What is the value of learning mathematics now that artificial intelligence (AI) can solve almost all the questions we throw at it? Will AI change the way we think of mathematics and the way we teach and learn it? Will learning mathematics even remain relevant in 10 years from now, in an age when AI will surely play a key role? These are all valid questions, regardless of whether one is a mathematician or an educator or not. Finding definite answers to such questions is a key challenge in times like these; governed by uncertainty, by the fact that many questions have no clear answers, and that most answers are questionable. It is challenging yet exciting at the same time though, just like mathematics! It is now that our human intelligence and character need to be fully activated to ensure we proceed confidently, and try to find answers that allow us to prosper as a civilisation. So, I cannot answer these big questions on behalf of everyone, but what I can and will do here is share my personal thoughts on mathematics in the age of AI, as a mathematician, an educator, and a human.
First, mathematics is here to stay, no matter how AI proliferates. Not necessarily as it is now, not necessarily as an independent subject, not necessarily in its current shape and form, but as something to be embraced and experienced, and as something to be learned as the Greek origins of the word imply. How will it be learned though? Not necessarily the same way as now, not necessarily in classrooms, not only through textbooks and assessments, and probably not just facilitated by a human. I am convinced that teaching and learning in general will change in theory and practice, sooner or later, as a result of AI and all the emerging technologies, all the new discoveries in various fields, and our own evolving understanding of the concepts of teaching and learning. A change in education models will surely accompany this, whether we are in favor of it or not. I am also certain that if we manage to perceive AI and technology as catalysts and not threats, and if we truly believe in our human intelligence and resilience and our ability to adapt while protecting our human characteristics, we can then conceive new resilient learning and education models, mostly human-led and AI- and technology-powered as a starting point, that take us to new levels of innovation, efficiency and progress.
Will mathematics remain relevant? Yes. Always. But again, what do we mean by mathematics and relevance here? Mathematics is not just a book or an assessment. It is more than a grade or a degree. Mathematics is relevant for how sublime it is, and we need to accentuate that. What do I mean by sublime? First, from a philosophical and aesthetic perspective, mathematics induces awe and exceeds the realm of senses; its infinite horizons and abstract beauty align with the scale and magnificence the word induces among philosophers, from Burke and the vastness beyond comprehension to Kant and triumph of reason. This somehow resonates with Kurt Gödel who said: "Either mathematics is too big for the human mind or the human mind is more than a machine."1 For me personally, mathematics is SUBLIME in the way it embodies seven aspects: it is stimulating, ubiquitous, borderless, limitless, intriguing, monumental, and enduring. How, you ask?
Stimulating: Mathematics is not just memorising and applying formulas, sketching a graph, solving an equation, or finding the area of a 2D-shape, and we need to ensure it is not perceived as such. Nothing activates critical thinking, reasoning and problem-solving skills like mathematics. It pushes us to think about everything we deal with in real life from finances and measurements to more impalpable topics like the origin of the universe and the future of human intelligence, and even infinity and beyond. But not just that, mathematics teaches us to persevere while remaining patient, and to dream big while remaining humble. Maryam Mirzakhani was spot-on when she declared that “the beauty of mathematics only shows itself to more patient followers”.2 What is more humbling than the ‘simple’ prime numbers that still offer the most complex and challenging uncharted ramifications. Patience, and maybe AI, can help us explore these!
Ubiquitous: Mathematics is not just a ‘subject’ or a ‘course’ to be taught at school. While it has been tagged as such by many for a long time, for structural and practical reasons mainly, it might be time we rethink that. Purists may say that keeping it “independent” is an explicit acknowledgement of its importance, but is it? Maybe education systems should rethink how subjects are packaged and how content is delivered. Mathematics can (should?) be the element that beats the long-standing stagnation in the existing education models, which traditionally segregate subjects, with the help of new technologies, drawing upon how mathematics beats within so many other subjects that rely on mathematical concepts and reasoning to survive. Who said mathematics and computer science are totally independent, for example? Same for mathematics and physics or mathematics and medical sciences. Is teaching mathematics in context nowadays better or worse for learners? Joseph Fourier said: “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them”.3 What if mathematics gets taught, at least partially, in that way; following a unifying approach?
Borderless: Mathematics is not to be confined in curricula and textbooks; it is not to be framed as a collection of topics and domains or a list of theorems and formulas. It is and should be free to roam, as Georg Cantor wanted it to be when he said “the essence of mathematics lies precisely in its freedom”.4 One of the beautiful things about technology, and mainly AI, when used carefully, is that it allows us to break human-made borders between education resources and brings almost everything to our fingertips: we can now access real-time data related to a medical condition for use in a statistics course; we can create a mathematical model for climate change based on relevant and recent information; we can visualise and touch 3D graphs and interact with all sorts of holograms; we can juggle equations and play with charts in various ways; and much more. When all of this is available, why stick to the book and the blackboard? Technology allows us to amplify how free-spirited mathematics can and should be. After all, mathematics is not just something we learn: it is something we live, and that should be at the core of how mathematics is experienced.
