# Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning

Simon Frieder, Jonas Bayer, Katherine Collins, Julius Berner, Jacob Loader, András Juhász, Fabian Ruehle, Sean Welleck, Gabriel Poesia, Ryan-Rhys Griffiths, et al.

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Simon Frieder, Jonas Bayer, Katherine Collins, Julius Berner, Jacob Loader, et al. Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning. 2025. hal-05425435

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Data for Mathematical Copilots:
Better Ways of Presenting Proofs for Machine Learning

Simon Frieder*,¹, Jonas Bayer², Katherine M. Collins², Julius Berner⁴, Jacob Loader²,
András Juhász¹, Fabian Ruehle⁸, Sean Welleck⁷, Gabriel Poesia⁹, Ryan-Rhys Griffiths¹⁰,
Adrian Weller², Anirudh Goyal¹¹, Thomas Lukasiewicz³,¹, and Timothy Gowers²,⁶

¹University of Oxford

²University of Cambridge

³Vienna University of Technology

⁴Caltech

⁶Collège de France

⁷Carnegie Mellon University

⁸Northeastern University

⁹Stanford University

¹⁰FutureHouse Inc.

¹¹Meta

Abstract

The suite of datasets commonly used to train and evaluate the mathematical capabilities of AI-based
mathematical copilots (primarily large language models) exhibit several shortcomings. These limitations
include a restricted scope of mathematical complexity, typically not exceeding lower undergraduate-level
mathematics, binary rating protocols and other issues, which makes comprehensive proof-based evalua-
tion suites difficult. We systematically explore these limitations and contend that enhancing the capa-
bilities of large language models, or any forthcoming advancements in AI-based mathematical assistants
(copilots or “thought partners”), necessitates a paradigm shift in the design of mathematical datasets and
the evaluation criteria of mathematical ability: It is necessary to move away from result-based datasets
(theorem statement to theorem proof) and convert the rich facets of mathematical research practice to
data LLMs can train on. Examples of these are mathematical workflows (sequences of atomic, poten-
tially subfield-dependent tasks that are often performed when creating new mathematics), which are
an important part of the proof-discovery process. Additionally, we advocate for mathematical dataset
developers to consider the concept of “motivated proof”, introduced by G. Pólya in 1949, which can serve
as a blueprint for datasets that offer a better proof learning signal, alleviating some of the mentioned
limitations. Lastly, we introduce math datasheets for datasets, extending the general, dataset-agnostic
variants of datasheets: We provide a questionnaire designed specifically for math datasets that we urge
dataset creators to include with their datasets. This will make creators aware of potential limitations of
their datasets while at the same time making it easy for readers to assess it from the point of view of
training and evaluating mathematical copilots.

<footnote>*Corresponding author: simon.frieder@cs.ox.ac.uk. Contributions: S.F. was the project lead, outlined the vision, content and individual sections, and wrote the paper with significant assistance from Jo.B. on all aspects of formal datasets, K.M.C. for the datasheets section, Ju.B., A.J, F.R. and G.B. for workflows, J.L. and T.G. for the motivated proof section, and S.W. on the recommendation subsections, and Ju.B. for the table. R.R.G., A.G., A.W., and T.L. provided general and strategic feedback.</footnote>
# 1 AI Systems for Mathematicians – Present and Future

There has been a rapid surge in research around AI for mathematics. For example, AlphaGeometry was shown to solve mathematical geometry problems on the level of the International Math Olympiad (IMO) [Trinh et al., 2024], Numina won the first AIMO Progress Prize,<fnref n="1" /> by providing a model capable of turning mathematical questions into executable Python code, and previously GPT-4 [OpenAI, 2023] was shown to attain, in some cases, the performance of an undergraduate university student [Frieder et al., 2023a]. High accuracies on standard mathematics benchmarks [Reid et al., 2024] suggest that many benchmarks may be close to being solved.

The rapid progress and proliferation of ideas in this space – while exciting and productive in many ways – has led to a series of problems due to a lack of alignment across research directions and a lack of coherence around clear goals. Problems include:

(1) some datasets being overstudied (e.g., the many versions of the GSM8K dataset [Cobbe et al., 2021]), whereas other data-related aspects (few datasets comprising advanced mathematics, or reflecting tool use in mathematics) are neglected;

(2) various workflows and reasoning modes of how professional mathematicians conduct their research are not represented in datasets;

(3) different ways to express the same mathematical content (in formal language, vs. in natural language) have not been unified;

(4) scalability of evaluation that aligns with the goals of the tools being developed.

Despite these problems, the landscape of machine learning models that have been devised (and embraced by parts of the mathematical community) consists primarily of models that **lie between the two categories below**, based on the amount of interaction with humans they are designed for:

1. **Highly specialized, "narrow" models** that fulfill all of the criteria of 1) being designed for a precisely defined mathematical domain or mathematical problem, 2) requiring domain knowledge to be used (such as formal languages as input), 3) that cannot explain themselves, and 4) do not allow free-form interaction. Examples of such specialized models vary widely in terms of how their foundations look like. Specialized models could rely on symbolic reasoners, such as the recent solver for elementary geometry (Newclid [Sicca et al., 2024] which supersedes AlphaGeometry [Trinh et al., 2024]) or for inequality problems ([Wei et al., 2024]). Olympiad-level mathematics problems are within reach of both. Alternatively, such specialized models could simply be regression models, relying on raw mathematical data to infer new relationships between mathematical objects, such as for knot theory or representation theory [Davies et al., 2021], or group theory [He et al., 2023]. Many further approaches exist [Romera-Paredes et al., 2024, Fawzi et al., 2022]. These systems typically require a significant amount of specialization by a user, such as mastering a formal input language (e.g., AlphaGeometry) or understanding the machine learning model itself (e.g., [Davies et al., 2021]) beyond the necessary mathematical domain knowledge.

2. **General purpose, "broad" models** that can interact with humans via natural language, understand input images, and use tools (being able to call, in particular, specialized models, which can be viewed as a particular kind of "tool", as we argue below), can provide general assistance to a mathematician (both with routine tasks, such as literature search, and domain-dependent tasks, such as counterexample search) and are not tied to a specific area of mathematics. These are what we refer to as **"Mathematical Copilots"**, as their goal is not simply to provide correct proofs but also to enhance the user's understanding of the results and guide them towards their own discoveries [Collins et al., 2024a]. Currently, the best candidates for these systems are large language models (LLMs), which we take to include multi-modal models that also admit images as components of the prompt.

<footnote n="1">¹https://aimoprize.com/updates/2024-07-20-progress-prize-results</footnote>
We contend that LLMs represent early predecessors of future general-purpose models (mathematical copilots), ultimately acting as mathematicians’ “thought partners” [Collins et al., 2024b]. Anecdotal evidence of such usefulness of current LLMs has been presented in various discussion forums,<fnref n="2" /> while noting the failures that LLMs often succumb to. However, there are concrete ways in which LLMs can be improved to become mathematical copilots: As indicated by the tool-integrated-reasoning approach, championed by models such as ToRA [Gou et al., 2024] or Numina,<fnref n="3" /> general LLMs can incorporate other symbolic systems to which they can delegate symbolic tasks to. The mixture-of-experts paradigm [Cai et al., 2024], on the other hand, highlights different ways in which LLMs can incorporate other LLMs as specially trained “modules” to which they can delegate mathematical tasks that are not symbolic in nature but still require particular mathematical skills. Such a hierarchical system can help mathematicians both on routine academic tasks (see [Frieder et al., 2023b] for a short, high-level overview) and specialized ones.

We envision that such models will require significantly less effort from users to use them, compared to specialized models, being self-explanatory, where natural language is the main mode of interaction. (While the outlined architectural approaches provide a viable step forward, in some cases, no data exists on which to train these models; see below.)

In addition, a third class of models will likely be developed:

3. **Fully automated, “universal” models** that can generate mathematical theorems and theories autonomously, merely by being directed to a mathematical result. We envision that these systems will be the true successors of automated theorem provers (ATPs) such as the Vampire [Riazanov and Voronkov, 2002, Kovács and Voronkov, 2013] or E [Schulz, 2002] ATPs. These systems have not been widely adopted by the mathematical community. While there have been attempts to imbue these systems with (non-LLM) machine learning techniques [Holden, 2021], these have not yet dramatically increased their performance. Nonetheless, once general-purpose models have been established, it is conceivable one could put these in automation loops to obtain “AI mathematicians”. For the domain of AI, this has already been recently investigated in the form of an “AI Scientist” [Lu et al., 2024a] that autonomously generated machine learning research articles. These are systems that conceivably will require minimal human input, merely guiding them towards certain theorems that one wants to see analyzed (proved, refuted, with comments on whether slightly different variations of a theorem, with slightly altered hypotheses, admit a proof).

We focus in this article on **general purpose models**, as outlined above, which are **mathematical copilots**. LLMs are currently the systems that hold the most promise to become, given a better data foundation, general purpose models. Fully automated, universal models are currently out of reach, as they are contingent on the former category of models. These are the systems that are ultimately sought when one speaks of “AI Mathematicians” [Bengio and Malkin, 2024]. However, without the proper datasets, it is unlikely that these will come into existence. Hence, our focus is on the **data that is needed to arrive at general-purpose models, the mathematical copilots**. We have deliberately avoided drawing a hard boundary among the properties a model must fulfill to be placed in one of these three because the boundaries will be fluid, as the levels of automation and required human interaction are continuous scales.

Currently, the existing datasets only support highly specialized models and do not put LLMs on a trajectory to achieve general-purpose models. In this article, we highlight the limitations of current datasets (Section 3, which is retrospective) and necessary changes in terms of data that will be needed to train the next generation of LLMs that approximate what we outlined general-purpose models to be (Section 4, which is prospective).

Our main reader audience consists of machine learning researchers, as we intend this article to raise awareness of these issues. Automated theorem provers (ATPs) and interactive theorem provers (ITPs), which have a decades-old history [Harrison et al., 2014], also had the ambition of realizing the goal of implementing

<footnote n="2">²https://www.reddit.com/r/math/comments/14p6j5c/terence_tao_on_using_gpt4_to_help_with Cole/</footnote>

<footnote n="3">³https://huggingface.co/AI-MO/NuminaMath-7B-TIR</footnote>
mathematical copilots. Unfortunately, history has shown that if the computer science community and the
mathematical community do not operate in lockstep, there is a risk of obtaining systems that will not be of
interest to practicing research mathematicians; maybe controversially, Blanchette et al. [2012] mentions that
certain efforts to make these systems more user-friendly have perhaps reached a “plateau”. (Nonetheless,
ATPs and ITPs have, in other domains, contributed significantly to driving the field of software automation
forward.)

One of the goals of this paper is to make sure that the renewed effort by computer scientists and machine
learning researchers to provide mathematicians with tools that make their trade easier this time is on a track
to converge to the needs of mathematicians. We hope that one of the first manifestations of this will be in
the form of datasets that better capture the various rich facets of mathematical research so LLM creators
can provide scores on such benchmarks that are more informative for mathematicians regarding how useful
the LLM is for actual daily mathematical practice. While the most-used datasets, GSM8K and MATH, are
indicative of LLMs' reasoning abilities, they are largely irrelevant to mathematical practice.

2 An Overview of Data-Related Issues

Without using tools specialized for mathematics, the current general models, LLMs, simply by well-designed
training methodologies and trained on mathematical data, have demonstrated unprecedented capabilities in
generating humanlike mathematical text, solving complex problems, and even engaging in creative problem-
solving. For example, GPT-4 has performed promisingly on undergraduate-level mathematics formulated
in natural language, as pointed out by Frieder et al. [2023a]. Math-Specialized Gemini 1.5 Pro [Reid et al.,
2024], a commercial model by Google not available to the public, has been reported to have an accuracy of
over 90% on the MATH dataset [Hendrycks et al., 2021], one of the most widely-used datasets for testing
mathematical ability. This score has recently been replicated by an open-weight model, QwQ.<fnref n="4" /> Attaining
such a high score is equivalent to achieving the ability of an IMO gold medallist (according to [Hendrycks
et al., 2021]). However, to date, no reports have been made where the mathematical benchmarks of QwQ
are tested for contamination; not undertaking tests may lead to this score not being reproducible on a
different dataset from the same difficulty and problem type distribution [Yang et al., 2023, Xu et al., 2024].
Further notable general-purpose LLMs are open-weight models with strong reported performance on baseline
datasets, such as MATH and GSM8K [Cobbe et al., 2021], the DeepSeek family of models [Shao et al., 2024,
Liu et al., 2024, Zhu et al., 2024, Xin et al., 2024], and the Qwen family of models [Bai et al., 2023, Yang
et al., 2024a, Hui et al., 2024, Yang et al., 2024b]. Other LLMs focus more on specific abilities such as
MathPrompter [Imani et al., 2023], which associates a confidence value to arithmetic problem solutions,
MathVista [Lu et al., 2024b], which solves geometric reasoning problems, WizardMath [Luo et al., 2023], or
Llemma [Azerbayev et al., 2024], which was specialized through further pretraining on a math corpus, and
is both open-weight and open-source. A special use case where LLMs have found success is in interactive
theorem provers (ITPs) such as Isabelle [Nipkow et al., 2002] or Lean [de Moura and Ullrich, 2021], where
they are used to prove theorems most prominently in the context of generating proofs of given formal
statements [First et al., 2023, Zheng et al., 2023, Wang et al., 2024a], autoformalization [Szegedy, 2020,
Jiang et al., 2023], or providing code snippets of formal mathematics [Song et al., 2024a]. For a survey of
deep learning approaches on both formal and informal (natural language) mathematics, we refer to Lu et al.
[2023], and for more recent information on language models, including their performance on mainstream
datasets, we refer to Zhao et al. [2023].

