[This is a guest post by Karthik C. S., Amit Sahai, and Jie Xue. This blog post was initially written in a different file format and converted using AI. — T.]
This is a good time for mathematicians to think carefully about the purpose of their work. The development of artificial intelligence brings that question before us with unusual force. Much of the discussion concerns which tasks machines will perform and which abilities will remain distinctively human. But our response also depends on a prior question: what do we understand mathematical work to be, and what gives it its worth? If our answer has become inseparable from the production of results and the demonstration of individual ability, the present uncertainty may reveal a difficulty in our conception of the work itself.
There are broadly two perspectives on mathematics. According to one, we invent it as we construct it: we introduce concepts, choose axioms, and develop theories. According to the other, mathematics exists independently of these activities, and our task is to discover it. These perspectives need not disagree about every aspect of mathematical practice. We plainly develop formalism and arguments. The deeper question is whether the truths we come to understand owe their existence to these acts, or whether our constructions enable us to recognize something that does not depend on us.
While discovery of mathematical truth can sound passive, a paleontologist searching for fossils must decide where to excavate, develop techniques, and recognize the significance of incomplete remains. The fossils existed long before the investigation, but finding them may require extraordinary imagination. Their prior existence gives direction to the investigator’s activity. Similarly, a mathematical discovery may require the invention of an entire language and framework through which a previously unnoticed relationship becomes intelligible. The discovery may require great creativity, but that does not make the truth discovered a creation of the individual. We encounter this distinction whenever our constructions surprise us. A definition introduced for one purpose reveals an unexpected connection elsewhere. A conjecture that seemed to organize our examples admits a counterexample. Two theories developed through different questions turn out to express a common structure. In each case, our thought encounters something it did not freely determine. We can revise our definitions or abandon our conjecture, but we cannot make the original argument valid merely by wishing it so.
The Platonic perspective gives a natural account of this independence. On this view, there were infinitely many primes before anyone proved that fact. Mathematical truth neither waits for our recognition nor receives its authority from it. We may choose how to describe a structure, but we do not choose its properties in the same way. This perspective also changes the place of the mathematician. Our intelligence becomes a means through which an order independent of us can be understood. The significance of the work lies in its relation to that order.
The distinction between a correct derivation and an understanding of its significance is equally important. An axiomatic system permits indefinitely many deductions, but the ability to produce them does not by itself supply a worthy direction of inquiry. Mathematical statements may be formally correct without being enlightening. We can introduce countless definitions and prove countless statements about them. What makes one definition illuminating? Why does one theorem disclose something fundamental while another merely adds to the available material? These questions concern the relation of a particular investigation to the principles it brings into view. Mathematical taste has a place here, provided we understand it as more than individual preference. We value a theory when it reveals a unity among things previously understood separately, or enables us to recognize why a phenomenon occurs. Although mathematical truths may exist independently of us, their discovery can transform our understanding and enrich our experience of the world. The ability to carry out longer chains of reasoning does not by itself explain such understanding. Reasoning serves understanding by articulating relationships and making their justification explicit. However, the success of reasoning does not mean that all knowledge can be reduced to it. Nor does our ability to reason more extensively determine what is worth knowing or why we seek knowledge.
Mathematics is only one field of knowledge. Not everything can be understood mathematically, nor is mathematical achievement the ultimate purpose of intellectual life. Yet we increasingly measure intellectual life through publications, citations, awards, and other indicators of performance. While useful, these measures become problematic when they define the worth of the work itself. We begin to see mathematicians as collections of productive abilities, and mathematics as merely their output. From this perspective, anxiety about being replaced by AI is natural: a machine that performs these tasks better appears to threaten our worth. But the problem began before AI arrived. We had already come to understand ourselves in terms suited to machines. AI merely exposes the consequences of this view.
The response cannot simply be to seek refuge in distinctively human abilities such as creativity, intuition, or taste. If we value these abilities primarily because they give us an advantage over machines, we remain trapped in the same conception of intellectual worth. Why should the value of our pursuit depend on being better at it than someone or something else?
Mathematics offers an opportunity to reconsider this whole orientation. Its truths do not become more true through the distinction of the person who expresses them. In understanding a proof, we bring our thought into conformity with a relationship whose validity does not belong to us. The particular path by which we arrive may bear the marks of our individuality. What is understood is available beyond that individuality. We can allow this experience to loosen the belief that knowledge has its highest value as an achievement attributable to ourselves.
Egoless mathematics begins with taking that possibility seriously. Humility then reaches beyond modesty about our accomplishments or generosity in sharing credit. It concerns the place from which we pursue knowledge. We may seek truth to serve our existing purposes, or allow what we discover to change those purposes themselves. If we approach truth on the condition that its discovery must preserve our importance, we have already made it subordinate to ourselves. Accomplishing this reordering can be demanding, as intellectual sophistication can furnish elaborate defenses of the very attachments that prevent us from understanding. What matters is whether our abilities remain answerable to truth, including when it requires us to reconsider what we have been using those abilities for.
From this perspective, the possibility of mathematical discoveries through AI need not be a source of sadness about our purpose. A truth has become accessible; its worth is not diminished by the absence of our authorship. We need not immediately turn the event into a judgment about our relative importance. The search can continue without requiring every advance to vindicate the seeker. Indeed, the demand for such vindication has always burdened the pursuit, whether the competing claim came from another mathematician or from a machine.
The opportunity before us is therefore more substantial than a rearrangement of mathematical labor. It invites us to reconsider what gives an intellectual life its worth, and to let the pursuit of truth shape not only our work but also how we approach life.
This may sound like asking us all to become saints. Insofar as it asks us to grow in virtue and relinquish our attachment to our own importance, that impression is not far from the truth. But virtue is not something to cultivate for the admiration of others. It is necessary for genuine joy, even in our work. The satisfaction of solving a difficult problem or winning a prize can be fleeting when it rests on our standing relative to others. The joy of understanding is different. Something previously obscure becomes clear, and we recognize a beauty that does not diminish when someone else discovers it first. We can share this joy without depriving anyone of it. Growing in virtue allows us to experience it more fully, free from the need to establish our own importance.
This perspective may also help reconcile two positions within the mathematical community regarding AI. Some are comfortable with or even enthusiastic about using AI, but wish to develop appropriate processes and principles for its use. Others prefer not to use AI at all. Both positions have their merits. Egoless mathematics offers a way of reconciling them by introducing a higher consideration: both approaches can reflect the same underlying commitment to mathematical truth, regardless of how we choose to pursue it. Our aim here is not to settle particular questions of policy, but to develop a higher-level perspective that can guide our deliberations and ultimately our actions.
Seen in this light, we need not be afraid of what AI may achieve in mathematics. Mathematics was beautiful before any of us were born, and it will remain beautiful long after we are gone. Its worth does not depend on our accomplishments, nor does our joy in understanding it have to depend on being the ones who make the discoveries. Our purpose rests on our willingness to let truth govern what we seek and what we become.
We, the authors of this essay, are still far from having attained such egolessness. Nevertheless, we recognize it as an ideal toward which we wish to strive.
We conclude with a thought experiment. Ramanujan is reported to have said, “An equation has no meaning for me unless it expresses a thought of God” (Robert Kanigel, The Man Who Knew Infinity, 1992). If he were alive today, witnessing the advances of AI in mathematics, would he be perturbed by what they might mean for his purpose as a mathematician?
Statement of AI Usage: While the ideas expressed here are entirely our own, we gratefully acknowledge the assistance of GPT Astra 6.0 and GPT 6.1 Sol in drafting this post.
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