# A Modern Retrospective on Probabilistic Numerics

## TL;DR

This paper reviews the development of probabilistic numerics, tracing its historical roots, key contributions, and current state, emphasizing its maturation and future challenges in the context of applications and formal frameworks.

## Contribution

It provides a comprehensive retrospective on probabilistic numerics, connecting early ideas to modern formal treatments and highlighting its evolution and current research landscape.

## Key findings

- Historical roots traced to 1960s contributions
- Probabilistic numerics has matured through paradigms like average-case analysis
- Identifies future challenges in the research field

## Abstract

This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul'din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.