# A Triumvirate of AI Driven Theoretical Discovery

Yang-Hui He<EMAIL_ADDRESS>
###### Abstract

Recent years have seen the dramatic rise of the usage of AI algorithms in pure
mathematics and fundamental sciences such as theoretical physics. This is
perhaps counter-intuitive since mathematical sciences require the rigorous
definitions, derivations, and proofs, in contrast to the experimental sciences
which rely on the modelling of data with error-bars. In this Perspective, we
categorize the approaches to mathematical discovery as “top-down”, “bottom-up”
and “meta-mathematics”, as inspired by historical examples. We review some of
the progress over the last few years, comparing and contrasting both the
advances and the short-comings in each approach. We argue that while the
theorist is in no way in danger of being replaced by AI in the near future,
the hybrid of human expertise and AI algorithms will become an integral part
of theoretical discovery.

Key points:

  * •

Bottom-up mathematics is building statements from axioms and definitions.

  * •

Meta-mathematics is to treat all statements and their derivations as a
language.

  * •

Top-down mathematics is theoretical research guided by intuition, experience
and pure data.

  * •

Special consideration should be given to pure, theoretical, noiseless data,
which can lead to profound conjectures.

  * •

AI has made significant advances in all these approaches of mathematical and
theoretical research, but the human expert will not be replaced any time soon.

  * •

Humans and AI will work in tandem for theoretical discovery.

##  1 Introduction

In a recent conversation, a friend who is an eye doctor recalled that when he
was a trainee ophthalmologist, his mentor was a world-leading specialist who
could recognize a diseased eye at a single glance from a wall of images. Yet,
when pressed, his mentor often could not give more precise an answer than
dismissing it as gut-instinct. This, we concluded, is the fundamental reason
why Ariticial Intelligence (AI) is taking over so many disciplines of human
endeavour.

For practical purposes, the vast majority of human activity falls under the
adage of “if it ain’t broke, don’t fix it”. A medic’s primary goal is to cure
the patient; understanding the mechanisms of the disease is secondary. ChatGTP
can perform as well as an average undergraduate student in an exam not because
a large language model (LLM) can have true comprehension but because for all
but the very top students in the world the goal is to pass the test, not to
understand the material. Even (or perhaps especially) in the creative arts, a
good definition of genius would be the mastery of style (via supervised
learning of preceding samples) and the deviation therefrom in an inexplicable
way. Deep neural networks can mimic so much of human activities because they
more often than not simply requires black-boxes: learning from trial-and-
error, and performing to within a margin of error.

This can never be so for the scientist. The raison d’être of this tiny
percentage of humanity is to understand and to question. Such a compulsion for
explainability is especially true for the theorist. Here and throughout, we
will use the term theoretical science to include both (1) pure mathematics and
(2) the development and testing of theories and hypotheses using mathematics,
exemplified by theoretical physics. In other words, we will adhere to the
British academic convention and consider the likes of theoretical physics or
theoretical computer science as sub-disciplines of mathematics.

In some sense, the desire for interpretability and explicability is the very
reason why the field of AI-assisted theoretical discovery has been a relative
late-comer to fundamental science. While as far back as the 1990s experimental
physicists at CERN, in searching for new particles, were amongst the first
scientists to use AI [110], it wasn’t until 2017-8 that machine learning
techniques emerged in theoretical physics [65, 21, 90, 117, 97] and pure
mathematics [68, 20].

The past five years has seen tremendous progress in AI-driven theoretical
investigations. We give but a few of the representative examples out of the
hundreds of papers that have come to define this exciting field. In
theoretical physics, these have included particle phenomenology from string
theory [40, 103, 29, 107, 116, 94, 16, 109, 1], establishing dictionaries
between field theory and deep-learning [64, 88, 45], theoretical cosmology
[97, 83, 115], quantum field theories [24, 25, 84, 63, 62, 92], and uncovering
fundamental symmetries [25, 52, 91, 125, 31, 98, 96]. In parallel, in pure
mathematics, these have included algebraic geometry [65, 68, 30, 87, 5, 75,
59, 44, 14, 13, 27], algebraic structures and representation theory [72, 10,
35, 39, 77], symbolic algebra and computation [134, 43, 93, 42, 108, 33],
differential and metric geometry [6, 82, 7, 95], number theory [4, 70, 74, 73,
2], graph theory and combinatorics [76, 8, 9, 13], as well as knot theory [61,
32, 35]. As to the question of AI’s role in our civilization, there is a
mixture of optimism [28, 80, 132, 54, 120, 60, 48, 12] and anxiety [119].

The purpose of this Perspective is to attempt a balanced overview of the
advances that have been made in the last few years [68, 99, 135, 129, 66, 60,
128], whilst bearing in mind the concerns and limitations [58, 41, 46, 89]. We
shall caution against both euphoric over-enthusiasm and unwarranted fear, and
convey that the future of the mathematical sciences will be an inevitable and
hopefully harmonious union between the human and AI.

###  1.1 What is the AI Mathematician?

Regarding the issue of AI-driven theoretical discovery, the natural question
arises as to “What is the AI mathematician”? First, we need to ask a more
basic question: “What is mathematics?”, or perhaps on a more pragmatic level,
“What is a mathematician?”. One could look at it in three ways [67], on which
we will expound in detail, in light of how AI has been instrumental to each
over recent years. Not delving into the depth of the philosophy of mathematics
and the philosophy of science, one can loosely categorize as follows:

I. Bottom-Up:

One can think of mathematics as being built from foundational axioms, where
all theorems and equations are constructed from the roots up using logic. This
is what is known as Hilbert’s Formalism Programme [133]. We will refer to this
approach as “bottom-up” to reflect the rigorous nature of theoretical
research.

II. Meta-Mathematics:

Closely related to the bottom-up approach is the Logicism of Russell-Whitehead
[118] and ultimately Wittgenstein [130]. We will think of this as “Mathematics
as Language”, where one considers any proposition as a set of symbols, led to
by sequences of symbols that one calls proof or derivation. We will refer to
this as meta-mathematics in the sense of looking at the problem from a
distance, perhaps more as a linguist or computer-scientist.

III. Top-Down:

The practicing theorist often experiments and conjectures before tackling a
proof or derivation. This is somewhat in the spirit of Brouwer’s intuitionist
approach to mathematics [19] where one factors in the human element. We will
refer to this as “top-down”, where an over-arching view, based on experience
and speculation, guides one towards a problem.

The above is the “Triumvirate” in the title. We shall discuss how each has
witnessed dramatic advancement in the last few years.

##  2 Bottom-Up Mathematics

Hilbert’s “Wir mussen wissen. Wir werden wissen.” (We must know. We will
know.) is a famous declaration that should be considered in conjunction with
the Principia Mathematica of Russell-Whitehead, in a tradition that dates back
at least to debates with Frege [49], or to the binary machines of Leibniz, or
indeed to the Elements of Euclid. The Programme of building up mathematical
truths from the ground up received a fatal blow from the Undecidability and
Incompleteness Theorems of Gödel [55], Church [26] and Turing [124] by the
1930s.

