# AI for Mathematics: A Cognitive Science Perspective

Cedegao E. Zhang*
MIT BCS
<EMAIL_ADDRESS>
Katherine M. Collins*
University of Cambridge
<EMAIL_ADDRESS>
Adrian Weller
University of Cambridge
Alan Turing Institute
<EMAIL_ADDRESS>Joshua B. Tenenbaum
MIT BCS
<EMAIL_ADDRESS>
###### Abstract

Mathematics is one of the most powerful conceptual systems developed and used
by the human species. Dreams of automated mathematicians have a storied
history in artificial intelligence (AI). Rapid progress in AI, particularly
propelled by advances in large language models (LLMs), has sparked renewed,
widespread interest in building such systems. In this work, we reflect on
these goals from a cognitive science perspective. We call attention to several
classical and ongoing research directions from cognitive science, which we
believe are valuable for AI practitioners to consider when seeking to build
truly human (or superhuman)-level mathematical systems. We close with open
discussions and questions that we believe necessitate a multi-disciplinary
perspective—cognitive scientists working in tandem with AI researchers and
mathematicians—as we move toward better mathematical AI systems which not only
help us push the frontier of the mathematics, but also offer glimpses into how
we as humans are even capable of such great cognitive feats.

**footnotetext: These authors contributed equally to this work.

##  1 Introduction

Building computational systems that understand and practice mathematics at the
level of human mathematicians has been a long-standing aspiration of
artificial intelligence (AI) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15]. The rise of large language models (LLMs) has sparked imaginations that we
are closer than ever to attaining, or surpassing, human-level performance on a
range of tasks [16, 17, 18]. Yet, simultaneously, something seems amiss:
despite these models achieving tremendous performance in many realms of human
expertise (e.g., medicine, law, creative writing), the performance of these
models on mathematics specifically lags behind [19, 20, 21, 22]. There are
many efforts to improve the mathematical problem-solving capabilities of LLMs,
such as adjusting the training data and feedback strategies [23, 24, 25, 26,
27], equipping models with expanded background knowledge at inference-time
[28], or composing LLMs with existing computational mathematics systems [29,
30, 31, 32, 33]. Recent efforts to build in principles from cognitive science,
e.g., the importance of learning abstractions, have also seen success [34,
35]; however, we believe that the broader AI-mathematics community still has
much to draw from cognitive science: in the questions we ask and the methods
by which we approach such challenges.

First, we believe it is essential to reflect on what goals we are even trying
to pursue. What does it mean for AI systems to excel at mathematics at or
beyond a human level? Is simply excelling at a suite of standard benchmark
datasets sufficient? While there is no doubt value in benchmarks to spur
progress, humans—and human mathematicians—are capable of so much more than
what can be captured in a static benchmark. We are capable of intuitions and
judgments [36], of reasoning about the world as world [37], of seeking deeper
explanations and understandings of results [38], of flexibly developing new
problem solving tactics and not just solving new problems, but posing them too
[39, 40]. How then should we proceed to develop human-level AI mathematicians?
In the rest of this position paper, we argue that perspectives from cognitive
science have a lot to offer in this new age of LLMs. Cognitive scientists, AI
researchers, and mathematicians can productively contribute together to this
vision towards growing flexible, automated mathematicians that help us push
the frontiers of mathematical knowledge and reflect back on how we are even
capable of remarkable achievements of mathematical cognition [41].

##  2 Looking to cognitive science

We now call attention to several classical and active research directions
within cognitive science which we believe hold value for those building
mathematical AI systems.

