On the distribution of time-to-proof of mathematical conjectures

TL;DR

This paper investigates the distribution of time-to-proof for mathematical conjectures, exploring whether it can reveal insights into mathematical productivity and conjecture resolution timelines, but finds data limitations and non-stationarity hinder definitive conclusions.

Contribution

It proposes a methodology combining recurrent process mathematics with empirical data to analyze conjecture proof timings, highlighting challenges in modeling due to non-stationarity and dataset incompleteness.

Findings

Data suggests an exponential proof rate of 0.01/year

Average conjecture proof time estimated at 100 years

Current data cannot definitively determine the true distribution

Abstract

What is the productivity of Science? Can we measure an evolution of the production of mathematicians over history? Can we predict the waiting time till the proof of a challenging conjecture such as the P-versus-NP problem? Motivated by these questions, we revisit a suggestion published recently and debated in the "New Scientist" that the historical distribution of time-to-proof's, i.e., of waiting times between formulation of a mathematical conjecture and its proof, can be quantified and gives meaningful insights in the future development of still open conjectures. We find however evidence that the mathematical process of creation is too much non-stationary, with too little data and constraints, to allow for a meaningful conclusion. In particular, the approximate unsteady exponential growth of human population, and arguably that of mathematicians, essentially hides the true distribution. Another issue is the incompleteness of the dataset available. In conclusion we cannot really reject the simplest model of an exponential rate of conjecture proof with a rate of 0.01/year for the dataset that we have studied, translating into an average waiting time to proof of 100 years. We hope that the presented methodology, combining the mathematics of recurrent processes, linking proved and still open conjectures, with different empirical constraints, will be useful for other similar investigations probing the productivity associated with mankind growth and creativity.