This paper argues that the current dataset landscape does not support the advancement of such general-
purpose AI systems, in particular LLMs, towards a level of mathematical performance that makes them
usable as daily mathematical thought partners [Collins et al., 2024b] that help mathematicians push the
boundaries of what is known, and capture the richness of mathematical reasoning and invention that human
mathematical minds are capable of [Zhang et al., 2023a, Dehaene, 2011, Feigenson et al., 2004]. Our focus thus

<footnote n="4">⁴https://qwenlm.github.io/blog/qwq-32b-review/</footnote>
includes, in particular, research-level mathematics, which involves many steps beyond deriving or formalizing
proofs – the aspects of mathematical practice that are currently most strongly represented in data.

The spectrum of tasks a copilot can assist with is not solely related to mathematics: programmers presently
carry out programming tasks, such as fixing bugs or other GitHub issues [Jimenez et al., 2024], using
various code-generating copilots, such as the GitHub Copilot [Chen et al., 2021]; the emerging Lean Copilot
ecosystem [Yang et al., 2024c, Song et al., 2024b] assists users in the task of formalizing a mathematical
theorem; Tutor CoPilot [Wang et al., 2024b] provides real-time guidance to human tutors that teach school
students mathematics at the level of math word problems. Yet, mathematics, with its rich set of domain-
dependent workflows, tools, and modes of thinking, requires copilots with exceptionally broad capabilities. A
mathematical copilot that is useful to a mathematician must significantly exceed the capabilities of the three
exemplary copilots mentioned above, which are systems that are designed to help with a narrow technical
set of tasks.

*Our paper is split into three main sections: In the ensuing Section 3, we review past datasets that may have limitations of various kinds. We split these into two further categories, relating to natural language and formal language datasets.*

To give an overview of our contributions in that section, current natural language datasets typically model a question-response interaction, such as posing a problem and writing a solution. We identify four dimensions where such datasets frequently fall short of being able to assess advanced mathematics:

*   **Difficulty:** Most datasets are situated on a level of difficulty at or below the level of an undergraduate degree in mathematics, or impossibly hard, with few datasets at an *appropriate* level of difficulty, from which model creators can better learn about the strengths and weaknesses of their system.

*   **Binary evaluation:** Benchmarks typically only support assessing *if* the output was correct or incorrect, but in case of errors, do not provide information about *how* the output failed to be correct, which is crucial to support.

*   **Lack of trusted automation evaluation:** Benchmarks rely on checking a final answer (generally, by keyword matching), imperfect model-based evaluation, or provide no automatic evaluation at all.

*   **Standardized Interaction Modes:** Often, natural language datasets only contain question-answer mathematical problems. The daily experience of a mathematician is much richer than that [Zhang et al., 2023a], since to prove/search for a theorem, all kinds of “lateral” modes of thinking have to be applied. Examples are intermediate conjecture formulation, (counter)example search, and finding analogies with other theorems. These are typically not found in textbooks, so there is no representation of these in terms of data. We point to Collins et al. for an exposition of the problems of static evaluation for natural-language mathematics problems in practice.

*   **Contamination:** Many datasets released solely for evaluation and benchmarking purposes have a short half-life, as, once released, they are easily leaked into datasets that will be used as training data for future models.

For the other category, datasets around formal mathematics frequently suffer from other issues than those
listed above; for instance, datasets for formal mathematics often come at a range of difficulties, from
competition-level problems [Zheng et al., 2022] to undergraduate textbooks [Azerbayev et al., 2023], to
some results that are at the mathematical state-of-the-art [Scholze, 2022, Bordg et al., 2022]. Moreover,
formalized mathematics evaluation can be easily carried out in a safe and automated manner since proofs
can be checked by the ITP.

*   **Tool misalignment:** Various automation tools exist that aid in proving small, intermediate lemmas arising in the formalization of more comprehensive results. This creates incentives to change a proof's structure to maximize the use of these tools, potentially resulting in less readable and "human looking" proofs.
*   **Non-trivial data duplication:** In mathematics, it is common that one theoretical concept has several formal representations. For example, the concept of a graph can be described formally in multiple ways. Formal libraries typically feature a variety of representations for the same mathematical concept, some being almost exact copies of each other. Such near-duplication can lead to downstream issues when the library is both used as training and test data in machine learning approaches.

*   **Standardized interaction modes.** (Analogous to natural-language, but with different characteristics) Current formal benchmarks focus on proving theorems in isolation or translating individual statements into formal code. These tasks do not capture all aspects of advanced mathematical practice.

These limitations have various consequences: For example, evaluating progress of the mathematical capabili-
ties of AI systems, particularly for advanced mathematics, is difficult. For instance, improved accuracy on a
standard problem-solving dataset may not provide a meaningful signal about what has improved, while per-
forming well on an existing formal benchmark may not accurately indicate where models currently struggle.

In the next section, Section 4, we discuss how certain aspects of the proof creation can be mapped to datasets. We argue that such datasets that describe intermediate stages that mathematicians go through in the long process of devising proofs are potentially highly valuable to support mathematical copilots that assist humans in these tasks. The overt focus of current datasets on result (the final proof), rather than the process that led to that proof, severely diminishes their usefulness for training LLMs to be used as mathematical copilots. While from a perspective of pure proof search, results-based datasets are justified, it is unlikely that mathematics will soon be reduced purely to proof search, as there are several mathematical activities (such as finding the “right” set of definitions, that makes a mathematical theory clear and easy to follow) that are not solely about proof search. Such datasets that describe these intermediate stages do not exist at all currently! The GHOSTS dataset, to our knowledge, is the only one that preliminarily investigates how well LLMs can engage in literature search. Thus, in Section 4.1, we take a more holistic level to describe how mathematical workflow might be described by datasets. This problem is essential, as a lot of mathematical activity is based on chaining together complex workflows. There is currently no clear way of measuring the advances of AI systems in tackling the full spectrum of mathematical workflows. Such data about intermediate stages in the proof discovery process is often not explicitly contained in textbooks and articles – the data sources on which current LLMs are trained. In Section 4.2, novel ways of capturing data that may contain this information are described. Finally, in Section 4.3, we discuss a stricter proof structure called a “motivated proof”, requiring the author to explain the origin behind each step of a proof. We then argue for an evaluation benchmark enforcing this proof structure from the perspectives of end-user utility and evidence of LLM reasoning rather than LLM memorization (coming up with a motivated proof, given an unmotivated one, requires reasoning and deeper proof understanding) and speculate on paths toward designing such a benchmark.

Lastly, in Section 5, we provide an extension of the well-known “Datasheets for dataset” questionnaires [Gebru et al., 2021], specifically aimed at mathematical datasets, as these come with their own specific characteristics, not covered by the original datasheets. We hope that our questionnaire facilitates better cross-disciplinary collaboration and alignment between AI practitioners and mathematicians.

We note that in this article we do not focus on datasets that are designed primarily to be consumed by the current LLM technology stack, for example, various instruction tuning datasets, such as the OpenMathIn-struct dataset [Toshniwal et al., 2024]. These are often based on existing datasets (GSM8K and MATH in the case of OpenMathInstruct) and thus could be described as *derived* datasets. Rather, our focus is on assessing how much and how well *primary* datasets describe various aspects of *doing mathematics*.

3 The Past – Common Pitfalls in Current Datasets

In this section, we discuss several issues with current datasets. Note that our limitations are with respect
to the landscape of AI for mathematics at the time of writing (Dec 2024); it is possible that some of these
<table><caption>Table 1: Selected datasets for evaluating the mathematical capabilities of LLMs.</caption><thead><tr><th>Dataset</th><th>Reference</th><th>Size</th><th>Field</th><th>Description</th></tr></thead><tbody><tr><td colspan="5" style="text-align:center; font-weight:bold;">Elementary to Middle School Level</td></tr><tr><td>Commoncore / MultiArith</td><td>Roy and Roth [2015]</td><td>600</td><td>Basic arithmetic</td><td>Word problems with two steps</td></tr><tr><td>MAWPS</td><td>Koncel-Kedzierski et al. [2016]</td><td>3320</td><td>Basic arithmetic and algebra</td><td>Word problems requiring systems of up to two linear equations (includes MultiArith)</td></tr><tr><td>Math23K</td><td>Wang et al. [2017]</td><td>23161</td><td>Basic arithmetic</td><td>Word problems in Chinese</td></tr><tr><td>ASDiv</td><td>Miao et al. [2020]</td><td>2305</td><td>Basic arithmetic</td><td>Word problems (selected from the internet)</td></tr><tr><td>SVAMP</td><td>Patel et al. [2021]</td><td>1000</td><td>Basic arithmetic</td><td>Word problems requiring up to two steps (derived from ASDiv)</td></tr><tr><td>GSM8K</td><td>Cobbe et al. [2021]</td><td>8500</td><td>Basic arithmetic</td><td>Word problems requiring 2-8 steps to solve</td></tr><tr><td>GSM8K-symbolic</td><td>Mizadeh et al. [2024]</td><td>5000</td><td>Basic arithmetic</td><td>Variants of GSM8K problems using symbolic templates</td></tr><tr><td colspan="5" style="text-align:center; font-weight:bold;">High School to Intermediate Level</td></tr><tr><td>Dolphin18K</td><td>Huang et al. [2016]</td><td>18460</td><td>Basic arithmetic and algebra</td><td>Problems and answers from the mathematics category of Yahoo! Answers</td></tr><tr><td>AQuA</td><td>Ling et al. [2017]</td><td>100000</td><td>Basic arithmetic and algebra</td><td>Algebraic multiple-choice word problems (similar to GMAT and GRE)</td></tr><tr><td>MATH</td><td>Hendrycks et al. [2021]</td><td>12500</td><td>Problem-solving</td><td>Problems from high school math competitions with step-by-step solutions</td></tr><tr><td>NumGLUE</td><td>Mishra et al. [2022a]</td><td>101835</td><td>Arithmetic reasoning</td><td>8 different mathematical reasoning tasks (including previous datasets).</td></tr><tr><td>GAOKAO (Math)</td><td>Zhang et al. [2023b]</td><td>844</td><td>Various high school math topics</td><td>Questions from the Chinese College Entrance Exam (single-choice, cloze, open-ended)</td></tr><tr><td>Hungarian High School Finals</td><td>Paster [2023]</td><td>33</td><td>Problem-solving</td><td>Problems from the 2023 Hungarian National High School Finals in mathematics</td></tr><tr><td>MATH 401</td><td>Yuan et al. [2023]</td><td>401</td><td>Advanced arithmetic and computation</td><td>Arithmetic expressions and numerical results</td></tr><tr><td>TAL-SCQ5K</td><td>TAL Education Group [2023]</td><td>5000</td><td>Various</td><td>Multiple-choice competition questions from elementary to high school levels in English and Chinese</td></tr><tr><td>AGIEval (Math)</td><td>Zhong et al. [2024]</td><td>1943</td><td>Various</td><td>Problems (in English and Chinese) derived from college admission tests (GRE, Gaokao, SAT) and high school competitions (from MATH and AQuA)</td></tr><tr><td colspan="5" style="text-align:center; font-weight:bold;">College Level</td></tr><tr><td>TheoremQA</td><td>Chen et al. [2023]</td><td>800</td><td>Various (including algebra, number theory, graph theory, information theory)</td><td>Questions (from the internet and textbooks) and expert answers based on 354 theorems</td></tr><tr><td>ARB</td><td>Sawada et al. [2023]</td><td>234</td><td>Various</td><td>Problems from university exams and competitions at the undergraduate level</td></tr><tr><td colspan="5" style="text-align:center; font-weight:bold;">Graduate to Research Level</td></tr><tr><td>NaturalProofs</td><td>Welleck et al. [2021]</td><td>25271</td><td>Various (focus on algebraic geometry)</td><td>Theorems with proofs and reference graphs from ProofWiki, the Stacks project, and textbooks.</td></tr><tr><td>CHOSTS</td><td>Frieder et al. [2023a]</td><td>709</td><td>Various</td><td>Problem-solving, theorem-proving, and retrieval tasks derived from textbooks, StackExchange, and existing datasets (MATH and Symbolic integration)</td></tr><tr><td>FrontierMath</td><td>Glazer et al. [2024]</td><td>> 35 (unknown)</td><td>Various</td><td>Challenging problems with automated verification created by mathematicians</td></tr><tr><td colspan="5" style="text-align:center; font-weight:bold;">Olympiad Level</td></tr><tr><td>miniF2F</td><td>Zheng et al. [2022]</td><td>488</td><td>Problem-solving, algebra, number theory</td><td>Formalized Olympiad-type problems from AIME, AMC, IMO, as well as high school and undergraduate courses (also derived from MATH)</td></tr><tr><td>OlympiadBench</td><td>He et al. [2024]</td><td>8476</td><td>Various</td><td>Problems and solutions from mathematics and physics competitions and the Chinese College Entrance Exam (in English or Chinese and partially with images)</td></tr><tr><td>IMO Small Challenge</td><td>Frieder et al. [2024]</td><td>100</td><td>Combinatorics</td><td>Selected problems and solutions from IMO and BWMC</td></tr><tr><td>IMO-AG-30</td><td>Trinh et al. [2024]</td><td>30</td><td>Geometry</td><td>Plane Euclidean geometry problems from IMO in JGEX formal language</td></tr><tr><td colspan="5" style="text-align:center; font-weight:bold;">Various / Mixed Difficulty</td></tr><tr><td>Mathematics</td><td>Saxton et al. [2019]</td><td>Generated</td><td>Arithmetic, algebra, probability, calculus</td><td>Sequential questions generated with modular structure</td></tr><tr><td>MMLU (Math)</td><td>Hendrycks et al. [2020]</td><td>960</td><td>Elementary, high school, and college mathematics, and abstract algebra</td><td>Multiple-choice questions collected from online sources ranging from elementary to college (similar to GRE) mathematics</td></tr><tr><td>INT</td><td>Wu et al. [2020]</td><td>Generated</td><td>Inequalities, General Mathematics</td><td>Generated formalized (in-)equality theorems based on ordered field axioms</td></tr><tr><td>AMPS</td><td>Hendrycks et al. [2021]</td><td>≈ 5100000</td><td>(Linear) algebra, calculus, statistics, geometry, and number theory</td><td>Problems and step-by-step solutions from Khan Academy (> 100k) and generated using Mathematica (≈ 5M)</td></tr><tr><td>Lila</td><td>Mishra et al. [2022b]</td><td>132239</td><td>Arithmetic, (linear) algebra, calculus, statistics, geometry, and number theory</td><td>Derived from 20 existing datasets (including NumGLUE, SVAMP, MultiArith, Dolphin18K, Mathematics, AMPS, GSM8K, MATH) with numbers or expressions as outputs</td></tr><tr><td>CMMLU (Math)</td><td>Li et al. [2023]</td><td>499</td><td>Various</td><td>Multiple-choice questions (in Chinese) from elementary to college level collected from freely available resources</td></tr></tbody></table>
limitations are addressed. Indeed, we hope that they *are* addressed swiftly. We first list issues that are common to both natural-language datasets and datasets comprising formalized mathematics. Then, we list issues that are specific to either natural-language datasets or formal datasets.

