We are mathematicians: we seek answers to problems that, at present, have no solution. To do so, we study the work of other colleagues, adapt it to the context we are interested in and, starting from there, look for new solutions. Now, the problem to be solved is the rethinking of our own profession. To address it, the most natural comparison is with the other scientific disciplines. Why not draw inspiration from the other sciences to understand how to reshape our work?
The content of this contribution is essentially speculative, and I am aware that my limited experience as a PhD student is not sufficient to understand all the dynamics involved in the practice of our profession. Nevertheless, LLMs require us to rethink the academic system of pure mathematics from the ground up and, however frightening this may be, we have to start somewhere.
I will begin from the assumption, which I believe to be true, that the integration of LLMs into the work of mathematicians is inevitable. I will not discuss here the reasons for this starting point, which, in any case, is reached both by the most progressive and, regretfully, by the most reactionary (AI dissenters). By this, I do not mean to ignore the questions concerning the economic and environmental impact of using LLMs; I simply have nothing to add to what has already been said on this blog. I will therefore allow myself to move the debate towards a subsequent stage: how should we rethink the research, communication and training of mathematicians in the age of LLMs? Given the speed at which events are unfolding, standing still means allowing things to proceed according to the will and objectives of those who currently hold the reins, namely private companies. We must try to understand how to regain control of our field.
Inventing and interpreting, originality and credit
Ever since I was a child, I have always been fascinated by the relationship between inventing and interpreting. These are two different ways of having new ideas: interpreting requires paying attention, recognizing a macroscopic pattern from microscopic details, connecting the dots, knowing how to receive and take in; inventing means acting, intervening, devising stratagems to create new dots, adding material.
Invention and interpretation alternate in the practice of all scientific disciplines: we interpret what is known in order to formulate a new question, invent a new technique to answer it and, finally, interpret the answer obtained. However, theoretical mathematics has one distinctive feature: it is the only discipline (at least, the only one that comes to mind) in which, even today, each researcher is responsible for both phases. For this reason too, in mathematics we are accustomed to attributing greater moral value to invention, as the highest expression of originality. By contrast, in applied disciplines technology has progressively separated responsibility for invention from responsibility for interpretation: those who build the experimental apparatus are often not the same people who use it to conduct research, and therefore do not necessarily receive credit for the discoveries made by others on the basis of the data collected with that apparatus. In mathematics, by contrast, both the method and the result are credited to the person, or the few people, who proved a given theorem.
Today, for the first time, the integration of LLMs into pure mathematics is breaking this inseparability and forcing us to reconsider the relationship between invention, interpretation and originality. And this throws us into crisis. Deprived by LLMs of the primacy of invention, many mathematicians feel that they are also being deprived of the credit that we have always associated exclusively with invention. The reading I would like to propose, however, is that originality and credit are not the exclusive prerogatives of invention, but pertain equally to the interpretative contribution. In this sense, the work of the LLM-assisted mathematician becomes much more similar to that of those working in other experimental disciplines and, in my view, it is precisely from this comparison that we can draw inspiration to reimagine our profession.
The role of the mathematician
The shock we have been experiencing in recent months stems from the fact that, for the first time in history, mathematicians have access to an immense quantity of results that we will call primary: those that an LLM can provide and whose production, until now, might have required an amount of work comparable to that of a PhD thesis.
As we know well, however, there is a major difference between data and information. I recently visited a friend, an experimental physicist who works at CERN: he is currently working on a component of the LHCb detector, specifically a particular photosensor. The detector contains photosensors that produce the equivalent of tens of thousands of electronic output signals. My friend told me that it was not possible to know in advance how to reconstruct the overall output image from the individual signals: this would have required knowing the exact individual path of every cable, which was impossible given the complexity of the technical apparatus. Someone therefore had to reconstruct on a computer the correspondence between the individual signals and the overall image. The information was already there, but it remained inaccessible until it was reorganised.