However, to quote Prof. Minhyong Kim in a private communication, “the
practicing mathematician rarely contemplates whether your daily proposition is
provable or not”. In other words, the space of decidable and interesting
statements are so vast that one could first focus on these. Thus, despite the
logical impossibility of building all mathematical statements bottom-up,
theorists certainly never stopped pursuing proofs for countless propositions.
This lead to the modern day answer to Russell-Whitehead’s Principia
Mathematica [118]: the Automated Theorem Proving programme (ATP). Arguably the
first AI system for mathematics - or indeed, the first AI system - was the
Logic Theory Machine [104] of Newell-Simon-Shaw of 1956, which was an early
computer system 111Interestingly, this was around the same time as the
emergence of the first trainable neural network [114].  designed for, and
succeeded in, proving a number of propositions of the Principia.

It has been some 70 years since this first AI-for-mathematics system and much
progress has been made. Over the second half of the twentieth century, it
became clear that an increasing number of proofs of fundamental results in
mathematics are impossible without the computer. These have ranged from
situations where key steps reduce to extensive brute-force computation, such
as in the four-colour theorem, to more extreme circumstances where it takes
longer than a human life-span to go through all the details, such as the
classification of simple finite groups. Dependence of the human theorist on
machines have prompted such influential figures as Terrance Tao [112] and
International Congress of Mathematicians addresses [34] to seriously consider
the future of mathematics.

While the first proof-assistant appeared in the 1970s, [37], Isabelle/HOL
[105], Coq [15], Agda [126], and Lean [38] softwares are spear-heading the ATP
programme in this century. One notable direction well under way is the Xena
project (https://xenaproject.wordpress.com/) headed by Prof. Kevin Buzzard to
formalize all (every statement and every step of proof) of undergraduate-level
mathematics into Lean. More recently and more non-trivially, Gowers, Green,
Manners and Tao [56] used Lean’s MathLib library to prove the Polynomial
Freiman-Ruzsa conjecture. Over several private conversations with Buzzard and
Davenport, we are still far from having established anything close to a full
database of all of contemporary mathematics in Coq or Lean format222Only
earlier this year was a new project launched to formalize all the requisite
pieces to Wiles’ proof of Fermat’s last theorem (UK EPSRC Grant
EP/Y022904/1)., let alone have AI automation on selecting correct proof
strategies given a proposition or conjecture. Nevertheless, given such a
database [106, 100], there inevitably will be a plethora of research devoted
to mining this data for new theories. This brings us to the next point.

##  3 Meta-Mathematics

From the Principia to the advancement of computer science - or indeed, from
Euclid’s Elements or Galileo’s Il Saggiatore \- there has been a tradition of
viewing mathematics as a language [53]. Indeed, Natural Language Processing
(NLP) is rooted in Turing’s original proposal of his famous eponymous test
[123]. AI and the internet have propelled NLP to the era of the Large Language
Model (LLM). Indeed, openAI’s ChatGTP (or its counterparts such as google’s
Gemini) has passed the Turing Test [17]. The important point here, of course,
is that LLM has no understanding of the underlying material, it is merely
grouping together words in the right order based on large corpora of
statistical samples. The philosophy 333Perhaps for this very reason of whether
there is understanding of the underlying mathematics that we have chosen to
call this direction “meta-mathematics”. of “understanding” aside, it is
indubitable that ChatGTP has been transformative in mimicking human
communication.

One of the earliest experiments [71] on LLM for theory was the application of
the Word2Vec [101] neural network (perhaps the most basic LLM technique)
applied to the titles of several sections of the ArXiv (www.arxiv.org), the
most comprehensive repository for contemporary research in mathematics and
theoretical physics. Perhaps more interesting than in retrieving seemingly
sensible linguistic identities (e.g., ‘string-theory + Calabi-Yau = M-theory +
G2’), was a comparison with viXra (www.viXra.org), the repository of fringe
ideas not accepted by main-stream science. From the titles alone, one could
significantly distinguish (from the confusion matrix) different sub-fields of
theoretical physics (high energy theory, high energy phenomenology, general
relativity/quantum cosmology, etc) in arXiv, whereas in viXra this is not so.
In other words, the syntax of proper theoretical science is more self-
consistent than that of fringe science even at the level of titles444 If one
enriched the data, and included abstracts, application of Word2Vec on papers
in material science have uncovered new chemical reactions [122]. .

Today, Word2Vec has been superseded by transformers which are the preferred
architectures for NLP, and the programme of LLM for mathematics is blooming
[86, 81, 127, 113, 121, 3, 50]. Notably, in parallel to the aforementioned
OpenAI and MetaAI projects, DeepMind’s AlphaGeo [121] has recently been able
to generate correct, human-understandable proofs for Olympiad level problems
in Euclidean geometry. It is clear, when and if we do have a linguistic
database of all contemporary mathematics - which is certainly many decades in
the future - the LLM approach of AlphaGeo on this vast data should produce new
mathematics.

##  4 Top-Down Mathematics

Everything we have said so far has to do with building correct mathematical
statements. But frequently one has no idea what statement one should try to
show. Indeed, how does the practicing mathematician actually work? In many
ways, our papers are written backwards. From day to day, we doodle on paper
and on board, experimenting with ideas, mistakes, and expressions, until
something sensible comes out. Then, we go back and formalize with definitions
followed by theorems and derivations that lead to logical conclusions. Thus,
journal papers in mathematics and theoretical physics look “bottom-up” even
though the discovery process is quite the opposite. Historically, this is even
more true for some of the greatest theoretical discoveries. Newton and Euler
were freely manipulating formal expressions in calculus, centuries before a
proper notion of analysis and convergence; Galois showed the unsolvability of
the quintic by radicals by seeing the structures of permutation groups, before
the definitions of groups and fields that we are taught today; theoretical
physicists freely manipulate Feynman integrals to give results that agree with
experiment to astounding accuracy, when we still do not have a mathematically
rigorous formulation of quantum field theory etc.

In a recent AI safety conference, a policy maker jokingly said to me that
mathematicians are high on the list of jobs being replaced555The eye doctor
mentioned earlier confessed to me that he will not take on any further
trainees in diagnosis; that will soon be entirely taken over by AI. In her
mind, a human mathematician is a bottom-up Logical Theory Machine, building
sentences (proofs and derivations) from definitions. In reality, actual
mathematical research is based on a combination of inspiration, intuition, and
experience. To contrast with the almost dry narrative of “bottom-up”, this
almost fuzzy approach we will call “top-down”. Of course, at the end of the
day, all statements must be rigorous and any fuzziness and inaccuracies must
be distilled out (see nice recent Perspective [60]). It is this direction
which we now discuss in detail, with illustrative examples.