# 3.1 Common Issues

## 3.1.1 Difficulty

The difficulty of mathematical problems typically varies along at least two salient dimensions: level of abstraction and proof (or solution) sophistication. Abstraction varies from simple “one-layer” definitions, such as those that involve properties of topological spaces, to those in areas such as algebraic geometry, where definitions of highly intricate mathematical objects involve several layers.

Proof sophistication is related to problem-solving ability and is often tested (for humans and AI systems alike) in mathematical competitions, such as the International Mathematical Olympiad (IMO). Typical for these is that the statements to be proved are elementary, but highly ingenious proof techniques need to be used to arrive at the solution.

Several datasets have been proposed which explore proof sophistication, such as the OlympiadBench [He et al., 2024], the IMO Small Challenge, or the IMO-AG-30 dataset [Trinh et al., 2024], but very few existing datasets reach a level of difficulty that is more advanced than that of an undergraduate degree in mathematics in terms of level of abstraction. For a (non-exhaustive) overview of datasets, ordered by their difficulty, see Table 1.

For example, MATH focuses on high-school competition problems, while the recent TheoremQA [Chen et al., 2023] and Advanced Reasoning Benchmark (ARB) [Sawada et al., 2023] are upper-undergraduate level. Many commonly used datasets test mathematics below the undergraduate level, such as GSM8k and the 23 tasks in the Lila benchmark [Mishra et al., 2022b].

Two datasets that go beyond upper-undergraduate level are the GHOSTS dataset [Frieder et al., 2023c], as well as the NaturalProofs(-Gen) [Welleck et al., 2021, 2022] datasets, which contain problems, theorems, and proofs on a wide range of more advanced mathematical topics. However, these suffer from a lack of automated evaluation and contain many problems that could, in principle, be in the training corpora of modern LLMs, as we discuss later. FrontierMath [Glazer et al., 2024] is also a more advanced dataset, but is currently not publicly accessible.

Most current LLMS (e.g., the Qwen [Yang et al., 2024a] family of models, or commercial models released by OpenAI and Google) are evaluated on the MATH dataset [Hendrycks et al., 2021] and the GSM8K dataset [Cobbe et al., 2021], with a recent trend of some LLMs using held-out datasets such as the Hungarian National Finals high-school exam as an additional test of generalization [Paster, 2023]. The MATH dataset uses five categories of difficulty and is strictly more difficult than GSM8K. Nonetheless, state-of-the-art LLMs have been found to reach an acceptable performance on it [Frieder et al., 2023a]. For instance, Gemini reaches a performance of about 50%.<fnref n="5" />, while solving almost all problems on GSM8K. Progress in LLMs is proceeding at a rapid pace, and the “Math-Specialized” version of Gemini 1.5 reaches (at best) a score of 91% on MATH, which is representative of the current state of the art (although, due to the closed nature of this line of research, it is difficult to place this achievement into context, such as by analyzing how similar training data has been to the MATH dataset, and how much was needed).

While this performance may appear impressive, neither of these datasets covers a high degree of difficulty in terms of abstractness and proof sophistication. Hence, the current performance of LLMs on these datasets is not yet indicative of advanced mathematical understanding, which is relevant for professional mathematicians.

<footnote n="5">⁵https://blog.google/technology/ai/google-gemini-ai/performance</footnote>
This is particularly striking in the case of AlphaGeometry, which can solve 25 geometry problems from the IMO out of a test set of 30 problems [Trinh et al., 2024] (and 213 out of 231 on a larger benchmark). We argue that the accompanying dataset, while indeed difficult in terms of proof sophistication, is not sufficiently large to be able to accurately compare AlphaGeometry with other models.

**Recommendations.** We believe that it is crucial to introduce more difficult datasets into the evaluation repertoire of math-AI. And importantly, the skill level of these datasets should be *graded*, such that we can form a nuanced characterization of model capabilities – this can be achieved, in part, by interpolating difficulty level between existing datasets and those representative of advanced mathematics. This includes datasets on graduate-level mathematical domains, such as functional analysis (to measure handling of arguments involving high levels of abstraction), partial differential equations (to measure handling of intricate computational arguments), as well as other domains that come with their own “mathematical flavor”. At the same time, more comprehensive datasets illustrating and measuring advanced problem-solving skills are needed. The miniF2F [Zheng et al., 2022] dataset and the GHOSTS dataset are just the first steps in this direction. For natural language datasets, we note that simply scraping many competition-level problem sets, coupled with an evaluation principle that only verifies a “final answer”, is not sufficient to ensure that systems can progress to becoming useable mathematical copilots. Such problem sets, even if some of them ultimately are only math word problems, require expert, specialized knowledge and training for humans to be proficient. Without a detailed evaluation protocol (see Section 3.1.2), not much insight will be gained into why poor performance was obtained - and, conversely, human evaluation will be very costly (see Section 3.2.1).

## 3.1.2 Binary Benchmarks

Most datasets tailored for machine learning, except GHOSTS and NaturalProofs-Gen, including the dataset for AlphaGeometry, use a simple correct-incorrect rating scheme. The given feedback indicates *if* a system fails to solve a problem correctly, but without any indication of *how* it fails. Binary evaluation thus provides limited information on how to improve or interpret performance.

GHOSTS and NaturalProofs-Gen address this problem by using labels that are attached to language model outputs, which provide a classification of (mathematical) errors. For instance, GHOSTS distinguishes two types of labels, called “error codes” and “warning codes”, which make up, in total, 15 error or warning labels, while NaturalProofs-Gen distinguishes between 11 types of errors.

A small number of datasets use non-binary error rating schemes, to our knowledge these are only TheoremQA (3-point scheme) and ARB (5-point).

Finally, formalized mathematics offers feedback from the interactive theorem prover (ITP) when a system fails to produce a correct proof (one case being timeouts). However, this feedback is more akin to a compiler error than high-level feedback on the mathematical reasoning process. We speculate that this feedback may be useful but still limited in its ability to diagnose AI systems.

**Recommendations.** We believe that it is crucial to develop datasets that support multi-aspect feedback. Relying on a single numerical representation, in the face of the diversity of mathematical failure modes, risks providing a signal that is too weak to be suitable for learning or evaluation – which is especially important to guard against when we consider deploying systems alongside people, who may care not only that a model is correct but also that the response is appropriately helpful [Collins et al., 2024a].

One of the reasons that AI researchers may have avoided multi-aspect feedback, particularly subjective feedback, is for fear of the difficulty of obtaining consistent human annotations. While annotator disagreement is a challenge, we suggest that the math-AI community look to the ways in which computer vision researchers have worked on developing new theoretical and empirical tools which *lean in* to the diversity of human responses [Uma et al., 2021, Collins et al., 2022, Sucholovsky et al., 2023, Gordon et al., 2022]. Appropriately interleaving formality and subjectivity – especially for advanced mathematics, where there is a limited pool
of possible human annotators – is a challenge, but one which demands cross-disciplinary collaboration and
is ripe for future work.

As a middle-ground, we also recommend considering evaluation methods that do not assign numerical ratings at all – but rather annotate the outputs of AI systems with error codes and warning codes, then derive ratings from the codes directly (where no error codes mean a perfect rating).

3.1.3 Standardized Interaction Modes

Almost all datasets to date that are designed to evaluate LLMs capture only textbook-like questions. Moreover, owing to issues of automatic evaluation described in Section 3.2.1, the majority of datasets focus on a smaller subset of questions with a unique, typically numerical, answer. While such questions might span a range of different topics, from numerical (e.g., GSM8K, problems of level 1-2 from MATH, MultArith) and symbolic evaluations (the dataset associated with the work of [Lample and Charton, 2020]) to somewhat complex problem-solving tasks (problems of level 4-5 from MATH), they do not come close to covering all aspects of the mathematical tasks a mathematician might encounter in their daily practice. We believe that any future mathematical assistant should provide support for these tasks.

In the following list, we describe different dimensions of interaction modes for natural-language mathematics (both question types as well as session types), across which we believe that mathematical assistants should be evaluated.

Question Types.

1. **School/University Curriculum-Like Questions** that are encountered in educational settings and have well-defined answers. This includes questions with arithmetic, symbolic content, math word problems, as well as problem-solving and proofs.

2. **Proof-Specific Questions** as the concept of a mathematical proof is varied and gives rise to a host of specific questions about proofs, which go beyond the type of questions related to proofs from the previous point. These are advanced questions concerning matters such as: Establishing whether a proof is effective or not, in the sense of proving a general statement about a mathematical object merely by proving its existence as opposed to producing a witness or an algorithm that computes; investigating what distinct proof strategies might be used to prove a certain statement; exploring the distinction between whether an “elementary” or an “advanced” proof is given.

3. **Mathematical explanations and intuition**, e.g., by providing (counter-)examples. Such questions are open-ended and have potentially (infinitely) many correct answers. This freeform exploration is crucial in the process of establishing the truth value of a given (advanced) mathematical statement, where one swings between attempting a direct proof of the statement and finding a counterexample to the statement.

4. **Retrieval Tasks** for definitions and mathematical facts, either from name to statement or reverse (“mathematical search engine”). This can range from a well-defined answer (e.g., “*Define this mathematical object!*”) to open-ended questions (“*what are the most important theorems in a certain field?*; *which results are relevant to prove a certain statement?*; *has a form of this statement already been proved?*”). This tests the capabilities of mathematical assistants to provide high-level explanations similar to those of a human expert in a certain mathematical field.

5. **Informal Proof Completion**, where a proof (or generally a mathematical statement) needs to be checked or completed. In the best case, the mathematical assistant should be able to provide a correction and fill in gaps. In such functionality, mathematical assistants can be running in the background to act as “co-pilots” and “mathematical grammar-checkers” — as the current generation of copilots<fnref n="6" /> does successfully for programmers.

<footnote n="6">⁶Such as GitHub’s Copilot: https://github.com/features/copilot</footnote>
6. **Mixed-mode interactions**, where translation between formal and informal mathematics needs to be performed. These are particularly important for the combination of ITPs and general-purpose LLMs, which has recently picked up interest (e.g., [Jiang et al., 2023, First et al., 2023, Frieder et al., 2023c]).

To our knowledge, GHOSTS is the only natural-language dataset that focuses on all of the first four items from above. Accompanying this variety of questions (*what* is asked) are session types that describe *how* the questions are posed.

## Session Types.

1. **Zero-Shot**, where the model is prompted without further information.

2. **In-Context Learning**, where additional background information is provided, ranging from clarifications of the notation to context (e.g., the chapter of the book or paper), to similar questions with corresponding proof.

3. **Interactive**, where the prompter (whose mathematical abilities can span a wide spectrum) solves mathematical questions together with the language model in an interactive way as in Collins et al. [2024a].

In practice, typical mathematical workflows require a mixture of different question and session types. Even if a question is not open-ended, the amount of mathematical detail and rigor the answer should contain will depend on the user's background. In part, this should be deduced from the way the question is phrased. Currently, no datasets offer support for this.

We note that with the exception of Collins et al. [2024a] and the collection of CheckMate, almost all datasets are zero-shot. Interactive training modes are extensively employed in a formal language setup.

For formal mathematics, different considerations apply, as outlined below. While modern LLMs can, to a degree, perform the tasks below, they are not specialized for these tasks:

* **Refactoring** formal mathematics to yield nicer-looking proof is a significant undertaking. A tool that takes as input a formal proof and refactors it along desired criteria would be a welcome addition to the toolbox of researchers working in formalizing mathematics. To our knowledge, the only work to date that attempts this is the *ImProver* [Ahuja et al., 2024] for Lean, which allows formal proof creation of specific length.

* **Interoperability** between different libraries of formal proofs (such as Lean's Mathlib library and Isabelle's AFP) is a long-term (but still largely unachieved<fnref n="7" />) goal. Currently, it takes intimate knowledge of the various formal libraries in order to assess which library, and ensuing ITP, would best be used to formalize a given mathematical theorem. A tool that can assess the suitability of a certain ITP and its library before starting the formalization process could make formalization more accessible.