The fact that something is theoretically there, within reach, does not mean that it can be used. Extracting information from data requires effort. No experimental physicist would publish a paper presenting only a set of data: it is necessary to draw conclusions, indicate a direction, and understand the applicability and limitations of one’s work. And here we return to mathematics. The individual results that can be obtained through LLMs (assuming that their correctness has been verified) are the data: it is up to mathematicians to interpret them, organise them into a theory and derive new directions of inquiry from that theory. In other words, the processes of digestion and canonisation that are now at the centre of the debate become essential. The fact that LLMs make mathematical results more accessible does not mean that human intervention is no longer necessary. Truth does not exist if no one knows it; mathematics does not exist if humans do not study and understand it. As long as there is interest in knowing and using mathematics, human beings capable of understanding and handling it will be needed. It is possible that an increasing number of career opportunities will open up outside academia, but the “classical” profession of the mathematician will remain.
Having established this, I would like to focus on three aspects, research, training and publication, and put forward some preliminary ideas about how they might be rethought.
1. Research
Method. At CERN, the apparatuses are periodically switched off, and the data collected are then analysed. Similarly, we need to find a personal balance between phases in which LLMs are used and those devoted to processing, digesting and reformulating the information obtained. It should also be noted that a measured use of LLMs will be necessary not only methodologically, but also ethically, given their enormous environmental and economic impact.
Objectives. Having many “primary” results available allows us to be more ambitious and pursue broader objectives. I imagine that, once the race for low-hanging fruit is over and we have come back down to earth and resumed directing research judiciously, new projects (and the resulting publications) will no longer concern only “Result X in context $Y_i$”, but rather “Result X in the area of mathematics Z”, where $Z \supseteq (Y_i)_i$.
Collaboration. Our work will become increasingly collaborative and will aim at increasingly ambitious goals, in proportion to the quantity of primary results obtainable through LLMs. All this will open the door to significant interdisciplinarity, both between different areas of theoretical mathematics and between pure mathematics and applied fields. Having large quantities of results at our disposal entails the need to manage them organically, just as the LHCb detector consists of numerous components whose different aspects are studied by hundreds of physicists. Just as already happens in certain areas of applied mathematics and the experimental sciences, pure mathematics will increasingly see papers written by teams of five to ten co-authors, rather than the two to four authors currently more typical. The Polymath Project and the Equational Theories Project already provide examples of collaborations on an even larger scale. This will also require us to rethink the system for attributing credit, bringing it closer to that of the experimental sciences: the number of individual results will count for less, while making each person’s contribution within a collective project recognisable will become more important.
Institutions. The main LLM-based systems in use today are proprietary technologies, developed and controlled by private companies, and have a gigantic socioeconomic impact. Faced with these conflicts of interest, I think again of CERN: experiments often have immense costs, and the construction of the experimental set-up necessarily also requires the contribution of private companies. The essential ingredient becomes collaboration between the public and private sectors, coordinated by international institutions, on neutral ground and with clear and shared objectives. Although imperfect, this model would at least provide an excellent source of inspiration for countering the current attack on open-source research.
2. Training
By training, I mean primarily three aspects:
- school or undergraduate education;
- the training of PhD students;
- the ongoing self-training that a mathematician undergoes whenever they learn how to solve a problem.
I will not go into the first point, which is significantly different from the other two and on which, at present, I have no proposals. As for the latter two, the ideas I have in mind are much less concrete, especially because I myself, as a PhD student, am still trying to understand how to train myself.
As regards the training of a PhD student, it is clear that it can no longer be based solely on solving problems of the kind and level of difficulty so far considered usual: not only because an LLM can solve them much faster, but above all because students need to be trained to adopt a much broader and more interdisciplinary approach to research.
- Schematically, the strategy could be as follows:
- obtain primary results through LLMs;
- understand, process, digest and connect them across different areas;
- acquire an in-depth knowledge of the state of the art in one’s own field;
- where appropriate, combine one’s results with those of other PhD students in the same group, to whom the PI can assign complementary or adjacent projects, or with those of one’s supervisor or other collaborators.