Perhaps contrary to common conception, an indispensable component to even the
purest mathematical discovery is data. This is not experimental data in the
sense of, e.g., particle trajectories from CERN, with errors and variance, but
results of classifications and computations, e.g., tables of characters of
finite groups. This “pure data” is exact and without statistical error and
shed light on the underlying theory. To quote Vladimir Arnol’d, “mathematics
is the part of physics where experiments are cheap.” Indeed, while AI has long
been instrumental in the scientific big data revolution [57, 79], its
application to pure mathematical data is new [68, 67].

The best neural network of the nineteenth century is undoubtedly the brain of
Gauss. Confronted with the ancient problem of finding patterns in primes which
date back at least to Euclid, the sixteen-year-old defined the prime counting
function which gives the number of prime numbers not exceeding a positive real
$x\in\mathbb{R_{+}}$,

$\pi(x)=\\#\\{p\leq x:p\mbox{ prime }\\}\ .$ (1)

Gauss consulted tables available at the time and computed tens of thousands
more (by hand!) and simply plotted $\pi(x)$. With divine inspiration, he
conjectured that $\pi(x)\sim x/\ln(x)$. This profound observation had to wait
for the establishment of complex analysis by Cauchy and Riemann in order to be
proven by Hadamard and de la Vallée Poussin at least 50 years later. It is now
known as the Prime Number Theorem, a cornerstone of mathematics.

In the twentieth century, Birch and Swinnerton-Dyer plotted, using the
earliest computers of the 1960s, ranks and other quantities for elliptic
curves, and conjectured that the order of vanishing of the L-function $L(s)$
for the curve at $s\rightarrow 1$ equals to the rank. This observation is the
the now celebrated BSD Conjecture that bears their name; it is a Millennium
Prize problem [22] and central to modern mathematics.

The above are but two of the countless examples where experimenting with pure
data can lead to profound results. They illustrate the importance of
conjectures. In theoretical research, finding the interesting problem is
vital, and is time and again guided by the almost undefinable process of
intuition. G. H. Hardy’s definition of mathematics is succinct: “A
mathematician, like a painter or a poet, is a maker of patterns.”. Even a
theoretical physicist or mathematical biologist, whose principle motivation
comes from real world data and observations, would first distill the problem
from Nature into a mathematical problem. Then it again becomes a mathematical
game of pattern spotting, from graphs and plots, to formal symbols. But here
is an undeniable point: if there is one thing that AI can do better than
humans, it is pattern recognition, especially when the data is in high
dimension.

###  4.1 Playing with Binary Sequences

Let us perform the following simple experiment. Given (i) the sequence { 0, 0,
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 } and asked what the next
number is, any human would instantly say 0. One way to describe it is the
sequence of whether $n$ divides 3, for positive integers
$n\in\mathbb{Z}_{>0}$. Now, (ii) try { 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1,
0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0,
0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 }. An inspired person might, after some
experimentation, conclude that the next number is 1; this is the sequence of
PrimeQ, whether the $n$-th positive integer is prime or not. Next, (iii) try {
1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1,
0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1}.
The person will be rather hard pressed to guess 666I have tried this on the
audience in numerous talks over the last few years.. This is the sequence of
whether the $n$-th positive integer has a even (0) or odd (1) number of prime
factors counted with multiplicity, a shifted version of the so-called Möbius
mu-function. Uncovering its patterns would have incredible repercussions for
mathematics: there are equivalent formulations of the Riemann Hypothesis in
terms of this sequence.

What if we gave the sequence to AI? For instance, what about a supervised
machine learning algorithm? In order to establish a reasonable training set,
one could chose the following representation (and indeed the choice of
representation is extremely important!). Take one of the above infinite
sequences $\\{a_{i}\\}_{i=1,2,3,\ldots}$ and a sliding window of length $N$.
In other words, consider a set of sequences $\\{\\{a_{i}\\}_{i=1,2,\ldots,N}\
,\\{a_{i}\\}_{i=2,3,\ldots,N+1}\ ,\ldots\\{a_{i}\\}_{i=k,k+1,\ldots,N+k-1}\\}$
for some $k$. Here, $k$ will be taken to be sufficiently large (say 100,000)
to create a decent data size, and $N$ will be taken to be sufficiently large
(say 100) to give enough features. After all, mathematical data is cheap! We
can then consider each of the finite sub-sequences as a single vector in
$\mathbb{R}^{N}$ and label it by the next number outside the window:

$\\{(a_{i})_{i=1,2,\ldots,N}\longrightarrow a_{N+1}\
,\quad(a_{i})_{i=2,3,\ldots,N+1}\longrightarrow a_{N+2}\ ,\quad\ldots\
,\quad(a_{i})_{i=k,k+1,\ldots,N+k-1}\longrightarrow a_{N+k}\\}\ .$ (2)

Note that we have chosen our sequences judiciously to standardize everything
into a binary classification problem of binary vectors of dimension $N$. The
question is then: having seen $k$ labelled samples, how well will the ML
algorithm predict on $k^{\prime}$ unseen vectors. This familiar supervised ML
paradigm can be then compared to the human eye.

One can readily check that with the most basic ML algorithms suited for this
problem, such as decision trees, support vector machines, relatively shallow
feed-forward neural networks with ReLU activation functions, etc., on Eq (2)
applied to our three sequences. For (i) one very quickly reaches 100% accuracy
for any of the ML strategies. On (ii) one reaches 777Note, in this case
because of the increasing rarity of primes - approximately by a factor of
$x/\ln(x)$ due to the Prime Number Theorem - we scale the window size
accordingly. about 80%, while on (iii) one struggles to find any AI algorithm
that would beat 50%. What this means is that (i), a trivial problem for the
human eye, is as trivial for the AI; for (ii) the AI might be finding some
version of the Sieve of Eratosthenes for checking PrimeQ; and for (iii) the AI
is not beating a random guess. Of course, should one find an algorithm which
does, then one might be well under way in finding a new approach toward the
Riemann Hypothesis!

Many of the papers referenced in the introduction employ similar ideas to the
previous paragraph, but to much more sophisticated situations. Indeed, what
(computable) mathematics, in the sense of a Turing Machine, does not fall into
some version of Eq (2)?! We can make the situation even more visual and suited
to AI by wrapping each $N$ vector into an $m\times n=N$ matrix, which can be
interpreted as a pixelated image where 1 is black and 0 is white. For
instance, suppose $N=100$, the first vector for situation (ii) can be wrapped
into a $10\times 10$ matrix, together with a label 1 (since 101 is prime):
something like
$\begin{array}[]{c}\includegraphics[width=28.45274pt]{primePict.png}\end{array}\longrightarrow
1$. Take, as a much more elevated example, the problem of computing a
topological invariant for a manifold in algebraic geometry (which involves
rather advanced calculations). Yet, one can represent the manifold as a
pixelated image by tensorizing the multi-degree information of the manifold as
an algebraic variety [65, 68]. In a similar manner one could fashion any
mathematical computation as an image recognition problem. Learning and gaining
experience and intuition by a plethora of calculations - as mathematicians and
theorists do during their careers - is in analogy to training a neural
network. We could summarize this paradigm as

> Bottom-up (and meta-) mathematics is language processing while top-down
> mathematics is image processing.

