Communications of the ACM,
Vol. 59 No. 4, Pages 13-15
10.1145/2892710
Over the past two decades, mathematicians have succeeded in bringing computers to bear on the development of proofs for conjectures that have lingered for centuries without solution. Following a small number of highly publicized successes, the majority of mathematicians remain hesitant to use software to help develop, organize, and verify their proofs.
Yet concerns linger over usability and the reliability of computerized proofs, although some see technological assistance as being vital to avoid problems caused by human error.
1 Comments
Scott Cotton
As a practitioner using proof tools I appreciate in particular 2 (at once somewhat contradictory, somewhat synergistic) sides to this discussion. First is the distinction between mathematical knowledge, i.e. understanding by people, and the existence of a (finite) formal proof or tools to find and check them. In the end, I believe the former is much more valuable than the later and the most fruitful path may be to keep our eyes open for how formal proofs augment our understanding of mathematical problems, and turn away from blind reliance on formal proofs for correctness. For example, formal proofs (and counterexamples) can increase our confidence in results, but then those results may become banal, so we can turn toward more interesting problems. Searching the space of formal proofs may help our understanding about a given system, or it may find a trace to analyse. Perhaps searching the theory of ZFC for small sentences that have no short proof would be interesting, because it might find something counterintuitive but true, or something otherwise somewhat questionable.
The second side is simply that humans are in some ways much more error prone and our ideas fundamentally limited by experience than computers. As a programmer, I know that people make logical mistakes all the time, and as a scholar I know mathematical (and computing) history is full of flaws and misconceptions and a myriad of other ideas which, over time, have become trivial.
So in a sense, computer based reasoning is more authoritative, but more banal.
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Scott Cotton
As a practitioner using proof tools I appreciate in particular 2 (at once somewhat contradictory, somewhat synergistic) sides to this discussion. First is the distinction between mathematical knowledge, i.e. understanding by people, and the existence of a (finite) formal proof or tools to find and check them. In the end, I believe the former is much more valuable than the later and the most fruitful path may be to keep our eyes open for how formal proofs augment our understanding of mathematical problems, and turn away from blind reliance on formal proofs for correctness. For example, formal proofs (and counterexamples) can increase our confidence in results, but then those results may become banal, so we can turn toward more interesting problems. Searching the space of formal proofs may help our understanding about a given system, or it may find a trace to analyse. Perhaps searching the theory of ZFC for small sentences that have no short proof would be interesting, because it might find something counterintuitive but true, or something otherwise somewhat questionable.
The second side is simply that humans are in some ways much more error prone and our ideas fundamentally limited by experience than computers. As a programmer, I know that people make logical mistakes all the time, and as a scholar I know mathematical (and computing) history is full of flaws and misconceptions and a myriad of other ideas which, over time, have become trivial.
So in a sense, computer based reasoning is more authoritative, but more banal.