**Recommendations.** We advise exploring more diverse mathematical interaction modes – across a variety of question and session types – as outlined here, both for formal as well as informal datasets. Our list is not exhaustive and can also be broken down to a more detailed level. In particular, oral, mathematical communication is not well mapped to data (see Section 4.2). Mathematical collaboration often involves an exchange of very high-level ideas at a rapid pace. Specific, wide-ranging recommendations can be found in Section 4. We believe capturing this in data will help to lead to more widespread mathematical assistants. Finally, we believe that input from research in mathematical education must be taken into account to derive curated datasets that exhibit richer interaction modes.

<footnote n="7">⁷A score of overlap by assessing how many theorems out of 100 theorems have been formalized in different ITPs is provided here: https://www.cs.ru.nl/~freek/100/</footnote>
### 3.1.4 Contamination

When releasing a dataset, control is lost over whether the dataset ends up as training data for machine learning models. For several of the state-of-the-art models, such as GPT-4 [OpenAI, 2023], that perform well on benchmarks, no information is available in the ensuing on whether efforts were made to ensure that training datasets were decontaminated and no datapoints from the evaluation benchmark were included in the training set. For Gemini 1.5 [Reid et al., 2024], Qwen2 [Yang et al., 2024a] and DeepSeekMath [Shao et al., 2024], only *n*-gram decontamination approaches have been tested. For complex datapoints, such as math, *n*-grams are unfortunately not always sufficient to ensure the training dataset is clean. There is evidence that suggests that several such modern models are already contaminated [Xu et al., 2024]. While mitigating approaches have been proposed, such as generating new data, either from scratch [Mishra et al., 2024] or by using existing datasets as seeds [Zhou et al., 2024], it is not clear whether these can scale to more sophisticated mathematics, as the methods were tested using mathematics on the level of grade-school, and it is unclear how well they would scale to much higher levels of mathematical abstractions and problem-solving difficulty (the two main metrics of difficulty, as outlined in 3.1.1).

**Recommendations.** If the dataset is sufficiently large, it is advisable to keep a part of the dataset hidden from the public so that the dataset creators can compare how well newly-released, publicly accessible open-weight models score in the public versus the hidden part of the dataset. Diverging scores on two splits of the dataset can indicate contamination. Care must be taken that datapoints from the hidden datasets are from the same distribution compared. For mathematics, this means at minimum: Same domain, difficulty range, and proof technique. This can be challenging: Given a specific problem *P*, it takes effort to source a new problem *P*\*, that on all relevant metrics (including the mentioned ones) is similar to *P*, but at the same time is not completely analogous to *P* – as it would be by, e.g., merely changing items in the problem statement that have a negligible on the proof (this is an approach followed by GSM8K-Symbolic dataset [Mirzadeh et al., 2024], which nonetheless turned out to be challenging for LLMs, highlighting their current limited reasoning capabilities). The approach advocated here has limitations for models released as-a-service [La Malfa et al., 2024], via APIs or GUIs, as running the hidden dataset through risks it automatically being used at a later stage as training data for those models.

## 3.2 Distinct Issues

This section will detail issues that specifically appear with either natural language or formal language datasets, or mixed datasets. The most prominent difficulty with natural language datasets is the lack of trusted automatic evaluation as will be explained in Section 3.2.1.

A natural way to circumvent the difficulty of evaluating natural language proofs is by making use of formal language datasets. Instead of having an LLM generate natural language mathematical proofs, one can train it to generate proofs in a verifiable formal language such as Lean or Isabelle.

Therefore, existing libraries of formal proofs can play a crucial role in the development of machine learning-based automatic theorem provers. Sections 3.2.2 and 3.2.3 will provide details on how some of the largest existing collections of formal proofs have been and are being created. This is followed by an analysis of how this process affects the use of formal proof libraries as evaluation datasets; with some remarks relevant to using formal libraries for training.

More concretely, Isabelle's Archive of Formal Proofs (AFP) and Lean's Mathlib will be investigated. This choice is due to the authors' familiarity with these libraries but many of the points mentioned below will also be valid (at least to some extent) for other systems.

One particular issue relevant to all datasets is their scale. The lack of a large-scale dataset consisting of pairs of formal and informal mathematics is a significant bottleneck for autoformalization. Current datasets, such as miniF2F are on the order of a few hundred datapoints, which are enough for assessing autoformalization, but not for training models to support autoformalization. A recent effort [Ying et al., 2024] scales this to
57k datapoints of pairs. Nonetheless, this scale is not yet comparable to the scale at which LLMs can be trained on informal mathematics.

### 3.2.1 Lack of Trusted Automatic Evaluation

The easiest way to automate the evaluation of mathematical prompts is to formulate them in a way such that the answer can be represented as a single token (e.g., a number or a mathematical term). Automatic evaluation can then be performed by keyword-matching the token to the gold-truth answer. A slightly more advanced variant is pursued by the MATH dataset, which allows a reasoning section, but encloses the final answer in a \boxed environment (and only this is keyword-matched).

No large machine learning dataset on mathematics currently exists that allows an arbitrary proof of an AI system to be checked for correctness against the gold-standard proof in the evaluation dataset. A proof-of-concept, highlighted by the IMO Small Challenge<fnref n="8" />, shows how partial automation might be achieved, where a necessary test for correctness is carried out using detailed proof annotation. This builds on the idea that a system like an LLM, combined with a deterministic form of matching, can at least exclude incorrect proof candidates. If the gold truth is sufficiently rich, the annotations help the LLM break down a proof and assist it in understanding its main features (accepting that subtler points of proofs may still be currently out of reach for LLM understanding).

Some success has been attained in using an LLM to grade (or teach) another LLM [Eldan and Li, 2023, Mukherjee et al., 2023]. Yet, for mathematics, we believe that this is not necessarily the right approach forward yet. The current generation of models do not have a sufficiently high performance on mathematics to be used as graders: for tasks that would often be deemed simpler than mathematics, this approach can fail [Wang et al., 2023], although for solving programming puzzles this approach was successful [Haluptzok et al., 2023]. Anecdotal evidence suggests that adapting the approach to proofs is challenging because of the multitude of potential proof variations for a single statement, which gives rise to a diverse set of textual expressions.

At present, grading can, at best, be automated by the use of detailed human annotation for each problem in combination with LLM assistance based on such annotation.

**Recommendations.** In general, we recommend developing evaluation methodologies that allow for natural language proofs to be checked for correctness. This is difficult in general. A middle ground is to include as much metadata as possible when annotating datasets with manual effort, with a view towards supporting LLMs in using this metadata to assess proof candidates. Second, we recommend more research that quantifies the degree to which LLMs can evaluate mathematics. This will help in making objective statements about how good or how poor LLMs are at evaluating different kinds of mathematics, and help in identifying areas for improvement.

### 3.2.2 Tool Misalignment

As writing formal proofs imposes slightly different challenges to writing informal proofs, it is important to distinguish mathematical proficiency from proficiency in using tools provided by the formal environment.

One cause of this discrepancy is that side conditions deemed trivial in conventional mathematical literature have to be formalized as diligently as the rest of the mathematical text. On the other hand, proving many statements considered routine by mathematicians can be mechanized.

For example, proving that a function is continuous often corresponds to choosing the correct subset of a few relevant lemmas and this can be achieved with a simple search algorithm, such as Lean's continuity tactic.

Next to this domain-specific formalization aid, there are also general-purpose automation tools that can be used to formalize certain simple statements. For example, Isabelle's Sledgehammer tool [Paulson and

<footnote n="8">⁸www.imo-small-challenge.io</footnote>
Susanto, 2007, Meng and Paulson, 2008] translates the given statement so that it can be understood by a number of external automatic theorem provers (ATPs). The ATPs, in turn, attempt to prove the statement and send the proof back to Isabelle.

**Automation and Proofs** Naturally, the existence of automation affects how the task of formalization is approached: Without automation, formalization would consist of translating a mathematical proof step-by-step into the ITP's logic. But having access to advanced tactics and general-purpose ATPs, one only needs to repeatedly break down the proof into smaller pieces until these can be tackled by automation.

Eventually, this can lead to proofs being formalized in a way that is not very close to the original natural language proof. For example, elementary homework-style problems often consist of long calculations that might correspond to one single tactic invocation in the formal system. Also, in more advanced topics, the automation of a theorem prover can make the proof less readable or skew its focus. For example, natural language proofs typically make it very clear which definitions need to be unfolded and when. In Lean, such unfolding does not have to be made explicit, which can widen the gap between formal and informal.

The impact of these mechanisms needs to be kept in mind when formal datasets such as Lean's Mathlib or Isabelle's AFP are used for LLM evaluation. In particular, it is likely that current datasets overly reward models for their ability to deploy the language-specific automation, rather than complete "understanding" of the underlying mathematics. As some evidence of this, Hu et al. [2024] show that LLM performance drops significantly when common automation tactics are turned off, even for theorems that have human-written proofs that do not rely on the automated tactics.

**Strong automation and the Archive of Formal Proofs (AFP)** These matters are especially significant with the Isabelle AFP because of the powerful general-purpose automation that has been widely used in its development over the last decade. It can therefore be expected that a large subset of theorems in today's AFP have been formalized by repeatedly breaking them down into smaller subproblems until Sledgehammer could find their proof.

A natural way to turn the AFP into a proof dataset is by splitting up all formal proofs into their individual steps [Jiang et al., 2021]. However, by the above characterization, such a dataset might be skewed towards being tractable for automation because of the particular way in which AFP theorems have come into existence. This can make it hard to assess the level of difficulty in the problem set.

Especially when equipping LLMs with the ability to access existing automation, careful evaluation is essential. It is important to determine whether the model genuinely “understands” complex mathematical concepts and can scale this understanding up to more difficult problems; the alternative being that the model merely excels at finding “low-hanging fruits”, which, when combined with non-ML automation, might appear deceptively impressive.

We would like to stress that we believe that the use of formal mathematics datasets such as the AFP is a valid evaluation tool. The results of recent publications [First et al., 2023, Jiang et al., 2022, Mikula et al., 2024] are especially promising, and we are looking forward to their continuation. Nevertheless, we believe that future evaluation should carefully consider the impact of existing automation in ITPs on their libraries.

**Recommendations.** The available formal mathematical libraries that include deep, research-level results have generally been created using a significant amount of automation. Researchers could create small, specific additional test sets of proofs that are formalized without automation and evaluate mathematical assistants on these.

### 3.2.3 Non-Trivial Data Duplication

Next to these aspects related to the nature of theorem provers, there are also challenges intrinsic to mathematics itself. In particular, mathematics has a self-similar structure: there are profound connections between
areas that are seemingly disconnected, and many mathematical statements have several related versions, depending on the perspective and level of generality chosen. This section will detail some of the inherent challenges this creates for the construction of formal mathematical libraries and then consider the impact of using them as datasets.

Ideally, a formal library should have as little duplication as possible to allow for a smoother user experience and to reduce the amount of code maintenance required. At the same time, this ideal can seldom be upheld fully since formalizing theorems only in their most general form is intractable. Some level of non-trivial duplication is therefore accepted and common in formalization.

As an example, Lean's Mathlib often provides several versions of a lemma,<fnref n="9" /> which makes it more convenient to use the library. Instead of having to "import" a lemma and then transform it into the right format, one can directly refer to commonly used variants. For example, the statement that

$$a + b + (c + d) = a + c + (b + d) \qquad (1)$$

in a commutative semigroup is explicitly formalized even though it could easily be derived from associativity and commutativity whenever needed. Furthermore, there is support for automatically generating lemmas, e.g., to generate the additive version of an abstract algebra statement given in terms of multiplication.

Next to this, the rules and customs of how a collection of formal proofs is built and maintained have a significant impact on its properties as a machine-learning dataset. Compared with Mathlib, the Isabelle AFP is more static and aims at being archival, i.e., providing long-lasting support for its content and not removing previously defined concepts. This makes duplication inevitable as soon as parts of the library go through a larger restructuring or even redevelopment. For example, there are two separate algebra libraries for Isabelle, one in the AFP and a second one using a different formalization approach in the Isabelle HOL-Library, which often gets used in conjunction with the AFP.

As a further example, consider the AFP section on category theory. Currently, there are at least five separate formalizations of category theory at various stages of development with all of them taking different approaches<fnref n="10" />. This is likely to be for historical reasons, but also because exploring different representations of category theory is an interesting topic in mathematical foundations.

Naturally, if duplication is present in the formal library, a lot of care becomes necessary when designing machine-learning datasets from it. If data is just randomly split into a training and validation set, then leakage can occur on one hand. Of course, the duplication will typically be on the level of mathematical concepts and not exist as a verbatim doubling of formal language code. Nevertheless, disregarding the issue of duplication makes it hard to gauge to what extent a model manages to produce formal proofs from the ground up. Possibly, parts of the model's performance simply stem from learning how to translate between different representations of the same mathematical concept. On the other hand, duplication brings to the forefront the issue of data representation, as machine learning models will only perform well on that representation on which they have been trained, which is in contrast to human mathematical reasoning, which, to a degree, is robust against different formal representations of the same mathematical objects.

Translation of formal representations constitutes a highly interesting and relevant skill in its own right. Nevertheless, good test design should enable researchers to judge to which degree the respective skills are achieved.