###  4.2 The Birch Test

The key steps to top-down mathematics are (1) identifying the problem and (2)
identifying a strategy to attack the problem. Both depend on experience, with
a healthy dose of intuition. While LLMs applied to the likes of Lean’s MathLib
[38] are making baby-steps in (2), step (1) is formally known as “Conjecture
Formulation”, exemplified by the aforementioned cases of Gauss, Birch,
Swinnerton-Dyer. Can AI assist in telling a good conjecture from a useless
one? Which patterns found from mathematical data lead to interesting as
opposed to trivial mathematics? More recently, this AI-guided “Conjecture
Formulation” has been given much systematic thought [131, 67, 35, 51, 111,
102, 11, 36].

In a six-month workshop in Cambridge in 2023 which I helped to co-organize
[23], together with Professor K. Buzzard et al., we wanted to give some
criteria on AI-driven theoretical discovery, and in particular on AI-assisted
conjectures. Since chatGTP has passed the Turing Test, we wanted to up the
ante. Inspired by a talk given by Birch [18], we called it the Birch Test
[69]. The AI-assisted discovery must be such that

(A) Automaticity:

it is completely made by AI from pattern-spotting, without any human
intervention;

(I) Interpretability:

any statements - conjectures or conclusions - be precise to a human
mathematician, who cannot distinguish it from one given by a human colleague;

(N) Non-Triviality:

it is non-trivial enough that the community of human experts will work on it.

To be fair, these are very stringent criteria (one needs to have a high bar in
honour of Birch!) and so far no AI-assisted theoretical discovery has passed
all three parts of the test. We now highlight with some examples where they
succeed and fail.

Take the early experiments of obtaining topological invariants by deep-
learning algebraic varieties [65, 68]. They have been improved to $>99.9\%$
accuracy [44], which hint toward underlying and yet unknown structures in
algebraic geometry that facilitate calculations without recourse to standard
and computationally expensive sequence chasing. These results suffer from the
typical problem of deep neural networks: there is no interpretible formula one
could extract. Thus, they fail Birch Test (I). A better situation [72] is
where a support vector machine found a separation between simple and non-
simple finite groups by plotting the Cayley multiplication tables. However,
the hypersurface of separation is so complicated and furthermore deforms as
more samples of groups were added that Birch Test (I) is still not passed. The
knot invariant relations found by saliency analyses [35] and the Reidemeister
moves untangled for extremely complicated knots [61], though novel,
interesting, and precise, were either readily proven or have not become
sufficiently influential in the field; thus they fail Birch Test (N).
Likewise, the continued fraction identities found by the Ramanujan machine
[111], or the physical conservation laws found by AI-Feynman [125] also belong
to this category. Even the faster matrix multiplication algorithm found by
DeepMind [47] was shortly thereafter beaten by human researchers [85].

The closest any AI-guided theoretical discovery to passing the Birch test so
far, which has been precise enough and propelled a community of human experts
to work on, is the Murmuration Conjectures in number theory [78]. This
discovery has passed (I) and for the first time passed (N). However, it fails
(A) in that humans intervened in the process by digging under the hood:
surprised by why AI was doing so well at distinguishing ranks of elliptic
curves in the context of BSD, the researchers had to home in on a principal
component analysis, and then look at the weight matrices in order to extract
an unexpected formula.

##  5 Prospectus

The human theorist is not in danger of being put out of the job in the
foreseeable future. From the lack of a complete bottom-up MathLib database
[38] for all of mathematics, to the challenges LLMs would face given the vast
search space of proof strategies even with such a database, to the exacting
requirements of the Birch Test in top-down mathematics, we are far from
automating theoretical discovery.

Nevertheless, it is undeniable that AI is beginning to play and will continue
to play a pivotal role in partnership with the human mathematician and
theoretical scientist. In the nineteenth century, Gauss’s intuitions alone
were good enough to spot patterns that led to such profound results as the
Prime Number Theorem. In the twentieth century, computer experimentations were
needed along-side the insights of Birch and Swinnerton-Dyer to raise the BSD
Conjecture. Now, in the twenty-first century, AI will work hand-in-hand with
human experts to find new insights, conjectures, as well as strategies for
derivations and proofs.

## Acknowledgements

I am most grateful to Dr. Ananyo Bhattacharya, Prof. Alexander Kosyak and Dr.
Melissa Duncan for many valuable comments on the draft. I would like to lend
this opportunity to thank my many collaborators over the past seven years on
AI-assisted mathematics, for the great fun and friendship: Daattavya Aggarwal,
Laura Alessandretti, Guillermo Arias-Tamargo, Anthony Ashmore, Jiakang Bao,
Andrea Baronchelli, Per Berglund, David Berman, Kieran Bull, Lucille Calmon,
Hengyu Chen, Siqi Chen, Andrei Constantin, Pierre-Philippe Dechant, Rehan
Deen, Stavros Garoufalidis, Elli Heyes, Edward Hirst, Johannes Hofscheier,
Juan Ipiña, Vishnu Jejjala, Alexander Kasprzyk, Minhyong Kim, Shailesh Lal,
Kyu-Hwan Lee, Seung-Joo Lee, Jianrong Li, Andre Lukas, Suvajit Majumder,
Challenger Mishra, Gregg Musiker, Brent Nelson, Andrew Nestor, Thomas Oliver,
Burt Ovrut, Toby Peterken, Stephen Pietromonaco, Andrey Pozdnyakov, Dmitrii
Riabchenko, Diego Rodriguez-Gomez, Henrique Sá Earp, Max Sharnoff, Tomás
Silva, Eldar Sultanow, Yuxuan Xiao, Shing-Tung Yau, and Zaid Zaz.

The resaerch is funded in part by STFC grant ST/J00037X/2 and the Leverhulme
Trust for a project grant.

## Competing interests

The author declares no competing interests.

## Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.