Limitless: Mathematics is not a finite set of components. It kept on giving for centuries and keeps amazing us day after day, and this will be the case for an infinite amount of time! Speaking of infinity, who can think of different ‘sizes’ of infinity but a mathematician? That being said, new technologies should be seen and used as tools in support of this journey of ours to ensure more brains are triggered, more mysteries are unraveled, more conjectures are proven, and even new domains and fields are created and explored, just like with DeepMind. After all, keep in mind that there are still many unsolved problems in mathematics that require deep thinking and much perseverance, so why not let AI help us tackle these quests while we move on to think of new mounts to climb? Andrew Wiles once said: “The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.”5 Mathematics builds on itself, and the more we discover the more there will be left to uncover.
Intriguing: Mathematics is not boring and should not be so. It never ceases to push intelligence, human or artificial, to the limit, as it tackles a combination of structured and irregular numbers and objects, as it deals with systematic and chaotic systems and models, and as it includes logical and counterintuitive theorems and results. How fascinating to study a Koch snowflake, with its infinite length enclosing a finite area? How captivating to learn about Gabriel’s horn, with its infinite surface area and finite volume? This is the never-ending generosity of mathematics, always breaking the norms of reason and common sense, while being the guardian of logic! Mathematics is that magical mixture of certainty and conjectures: a special key that opens the most basic locks and the most challenging ones at once; it helps you add two digits and has the power to reveal the mysteries of the universe. But while this is so, we should not forget that there is still a stigma associated with mathematics for many, from numeracy to advanced level topics, and the fact that it is not presented and delivered as fun, attractive, and useful for all, adds to this stigma. We need to keep that gripping aspect of mathematics alive and to eliminate the stigma, and technology can assist with that.
Monumental: Mathematics is not just a supporting actor in a play led by other ‘fields’. Mathematics always had a tremendous role to play in humanity’s flourishing, and this has been the case to a massive extent from the beginning of times. Imagine a world without ‘0’, without ‘pi’, without ‘x’, without ‘i’, without ‘e’! All tiny on paper but huge in impact on human civilisation, and not only theoretically or in closed labs. Mathematics, whether explicitly or implicitly, even makes our world more beautiful to look at and live in; imagine architecture without the golden ratio, or nature without fractals and the Fibonacci sequence. This impact of mathematics will be further amplified with AI and new technologies; new discoveries, new proofs, new unexplored areas. But maybe it is time we rebrand mathematics to highlight its role and impact even more? If you ask a student nowadays if they prefer taking a course titled “Linear Algebra and Statistics” or one titled “Machine Learning and AI”, I believe I know what most of them would choose (someone to do a survey?), because mathematics is not marketed as a key component to all the transformations we are now witnessing in technology, nor to so many discoveries. Maybe the status of mathematics and its stardom is not debatable among scholars, but that is not necessarily the case in society or among students; this should change.
Enduring: Mathematics is not confined in an era or a timeline. It keeps on giving and history provides numerous examples of this. Mathematics evolves and expands, but it never stops being vital. Mathematics is always there, with its facts guiding how everything works whether we feel it or not, whether we know they are facts or not. As Erwin Schrödinger eloquently put it: "A mathematical truth is timeless, it does not come into being when we discover it".6 Mathematics has always been a companion and a guide to discoveries and progress; from Thales, Euclid, and Hypatia; to Al Khawarizmi, Al-Battani and Al-Kashi; to Euler, Gauss, and Germain; to Ramanujan, Turing, and Mandelbrot; the list is long. Mathematics will remain alive as long as humanity exists, and beyond. It will remain relevant as long as we have problems to solve, patterns and relationships to understand, and things to measure and compare. Mathematics will remain a core element of this universe as long as curiosity is burning.
So, I say it is time to bring SUBLIME back to Mathematics. What do you think? .
References:
- Gödel, K. (1995). Some basic theorems on the foundations of mathematics and their implications. In S. Feferman, J. W. Dawson Jr., W. Goldfarb, C. Parsons & R. M. Solovay (Eds.), Kurt Gödel Collected works: Volume III. Unpublished essays and lectures. Oxford University Press. (Original lecture presented in 1951).
- Mirzakhani, M. (2014, August 13). Interview. Maryam Mirzakhani: “The more I spent time on maths, the more excited I got”. The Guardian.
- Fourier, J. (1822). Preliminary discourse. Théorie analytique de la chaleur (p.7).
- Cantor, G. (2021). Cantor’s Grundlagen “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (Foundations of a general theory of sets) (Section 8) (J. R. Meyer, Trans.). Logic and Language. (Original work published 1883).
- Wiles, A. (2000, November 1). Andrew Wiles on solving Fermat. NOVA.
- Schrodinger, E. (1967). What is life? and mind and matter (p. 154). Cambridge University Press. (Original works published in 1944 and 1958).
Join the conversation: You can tweet us @CambridgeMaths or comment below.