If intuitionism has been characterized with a certain propriety as revolutionists who overturned the ancien régime, Hilbert might be compared with Napoleon who, without regard for considerations of legitimacy, established, through a brilliant political stroke, a new order whose success is the substitute for legitimacy.
Kurt Grelling, 1928And yet when one hears one of the greatest living mathematicians calmly telling the world that a considerable part of our analysis is devoid of proof, if not nonsense, and when one beholds the mighty efforts which the champions of Weierstrass are making to repel these attacks, it is reasonable, in view of such facts, to ask ourselves, ‘Is all well?’
James Pierpont, 1928
Taken from “Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer” by Dirk van Dalen
The discipline of Mathematics actualizes itself in the interstitial spaces between our research: in the coffee breaks during conferences, in the asides and footnotes in monographs, and in forum posts of mathematical communities. In those spaces we often hear and refer to the great crises of the past, of the crisis in euclidean geometry, of the crisis in foundations, of the crisis of formalism and constructivism. We now find ourselves in a new crisis which dominates the chatter of the collective coffee rooms of the world. Mathematics has been able to reconstitute itself from those past crises into a singular discipline; however I believe today the divisions are too deep to recover. I make a case for splitting the discipline into two: A positivist mathematics and a humanist mathematics, each with its own journals, institutions, faculty and norms.
There have always been two parts to our discipline. The first part, the positivist aspect of mathematics, entails formulating and proving theorems as its ultimate end goal. The second part, the humanist aspect of mathematics, entails the formulation of mathematical beauty and maturity as its ultimate end goal. These two parts have historically gone along hand-in-hand, with few eruptions. The humanist tradition of mathematics has been fruitful for the positivist end goals, and those positivist end goals have given direction and purpose to the humanist tradition. The current advances in AI technology now have broken this fruitful cooperation. We are reaching a stage where machines can now prove theorems, and we are not too far from the day where an engineer or scientist, needing grounding for a new theory or invention, can ask a machine to formulate the appropriate mathematical background and answer the relevant theorems.
There is of course, still a great need for humans in positivist mathematics: we still need humans to clean up proofs, to guide research directions, to act as the interface between the machines and their application. This is anathema to most humanist mathematicians: no longer would we be building towards mathematical beauty, but rather our discipline would be in some ways, ancillary to a greater project taken over by the machines. Personally, I fall in the humanist camp, but I think it is wrong to dismiss positivism entirely. The goal of positivist mathematics is certainly great: to facilitate the production and maintenance of the book of all the world’s greatest theorems. However, this goal is fundamentally incompatible with the way many of us want to do mathematics.
Some people wave away this incompatibility, arguing that we can square the difference: let humans work in theory formulation, and leave the rest up to the machines. It must be understood that this is already the positivist position. Indeed, it admits that our goal is the formulation of important theorems, and that our discipline should adapt itself to whatever new technologies are possible to achieve that goal. No one knows how far this technology will develop, but this attitude would see it that mathematicians continue to adapt as far as possible to those demands.
On the other hand, in the humanist viewpoint, we may see this as fundamentally absurd. The goal of humanist mathematics is the general development of the intellect. In this sense it shares allegiances with the arts and the humanities, even as it sometimes aids in the work of scientific discovery. Do we stop playing the piano just because digital synthesizers exist? Do we stop painting just because we have photography? Creating the most perfect, realistic image possible to capture the world is certainly a noble pursuit, but we should not let that stop us from engaging in the enterprise of painting as self-expression. By splitting mathematics in twain, both the positivists and humanists will find themselves more free. The positivists will be more free to pursue mathematics as a pure science, and the humanists will be more free to pursue mathematics as a form of beautiful self-expression.
To those reading this, imagine yourself on a hiring committee weighing two different job candidates. The first applicant has 30 papers. The papers are generally rough to read but answer some of the most important questions in the subfield. All their papers make very heavy use of the most advanced LLM technologies. They may answer poorly when asked to recall and explain much of their work but clearly understand intimately how to use LLMs to generate useful, important and interesting results. The second applicant has 5 papers, all human written. The questions they answer are small, some questions they answer are even already known! But the exposition in their papers is stunning and beautiful. When you interview them they are lively and demonstrate deep understanding and love of their work. Which candidate do you hire? The answer to this question determines where you fall along the positivist-humanist axis. It seems clear to me that these two candidates are working in fundamentally different disciplines and traditions, and that treating them in the same pool is a mistake.
If Mathematics does split along this division, it will be with some inevitable strife. Determining which departments, journals, conferences and institutions value the positivist endeavour and which ones value the humanist endeavour may require some bitter conflict. However, the disciplines that emerge from this conflict, Positivist Mathematics and Humanist Mathematics, will be stronger for it. If we do not make these decisions consciously, they will be made unconsciously for us. If we do not split the discipline in two, one half of the discipline will inevitably dominate the other, and we will all be worse off for it. After all of the great crises of mathematics in which it survived as a whole, we now find ourselves in an entirely unprecedented situation. We either schism or die.
Received 12 October 2026.
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