**Recommendations.** Following the discussion above, we recommend developing methodologies that ensure that train/test splits adequately measure a model's ability to generalize. One direction is to test on new formalizations that occur after a model has been trained, as was explored in Hu et al. [2024]. Nevertheless,

<footnote n="9">⁹We note that this facility is not exclusive to Lean; e.g., Isabelle provides several versions of a lemma.</footnote>

<footnote n="10">¹⁰cf. https://www.isa-afp.org/topics/mathematics/category-theory/ not counting generalizations of category theory and continuations of previous developments.</footnote>
care should be taken to ensure that the mathematics being formalized does not occur in previous projects. Hence, we particularly recommend evaluating on domains that have not previously been formalized.

More broadly, the ideal evaluation of digital mathematical assistants should include case studies in which previously unformalized proofs get formalized using the assistant. There are various ways to realize this. On a small scale, researchers can qualitatively evaluate the LLM by formalizing some reasonably difficult, previously unformalized proof. On a larger scale, the assistant can be provided as a plugin to theorem provers, which a large number of people working in formalization can then use in real-world scenarios. If there are enough users, such a setup could even use A/B testing for multiple architectures of mathematical assistants.

# 4 The Future – Novel Datasets to Support Mathematical Copilots

In this section, we cover various aspects of mathematical research practice that, to date, are not covered at all by any of the existing datasets. This contrasts with the previous section on mathematical datasets that covered parts of mathematical practice, albeit with the noted limitations. Current datasets exclusively are focussed on publishing the results rather than the intermediate steps that mathematician goes through in the process of devising a proof. The different types of workflows, the proof-counterexample dialectic, and proof transfer,

etc., are all examples of intermediate processes that do not have clear representations in data that can be used to fine-tune LLMs. Contrasting with the previous section, where we made specific recommendations on how to improve existing datasets, in this section, we do not make specific recommendations, as our recommendation is to simply devise such datasets. While a few inroads have been made into non-proof-based datasets, see Section 3.1.3 these are still very much at their inception, and, to our knowledge, the GHOSTS dataset is the only attempt to date to try to “data-ify” some aspects (solely related to mathematical literature search) of daily mathematical process.

## 4.1 Mapping Mathematical Workflows to Data

The previous sections have shown that existing datasets and benchmarks only deal with specific mathematical tasks. In particular, they focus mostly on question answering and theorem proving, with solutions being presented in a streamlined way (as is typical for results in mathematical textbooks). In consequence, they do not sufficiently cover all steps in typical mathematical workflows, such as surveying mathematical topics, gathering related results, establishing high-level proof strategies and intuition, refactoring proofs, or carrying out field-specific routines. Yet, we would like to evaluate and train mathematical copilots on these tasks, too, in order to obtain full-spectrum assistance across all facets of mathematical research practice. We note that datasets used to pre-train LLMs exhibit broader coverage, e.g., of mathematics-related Q&A communities, blogs, and educational material; however, it is hard to assess and control the quality of the included mathematical content. This results in a weak learning signal and, for a strong mathematical copilot, one likely requires fine-tuning on a comprehensive, high-quality dataset of mathematical workflow steps.

We advocate a more bottom-up approach that consists of creating a taxonomy of workflows, i.e., isolating and categorizing workflow steps. Specifically, many mathematical workflows can be modularized into smaller sequential steps, which themselves can be ordered along different dimensions, e.g., their mathematical depth, level of abstraction (from general, high-level proof techniques and principles to specialized, problem-specific approaches), mathematical subject, how well a symbolic approach might handle them, whether the workflow step is strictly mathematical or meta-mathematical (e.g., literature search). With this in view, we distinguish between general *global* workflows, which are used across mathematical fields, and field-specific *local* workflows. For instance, we refer to the *Tricki*<fnref n="11" /> for a collection of problem-solving techniques that can be viewed as abstract workflows. General proof techniques, such as linearization and fixed point theory, can be viewed as

<footnote n="11">¹¹https://www.tricki.org</footnote>
global workflows. However, we note that specific instantiations of such techniques, e.g., related to dynamical
systems, can also be classified as local workflows. Examples of even more local workflows include approaches
tailored to specific kinds of partial differential equations, e.g., the method of characteristics, maximum
principle, energy estimates, or Green's functions, as these methods frequently occur as individual steps in
larger chains of arguments in research-level mathematics.

There is no hard boundary for transitioning between local and global workflows. For example, the “routine”
task in analysis to upper bound an expression efficiently, as opposed to evaluating it exactly, is found
both in elementary contexts, such as proving inequalities such as versions of arithmetic mean-geometric
mean inequality, up to more advanced cases, such as for ordinary differential equations (ODEs) or partial
differential equations (PDEs), where a routine task is that of applying an “energy method” or “Lyapunov
functional method”. We, therefore, deliberately do not draw a specific boundary of when to categorize a
workflow step as local vs. global.

While creating a complete taxonomy of workflow steps is beyond the scope of this article, we want to examine
selected workflows and their steps (both local and global) in order to highlight issues in translating these to
data and specific challenges. We note that these are merely illustrative examples meant to show how first
steps could be undertaken to generate datasets that support such workflows.

While a workflow (indicated with arrows below) can, in principle, easily be mapped to data by converting
it to *n* tuples of datapoints (where *n* denotes the number of workflow steps) and filling text in between, in
practice, in certain cases, issues may arise. We give examples of both global and local workflows below and
highlight both examples where these can be transformed to data in a straightforward manner, as well as
cases in which this transformation is more difficult. We urge the community to explore such translation of
workflows to data further.

**Global workflows** Literature search is one of the most general workflows across the sciences, having par-
ticularities in how it takes place in mathematics. Prior work can be related to the problem a mathematician
has at hand in many ways. One might, for instance, want to know what is known about a particular math-
ematical object at hand. The context of a problem might have many options for what to look for, so the
first step in the workflow might be to identify an object that has a good chance of having been seen before.
One example is integer sequences: perhaps the solutions to a combinatorial problem are computable in small
cases, and it is often useful to know if the resulting sequence has been encountered in the context of other
problems (there might be many ways to get a sequence: for instance, we might have to choose variables to
fix and then only vary one parameter of the problem). For this particular case, the Online Encyclopedia of
Integer Sequences¹² is a well-known, efficient, specialized tool, with many pointers to the literature, and the
ability to recognize partial matches (e.g., perhaps one's sequence is coarser than the one from relevant prior
work). In case there are no matches, we can try to repeat the process with other candidate objects; in case
there are, the challenge turns to assess whether any of the results seem meaningfully related to the current
case (and given only a few integers, there are often too many sequences containing them). For instance, the
sequence 1, 4, 44 appears in path-counting problems in some lattices, or in a few number-theoretic contexts
(like products of odd-indexed Lucas numbers), as well as many other contexts, and more context that was not
included during the search might be relevant in identifying which of these sequences might be meaningfully
related to the current problem. We could broadly summarize this workflow as:

*Identify objects to search for → find related work describing the object → assess potential relation to current context*

We might expect LLMs to help generalize this process beyond what specialized tools, such as the OEIS,
are capable of. Most mathematical objects are significantly harder to describe than integer sequences: for
instance, one might have a particular topology on a function space that might be unique to the current
context, but something of the same “shape” might have been seen before. The assistant would have to

<footnote>12https://oeis.org/</footnote>
recognize and know how to describe this shape, and identify matches across potential differences in what definitions are implicitly or explicitly used, as well as potential equivalent definitions that might cause a description in the literature to differ on the surface (e.g., mention “accumulation points” vs “limit point”).

Examples of this workflow for training and evaluation might be automatically extractable in a post-hoc fashion from the mathematical literature itself: one might be able to look at how previous work is referenced in existing proofs as a source of examples of relevant previous work (other matches for the same object that are not the paper that was cited are most likely examples of spurious matches).

**Local workflows** We list below examples from two subfields of mathematics (differential equations and knot theory/low-dimensional topology).

* Suppose, for instance, one has a first-order ODE $\partial_t u = F(u)$ with some initial data $u(0) = u_0$ and wants to know how the solution grows in time. A standard technique is to introduce a key functional $E(u)$ of the solution (often something like an “energy” or “Lyapunov functional”) and then compute the derivative $\partial_t E(u)$ using the ODE and the chain rule (for PDEs, one often has to also perform several times integration by parts). Then, one bounds this derivative as best one can.

The goal is to reach some differential inequality of Gronwall type, e.g., $\partial_t E(u) \le CE(u)$, so that a Gronwall-type lemma may be applied (but it is not always precisely the Gronwall lemma, but something similar).

Summarizing, the workflow is thus of the form:

ODE/PDE → find functional → compute derivative of functional → bound derivative → reach
Gronwall-type inequality

We note that individual steps of this may be solved using symbolic-numeric methods, which opens the possibility of using a tool-integrated-reasoning (TIR) approach to enable an LLM to discharge these proof steps to symbolic or numeric tools to, e.g., compute derivatives symbolically. Thus, one option to encapsulate these workflow steps is in datapoints whose core form is

(ODE/PDE, functional, functional derivative, derivative bound, Gronwall-type inequality)

and where there is filler text between the raw mathematical objects (i.e., the ODE, the functional, etc.). The raw mathematical objects should be symbolically or numerically generated so that custom filler text can be added in between them, and an LLM can observe a range of ways in which to speak and textually connect a fixed tuple of mathematical objects.

What one typically wants, as a mathematician, in the case of this workflow, is to try various guesses for the functional. If a TIR approach is used, this data representation of this workflow should specifically offer “entry points” to make it easy for an LLM to accept arbitrary input at certain places and use tools to automatically run the essential parts of the workflow. An LLM is then much better equipped to run the workflow steps and produce an estimate, either automatically or semi-automatically. One can preliminarily do this already through conversation with current state-of-the-art LLMs, but with a lot of mistakes on the LLM’s part, which a stronger grounding in data would help to correct.

We now consider typical workflows in low-dimensional topology, which is an area of mathematics that has a unique flavor and uses tools from geometry, algebra, PDEs, group theory, combinatorics, and mathematical physics. Hence, some of these workflows are more challenging to convert to data. Low-dimensional topology is the study of $n$-manifolds (certain topological spaces that are locally homeomorphic to $\mathbb{R}^n$) of dimension $n \le 4$. A knot is a simple closed curve embedded in $\mathbb{R}^3$. Knot theory plays a fundamental role in low-dimensional topology as every 3- and 4-manifold can be represented by a framed link; i.e., a collection of knots labeled by integers, called a Kirby diagram [Kirby, 1978]. We refer the reader to Juhász [2023] for more detail on low-dimensional topology and knot theory and explanations of the mathematical terms used below.
* A central type of question in this area is classification, which requires being able to show whether two objects (e.g., knots or manifolds) are equivalent. The first step is finding a suitable representation. In case of knots, this could be a projection to the plane, called a knot diagram, a closure of a braid (a number of parallel strands running around a central axis), or a grid diagram. Knot diagrams can be encoded numerically as PD, DT, or Gauss codes, and braids as braid words. Manifolds can be represented using Kirby diagrams, as triangulations, branched covers along links, or as geometric objects (e.g., hyperbolic 3-manifolds).

To show that two representations of the same knot or manifold are equivalent, one finds a sequence of certain moves connecting them, which can be Reidemeister moves [Reidemeister, 1927] in case of knot diagrams, Markov moves for braids, or Kirby moves in case of Kirby diagrams. These search problems lend themselves to techniques such as reinforcement learning; see [Gukov et al., 2021]. The workflow steps can be encoded by a sequence of representations of the object, such that consecutive representations are related by one of the standard moves. Software packages such as SnapPy [Culler et al., 2024] can be used to check whether these moves are valid and can also list valid moves. We can hence represent the workflow as follows:

* choose type of representation and set of moves → find representations of two mathematical objects → repeatedly apply moves to first object → representation of second object.

* To show two objects are inequivalent, one defines invariants, which are typically algebraic objects (numbers, polynomials, or groups) that are unchanged by the above moves. These moves often also capture important topological properties of these objects. The most classical invariants are homology groups and the fundamental group. The Alexander polynomial is a knot invariant derived from the fundamental group of the knot complement. Representations of the knot group give rise to the more modern twisted Alexander polynomials. A recent knot invariant rooted in representation theory is Khovanov homology [Khovanov, 2000]. Floer homology [Ozsváth and Szabó, 2004] and gauge theory [Witten, 1994] give rise to highly sophisticated knot and 3- and 4-manifold invariants. These invariants often give lower bounds on hard-to-compute topological quantities such as the 3- or 4-genus of a knot. SnapPy is capable of computing many of these knot invariants when run in SageMath [The Sage Developers, 2024]. A typical workflow would thus be as follows:

* pair of mathematical objects → choose suitable invariant → compute invariant for objects → show invariants are inequivalent.

* To define a Floer-theoretic invariant, we construct a chain complex whose boundary map counts some pseudo-holomorphic curves with certain Lagrangian boundary conditions in some symplectic manifold. To show these counts are finite, one has to prove transversality results for the moduli spaces using difficult methods from PDEs, then obtain a formula for the dimension of the moduli spaces and compactify the moduli spaces. The next step is showing the boundary map squares to zero, in which case we have a chain complex, or sometimes we end up with a more complicated algebraic structure, such as an $A^\infty$-module or a differential graded algebra. One often has to use special coefficient systems, such as a Novikov ring, and deal with bubbling phenomena. There are many choices that go into the construction of our chain complexes, so one has to prove independence of these choices up to chain homotopy equivalence, which often relies on continuation maps and pseudo-holomorphic polygon counts. One then extracts more tractable invariants from the chain homotopy type using algebra, such as taking homology. Different invariants are often related by spectral sequences, defined using filtrations of the chain complexes. A similar workflow exists for gauge-theoretic invariants. More sophisticated invariants can be obtained by exploiting certain symmetries, such as $\mathbb{Z}_2$ or $\mathrm{Pin}(2)$. In another popular direction, one can sometimes construct a homotopy type from chain complexes; see [Lipshitz and Sarkar, 2018]. Certain constructions from algebraic topology can then be invoked to provide additional algebraic structure, such as Steenrod operations. We can hence represent the high-level workflow for defining a Floer-theoretic topological invariant as follows:
associate a symplectic manifold and Lagrangian submanifolds to our topological object → decide what pseudo-holomorphic curves to count → prove transversality of the moduli spaces → find dimension formula for moduli spaces → compactify moduli spaces → choose coefficients and the right algebraic structure → extract invariant using algebra → prove independence of choices.

