Apriori Knowledge in an Era of Computational Opacity: The Role of AI in Mathematical Discovery

The current state-of-the-art in artificial intelligence is impressive, especially in terms of mastery of language, but not so much in terms of mathematical reasoning. What could be missing? Can we learn something useful about that gap from how the brains of mathematicians go about their craft? This essay builds on the idea that current deep learning mostly succeeds at system 1 abilities -- which correspond to our intuition and habitual behaviors -- but still lacks something i...

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003).

It has been suggested that large language models such as GPT-4 have acquired some form of understanding beyond the correlations among the words in text including some understanding of mathematics as well. Here, we perform a critical inquiry into this claim by evaluating the mathematical understanding of the GPT-4 model. Considering that GPT-4's training set is a secret, it is not straightforward to evaluate whether the model's correct answers are based on a mathematical under...

As artificial intelligence (AI) gains greater adoption in a wide variety of applications, it has immense potential to contribute to mathematical discovery, by guiding conjecture generation, constructing counterexamples, assisting in formalizing mathematics, and discovering connections between different mathematical areas, to name a few. While prior work has leveraged computers for exhaustive mathematical proof search, recent efforts based on large language models (LLMs) asp...

We review the recent programme of using machine-learning to explore the landscape of mathematical problems. With this paradigm as a model for human intuition - complementary to and in contrast with the more formalistic approach of automated theorem proving - we highlight some experiments on how AI helps with conjecture formulation, pattern recognition and computation.

We currently witness the spectacular success of artificial intelligence in both science and public life. However, the development of a rigorous mathematical foundation is still at an early stage. In this survey article, which is based on an invited lecture at the International Congress of Mathematicians 2022, we will in particular focus on the current "workhorse" of artificial intelligence, namely deep neural networks. We will present the main theoretical directions along wit...

Philosophers have recently focused on critical, epistemological challenges that arise from the opacity of deep neural networks. One might conclude from this literature that doing good science with opaque models is exceptionally challenging, if not impossible. Yet, this is hard to square with the recent boom in optimism for AI in science alongside a flood of recent scientific breakthroughs driven by AI methods. In this paper, I argue that the disconnect between philosophical p...

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful...

A new viewpoint of the G\"odel's incompleteness theorem be given in this article which reveals the deep relationship between the logic and computation. Upon the results of these studies, an algorithm be given which shows how to search a proof of statement in first order logic from finite concrete examples, and an approach be proposed to improve searching mathematical proof by neural network.

Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof assistants now make it possible to encode mathematical knowledge in digital form. This article enumerates some of the ways that these and related technologies can help us do mathematics.