Why Professor AI Should Not Get Tenure: A Philosophical Perspective
Les articles de la SCHPM présente des travaux de recherche en histoire et en philosophie des mathématiques à la communauté mathématique élargie. Les auteurs sont membres de la Société canadienne d’histoire et de philosophie des mathématiques (SCHPM). Vos commentaries et suggestions sont le bienvenue; ils peuvent être adressées vers le rédacteur:
Amy Ackerberg-Hastings, chercheuse indépendante (aackerbe@verizon.net)
Nicolas Fillion, Simon Fraser University (nfillion@sfu.ca)
The popularity of artificial intelligence (AI) these days is fairly ubiquitous. From its use in political campaigns to saving authors the need to write their own text, it is impossible to get away from those singing its virtues. One of the earliest advocates of AI was the British mathematician and pioneer of computer science Alan Turing. Perhaps his most famous article on the subject was ‘Computing Machinery and Intelligence’, which appeared in Mind in 1950 [6]. If one looks at the kind of evidence Turing gave there in defense of the possibility of AI, some of it may be turned against the claim that artificial intelligence is entitled to assert that it is a mathematician.
As a reminder, Turing gives three tasks which might be assigned to a machine or a human being, and he indicates the way in which the former might plausibly claim to be the latter. The first task is to write a sonnet on the subject of the Forth Bridge. The cagey computer could respond plausibly, ‘Count me out on this one. I never could write poetry.’ The second is to add 34957 to 70764. Turing indicates that the machine could wait thirty seconds to simulate a human reaction and then supply the answer ‘105621’. The third comes in two parts, the first asking if the machine plays chess and, on receiving an affirmative answer, posing a problem about what to do in a certain position. Although Turing doesn’t quite specify the position, the machine (after a fifteen-second pause) responds with the correct mating move.
Figure 1. A request to an AI image generator for “Alan Turing using artificial intelligence” reinforces that the technology still has a ways to go. Microsoft Bing Image Creator.
Seventy-five years after Turing’s article, it is possible to go through each of these tasks and see how machines have progressed in being able to carry them out. If we start with the third, there is no doubt that the success of Deep Blue in defeating Garry Kasparov in a match in the 1990s put the chess world on notice that human chess players were going to have to recognize human limitations. There had been chess-playing programs of varying degrees of strength, but International Master David N.L. Levy of Scotland had been making bets at five-year intervals that no machine was going to be able to beat him and winning each time. Deep Blue provided the evidence that Levy’s success had run out.
It is worth mentioning that that was not the end of the story for computer chess. A major project to produce a Go-playing program that was capable of giving problems to leading Go players produced a program called AlphaGo, and its success in a game with a strong opponent generated a good deal of interest. Most previous programs had been based on using the computer’s ability to analyze enormous numbers of positions in a short time, while AlphaGo took advantage of a learning algorithm based on the machine’s playing an enormous number of games against itself. Since Go is more complex than chess, it would not have been surprising for a version of the program to have an immense edge over chess-playing programs, and so it proved. AlphaZero, the chess version of AlphaGo, became the envy of the international chess-playing community. Turing’s modest request has been transmuted into an industry.
The second of the tasks might have seemed to call on the computational strength of a machine. What was clever about Turing’s proposed response was not just the time involved, but the fact that the answer given was one digit off from the correct ‘105721’. What Turing was attempting to demonstrate was that machines could imitate human intelligence, and human beings are liable to mistakes. One is reminded of the short story by Lord Dunsany in which the computer demonstrates its capacity to imitate a human chess player by cheating [5]. It is just as well, in view of the universal dependence on computers to carry out complicated calculations, that the machines are being programmed only to match the capacity of human beings.
The answer for the first task depends for its plausibility on a human reluctance to claim the ability to write poetry. It might be argued that these days machines do generate rhyming verse in quantity. What might be lacking is an aesthetic quality that readers would hope to get from genuine poetry. If all one is trying to do is come up with the text for the interior of a greeting card, AI has achieved a useful status.