###  2.1 Sample-efficient learning

One of the hallmarks of human cognition is our ability to learn new concepts,
knowledge, and problem-solving strategies, from little data [42, 43, 44, 45,
46, 47]. In mathematics, data paucity is a particular conundrum, e.g., it is
costly and difficult to obtain high-quality data on advanced topics, and few
texts may exist on the cutting-edge or more obscure branches of mathematics.
On the other hand, human mathematicians, from early learners to expert-level
mathematicians, do not need millions of examples to learn mathematical
concepts and problem-solving strategies. Yet, even though the rote number of
examples developing mathematicians may be exposed to is small, that does not
mean that a concept is grasped immediately upon exposure. It make take a human
multiple encounters with an example, extended time sitting and
thinking—squeezing out a tremendous amount from a handful of examples, e.g.,
through active engagement like self-explanation (see below)—to master a
concept or strategy, after which such knowledge can be readily generalized to
new situations [44, 48, 49, 50, 51].

###  2.2 Concepts, representations, and world models

It is inspiring to reflect on the sample-efficiency of human learning. If we
are to obtain or surpass such capabilities in AI systems, it is important to
examine how humans may achieve such efficiency in the first place. Towards
this end, we point to the rich cognitive science literature on concepts, their
representations, and how the human mind builds rich models of the world out of
concepts [52, 53, 54, 55].

In cognitive science, much research in cognitive science points to powerful
inductive biases gleaned through evolution: “core knowledge” [56, 57]. It has
been speculated that a core “number sense” [58] forms the foundation upon
which our mathematical prowess is built. Strong evidence points to two core
number systems—for reasoning about numerosity exactly, and approximately [49,
58, 59]. From these core knowledge systems, we can develop concepts [42, 54,
60]. Notably in mathematics, concepts have precise definitions, unlike other
abstract concepts such as justice and knowledge or everyday concepts like
chair. At the same time, mathematicians think about concepts more than in
terms of definitions; they can give examples and counterexamples, draw out
relationships between concepts, and so on—this type of conceptual richness is
compatible with the psychological theory of conceptual-role semantics [52].

So, what are the form(s) of these conceptual representations? Contemporary
cognitive science has provided strong evidence for that conceptual
representations may be modeled by “languages of thought” [61, 62, 63, 64],
which in mathematics, may be built over core geometric primitives [65].
Closely linked with “languages of thought” is the notion of a “world model”.
In AI, many have highlighted the importance of world models, although
researchers disagree about how to build such models within AI systems [66, 67,
68, 64]. It is generally accepted that a world model should support simulation
of possibilities, causal and counterfactual reasoning, and calibrated
judgements about belief and truth [43, 67, 68, 69, 70, 71, 72, 73, 74]. We
hypothesize that the intuitions that mathematicians acquire over years of
practice can be seen as forming world models of the mathematical universe.
Here, we use the famous “$\mathsf{P}=\mathsf{NP}$?” problem as an illustrative
example. Most people believe that $\mathsf{P}\neq\mathsf{NP}$. It seems that
much evidence of such strong beliefs over an unproven statement comes from
simulating what would happen if $\mathsf{P}=\mathsf{NP}$ or
$\mathsf{P}\neq\mathsf{NP}$. If the former is true, many counter-intuitive
consequence would follow, whereas we would not need to heavily adjust our
other beliefs about computation if the latter is true [75, 76, 77]. This kind
of simulation and argumentation, we suggest, may be powered by world models.

###  2.3 Goals, planning, and agency

Today, the dominant paradigm for large language models is a passive one: a
(very large) training corpus is provided to the model, and the model optimizes
some given objective function [78, 16]. At inference-time, a model is
presented with a problem (e.g., a translation or reasoning task) and tries to
make good predictions. However, this is not how humans think about or perform
problem-solving. Humans are planning agents with goals spanning across
different communities and timescales [79, 80]. When planning to achieve a
goal, we can flexibly divide a task into sub-goals, form and leverage
simplified abstract representations to inform planning, and replan [81, 82,
83, 84, 85]. Planning is crucial to success in mathematical reasoning.
Consider when a teacher gives a student a problem to solve; the student needs
to generate sub-goals and come up with strategies, such as looking up
definitions, consider examples, examine different cases, or simply look for
help. Moreover, mathematical cognition is not just about planning for set
goals, but inventing new goals, problems, and concepts [39, 40, 41, 51]. How
do some mathematicians form the goal of inventing new mathematics, and how do
they achieve it? Engineering and scientific insights on these
questions—drawing on cognitive science, AI, and mathematics—may drive a huge
leap forward towards creative AI mathematicians.