## References

  * [1] Steven Abel, Andrei Constantin, Thomas R. Harvey and Andre Lukas  “String Model Building, Reinforcement Learning and Genetic Algorithms”  In _Nankai Symposium on Mathematical Dialogues: In celebration of S.S.Chern’s 110th anniversary_ , 2021  arXiv:2111.07333 [hep-th]
  * [2] Steven A. Abel and Luca A. Nutricati  “Ising Machines for Diophantine Problems in Physics”  In _Fortsch. Phys._ 70.11, 2022, pp. 2200114  DOI: 10.1002/prop.202200114
  * [3] Janice Ahn et al.  “Large Language Models for Mathematical Reasoning: Progresses and Challenges”, 2024  arXiv:2402.00157 [cs.CL]
  * [4] Laura Alessandretti, Andrea Baronchelli and Yang-Hui He  “Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer”, 2019  arXiv:1911.02008 [math.NT]
  * [5] Ross Altman, Jonathan Carifio, James Halverson and Brent D. Nelson  “Estimating Calabi-Yau Hypersurface and Triangulation Counts with Equation Learners”  In _JHEP_ 03, 2019, pp. 186  DOI: 10.1007/JHEP03(2019)186
  * [6] Anthony Ashmore, Yang-Hui He and Burt A. Ovrut  “Machine Learning Calabi–Yau Metrics”  In _Fortsch. Phys._ 68.9, 2020, pp. 2000068  DOI: 10.1002/prop.202000068
  * [7] Anthony Ashmore, Lucille Calmon, Yang-Hui He and Burt A. Ovrut  “Calabi-Yau Metrics, Energy Functionals and Machine-Learning”  In _International Journal of Data Science in the Mathematical Sciences_ 1.1, 2023, pp. 49–61  DOI: 10.1142/S2810939222500034
  * [8] Jiakang Bao et al.  “Hilbert series, machine learning, and applications to physics”  In _Phys. Lett. B_ 827, 2022, pp. 136966  DOI: 10.1016/j.physletb.2022.136966
  * [9] Jiakang Bao et al.  “Polytopes and Machine Learning”  In _Math. Sci._ 01, 2023, pp. 181–211  DOI: 10.1142/S281093922350003X
  * [10] Jiakang Bao et al.  “Quiver Mutations, Seiberg Duality and Machine Learning”  In _Phys. Rev. D_ 102.8, 2020, pp. 086013  DOI: 10.1103/PhysRevD.102.086013
  * [11] Andrej Bauer, Matej Petković and Ljupčo Todorovski  “MLFMF: Data Sets for Machine Learning for Mathematical Formalization”, 2023  arXiv:2310.16005 [cs.LG]
  * [12] Yoshua Bengio and Nikolay Malkin  “Machine learning and information theory concepts towards an AI Mathematician”  In _arXiv preprint arXiv:2403.04571_ , 2024
  * [13] Per Berglund, Ben Campbell and Vishnu Jejjala  “Machine Learning Kreuzer-Skarke Calabi-Yau Threefolds”, 2021  arXiv:2112.09117 [hep-th]
  * [14] David S. Berman, Yang-Hui He and Edward Hirst  “Machine learning Calabi-Yau hypersurfaces”  In _Phys. Rev. D_ 105.6, 2022, pp. 066002  DOI: 10.1103/PhysRevD.105.066002
  * [15] Yves Bertot and Pierre Castéran  “Interactive theorem proving and program development: Coq’Art: the calculus of inductive constructions, https://coq.inria.fr/”  Springer Science & Business Media, 2013
  * [16] Martin Bies et al.  “Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory”  In _JHEP_ 01, 2021, pp. 196  DOI: 10.1007/JHEP01(2021)196
  * [17] Celeste Biever  “ChatGPT broke the Turing test-the race is on for new ways to assess AI”  In _Nature_ 619.7971, 2023, pp. 686–689
  * [18] B Birch  “Reminiscences from 1958-62, Cambridge talk https://www.sms.cam.ac.uk/media/3530787”
  * [19] Luitzen Egbertus Jan Brouwer  “Intuitionistic reflections on formalism”  In _originally published in_ , 1927, pp. 490–492
  * [20] Kieran Bull, Yang-Hui He, Vishnu Jejjala and Challenger Mishra  “Machine Learning CICY Threefolds”  In _Phys. Lett. B_ 785, 2018, pp. 65–72  DOI: 10.1016/j.physletb.2018.08.008
  * [21] Jonathan Carifio, James Halverson, Dmitri Krioukov and Brent D. Nelson  “Machine Learning in the String Landscape”  In _JHEP_ 09, 2017, pp. 157  DOI: 10.1007/JHEP09(2017)157
  * [22] James A Carlson, Arthur Jaffe and Andrew Wiles  “The millennium prize problems”  American Mathematical Soc., 2006
  * [23] A Castro et al.  “Black holes: bridges between number theory and holographic quantum information https://www.newton.ac.uk/event/blh/”
  * [24] Heng-Yu Chen, Yang-Hui He, Shailesh Lal and M. Zaz  “Machine Learning Etudes in Conformal Field Theories”, 2020  arXiv:2006.16114 [hep-th]
  * [25] Heng-Yu Chen, Yang-Hui He, Shailesh Lal and Suvajit Majumder  “Machine learning Lie structures & applications to physics”  In _Phys. Lett. B_ 817, 2021, pp. 136297  DOI: 10.1016/j.physletb.2021.136297
  * [26] Alonzo Church  “An unsolvable problem of elementary number theory”  In _Amer. J. Maths_ 58, 1936, pp. 345–363
  * [27] Tom Coates, Alexander M Kasprzyk and Sara Veneziale  “Machine learning the dimension of a Fano variety”  In _Nature Communications_ 14.1 Nature Publishing Group UK London, 2023, pp. 5526
  * [28] Timothy Cohen, Marat Freytsis and Bryan Ostdiek  “(Machine) learning to do more with less”  In _Journal of High Energy Physics_ 2018.2 Springer, 2018, pp. 1–28
  * [29] Alex Cole and Gary Shiu  “Topological Data Analysis for the String Landscape”  In _JHEP_ 03, 2019, pp. 054  DOI: 10.1007/JHEP03(2019)054
  * [30] Andrei Constantin and Andre Lukas  “Formulae for Line Bundle Cohomology on Calabi-Yau Threefolds”  In _Fortsch. Phys._ 67.12, 2019, pp. 1900084  DOI: 10.1002/prop.201900084
  * [31] Cristina Cornelio et al.  “AI Descartes: Combining data and theory for derivable scientific discovery”  In _arXiv:2109.01634_ , 2021
  * [32] Jessica Craven, Vishnu Jejjala and Arjun Kar  “Disentangling a deep learned volume formula”  In _JHEP_ 06, 2021, pp. 040  DOI: 10.1007/JHEP06(2021)040
  * [33] Lennart Dabelow and Masahito Ueda  “Symbolic Equation Solving via Reinforcement Learning”, 2024  arXiv:2401.13447 [cs.LG]
  * [34] James Davenport and Kevin Buzzard  “The Future of Mathematics; ICM panels & lectures”  In _Proc. ICM_ , 2018, pp. 1085–1110  WS
