Software Diagnostics and Observability Institute

Mathematical arguments and machine checked evidence

PDF: https://www.dumpanalysis.org/files/History_of_Diagnostics_Observability_...

In this book, we consider the history of diagnostics, observability, and debugging in formal sciences, from recorded calculation procedures and geometrical arguments to automated reasoning and machine-checked mathematics. We follow how mathematical work became available for inspection, how defects were exposed, and how proposed corrections were checked.

The history connects logic, probability and statistics, numerical analysis, information theory, control theory, optimization, and the foundations of computation. Proofs, residuals, error syndromes, state models, counterexample traces, and solver certificates provide different kinds of evidence. We examine what can be inferred from them, which assumptions they depend on, and where available observations leave alternatives indistinguishable.

A failed proof attempt does not necessarily refute a theorem. A successful calculation can answer the wrong question. Such distinctions guide our analysis of historical examples, including foundational paradoxes, the four-color problem, and large formal proofs. Connections to our pattern-oriented software diagnostics show how context, invariants, missing evidence, and changes between artifact versions can help organize an investigation.

Seven parts and 35 chapters are complemented by a chronology, glossary, evidence taxonomy, and 82 references. The book is intended for readers interested in the history of mathematical reasoning and its practical connections to diagnosis, verification, and debugging.