Your news feature “Does the proof stack up?” (Nature 424, 12; 200310.1038/424012a) — addressing the difficulties faced by mathematicians in verifying the computer-aided putative proof of Kepler's conjecture concerning the densest arrangement of spheres — contains the seemingly reasonable statement by one mathematician: “I believe in a proof if I understand it”. Yet it is well known that many a supposed understanding is a misunderstanding, and that there is no general rule guaranteed to distinguish between these opposite possibilities.

Arguments over the validity of mathematical proofs have raged for as long as mathematicians have been trying to prove conjectures. The history of this formal science is strewn with 'proofs' that have been accepted and later rejected, and 'proofs' that have been simultaneously accepted and rejected by different mathematicians.

The understanding of a computer program is no more reliable than the understanding of a mathematical proof.

Computers lure us into unmanageable complexity. It is, after all, one of their great assets that we can construct very long sequences of simple manipulations for a computer to execute at high speed and with unerring accuracy. However, this complexity (as with a list of alternative cases in a computer-aided proof) soon compounds, such that 'understanding' the program is impossible and a proof of the program is as complex as the program itself, or even more complex.

Scientists seek truths about the world (or about formal systems that mirror the world) and they do their best to confirm the validity of any new truth uncovered. But the reality is that (apart from certain limited totally abstract constructions) scientific truth can never be secure, validity can only be up to a certain point and formal verification is not a practical option. Scientific truths are best-guesses given all the available evidence, or, more pessimistically, they are potential errors as yet unmasked. In sum, computers add nothing new to the difficulties of proof and verification — they simply enlarge existing problems through the magnifying glass of complexity escalation.