22.1.6 AM and degrees of mathematical novelty

22.1.6 AM and degrees of mathematical novelty

The question “Could a machine invent prime numbers?” is a cleaner mathematical analogue of the Einstein test. Merely inventing a new word is not enough. The system must define a concept not explicitly supplied to it, recognize why the concept is interesting, connect it to existing operations, and use it to formulate consequences that were not built into the prompt.

Lenat’s Automated Mathematician, AM, is an important early case [ Lenat , 1976 ] . AM began with 115 incomplete modules for elementary set-theoretic concepts and roughly 250 heuristic rules. It was not given numbers or proof as primitives. Guided by an agenda and concept-interest heuristics, it reconstructed natural numbers, multiplication, factors, and prime numbers. It then formulated familiar claims including unique factorization and Goldbach’s conjecture. AM was designed as a concept and conjecture generator rather than a theorem prover, so these claims were not accompanied by proofs.

This is genuine evidence against an overly simple identification of machine novelty with statistical compression. AM enlarged a structured vocabulary, created active concept modules, and explored their relations. Its search was guided by mathematical heuristics and estimates of interestingness, not by a single code-length objective. From the system’s perspective, primes were a new concept even though they were a rediscovery from the human perspective.

The example also qualifies the claim. AM’s mathematical seed was small, but its inductive bias was not: its supplied concepts, fixed facets, and hundreds of heuristics encoded substantial retrospective knowledge about productive mathematical search. Lenat reported that AM produced no mathematics that was wholly new to humankind on its own, and that its performance deteriorated as it moved farther from its initial conceptual base. Later methodological criticism likewise questioned how much of the reported behavior could be attributed to general principles rather than the program’s detailed engineering [ Ritchie and Hanna , 1984 ] .

LINCS can separate three levels that are easily conflated. Lexical novelty coins a name. Definitional extension adds a concept constructible from the current declaration, as primes can be isolated through factorization. Abductive declaration change introduces primitives or structural commitments not already generated by the available concept language. AM demonstrated substantial definitional extension; whether its heuristics constituted a mechanism for the third level remains a different question.

AM’s architecture also anticipates the LINCS separation between proposal and admission. A new concept module resembles an extension of a declaration; a conjecture declares a candidate composition; examples and counterexamples supply observations; and proof would provide a stronger admission certificate. AM was powerful on the proposal side and deliberately incomplete on the proof side. A LINCS mathematical-discovery system would retain that generative openness while requiring explicit transport, consistency, and proof obligations before a conjectural extension becomes admitted theory.