  * [35] Alex Davies et al.  “Advancing mathematics by guiding human intuition with AI”  In _Nature_ 600.7887 Nature Publishing Group, 2021, pp. 70–74
  * [36] Randy Davila  “Advancements in Research Mathematics through AI: A Framework for Conjecturing”, 2023  arXiv:2306.12917 [math.CO]
  * [37] Nicolaas Govert De Bruijn  “The mathematical language AUTOMATH, its usage, and some of its extensions”  In _Studies in Logic and the Foundations of Mathematics_ 133 Elsevier, 1994, pp. 73–100
  * [38] Leonardo De Moura et al.  “The Lean theorem prover (system description), https://leanprover-community.github.io/”  In _Automated Deduction-CADE-25: 25th International Conference on Automated Deduction, Berlin, Germany, August 1-7, 2015, Proceedings 25_ , 2015, pp. 378–388  Springer
  * [39] Pierre-Philippe Dechant, Yang-Hui He, Elli Heyes and Edward Hirst  “Cluster Algebras: Network Science and Machine Learning”  In _J. Comput. Algebra_ 8, 2023  DOI: 10.1016/j.jaca.2023.100008
  * [40] Mehmet Demirtas, Cody Long, Liam McAllister and Mike Stillman  “The Kreuzer-Skarke Axiverse”  In _JHEP_ 04, 2020, pp. 138  DOI: 10.1007/JHEP04(2020)138
  * [41] V.. Dolotin, A.. Morozov and A.. Popolitov  “Machine learning of the well-known things”  In _Theoretical and Mathematical Physics_ 214.3 Pleiades Publishing Ltd, 2023, pp. 446–455  DOI: 10.1134/s0040577923030091
  * [42] Weinan E  “Machine Learning and Computational Mathematics”  In _Communications in Computational Physics_ 28.5 Global Science Press, 2020, pp. 1639–1670  DOI: 10.4208/cicp.oa-2020-0185
  * [43] Matthew England  “Machine Learning for Mathematical Software”  In _Lecture Notes in Computer Science_ Springer International Publishing, 2018, pp. 165–174  DOI: 10.1007/978-3-319-96418-8˙20
  * [44] Harold Erbin and Riccardo Finotello  “Machine learning for complete intersection Calabi-Yau manifolds: a methodological study”  In _Phys. Rev. D_ 103.12, 2021, pp. 126014  DOI: 10.1103/PhysRevD.103.126014
  * [45] Harold Erbin, Vincent Lahoche and Dine Ousmane Samary  “Non-perturbative renormalization for the neural network-QFT correspondence”  In _Mach. Learn. Sci. Tech._ 3.1, 2022, pp. 015027  DOI: 10.1088/2632-2153/ac4f69
  * [46] Oscar Fajardo-Fontiveros et al.  “Fundamental limits to learning closed-form mathematical models from data”  In _Nature Communications_ 14.1 Springer ScienceBusiness Media LLC, 2023  DOI: 10.1038/s41467-023-36657-z
  * [47] Alhussein Fawzi et al.  “Discovering faster matrix multiplication algorithms with reinforcement learning”  In _Nature_ 610.7930 Nature Publishing Group, 2022, pp. 47–53
  * [48] Thomas Fink  “Why mathematics is set to be revolutionized by AI”  In _Nature, World View_ , 2024
  * [49] Gottlob Frege  “Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet”  H. Pohle, 1893
  * [50] Simon Frieder, Julius Berner, Philipp Petersen and Thomas Lukasiewicz  “Large Language Models for Mathematicians”, 2024  arXiv:2312.04556 [cs.CL]
  * [51] Pascal Friederich, Mario Krenn, Isaac Tamblyn and Alan Aspuru-Guzik  “Scientific intuition inspired by machine learning-generated hypotheses”  In _Machine Learning: Science and Technology_ 2.2 IOP Publishing, 2021, pp. 025027
  * [52] Yarin Gal, Vishnu Jejjala, Damián Kaloni Mayorga Peña and Challenger Mishra  “Baryons from Mesons: A Machine Learning Perspective”  In _Int. J. Mod. Phys. A_ 37.06, 2022, pp. 2250031  DOI: 10.1142/S0217751X22500312
  * [53] Mohan Ganesalingam and Mohan Ganesalingam  “The language of mathematics”  Springer, 2013
  * [54] Iulia Georgescu  “How machines could teach physicists new scientific concepts”  In _Nature Reviews Physics_ 4.12 Nature Publishing Group UK London, 2022, pp. 736–738
  * [55] Kurt Gödel  “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”  In _Monatshefte für mathematik und physik_ 38 Springer, 1931, pp. 173–198
  * [56] W.. Gowers, Ben Green, Freddie Manners and Terence Tao  “On a conjecture of Marton (see Quanta coverage in https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/)”, 2023  arXiv:2311.05762 [math.NT]
  * [57] Matthew J Graham et al.  “Machine-assisted discovery of relationships in astronomy”  In _Monthly Notices of the Royal Astronomical Society_ 431.3 Oxford University Press, 2013, pp. 2371–2384
  * [58] André Greiner-Petter et al.  “Why Machines Cannot Learn Mathematics, Yet”, 2019  arXiv:1905.08359 [cs.DL]
  * [59] Thomas W. Grimm, Fabian Ruehle and Damian Heisteeg  “Classifying Calabi–Yau Threefolds Using Infinite Distance Limits”  In _Commun. Math. Phys._ 382.1, 2021, pp. 239–275  DOI: 10.1007/s00220-021-03972-9
  * [60] Sergei Gukov, James Halverson and Fabian Ruehle  “Rigor with machine learning from field theory to the Poincaré conjecture”  In _Nature Reviews Physics_ Nature Publishing Group UK London, 2024, pp. 1–10
  * [61] Sergei Gukov, James Halverson, Fabian Ruehle and Piotr Sułkowski  “Learning to Unknot”  In _Mach. Learn. Sci. Tech._ 2.2, 2021, pp. 025035  DOI: 10.1088/2632-2153/abe91f
  * [62] Rajan Gupta, Tanmoy Bhattacharya and Boram Yoon  “AI and Theoretical Particle Physics”, 2023, pp. 465–491  DOI: 10.1142/9789811265679˙0026
  * [63] T.. Harvey and A. Lukas  “Quark Mass Models and Reinforcement Learning”  In _JHEP_ 08, 2021, pp. 161  DOI: 10.1007/JHEP08(2021)161
  * [64] Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka and Akio Tomiya  “Deep learning and the AdS/CFT correspondence”  In _Phys. Rev. D_ 98.4, 2018, pp. 046019  DOI: 10.1103/PhysRevD.98.046019
  * [65] Yang-Hui He  “Deep-Learning the Landscape”  In _Phys. Lett. B_ 774, 2017, pp. 564–568  DOI: 10.1016/j.physletb.2017.10.024