###  2.4 Cognitive limitations and resource-rationality

However, humans are far imperfect planners, and they may fail to execute the
plans we do embark upon. Mathematicians may become wedded to a particular
proof strategy only to realize it was misguided and need to backtrack, or
worse, could fall prey to functional fixedness [86, 87] and the sunk cost
fallacy [88]. Such instances put a damper in the notion that humans are
rational reasoners [89]. Cognitive scientists here too have developed rich
frameworks to reconcile such challenges. Rather, we may be viewed as rational
in light of resource constraints, i.e., “resource-rational” [90, 91, 92, 93].
This notion finds particular importance when thinking about humans and AI
systems. Fundamentally, humans and computational systems have different
resource limitations: computers are able to make calculations extremely fast,
are not constrained to the same limitations on working memory, and do not
succumb to daily inevitable fatigue that we humans do. When building
mathematical AI systems then, it is prudent to question whether we should be
designing AI systems to mimic human resource constraints [92]. If trying to
build a computational “thought partner” to complement humans and enable us to
explore greater mathematical depths than we have so far, for instance, by
making more calculations and proposing possible new patterns in troves of data
[15], then perhaps we do not want to curtail a model’s resources. However, one
could argue that perhaps, such resource limitations are not a failing, but
rather an advantage: for instance, empowering us to judiciously select which
problems to solve in the first place. Indeed, mathematics communities
(generally) do not waste too much time on problems that people believe to be
out of reach. Studying under what settings resource limitations on
mathematical cognition are advantageous, and when they are not, is a ripe
space for collaboration across cognitive science, mathematics, and AI,
particularly when thinking about making sensible use of limited resources even
present in large-scale AI systems [94, 95].

###  2.5 Communication and explanation

We close our tour of cognitive science insights to spark the imaginations of
those seeking to build mathematical AI by reiterating that mathematics is a
group activity consisting of communities, and development of knowledge in any
intellectual community depends on effective communication. We argue that a
cognitive perspective on communication is valuable for the math-AI community
for two core reasons. First, the output of our communication amongst each
other forms the bedrock of the data used to train LLMs. Second, insights from
cognitive science reveal that communication can spur learning for the
communicator [96]. We start by reflecting on the latter.

Ample evidence in cognitive science reveals the power of self-explanation for
improving learning and generalization [96, 97, 98, 99, 100, 101, 102].
Explanations can help the explainer identify abstractions to inform induction
[96, 99] and reveal gaps in one’s own knowledge [97], motivating information-
seeking to resolve such gaps [100]. At first glance, recent LLM research such
as chain-of-thought-prompting [103], “self-taught reasoning” [104], “self-
reflection” [105] could be viewed as self-explanation to improve reasoning,
but we encourage ruminating on the cognitive underpinnings. In fact, we argue
that these are not instances of self-explanation in the way that humans self-
explain. For humans, self-explanation is something that we want to do, because
understanding is intrinsically valuable [99, 106]. Thus, it is desirable to
not just have new prompting strategies leveraging explanations, but systems
designed with explanations at their core.

And what about communication to others? We externalize many of our inner
thoughts, whether that be writing out the steps of a new proof, drawing
diagrams to convey a concept, or debating with a friend what the largest
possible number is. These externalized thoughts and interactions increasingly
form the bedrock of training data for AI systems. Nonetheless, humans do not
communicate all of our inner thoughts; rather, we communicate what we believe
is essential to convey [107]—often requiring the listener to make inferences
about what the communicator intends to communicate (which may differ from what
they actually produced) [108, 109, 110]. Such communication frameworks may be
important for building mathematical AI systems that can adequately “read
between the lines” in the data available and recognize that when providing
mathematical assistance to humans, humans are capable of such inferences
(e.g., we do not always require overly verbose responses and in fact may find
it less helpful in mathematics [19]).