This will be a much less niche and fundamentally cross-sectoral form of training.
As regards what I have called the researcher’s ongoing self-training, this is, among all aspects of the debate concerning the use of LLMs, the one that seems to me the most subtle and complex. The answer I am currently giving myself is that the problems we will be called upon to solve with our intellect, once the use of LLMs is factored out, will simply be of a different kind. Before computers became widespread, being able to solve tedious integral calculations was an essential skill for an analyst and, indeed, there were formula books and manuals specifically devoted to that. Long before the advent of LLMs, this skill had already ceased to be indispensable. Likewise, it will be necessary to develop transversal, rather than merely vertical and specialised, skills: as stated above, interpretation and interdisciplinarity will no longer be subordinate to invention and specialisation. In this sense, ongoing self-training will take place exactly as it always has. Each of us knows when to stop and devote hours to understanding a theorem or technique that we may have been using for some time but have not, in essence, understood; this will not change. Will we lose some skills? Certainly, but we will train new ones.
Likewise, PhD students will learn the profession as they always have. It will be a different profession and they will be called upon to solve different problems, but they will learn through practice, as has always been the case.
3. Publication
Thanks to LLMs, the volume of results that mathematicians will be able to process and, therefore, publish is bound to increase. This will happen whether the rate at which papers are currently released remains unchanged or whether (in line with what was set out in section 1. Research) the focus progressively shifts towards broader projects. In the latter case, the timeframes and numbers of publications could return to levels comparable to those we have considered “normal” until now, but the scope of each individual work would increase. This will lead journals themselves to require every submission to be accompanied by a Lean certificate, on pain of an unmanageable workload for referees. This transition will not be immediate, but it will happen, and it will prevent referees from being forced to use LLMs to review results obtained through LLMs.
From the point of view of peer review, Lean certificates will relieve the referee of the responsibility of checking the correctness of a proof. At most, the referee will have to ensure that the informal text corresponds to the formalisation. The role of the referee will remain, but with a different task: to ensure that the paper under review has been digested and placed in context, and that it presents the results with the aim of making them understandable to third parties. If we focus on this aspect, however much an author may have used AI in producing the paper, AI is neither necessary nor sufficient for reviewing it. What the reviewer must assess is whether the author has succeeded in giving the results a recognisable angle, direction and perspective. Of course, this step too can be carried out with the assistance of AI, but it requires a human understanding of the very purpose of publication: to disseminate our work and present it to others, so that they can understand and use it.
When considered in relation to the likely increase in the scope of research projects, this paradigm shift does appear quite natural. Assessing the content of a paper will require greater critical engagement from the referee; this will be made possible in part by the fact that, conversely, their responsibility for checking the correctness of the results will be reduced.
Moreover, the need to rethink the publication system is an excellent opportunity to solve the problem of overcrowded journals. I have no concrete proposals in this regard, but journals too will have a responsibility to orient their editorial policies towards “broader” projects, in line with what was described in section 1. Research, so as to discourage the aim of publishing as many papers as possible in order to boost one’s personal metrics. Other journals or repositories could be devoted to “primary” projects generated purely through LLMs.
Finally, I would like to close the circle and return to the initial discussion of invention and interpretation: originality and, therefore, authorship are not exclusive prerogatives of invention. The fact that a paper is based largely on results obtained through LLMs does not preclude those results from being interpreted and connected across different areas in an original manner, worthy of credit and publication.
Conclusion
I realise that, to some, this post will seem like the ravings of an inexperienced person (which I certainly am) but we are at a critical stage for our discipline, and it is necessary to begin experimenting with courageous solutions and therefore, first of all, to formulate them. If anyone has already begun experimenting in one of the directions described above, or in others, I would be very interested to hear about their experience!
Received 27 August 2026.
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