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Problem-Solving Before Instruction PS-I: A Protocol for Assessment and Intervention in Students with Different Abilities

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Published on: September 11, 2021

Epistemic phase transitions in mathematical proofs.

Scott Viteri¹, Simon DeDeo²

¹Department of Computer Science, Stanford University, Stanford, CA 94305, USA; Social & Decision Sciences, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA.

|April 11, 2022

Summary

Mathematicians gain confidence in complex proofs through a cognitive process combining deduction and abduction. This mechanism allows for a rapid shift from uncertainty to certainty, even with minor errors in mathematical arguments.

Keywords:

Belief formation Mathematical cognition Networks Philosophy of mathematics Psychology of mathematics

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Area of Science:

Cognitive Science
Philosophy of Mathematics
Formalized Reasoning

Background:

Mathematical proofs represent highly explicit arguments, yet their complexity increases the probability of errors.
Understanding how mathematicians develop confidence in complex proofs is a fundamental question in cognitive science and the philosophy of mathematics.

Purpose of the Study:

To investigate the cognitive mechanisms underlying belief formation in mathematical arguments.
To identify how confidence in proofs transitions from uncertainty to certainty.
To analyze the impact of error rates on the acceptance of mathematical claims.

Main Methods:

Analysis of a dataset comprising 48 machine-aided proofs from the Coq formalized reasoning system.
Inclusion of 5 hand-constructed proofs, including historical examples like Euclid and Fermat's Last Theorem.
Modeling a cognitively-plausible belief formation mechanism integrating deductive and abductive reasoning.

Main Results:

Demonstrated an 'epistemic phase transition' where belief shifts dramatically from uncertainty to confidence.
This transition occurs at reasonable claim-to-claim error rates within the analyzed proofs.
The proposed belief formation mechanism explains how complex mathematical arguments are accepted.

Conclusions:

The study provides insights into how humans justify complex beliefs, using mathematical proofs as a case study.
Results inform the history and philosophy of mathematics regarding the understanding and acceptance of proofs.
The findings contribute to cognitive science by modeling a plausible mechanism for belief justification.