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

• URL: http://arxiv.org/abs/2403.15437v1
• Date: Fri, 15 Mar 2024 21:38:26 GMT
• Title: Apriori Knowledge in an Era of Computational Opacity: The Role of AI in Mathematical Discovery
• Authors: Eamon Duede, Kevin Davey,
• Abstract summary: We argue that if a proof-checker is attached to such machines, then we can obtain apriori mathematical knowledge from them. Many accept that we can acquire genuine mathematical knowledge of the Four Color Theorem from Appel and Haken's program.
• Score: 0.7673339435080445
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Computation is central to contemporary mathematics. Many accept that we can acquire genuine mathematical knowledge of the Four Color Theorem from Appel and Haken's program insofar as it is simply a repetitive application of human forms of mathematical reasoning. Modern LLMs / DNNs are, by contrast, opaque to us in significant ways, and this creates obstacles in obtaining mathematical knowledge from them. We argue, however, that if a proof-checker automating human forms of proof-checking is attached to such machines, then we can obtain apriori mathematical knowledge from them, even though the original machines are entirely opaque to us and the proofs they output are not human-surveyable.

Related papers

• Artifical intelligence and inherent mathematical difficulty [0.0]
  We first present an updated version of a traditional argument that limitative results from computability and complexity theory show that proof discovery is an inherently difficult problem. We then illustrate how several recent applications of artificial intelligence-inspired methods do indeed raise novel questions about the nature of mathematical proof.
  arXiv  Detail & Related papers  (2024-08-01T20:08:31Z)
• Machine learning and information theory concepts towards an AI Mathematician [77.63761356203105]
  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. This essay builds on the idea that current deep learning mostly succeeds at system 1 abilities. It takes an information-theoretical posture to ask questions about what constitutes an interesting mathematical statement.
  arXiv  Detail & Related papers  (2024-03-07T15:12:06Z)
• AI for Mathematics: A Cognitive Science Perspective [86.02346372284292]
  Mathematics is one of the most powerful conceptual systems developed and used by the human species. Rapid progress in AI, particularly propelled by advances in large language models (LLMs), has sparked renewed, widespread interest in building such systems.
  arXiv  Detail & Related papers  (2023-10-19T02:00:31Z)
• Algorithm-assisted discovery of an intrinsic order among mathematical constants [3.7689882895317037]
  We develop a computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields unify thousands of existing formulas, generate infinitely many new formulas, and lead to unexpected relations between different mathematical constants.
  arXiv  Detail & Related papers  (2023-08-22T23:27:47Z)
• A Survey of Deep Learning for Mathematical Reasoning [71.88150173381153]
  We review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. Recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning.
  arXiv  Detail & Related papers  (2022-12-20T18:46:16Z)
• JiuZhang: A Chinese Pre-trained Language Model for Mathematical Problem Understanding [74.12405417718054]
  This paper aims to advance the mathematical intelligence of machines by presenting the first Chinese mathematical pre-trained language model(PLM) Unlike other standard NLP tasks, mathematical texts are difficult to understand, since they involve mathematical terminology, symbols and formulas in the problem statement. We design a novel curriculum pre-training approach for improving the learning of mathematical PLMs, consisting of both basic and advanced courses.
  arXiv  Detail & Related papers  (2022-06-13T17:03:52Z)
• NaturalProofs: Mathematical Theorem Proving in Natural Language [132.99913141409968]
  We develop NaturalProofs, a multi-domain corpus of mathematical statements and their proofs. NaturalProofs unifies broad coverage, deep coverage, and low-resource mathematical sources. We benchmark strong neural methods on mathematical reference retrieval and generation tasks.
  arXiv  Detail & Related papers  (2021-03-24T03:14:48Z)
• Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes [0.5874142059884521]
  We present a computational model of mathematical reasoning according to which mathematics is a fundamentally process. We show that this framework gives a compelling account of several aspects of mathematical practice.
  arXiv  Detail & Related papers  (2020-10-28T19:43:14Z)
• A Finitist's Manifesto: Do we need to Reformulate the Foundations of Mathematics? [1.384477926572109]
  This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary paradise take heed. Much of mathematics relies upon (i) the "existence" of objects that contain an infinite number of elements, (ii) our ability, "in theory", to compute with an arbitrary level of precision, or (iii) our ability, "in theory", to compute for an arbitrarily large number of time steps.
  arXiv  Detail & Related papers  (2020-09-14T14:44:08Z)
• Generative Language Modeling for Automated Theorem Proving [94.01137612934842]
  This work is motivated by the possibility that a major limitation of automated theorem provers compared to humans might be addressable via generation from language models. We present an automated prover and proof assistant, GPT-f, for the Metamath formalization language, and analyze its performance.
  arXiv  Detail & Related papers  (2020-09-07T19:50:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.