Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes

• URL: http://arxiv.org/abs/2012.08298v1
• Date: Wed, 28 Oct 2020 19:43:14 GMT
• Title: Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes
• Authors: David H. Wolpert and David Kinney
• Abstract summary: 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.
• Score: 0.5874142059884521
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.

Related papers

• One Example Shown, Many Concepts Known! Counterexample-Driven Conceptual Reasoning in Mathematical LLMs [57.48325300739872]
  Leveraging mathematical Large Language Models for proof generation is a fundamental topic in LLMs research. We argue that the ability of current LLMs to prove statements largely depends on whether they have encountered the relevant proof process during training. Inspired by the pedagogical method of "proof by counterexamples" commonly used in human mathematics education, our work aims to enhance LLMs' ability to conduct mathematical reasoning and proof through counterexamples.
  arXiv  Detail & Related papers  (2025-02-12T02:01:10Z)
• Formal Mathematical Reasoning: A New Frontier in AI [60.26950681543385]
  We advocate for formal mathematical reasoning and argue that it is indispensable for advancing AI4Math to the next level. We summarize existing progress, discuss open challenges, and envision critical milestones to measure future success.
  arXiv  Detail & Related papers  (2024-12-20T17:19:24Z)
• Learning Formal Mathematics From Intrinsic Motivation [34.986025832497255]
  Minimo is an agent that learns to pose problems for itself (conjecturing) and solve them (theorem proving) We combine methods for constrained decoding and type-directed synthesis to sample valid conjectures from a language model. Our agent targets generating hard but provable conjectures - a moving target, since its own theorem proving ability also improves as it trains.
  arXiv  Detail & Related papers  (2024-06-30T13:34:54Z)
• 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)
• 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)
• Peano: Learning Formal Mathematical Reasoning [35.086032962873226]
  General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. We posit that central to both puzzles is the structure of procedural abstractions underlying mathematics. We explore this idea in a case study on 5 sections of beginning algebra on the Khan Academy platform.
  arXiv  Detail & Related papers  (2022-11-29T01:42:26Z)
• NaturalProver: Grounded Mathematical Proof Generation with Language Models [84.2064569475095]
  Theorem proving in natural mathematical language plays a central role in mathematical advances and education. We develop NaturalProver, a language model that generates proofs by conditioning on background references. NaturalProver is capable of proving some theorems that require short (2-6 step) proofs, and providing next-step suggestions that are rated as correct and useful over 40% of the time.
  arXiv  Detail & Related papers  (2022-05-25T17:01:18Z)
• 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)
• 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)
• Epistemic Phase Transitions in Mathematical Proofs [0.0]
  We show that under a cognitively-plausible belief formation mechanism, belief in mathematical arguments can undergo a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. Our results bear both on recent work in the history and philosophy of mathematics on how we understand proofs, and on a question, basic to cognitive science, of how we justify complex beliefs.
  arXiv  Detail & Related papers  (2020-03-31T18:39:56Z)

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.