  * [66] “Machine Learning in Pure Mathematics and Theoretical Physics”  World Scientific, 2023
  * [67] Yang-Hui He  “Machine-Learning Mathematical Structures”, 2021  arXiv:2101.06317 [cs.LG]
  * [68] Yang-Hui He  “The Calabi–Yau Landscape: From Geometry, to Physics, to Machine Learning”, Lecture Notes in Mathematics  Springer-Nature, 2018  DOI: 10.1007/978-3-030-77562-9
  * [69] Yang-Hui He and Mikhail Burtsev  “Can AI make genuine theoretical discoveries?”  In _Nature_ 625.7994 Nature, 2024, pp. 241–241
  * [70] Yang-Hui He, Edward Hirst and Toby Peterken  “Machine-learning dessins d’enfants: explorations via modular and Seiberg–Witten curves”  In _J. Phys. A_ 54.7, 2021, pp. 075401  DOI: 10.1088/1751-8121/abbc4f
  * [71] Yang-Hui He, Vishnu Jejjala and Brent D. Nelson  “hep-th”, 2018  arXiv:1807.00735 [cs.CL]
  * [72] Yang-Hui He and Minhyong Kim  “Learning Algebraic Structures: Preliminary Investigations”, 2019  arXiv:1905.02263 [cs.LG]
  * [73] Yang-Hui He, Kyu-Hwan Lee and Thomas Oliver  “Machine-Learning Number Fields”, 2020  DOI: 10.4310/MCGD.2022.v2.n1.a2
  * [74] Yang-Hui He, Kyu-Hwan Lee and Thomas Oliver  “Machine-Learning the Sato–Tate Conjecture”, 2020  DOI: 10.1016/j.jsc.2021.11.002
  * [75] Yang-Hui He and Seung-Joo Lee  “Distinguishing elliptic fibrations with AI”  In _Phys. Lett. B_ 798, 2019, pp. 134889  DOI: 10.1016/j.physletb.2019.134889
  * [76] Yang-Hui He and Shing-Tung Yau  “Graph Laplacians, Riemannian Manifolds and their Machine-Learning”, 2020  arXiv:2006.16619 [math.CO]
  * [77] Yang-Hui He, Vishnu Jejjala, Challenger Mishra and Max Sharnoff  “Learning to be Simple”, 2023  arXiv:2312.05299 [cs.LG]
  * [78] Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver and Alexey Pozdnyakov  “Murmurations of elliptic curves (see Quanta feature in https://www.quantamagazine.org/elliptic-curve-murmurations-found-with-ai-take-flight-20240305/)”, 2022  arXiv:2204.10140 [math.NT]
  * [79] Tony Hey, Keith Butler, Sam Jackson and Jeyarajan Thiyagalingam  “Machine learning and big scientific data”  In _Philosophical Transactions of the Royal Society A_ 378.2166 The Royal Society Publishing, 2020, pp. 20190054
  * [80] Matthew Hudson  “No coding required: Companies make it easier than ever for scientists to use artificial intelligence”  In _Science_ , 2019  DOI: 10.1126/science.aay9534
  * [81] M Hutson  “AI learns to write computer code in “stunning” advance”  In _Science_ , 2022
  * [82] Vishnu Jejjala, Damian Kaloni Mayorga Pena and Challenger Mishra  “Neural network approximations for Calabi-Yau metrics”  In _JHEP_ 08, 2022, pp. 105  DOI: 10.1007/JHEP08(2022)105
  * [83] Ryusuke Jinno  “Machine learning for bounce calculation”, 2018  arXiv:1805.12153 [hep-th]
  * [84] Bastian Kaspschak and Ulf-G. Meißner  “Neural network perturbation theory and its application to the Born series”  In _Phys. Rev. Res._ 3.2, 2021, pp. 023223  DOI: 10.1103/PhysRevResearch.3.023223
  * [85] Manuel Kauers and Jakob Moosbauer  “The FBHHRBNRSSSHK-Algorithm for Multiplication in $\mathbb{Z}_{2}^{5\times 5}$ is still not the end of the story”  In _arXiv:2210.04045_ , 2022
  * [86] Joanne T. Kim, Mikel Landajuela and Brenden K. Petersen  “Distilling Wikipedia mathematical knowledge into neural network models”, 2021  arXiv:2104.05930 [cs.LG]
  * [87] Daniel Klaewer and Lorenz Schlechter  “Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties”  In _Phys. Lett. B_ 789, 2019, pp. 438–443  DOI: 10.1016/j.physletb.2019.01.002
  * [88] Ellen de Mello Koch, Robert Mello Koch and Ling Cheng  “Is Deep Learning a Renormalization Group Flow?”, 2019  DOI: 10.1109/ACCESS.2020.3000901
  * [89] Alexander Kolpakov and Aidan Rocke  “On the impossibility of discovering a formula for primes using AI”  In _arXiv preprint arXiv:2308.10817_ , 2023
  * [90] Daniel Krefl and Rak-Kyeong Seong  “Machine Learning of Calabi-Yau Volumes”  In _Phys. Rev. D_ 96.6, 2017, pp. 066014  DOI: 10.1103/PhysRevD.96.066014
  * [91] Sven Krippendorf and Marc Syvaeri  “Detecting Symmetries with Neural Networks”, 2020  arXiv:2003.13679 [physics.comp-ph]
  * [92] Shailesh Lal, Suvajit Majumder and Evgeny Sobko  “The R-mAtrIx Net”, 2023  arXiv:2304.07247 [hep-th]
  * [93] Guillaume Lample and François Charton  “Deep learning for symbolic mathematics”  In _arXiv:1912.01412_ , 2019
  * [94] Magdalena Larfors and Robin Schneider  “Explore and Exploit with Heterotic Line Bundle Models”  In _Fortsch. Phys._ 68.5, 2020, pp. 2000034  DOI: 10.1002/prop.202000034
  * [95] Magdalena Larfors, Andre Lukas, Fabian Ruehle and Robin Schneider  “Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds”  In _Mach. Learn. Sci. Tech._ 3.3, 2022, pp. 035014  DOI: 10.1088/2632-2153/ac8e4e
  * [96] Pablo Lemos et al.  “Rediscovering orbital mechanics with machine learning”  In _Machine Learning: Science and Technology_ 4.4 IOP Publishing, 2023, pp. 045002
  * [97] Junyu Liu  “Artificial Neural Network in Cosmic Landscape”  In _JHEP_ 12, 2017, pp. 149  DOI: 10.1007/JHEP12(2017)149
  * [98] Ziming Liu et al.  “Machine-learning nonconservative dynamics for new-physics detection”  In _Physical Review E_ 104.5 APS, 2021, pp. 055302
  * [99] Pan Lu et al.  “A Survey of Deep Learning for Mathematical Reasoning”, 2023  arXiv:2212.10535 [cs.AI]
  * [100] MetaAI  “Teaching AI advanced mathematical reasoning, https://ai.meta.com/blog/ai-math-theorem-proving/”, 2022
  * [101] Tomas Mikolov, Kai Chen, Greg Corrado and Jeffrey Dean  “Efficient estimation of word representations in vector space”  In _arXiv preprint arXiv:1301.3781_ , 2013