It is apparent from the complexity of the workflow that this is very difficult to turn into data. As the number of workflow steps increase, the number of datapoints has to increase exponentially to cover all possible combinations between steps (although not all combinations may be possible in all cases, which may mitigate this issue).

* After an invariant is defined, we study what topological information it captures and derive applications. Here, it is important to know what applications are mathematically relevant. This workflow is very different from many other areas of mathematics, where tools are developed to tackle specific open problems, and should be compared with Section 4.3 on motivated proofs. This step requires sophisticated intuition and a global vision. One way to turn this into data is to define pairs

(invariant, topological application),

which could be used by a mathematical copilot to recommend potential applications of a given type of invariant. We want to choose applications that are related to problems from a list of important open questions.

* As many of the invariants are hard to compute, it is crucial to develop methods to compute them. For Floer-theoretic 3- and 4-manifold invariants, for example, experts use surgery formulas, spectral sequences, grading arguments, exact triangles, gluing formulas, or computations in explicit diagrams in simpler cases. The difficulty lies in the fact that one has to solve non-linear PDEs to obtain the pseudo-holomorphic curve counts contributing to the boundary maps in the chain complex, which is not algorithmic. Knot Floer homology now has a combinatorial definition and can be computed by SnapPy. It is often helpful to restrict attention to special classes of objects, such as to the class of alternating knots, 3-braid closures, etc., and perform computations or prove results for these. A schematic of this workflow is the following:

invariant → method of computation → class of objects to restrict to.

Hence, a potential data point could look like

(invariant, method of computation, class of objects).

Certain knot invariants are easy to define, but no algorithm is known to compute them, such as the unknotting number or the 4-ball genus, and topologists use computable invariants, many of them arising from knot Floer homology, to give lower bounds on these. The paper [Davies et al., 2021] describes a workflow for using supervised learning to find correlations between various invariants, which has led to an inequality relating the knot signature and hyperbolic invariants. Upper bounds can be obtained by performing certain moves on knot diagrams, possibly with the help of reinforcement learning or Bayesian optimization; see [Gukov et al., 2023] and [Applebaum et al., 2024]. For example, one could take (4-ball genus, lower bound from Rasmussen s-invariant, torus knots), which leads to a solution of Milnor's conjecture on the 4-ball genus of torus knots [Rasmussen, 2010].

* The classification of smooth 4-manifolds is still wide open. There exist 4-manifolds that are homeomorphic but not diffeomorphic, which are called exotic pairs. To construct these, experts use methods from algebraic and symplectic geometry, such as blow-ups, fiber sums when given Lefschetz fibrations, and other gluings, typically along 3-tori, such as knot surgery. To show two 4-manifolds are homeomorphic, one shows they have isomorphic intersection forms and invokes Freedman's theorem [Freedman, 1982], for which the fundamental group has to satisfy certain restrictions (e.g., being trivial). The intersection form and a presentation of the fundamental group can be read off a Kirby diagram. This fundamental group computation is often the most difficult step, and note that any finitely presented
group can arise as the fundamental group of a closed 4-manifold. Furthermore, there is no algorithm to decide whether a finitely presented group is trivial. So, some questions in low-dimensional topology are beyond the reach of computers. As mentioned above, one can use Kirby calculus to show if two 4-manifolds are diffeomorphic. Swenton's Knot-Like Objects software<fnref n="13" /> is capable of performing Kirby moves. To show they are not diffeomorphic, one usually distinguishes them using the gauge-theoretic Seiberg-Witten invariants. No algorithm is known to compute these, but they can be computed in some instances using Taubes' non-vanishing result for symplectic 4-manifolds [Taubes, 1994], together with various glueing results, such as Fintushel and Stern's knot surgery formula [Fintushel and Stern, 1998]. A schematic of a typical workflow in smooth 4-manifold topology is as follows:

*choose method of construction* → *construct pair of smooth 4-manifolds* → *find their Kirby diagrams*
→ *show they are simply-connected* → *show they have isomorphic intersection forms (hence*
*homeomorphic by Freedman)* → *try to show they are diffeomorphic using Kirby calculus; if this fails*
→ *compute Seiberg-Witten invariants to show not diffeomorphic.*

The last step, the computation of Seiberg-Witten invariants, is discussed in more detail in the previous bullet point. Computer algebra systems can be useful for the computation of the fundamental group and the intersection form.

3-manifold topology has a completely different flavour. Here, methods from geometric group theory, hyperbolic geometry, and combinatorial topology dominate. These combinatorial methods include triangulations and normal surface theory. Geometric group theory studies groups using their actions on metric spaces (e.g., on the Cayley graph). Also note Perelman's proof of the 3-dimensional Poincaré conjecture using the Ricci flow, rooted purely in geometric analysis; see [Morgan and Tian, 2007]. Well-developed computer packages exist to aid 3-manifold topologists, including the already mentioned SnapPy for hyperbolic 3-manifolds and Regina [Burton, 2004] for triangulations. We do not provide specific workflows in 3-manifold topology due to the diversity of techniques used.

**Further considerations.** An important aspect that needs to be considered when mapping mathematical workflows to data is that different representations of the same data can lead to different mathematical properties that can change the method of proof, as well as the foundational model most suited for the chosen representation.

For example, it is an open problem in knot theory whether there exists a polynomial-time algorithm that can detect whether a given knot, which is an embedded circle in $\mathbb{R}^3$, can be simplified (without breaking it) to a standard round circle. Instead of describing the knot as a curve in $\mathbb{R}^3$, knots are often specified in terms of their projection to two dimensions, and data representations of the knots keep track of which strands are above and which are below along the projection ray.

The above-mentioned question then becomes to find a sequence of deformations of the embedded circle such that its projection has no crossings. It is known [Kauffman and Lambropoulou, 2012] that for some data representations of the projected knot (such as Dowker-Thistlethwaite codes [Dowker and Thistlethwaite, 1983] with Reidemeister moves as allowed deformations [Reidemeister, 1927]), there exist examples where the number of crossings needs to be increased before all crossings can be removed. In another representation (grid representation with Dynnikov moves as allowed deformations), the number of crossings is monotonically decreasing [Dynnikov, 2006], but both the representation of the knot as a grid diagram and the carrying out of the simplification steps are more complicated. These distinct representations of the same mathematical object lend themselves to different foundational models [Gukov et al., 2021, Kauffman et al., 2022]; for example, braid word representations of knot projections are closer to natural language, whereas other representations are closer to vision tasks or graphs. The different representations also inform the type of algorithm that needs to be used. If the simplification is non-monotonic, a local search can get stuck in local minima.

<footnote n="13">¹³https://community.middlebury.edu/~mathanimations/klo/</footnote>
One common activity in mathematics, when confronted with a new statement of unknown truth, a conjecture, is to engage in the search for proof – or a counterexample. This comes with its own workflows. The search for counterexamples to conjectures is particularly useful for hard problems with a truth certificate that can be verified in polynomial time, such as NP-hard or NP-complete problems. The idea is to cast the search problem into a Markov Decision Problem (MDP) whose terminal states are counterexamples, and attempt to solve the MDP using data science techniques. In the past, deep reinforcement learning has proven to be a powerful tool. If the RL agent finds a solution to the MDP, its episodic rollouts provide truth certificates for the counterexample, thus establishing a verifiably correct proof by counter-example, see e.g. [Gukov et al., 2024] for a recent summary of these ideas and [Gukov et al., 2021, Wagner, 2021, Gukov et al., 2023, Charton et al., 2024] for some recent concrete applications.

## 4.2 Data Collection in Real Environments

To gather representative data on mathematical workflow steps, as well as other metamathematical items, such as proof heuristics, limitations of certain proof techniques, etc., it may be desirable to observe and absorb the full process of producing mathematics, including all data that is not. For an example of real-time narration of thought processes that arise when doing competitive problem-solving, see co-author Tim Gowers' series of YouTube videos.<fnref n="14" />

Unfortunately, many of these intermediate process steps become evident only implicitly from data sources. For example, at the elementary level, various workflows to solve various limits,

which require repeated applications of known theorems, such as de l'Hôpital's theorem, or related ones to resolve indeterminate cases, are distilled only by solving a large number of exercises; no explicit annotations for these workflows exist. On a more advanced level, these are often conveyed in blogs, talks, and oral discussions between mathematicians at conferences – and not in the typical data sources used to train LLMs.

In principle, one could implement longitudinal studies that track research projects from inception to completion. However, such data collection would require classifying, structuring, and recording different tasks throughout the mathematician's daily work. The apparent challenge is to scale such approaches while keeping the overhead as small as possible for the researcher whose work is tracked.

While such approaches are welcome to be tested, as a first alternative step, we recommend transcribing from online sources such as lectures, panel discussions, seminars, vlogs, etc. While such data only requires minimal need for postprocessing and is often already collected in the form of videos, it typically covers broader mathematical aspects than textbooks. However, while humans can frequently generalize from a single instance of a certain workflow, LLMs often require a dataset that contains sufficiently many examples. While data collection effort will elicit higher-quality mathematical data, the time lag between the time when new research discoveries are made and when these are represented in lectures can be significant. On the other hand, at venues such as conferences, a lot of information about mathematics is exchanged, but not recorded. While it would technically be possible to record a portion of the conversations that happen at a conference in a privacy-preserving manner, and this would be a great source of data, as it one would consistently have data that is at the forefront of research (unlike the mentioned panel discussions, where there is often a time lag), it would require a paradigm shift from conference participants to accept such data collecting measures. We urge further conversations about privacy-preserving ways of curating richer datasets on mathematical workflows, that minimally impinge on mathematicians' naturalistic practices.

## 4.3 Motivated Proofs

We do not doubt that there is substantial value in bespoke AI tools that can automate specific elements of a mathematician's workflow. However, if the goal is not only to provide correct proofs but also to enhance a

<footnote n="14">¹⁴https://www.youtube.com/@TimothyGowers0/videos</footnote>
user's mathematical understanding of the results [Zhang et al., 2023a] and guide new discoveries, we need new tooling – and we argue, new kinds of data. In particular, we argue for data which faithfully represents the process of proof discovery instead of proof exposition. Towards this goal, we introduce motivated proofs [Pólya, 1949, Morris, 2019], which contain and make transparent more of the proof discovery process. We argue for their value as a standard for LLM evaluation. We then provide some examples and discuss some preliminary observations about current LLM's ability to construct and identify motivated proofs. Finally, we speculate on a path towards large-scale evaluation of a model's ability to produce motivated proofs.

Broadly speaking, a motivated proof is one that makes clear to the reader where each step comes from. For example, many interesting proofs require one to find a mathematical object with certain properties. An unmotivated proof will simply specify the object and check that it has the desired properties, while a motivated proof will explain how to arrive at the object.

### 4.3.1 Examples of motivated proofs

We present two theorems here with examples of motivated proofs. Further examples can be found in Appendix A.

**Cantor's theorem** Cantor's theorem states that there is no surjection from a set $X$ to its power set $P(X)$. The proof proceeds by letting $f: X \to P(X)$ be a function and trying to find a subset of $X$ that is not in the image of $f$. From here, an unmotivated proof will simply exhibit a set, namely, $\{x \in X : x \notin f(x)\}$, and verify that it is not in the image of $f$, which turns out to be straightforward.

By contrast, a motivated proof will systematically search for the required subset of $X$. Not knowing which subset to take, we can treat the subset as an unknown, just as we do when solving an equation, and try to narrow down the possibilities. The most general subset of $X$ can be expressed as $\{x \in X : P(x)\}$ for some as yet unspecified property $P$. We now want to prove, for an arbitrary element $y$ of $X$, that $f(y) \neq \{x \in X : P(x)\}$. So we need either an element $x$ of $f(y)$ such that $\neg P(x)$ or an element $x$ of the complement of $f(y)$ such that $P(x)$. There are not many elements around, so trying $y$ is one of the first things to do, and then we find that we need either $y \in f(y)$ and $\neg P(y)$ or $y \notin f(y)$ and $P(y)$. And now the property $y \notin f(y)$ is forced on us as our choice of $P$, and we end up with the same set as before, but this time with its origin explained.

**Nilpotent units** Now we consider an early result from commutative algebra. Let $R$ be a commutative ring, and let $x \in R$ be nilpotent, then $(1+x)$ is a unit, meaning it has a multiplicative inverse. This is typically proved by naming an element $y = \sum_{k=0}^{r-1} (-1)^k \cdot x^k$ and verifying that this is a multiplicative inverse element by calculation. We regard this as unmotivated, since the discovery process of finding $y$ was left out.

For a motivated proof, we need to search for an inverse element. Not knowing which element to take, we parametrize the most generic element we can. Since the only known elements of $R$ are $1$ and $x$, the most generic element is an integer polynomial in $x$, $\sum_{k=0}^{m} a_k x^k$. For this to be a right inverse of $1+x$, we must have that $1 = (1+x)(\sum_{k=0}^{m} a_k x^k) = a_0 + \sum_{k=1}^{m} (a_k + a_{k-1}) x^k + a_m x^{m+1}$. For this to hold, we must eliminate all coefficients of the polynomial besides the constant term, which should be 1. This gives us that $a_0 = 1$, $a_k = a_{k-1}$ for $1 \le k \le m$ and $a_m = 0$. The first two equations give us that $a_k = (-1)^k$ for all $k$, but this contradicts the last equation.

