to deduce consequences (theorems). But possible theorems are conjectured by induction: intuition, guesswork, trial and error and experimentation have a central role.
The process of proving a theorem by systematic application of logical rules is simple to mechanise; computers are ideally suited to carry out symbolic manipulations that lead from the initial hypotheses to the conclusion, producing a proof.
However, the best proofs involve brave leaps of imagination based on intuition. These are far more difficult to reduce to blind, plodding rules. On the other hand, computers can analyse millions of published proofs – many more than a research mathematician – and uncover patterns of reasoning and successful strategies for progress. Ultimately, this incredible power is bound to win out over human limitations, just as it already has for chess and Go.
Views on the role of computers remain controversial. Undoubtedly, they have a role in checking proofs and in deciding on the validity of theorems submitted for publication. Many mathematicians are unenthusiastic about them, favouring the methods they have always used.
While they see a role for the more mundane stages in a proof to be mechanised, they doubt if a computer can ever generate interesting conjectures. They argue that this creativity is a fundamentally human activity, impossible to automate. My own feeling is that rapid advances are inevitable and the role of research mathematicians will change dramatically.
Registration for a UCD Zoom course, AweSums: Marvels and Mysteries of Mathematics, is now open at ucd.ie/lifelonglearning. Peter Lynch is emeritus professor at the school of mathematics and statistics, University College Dublin. He blogs at thatsmaths.com