  * [102] Challenger Mishra, Subhayan Roy Moulik and Rahul Sarkar  “Mathematical conjecture generation using machine intelligence”  In _arXiv:2306.07277_ , 2023
  * [103] Andreas Mütter, Erik Parr and Patrick K.. Vaudrevange  “Deep learning in the heterotic orbifold landscape”  In _Nucl. Phys. B_ 940, 2019, pp. 113–129  DOI: 10.1016/j.nuclphysb.2019.01.013
  * [104] Allen Newell and Herbert Simon  “The logic theory machine–A complex information processing system”  In _IRE Transactions on information theory_ 2.3 IEEE, 1956, pp. 61–79
  * [105] Tobias Nipkow, Lawrence C Paulson and Markus Wenzel  “Isabelle/HOL: a proof assistant for higher-order logic, https://isabelle.in.tum.de”  Springer Science & Business Media, 2002
  * [106] OpenAI  “Solving (some) formal math olympiad problems, https://openai.com/research/formal-math”, 2022
  * [107] Hajime Otsuka and Kenta Takemoto  “Deep learning and k-means clustering in heterotic string vacua with line bundles”  In _JHEP_ 05, 2020, pp. 047  DOI: 10.1007/JHEP05(2020)047
  * [108] Dylan Peifer, Michael Stillman and Daniel Halpern-Leistner  “Learning selection strategies in Buchberger’s algorithm”  In _International Conference on Machine Learning_ , 2020, pp. 7575–7585  PMLR
  * [109] Ricardo Perez-Martinez, Saul Ramos-Sanchez and Patrick K.. Vaudrevange  “Landscape of promising nonsupersymmetric string models”  In _Phys. Rev. D_ 104.4, 2021, pp. 046026  DOI: 10.1103/PhysRevD.104.046026
  * [110] D Perret-Gallix and W Wojcik  “New computing techniques in physics research: Proceedings of the 1st International Workshop on Software Engineering, Artificial Intelligence and Expert Systems in High-Energy and Nuclear Physics”, 1990
  * [111] Gal Raayoni et al.  “Generating conjectures on fundamental constants with the Ramanujan Machine”  In _Nature_ 590.7844 Nature Publishing Group UK London, 2021, pp. 67–73
  * [112] Siobhan Roberts  “AI Is Coming for Mathematics, Too.”  In _International New York Times_ International Herald Tribune, 2023
  * [113] Bernardino Romera-Paredes et al.  “Mathematical discoveries from program search with large language models”  In _Nature_ 625.7995 Nature Publishing Group UK London, 2024, pp. 468–475
  * [114] Frank Rosenblatt  “The perceptron: a probabilistic model for information storage and organization in the brain.”  In _Psychological review_ 65.6 American Psychological Association, 1958, pp. 386
  * [115] Tom Rudelius  “Learning to Inflate”  In _JCAP_ 02, 2019, pp. 044  DOI: 10.1088/1475-7516/2019/02/044
  * [116] Fabian Ruehle  “Data science applications to string theory”  In _Phys. Rept._ 839, 2020, pp. 1–117  DOI: 10.1016/j.physrep.2019.09.005
  * [117] Fabian Ruehle  “Evolving neural networks with genetic algorithms to study the String Landscape”  In _JHEP_ 08, 2017, pp. 038  DOI: 10.1007/JHEP08(2017)038
  * [118] Bertrand Russell and Alfred North Whitehead  “Principia Mathematica Vol. I”, 1910
  * [119] Adam Satariano and Megan Specia  “Governments Warn AI Poses Risk of ‘Catastrophic’ Harm.”  In _The New York Times_ The New York Times Company, 2023, pp. A6–A6
  * [120] Jeyan Thiyagalingam, Mallikarjun Shankar, Geoffrey Fox and Tony Hey  “Scientific machine learning benchmarks”  In _Nature Reviews Physics_ 4.6 Nature Publishing Group UK London, 2022, pp. 413–420
  * [121] Trieu H Trinh et al.  “Solving olympiad geometry without human demonstrations”  In _Nature_ 625.7995 Nature Publishing Group, 2024, pp. 476–482
  * [122] Vahe Tshitoyan et al.  “Unsupervised word embeddings capture latent knowledge from materials science literature”  In _Nature_ 571.7763 Nature Publishing Group UK London, 2019, pp. 95–98
  * [123] Alan M Turing  “Computing machinery and intelligence”  In _Mind_ LIX (236), 1950, pp. 433–460
  * [124] AM Turing  “On computable numbers, with an application to the Entscheidungsproblem. A correction”  In _Proc. LMS_ , 1938
  * [125] Silviu-Marian Udrescu and Max Tegmark  “AI Feynman: A physics-inspired method for symbolic regression”  In _Science Advances_ 6.16 American Association for the Advancement of Science, 2020
  * [126] Philip Wadler  “Programming language foundations in agda, https://github.com/agda/agda”  In _Formal Methods: Foundations and Applications: 21st Brazilian Symposium, SBMF 2018, Salvador, Brazil, Proc 21_ , 2018, pp. 56–73  Springer
  * [127] Zengzhi Wang, Rui Xia and Pengfei Liu  “Generative AI for Math: Part I – MathPile: A Billion-Token-Scale Pretraining Corpus for Math”, 2023  arXiv:2312.17120 [cs.CL]
  * [128] Alex Wilkins  “How AI mathematicians might finally deliver human-level reasoning”  In _New Scientist_ , 2024
  * [129] Geordie Williamson  “Is deep learning a useful tool for the pure mathematician?”, 2023  arXiv:2304.12602 [math.RT]
  * [130] Ludwig Wittgenstein  “Tractatus logico-philosophicus”, 1922
  * [131] Chai Wah Wu  “Can machine learning identify interesting mathematics? An exploration using empirically observed laws”  In _arXiv:1805.07431_ , 2018
  * [132] Yongjun Xu et al.  “Artificial intelligence: A powerful paradigm for scientific research”  In _The Innovation_ 2.4 Elsevier, 2021
  * [133] Richard Zach  “Hilbert’s program then and now”  In _Philosophy of logic_ Elsevier, 2007, pp. 411–447
  * [134] Wojciech Zaremba, Karol Kurach and Rob Fergus  “Learning to Discover Efficient Mathematical Identities”, 2014  arXiv:1406.1584 [cs.LG]
  * [135] Cedegao E. Zhang, Katherine M. Collins, Adrian Weller and Joshua B. Tenenbaum  “AI for Mathematics: A Cognitive Science Perspective”, 2023  arXiv:2310.13021 [q-bio.NC]