But we also know that $x$ is nilpotent, so if we have $r$ such that $x^r = 0$, then all coefficients from $x^r$ onwards can be ignored. This solves our issue, as we can let $m = r - 1$ to remove the $a_m = 0$ condition, and we are left with $a_k = (-1)^k$, so $(1+x)$ is a unit with inverse $\sum_{k=0}^{r-1} (-1)^k \cdot x^k$.

### 4.3.2 Motivated proofs as an evaluation metric

The standard for a proof's acceptance into the mathematical literature has historically been correctness rather than a completely motivated account. Moreover, mathematicians have been incentivized to condense
and refine their proofs for reasons such as page limits in journals, which has further widened the gap between *proof discovery* and *proof exposition*. To assess a model's capabilities in aiding proof discovery, we suggest that holding the proof to the standard of being motivated provides advantages over requiring only correctness.

As argued in [Pólya, 1949, Morris, 2019], a motivated proof is more informative to the reader, as it provides more insight into how the proof was discovered and how the reader might discover similar results themselves. More speculatively, a model capable of producing motivated proofs will likely generalize better to novel problems. The first reason for this conjecture is that most results in the literature are not fully motivated, and a model that is capable of recognising this is less likely to blindly recall its training data. Requiring proofs to be motivated also restricts the amount of brute-force search a model can perform, so performance depends more on reasoning ability and less on test-time computational power.

We will now briefly evaluate current LLMs' ability to generate motivated proofs, as well as to evaluate whether a given proof is motivated.

**Generation of motivated proofs** As a preliminary investigation, we prompted o1-review and Llama 3.2-90B to produce motivated proofs of five mathematical results. Our goal was not to use best-in-class LLMs since they typically differ only by a few percentages on common benchmarks; in contrast, we wanted to obtain a preliminary assessment of whether LLMs have the ability to work with the concept of a motivated proof at all. The results on which we tested the abilities to generate motivated proofs were selected to be elementary but to require a slightly non-trivial construction. After defining motivated proofs, we provided an in-context example of a motivated proof of the other four results and asked the model to generate a new motivated proof. If the model failed to provide a motivated proof, we would prompt it once to fix it, pointing out the unmotivated step. Each model was asked twice per problem, first with only examples of motivated proofs in the context, following a traditional chain-of-thought approach, then again with both examples and non-examples, mimicking contrasting chain-of-thought prompting. The problems and associated proofs we used in the prompts can be found in Appendix A.

In most responses, the model showed some evidence of “understanding” the concept of motivated proofs by avoiding unmotivated constructions. However, this often led to pseudo-motivations, where some justification was given before a construction, but it didn’t provide any useful substance besides obscuring the unmotivated step (and often appealed to some irrelevant fallacies). In the example below, the unmotivated step of Cantor’s theorem is relegated to the statement “An important question arises: Does $x$ belong to its own image $f(x)$?”.

**o1-review (excerpt)**

Our goal is to find a subset of $X$ that cannot possibly be the image of any element under $f$. To do this, we'll consider how elements of $X$ relate to their images under $f$.

**Exploring the Relationship Between Elements and Their Images:**

For each element $x \in X$, $f(x)$ is a subset of $X$. An important question arises: **Does $x$ belong to its own image $f(x)$?**

This leads us to categorize the elements of $X$ into two groups:

1. Elements that are **members** of their image: $x \in f(x)$.

2. Elements that are **not members** of their image: $x \notin f(x)$.

The other common issue was the model making completely unsound deductions when trying to motivate their construction, only to arrive at the correct result by chance. This behavior is consistent with the model not fully understanding the result but having seen it in their training data and attempting post-hoc rationalization. Below is an example:
**Llama-3.2 (excerpt)**

We want to find a subset $Y \subseteq X$ such that $\phi(x) = Y$ for some $x \in X$. However, we don't know which subset to choose.

Let's try to find a property that $Y$ should satisfy. Since $\phi(x) = Y$, we know that $x \in \phi(x) = Y$. But we also know that $x \notin \phi(x)$, because if $x \in \phi(x)$, then $\phi(x)$ would contain at least two elements: $x$ and $\phi(x)$. This suggests that $Y$ should be a subset of $X$ that contains some elements of $X$, but not all of them. In particular, $Y$ should not contain $x$ itself.

With this in mind, let's define $Y = \{x \in X \mid x \notin \phi(x)\}$.

Here, the contradictory claims that $x \in \phi(x)$ and $x \notin \phi(x)$ have no basis, and if true would imply an immediate contradiction, making the remainder of the proof redundant. Also, the condition for $Y$ is in fact the opposite of what is needed, but this error doesn't seem to affect the remainder of the model's reasoning.

Other problems provided more hopeful examples: a motivated proof for the large totients problem was successfully generated both times by o1-review and once by Llama.

Overall, o1-review seemed to outperform Llama on most problems. We also observed a slight improvement by using negative as well as positive examples to “explain” the concept. As an unfortunate side effect, the extra information also increased the tendency to “obscure” unmotivated arguments as discussed above.

**Recognition of motivated proofs** To conduct a larger scale investigation of a model's ability to generate motivated proofs, it would help if there was an existing “judge” model which could determine whether or not a given proof was motivated. In any case, recognition of motivated proofs represents an easier sub-goal towards generating motivated proofs.

We performed a small, preliminary investigation of whether current state-of-the-art models can reliably distinguish a motivated proof from an unmotivated proof. We stress that this is only a preliminary investigation and do not claim to draw any strong conclusion, but hope that this will inspire the community to carry out larger evaluations in this regard and work towards a standard of evaluating motivated proofs.

We first tested the model's ability to identify whether a given proof was motivated. We tested the five problems from the previous section, where each problem had three proofs to be evaluated individually. This included two human-written proofs, which we judged to be paradigmatic examples of motivated and unmotivated proofs, one unmotivated machine-generated proof and, where available, one motivated machine-generated proof. The unmotivated machine-generated proof was the first correct but unmotivated proof generated by Llama-3.2 in the previous section. For the machine-generated motivated proof, we took the response in the previous section that we judged to be most motivated, which was generated by Llama in the integer sum problem and o1-review for nilpotent units and large totients, while for the other problems no responses were deemed sufficiently motivated.

For some proofs where there were incorrect but largely inconsequential components, we manually altered some equations to enforce correctness and removed any opening or closing sentences that included the phrase “motivated proof” to avoid misleading the model. As with generation, we prompted the model by explaining the definition of a motivated proof and giving an example and non-example for each of the other problems, where the order of examples and non-examples was alternated to ensure that the model wasn't learning the ordering. Each model was given three attempts per proof per problem, and the results are recorded below. We use HM, HU, MM and MU as abbreviations for “human motivated”, “human unmotivated” and “machine unmotivated” respectively.
<table><thead><tr><th rowspan="2"></th><th colspan="4">o1aroo</th><th colspan="4">Llama-3.2</th><th colspan="2">Total</th></tr><tr><th>HM</th><th>HU</th><th>MM</th><th>MU</th><th>HM</th><th>HU</th><th>MM</th><th>MU</th><th>TP</th><th>TN</th></tr></thead><tbody><tr><td>Cantor's theorem</td><td>3/3</td><td>3/3</td><td>-</td><td>3/3</td><td>3/3</td><td>0/3</td><td>-</td><td>0/3</td><td>6/6</td><td>6/12</td></tr><tr><td>Small doubling</td><td>3/3</td><td>3/3</td><td>-</td><td>2/3</td><td>3/3</td><td>3/3</td><td>-</td><td>2/3</td><td>6/6</td><td>10/12</td></tr><tr><td>Integer sum</td><td>3/3</td><td>3/3</td><td>3/3</td><td>2/3</td><td>2/3</td><td>3/3</td><td>3/3</td><td>0/3</td><td>11/12</td><td>8/12</td></tr><tr><td>Nilpotent units</td><td>3/3</td><td>3/3</td><td>3/3</td><td>3/3</td><td>2/3</td><td>3/3</td><td>3/3</td><td>0/3</td><td>11/12</td><td>6/12</td></tr><tr><td>Large totients</td><td>3/3</td><td>3/3</td><td>3/3</td><td>3/3</td><td>3/3</td><td>2/3</td><td>3/3</td><td>1/3</td><td>12/12</td><td>9/12</td></tr></tbody><tfoot><tr><td>Total</td><td>15/15</td><td>15/15</td><td>9/9</td><td>13/15</td><td>13/15</td><td>8/15</td><td>9/9</td><td>3/15</td><td>46/48</td><td>39/60</td></tr></tfoot></table>

Under this setup, o1aroo performed strongly, while Llama was heavily biased towards accepting a proof as motivated. A possible explanation for Llama's poor performance in the MU set is that the machine-generated unmotivated proofs, unlike the human-generated ones, were attempts at generating motivated proofs, so these proofs contained some superficial features such as proof length and “chatty” language that the model had associated with motivated proofs.

In an effort to address this issue, we re-framed the experiment as a binary choice task. This was largely the same as above, but instead of being given a single proof to judge, the models were given two proofs and were told that one was motivated and one was unmotivated, and to judge which was motivated. Again we recorded the success rate over three valid attempts per (unmotivated,motivated) pair, per problem. The results are shown below.

<table><thead><tr><th rowspan="2">Unmotivated proof source</th><th colspan="2">o1aroo</th><th colspan="2">Llama-3.2</th><th rowspan="2">Total</th></tr><tr><th>Human</th><th>Model</th><th>Human</th><th>Model</th></tr></thead><tbody><tr><td>Cantor's theorem</td><td>3/3</td><td>3/3</td><td>3/3</td><td>3/3</td><td>12/12</td></tr><tr><td>Small doubling</td><td>3/3</td><td>3/3</td><td>3/3</td><td>3/3</td><td>12/12</td></tr><tr><td>Integer sum</td><td>6/6</td><td>6/6</td><td>6/6</td><td>6/6</td><td>24/24</td></tr><tr><td>Nilpotent inverses</td><td>6/6</td><td>3/6</td><td>0/6</td><td>2/6</td><td>11/24</td></tr><tr><td>Large totients</td><td>6/6</td><td>4/6</td><td>6/6</td><td>5/6</td><td>21/24</td></tr></tbody><tfoot><tr><td>Total</td><td>24/24</td><td>19/24</td><td>18/24</td><td>19/24</td><td>80/96</td></tr></tfoot></table>

This improved evaluation method narrowed the gap between the two models, especially when using the MU proof which Llama previously struggled with, providing some evidence that the binary choice strategy reduces the model's tendency to be misled by superficial features common to motivated proofs.

We stress that this is only a preliminary investigation, and that a larger investigation would be required to make strong claims about an LLM's ability to judge motivated proofs. We welcome the community to conduct a larger investigation, advising the following notes of caution based off our experience and some speculation:

* It is easier for models to decide between two proofs, which is motivated, than to make an absolute judgement of a single proof.

* Models can be sensitive to the order in which the proofs are presented, so it is important to permute these (including the in-context examples).

* Models can be misled by superficial features common to motivated proofs, so it is important that false examples also have these features. We achieved this to some extent by using an LLM's failed attempt at writing a motivated proof, but this could potentially also be achieved with care in a human-generated dataset.

**Recommendations** The existing mathematical literature, machine learning datasets and output evaluation standards are concentrated mostly on proof correctness, and our principal recommendation is to increase emphasis on proof motivation to better reflect the proof discovery process. One natural path forward would be to create a corpus of mathematical results with corresponding motivated and unmotivated proofs, and
then to conduct a detailed evaluation of current models' ability to distinguish these. Once it is ascertained
that models can judge motivation with high accuracy, one can design metrics for motivated proof generation.

# 5 Mathematical Datasheet for Datasets

We next propose an amendment to the datasheets for datasets (DfD) framework [Gebru et al., 2021], specific for datasets in mathematics. The DfD framework provides a comprehensive questionnaire that any dataset developer would benefit from answering; we encourage developers of new mathematical datasets to include all original DfD questions as well as our new set of questions outlined below, yielding *mathematical* datasheets for datasets (*mDfDs*). (We note that there have been other important calls for data labelling, e.g., data cards [Pushkarna et al., 2022] and data statements [Bender and Friedman, 2018]; we choose DfD as a starting point given its comprehensiveness and generality.)

Based on the survey of the landscape of mathematical datasets provided in this article, we propose the inclusion of the additional questions for any dataset involving mathematics, as outlined in Figure 1. We include two additional sets of questions to consider on a case-by-case basis, depending on whether the dataset includes formal or informal mathematics. If both, we encourage providing information for each set of questions. We strive for only light—but we argue critical—additions to the DfD framework to avoid excess burden on research teams.

# 6 Conclusion

We have outlined difficulties that pertain to natural language as well formal language mathematics datasets that we believe are present hindrances to the progress of AI systems towards becoming real *mathematical thought partners* that are as, if not more, useful to mathematicians as GitHub's Copilot<fnref n="15" /> is to programmers.

The advantages and disadvantages of natural language and formal language datasets are frequently complementary. What is easy in one representation of mathematics is often hard in the other. For example, automatic evaluation is easy in formal language but hard in natural language; representing rich interaction modes is often comparatively easy in natural language but harder to express in formal language.