The relevance of this to mathematics becomes a little clearer in looking at the computer’s ability to generate proofs. It has been able to do this for decades, as indicated by Daniel J. O’Leary’s ‘Principia Mathematica and the Development of Automated Theorem Proving’ (published in 1991 but written in 1984) [2]. On a larger scale, Alan Bundy’s 1983 The Computer Modelling of Mathematical Reasoning indicates strategies current at the time for achieving proofs in a variety of mathematical areas [1]. Since then, the computer has gone from strength to strength in generating proofs.
Figure 3. In 2017, the University of Illinois celebrated the 40th anniversary of the famous computer-aided proof of the Four Color Theorem by Kenneth Appel and Wolfgang Haken.
University of Illinois Urbana-Champaign, College of Arts & Sciences, Department of Mathematics.
Just as poetry, however, requires more than just the ability to rhyme, good mathematics requires the ability to appreciate elegance. When one is confronted by two proofs of the same result, one offering only a justification and the other offering an explanation, the mathematician should be able to recognize which one is the more genuine. When a computer is using a formalized version of an area in mathematics to generate proofs, it might be able to recognize brevity but not elegance.
This point is connected with the insistence by Sir Roger Penrose in his books The Emperor’s New Mind (Oxford, 1989) and Shadows of the Mind (Oxford, 1994), in the latter of which he responds to many criticisms of the earlier volume [3, 4]. He argues that more important than intelligence is consciousness, and consciousness requires the ability to use non-algorithmic processes. It is safe to say that Penrose’s second volume did not dismiss all of the objections that had been raised earlier, and his use of Gödel’s theorem also continues to be debated. If, however, one is inclined to think that elegance cannot be reduced to an algorithm, then perhaps the mathematician needs to be more than a machine.
Why should tenure not be granted to Professor AI? The machine may be able to produce any number of proofs that it can offer to a classroom. It will not be able to provide the kind of explanation either of why proofs are needed in the first place nor how to tell a good proof from a bad one. There may be many professions in which the progress of artificial intelligence has struck fear into the hearts of the practitioners. Even some in academia are talking of an early retirement as a way of avoiding having to deal with, say, ChatGPT. Mathematicians should not have to be similarly concerned, if their teaching at least involves more than getting students to memorize the quadratic formula.
Figure 4. Adam Sacks’s cartoon on page 52 of the 16 December 2024 issue of The New Yorker can also stimulate considerable discussion among mathematicians, as it did in MAA’s Connect community. Slide 19.
References
[1] Bundy, Alan. (1983) The Computer Modelling of Mathematical Reasoning. San Diego: Academic Press.
[2] O’Leary, Daniel J. (1991) Principia Mathematica and the Development of Automated Theorem Proving. In Perspectives on the History of Mathematical Logic, edited by Thomas Drucker, 47–53. Boston: Birkhäuser.
[3] Penrose, Roger. (1989) The Emperor’s New Mind. Oxford: Oxford University Press.
[4] Penrose, Roger. (1994) Shadows of the Mind. Oxford: Oxford University Press.
[5] Plunkett, Edward John Moreton Drax (Lord Dunsany). (1952) The New Master. In The Little Tales of Smethers and Other Stories, 138–148. London: Jarrolds Ltd.
[6] Turing, Alan. (1950) Computing Machinery and Intelligence. Mind 49, 433–460.
Thomas Drucker studied history of mathematics at Princeton under Michael S. Mahoney and at Toronto under Kenneth O. May. At the 2025 Joint Mathematics Meetings in Seattle, he delivered the Philosophy of Mathematics Special Interest Group of the Mathematical Association of America’s invited lecture, “From Computing Machinery and Intellligence to Snake Oil.” He retired from teaching at the University of Wisconsin–Whitewater in 2021.