##  3 Concluding remarks

#### Catalyzing community cross-talk

As we highlight, the cognitive science community has been studying topics
deeply relevant to mathematical AI. We hope our piece helps further expose AI
practitioners and mathematicians to what we believe are valuable terminology
and conceptual structures from cognitive science. Cognitive scientists too can
sharpen our theories from further exchanges across communities; we lay out a
few strategies to facilitate such conversations. First, accessibility of
higher-level mathematics is perhaps one of the most pernicious barriers to
effective collaboration across cognitive scientists and AI practitioners in
the space. Convenings designed to engage not just the AI and mathematics
community but also cognitive scientists would aid in building a shared
vocabulary across these communities. Second, there is a need for improved
research tools to empower the study of mathematics across our communities. For
more than a decade, cognitive scientists and AI practitioners alike have
benefited enormously from crowdsourcing platforms such as Amazon Mechanical
Turk [111] and Prolific [112]. However, at present, it is hard to find
targeted domain practitioners on such sites. We suggest that it would be
extremely valuable for the community to discuss the idea of a “Mechanical Turk
for mathematics”; i.e., a platform where AI and cognitive scientists can post
studies, questions, data gathering attempts about mathematics and
mathematicians and students can participate in them. Ideally, such an effort
could benefit all parties involved. Third, we note that a strong catalyst for
collaboration can be a shared goal [113]. We point to games as a sensible
playground which may appeal to mathematicians, cognitive scientists, and AI
practitioners. Games have been ripe grounds for study in both AI [114, 115,
116] and cognitive science [81, 117, 118, 119], and as recently exposed by
Poesia [120], aspects of mathematics itself may be cast in the language of
games. We see this as a particularly exciting framing that allows us to better
understand many aspects of mathematics with the help of mathematicians.

#### Looking forward

With the resurgence in interest around AI and mathematics, we emphasize the
value of engaging with the cognitive science community in the quest towards
more powerful automated mathematicians. Engaging across the mathematics,
cognitive science, and AI communities is paramount in even defining what this
quest is and where we intend to go. To close, we propose several directions of
inquiry that we think the nexus of the cognitive science, AI, and mathematics
communities are poised to address. For instance, advances in AI can serve as
tools to help us better understand the relationship between mathematical
problem-solving capabilities and the modalities of mathematical data—language
(natural and formal) alongside figures and diagrams; what makes a problem easy
or hard (and how this differs across humans and AI systems); what kinds of
prior knowledge, including human commonsense knowledge, is necessary to learn
mathematics; and what are the computational foundations of mathematical
insights. We believe that steps along these directions, taken across our
communities, can not only spur the development of truly powerful AI
mathematicians, but also shed light on what is so special about humans’ feats
of mathematical cognition—sparking efforts to improved tailored mathematical
education and push the boundaries of what we, jointly with AI systems,
understand about the wonderful world of mathematics.

## Acknowledgements

We thank Timothy Gowers, Gabriel Poesia, and Roger Levy for comments on
earlier drafts. We thank Noah Goodman, Raymond Wang, and Lionel Wong for
discussions related to this work. We also thank Albert Jiang, Mateja Jamnik,
the Human-Oriented Automated Theorem Proving System Team at Cambridge, and the
Spring 2023 GPS Seminar at MIT for conversations that inspired aspects of this
work. KMC acknowledges funding from the Marshall Commission and Cambridge
Trust. AW acknowledges support from a Turing AI Fellowship under grant
EP/V025279/1, The Alan Turing Institute, and the Leverhulme Trust via CFI. JBT
acknowledges funding from AFOSR Grant #FA9550-22-1-0387 and the MIT-IBM Watson
AI Lab.

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