We have identified several facets of mathematical practice that are currently not represented in the data used to design and evaluate the deployment of AI systems for mathematics, such as various workflows. We acknowledge that the aspects of workflows we highlight here likely do not capture *all* possible facets of mathematical practice - but we believe they represent an important start, offering an outline for what we may be able to curate to more human-compatible and explainable mathematical AI systems.

While our aim has been exclusively in mathematics, these approaches of mapping processes of scientific discovery to data are not restricted to mathematics and may be adapted to other scientific domains as well. Some of the workflow items, like those related to literature search are directly relevant to other domains. We urge the community to explore how intermediate steps in the process of scientific discovery look like in other areas of science as well.

Regarding purely proof creation, time will tell whether human-annotated proofs are necessary—or whether a purely formal approach together with ingenious proof search techniques will succeed. The comparatively short history of machine learning has taught us to expect the unexpected. In the short term, however, it is imperative to establish better datasets and novel benchmarks for mathematics (either in natural language or formal language), to go beyond the current ones that only test proof or result creation - and also test the ability to assess different mathematical workflows, to summarize mathematics, to explain limitations of proof techniques etc. This will support the next generation of machine learning models and AI tools that can help us discover more mathematics fast and, subsequently assist with any other scientific discipline that uses

<footnote n="15">¹⁵https://github.com/features/copilot</footnote>
Questions for all mathematics datasets:

* Does your dataset cover only end results (statements and proofs), or (also) intermediate proof representations (which ones)?

* What is the precise focus of the dataset (proofs, workflows of various kinds, reasoning arguments, heuristics, etc.)?

* If proofs are included, are they motivated? Who confirmed that they are motivated?

* What is the type (degree of abstraction vs. proof complexity) and level of difficulty (e.g., grade school, high school, undergraduate, IMO, a mix) of the mathematical items?

* Who determined the difficulty rating?

* What scale did you use to determine difficulty rating (e.g., 5-point Likert scale)?

* Wich MSC codes represent your dataset best? (List all applicable ones)

(Case-based) For datasets that include formal mathematics:

* Which formal mathematics language and corresponding ATP/ITP was used (e.g., Lean, Isabelle, Coq, Prover9, Vampire)?

* What version of it was used? (Ideally reference the specific commit – if available, such as in the case of Lean)

* Which dependencies are used? (That is, does the data set stand on its own, or does it depend on existing libraries of formal mathematics (e.g., Isabelle AFP or Mathlib)? If so, what version of the library are you using, and when was it last updated?)

(Case-based) For datasets that include informal mathematics:

* Which items (statements, proofs, workflow steps, etc.) are checked for correctness?

* For proofs: On what level of detail are your datapoints? (The level of detail can vary dramatically, from a high-level sketch of a proof, down to a “formalizable” proof that has all details so that its formalization does not need any filling of mathematical gaps.)

* What procedure is used to check for correctness?

* Who (and how many people) checked correctness?

* If any other annotations are made about the informal mathematics that are not covered by the previous questions (e.g., whether a proof is aesthetically-pleasing, whether a specific workflow representation was used), please provide information on how (and by whom) they were procured.

* What language is the informal mathematics written in (e.g., English, German, French, Japanese, Chinese)?

Figure 1: The questionnaire for mDfDs. We note that some of the questions are dependent on each other, so all need to be answered. E.g., for the first question, what counts as an “undergraduate” differs from country to country (and even from university to university). The second question is thus necessary to put the first in context.

mathematics as a foundation (e.g. systems biology that relies on ordinary differential equations, or physics that relies on several subfields of mathematics – and inspires new subfields at the same time). At the same time, better mathematical copilots may have strong educational benefits.

Having a deeper understanding of the processes by which one arrives at a proof (heuristics, workflows, etc.), which are all concentrated in the concept of a motivated proof, mathematical copilots can also teach the next generation of mathematical minds.

Even though the history of the concept of “proof” spans millennia, the story is ongoing, as currently, the search for a machine-learnable “proof data structure” is an important focus point and represents a new chapter in the ongoing story of what a proof really is.
Acknowledgements

Thomas Lukasiewicz was supported by the AXA Research Fund. Katherine M. Collins acknowledges support from the Cambridge Trust. The work of Fabian Ruehle is supported by NSF grants PHY-2210333, PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions), and startup funding from Northeastern University. Timothy Gowers would like to acknowledge generous support from the Astera Institute. We thank Terence Tao for useful remarks.
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# A Appendix: Problems for Motivated Proof Experiments

For the motivated proof experiments in 4.3, we used five problems including the two examples. These are presented how they were presented to the models, including the human-written motivated and unmotivated proof used for in-context examples.

**Cantor's theorem** There is no surjection from a set $X$ to its power set $\mathbb{P}(X)$.

**MOTIVATED PROOF:** Let $f: X \to \mathbb{P}(X)$ be our function, we need to find some $S \subseteq X$ not in the image of $f$. Not knowing which subset of $X$ to take, we can treat the subset as an unknown, just as we do when solving an equation, and try to narrow down the possibilities. The most general subset of $X$ can be expressed as $\{x \in X : P(x)\}$ for some as yet unspecified property $P$. We now want to prove, for an arbitrary element $y$ of $X$, that $f(y) \neq \{x \in X : P(x)\}$. To obtain our contradiction, we need either an element $x$ of $f(y)$ such that $\neg P(x)$ or an element $x$ of the complement of $f(y)$ such that $P(x)$. There are not many elements around, so trying $y$ is one of the first things to do, and then we find that we need either $y \in f(y)$ and $\neg P(y)$ or $y \notin f(y)$ and $P(y)$. The choice $P(x) = x \notin f(x)$ satisfies this, so the set $S = \{x \in X : x \notin X\}$ has the desired property.

**UNMOTIVATED PROOF:** Let $f: X \to \mathbb{P}(X)$ be our function, we claim that the set $S = \{x \in X : x \notin f(x)\}$ is not in the image of $f$, hence $f$ is not surjective. Suppose there exists $y \in X$ such that $f(y) = S$. But then $y \in f(y)$ iff $y \in S$ iff $y \notin f(y)$ by the definition of $S$, giving a contradiction.

**Small doubling** There exists a subset $A$ of the natural numbers with cardinality $n$, such that the set $A + A = a_1 + a_2 : a_i, a_j \in A$ has cardinality $2n - 1$.

**MOTIVATED PROOF:** We don't know which set to take for $A$ so we treat it as an unknown, just as we do when solving an equation, and try to narrow down the possibilities. Since we know that $|A| = n$, we can write $A = \{a_1, ..., a_n\}$ for $a_1, ..., a_n$ distinct, so we have $A + A = a_i + a_j : i, j = 1, ..., n$. This isn't immediately helpful since we don't know how many collisions there will be in $A + A$, but we try to narrow down the possibilities. The simplest way to distinguish natural numbers is by ordering them, so we assume without loss of generality that $a_1 < ... < a_n$. Applying this monotonicity to our sums, we deduce that $a_i + a_j < a_i + a_k$ whenever $j < k$.

Trying to distinguish as many elements as possible, we fix some $i$ and take this inequality to its logical conclusion, finding that $a_1+a_i < \dots < a_n+a_i$, or that we have $n$ distinct elements. However, we can further extend this chain to the left if $1 < i$ or the right if $i < n$, giving us the longer chain $a_1+a_1 < \dots < a_1+a_i < \dots < a_n+a_1 < \dots < a_n+a_n$, which has $2n-1$ distinct elements. If we want to have $|A+A| = 2n-1$, then these are all the elements of $A+A$, but we still have this free parameter $i$. Repeating the construction with some $j > i$ and observing the first syntactic different element, we notice that $a_1+a_{i+1} = a_2+a_i$, and isolating the variable $i$ we notice that $a_2-a_1 = a_{i+1}-a_i$. This defines an arithmetic progression with first element $a_1$ and common difference $a_2-a_1$, so let's see if that enough.

Suppose $A$ is an arithmetic progression of length $n$, then $A = a, a+d, \dots, a+(n-1)d$, then $A+A = 2a, 2a+d, \dots, 2a+(2n-2)d$ has size $2n-1$, completing the proof.

**UNMOTIVATED PROOF:** Let $A = \{1, ..., n\}$, which has cardinality $n$, then $A+A = \{2, ..., 2n\}$ has cardinality $2n-1$, completing the proof.

**Integer sums** There is a polynomial $P(x)$ with rational coefficients such that $P(n) = \sum_{k=0}^{n} k$ for all non-negative integers $n$.

**MOTIVATED PROOF:** We can write a generic polynomial as $P(n) = \sum_{j=0}^{m} a_j n^j$. However, since $m$ is unknown the equation is reasonably unwieldy. Consequently, we wish to find an upper bound on the degree of $P(n)$. We do this by applying the trivial bound, $P(n) = \sum_{k=0}^{n} k \le \sum_{k=0}^{n} n = n^2 + n$, and since this equation holds for arbitrarily large $n$, we have that $m = \deg P \le 2$, so we can write $P(n) = a_0+a_1n+a_2n^2$. To
find $a_0, a_1$ and $a_2$, we can substitute some small values of $n$, so we have $a_0 = P(0) = 0$, $a_0+a_1+a_2 = P(1) = 1$ and $a_0 + 2a_1 + 4a_2 = P(2) = 3$. Solving the resulting linear equation, we have that $a_0 = 0$, $a_1 = a_2 = 1/2$ so $P(n) = \frac{1}{2}(n^2 + n)$.

To confirm that this works for all $n$, we can use induction. We have checked the base case already so we only need to do the induction step, so suppose that $\sum_{k=0}^{n} k = \frac{1}{2}(n^2 + n)$, then we need to show that $\sum_{k=0}^{n+1} k = \frac{1}{2}((n+1)^2 + (n+1))$. In order to use the induction assumption, we split the LHS to give us $\sum_{k=0}^{n} k + (n+1)$ and substitute in the induction assumption. By expanding and simplifying, we find that both sides of the equation are equal, completing our proof.

**UNMOTIVATED PROOF:** We show that $P(n) = \frac{1}{2}n(n+1)$ works using induction. First we observe that $P(0) = 0 = \sum_{k=0}^{0} k$. Then suppose that $\sum_{k=0}^{n} k = \frac{1}{2}n(n+1)$, then we have that $\sum_{k=0}^{n+1} k = \sum_{k=0}^{n} k + (n+1) = \frac{1}{2}n(n+1) + (n+1) = (n+1)(\frac{n}{2} + 1) = \frac{1}{2}(n+1)(n+2)$, completing the proof.

**Nilpotent units** Let $R$ be a commutative ring, and let $x \in R$ be nilpotent. Then $(1+x)$ is a unit.

**MOTIVATED PROOF:** To show that $1+x$ is a unit, we need to find an inverse element. Not knowing which element to take, we parametrize the most generic element we can. Since the only known elements of $R$ are $1$ and $x$, the most generic element is an integer polynomial in $x$, $\sum_{k=0}^{m} a_k x^k$. For this to be a right inverse of $1+x$, we must have that $1 = (1+x)(\sum_{k=0}^{m} a_k x^k) = a_0 + \sum_{k=1}^{m} (a_k + a_{k-1})x^k + a_m x^{m+1}$. For this to hold, we must eliminate all coefficients of the polynomial besides the constant term, which should be 1. This gives us that $a_0 = 1$, $a_k = a_{k-1}$ for $1 \le k \le m$ and $a_m = 0$. The first two equations give us that $a_k = (-1)^k$ for all $k$, but this contradicts the last equation.

But we also know that $x$ is nilpotent, so if we have $r$ such that $x^r = 0$, then all coefficients from $x^r$ onwards can be ignored. This solves our issue, as we can let $m = r - 1$ to remove the $a_m = 0$ condition, and we are left with $a_k = (-1)^k$, so $(1+x)$ is a unit with inverse $\sum_{k=0}^{r-1} (-1)^k \cdot x^k$.

**UNMOTIVATED PROOF:** Let $r$ be such that $x^r = 0$ from nilpotency, and observe that $(1+x)(\sum_{k=0}^{r-1} (-1)^k \cdot x^k) = 1 + -x^r = 1$ by the formula for summing geometric progressions, hence $\sum_{k=0}^{r-1} (-1)^k \cdot x^k$ is an inverse of $1+x$ and $1+x$ is a unit.

**Large totients** We define Euler's totient function $\phi$ as follows. If $n = \prod_{i=1}^{m} p_i^{k_i}$ is its prime factorisation, then $\phi(n) = \prod_{i=1}^{m} p_i^{k_i-1}(p_i - 1)$. Show that for all $\epsilon > 0$, $\phi(n)/n$ can take values in $(1 - \epsilon, 1)$.

**MOTIVATED PROOF:** We first try to simplify $\phi(n)/n$. Expressing $n$ in its prime factorisation to match the definition of $\phi(n)$, we have $\phi(n)/n = \prod_{i=1}^{m} (p_i^{k_i-1}(p_i - 1)/p_i^{k_i}) = \prod_{i=1}^{m} (1 - 1/p_i)$. We notice that each factor is between 0 and 1, so the product is large where there are few factors so we let $m = 1$ (or equivalently, letting $n = p^k$), giving us $\phi(n)/n = (1 - 1/p)$. Since this is clearly less than 1, we only need to choose $p$ such that $1 - \epsilon < 1 - 1/p$, which simplifies easily to $p > 1/\epsilon$. Since there are infinitely many prime numbers, we can always choose such a $p$.

**UNMOTIVATED PROOF:** Let $p$ be the smallest prime number such that $p > 1/\epsilon$, which exists as there are infinitely many primes. Then we have $\phi(p)/p = 1 - 1/p$, and as $p > 1/\epsilon$, we see that $1 - \epsilon < 1 - 1/p < 1 - \epsilon$ as required.