The Existential Risk of Math Errors

submited by

Style Pass

2024-12-02 18:30:05

[For support of key website features (link annotation popups/popovers & transclusions, collapsible sections, backlinks, tablesorting, image zooming, sidenotes etc), you must enable JavaScript.]

How empirically certain can we be in any use of mathematical reasoning to make empirical claims? In contrast to errors in many other forms of knowledge such as medicine or psychology, which have enormous literatures classifying and quantifying error rates, rich methods of meta-analysis and pooling expert belief, and much one can say about the probability of any result being true, mathematical error has been rarely examined except as a possibility and a motivating reason for research into formal methods. There is little known beyond anecdotes about how often published proofs are wrong, in what ways they are wrong, the impact of such errors, how errors vary by subfield, what methods decrease (or increase) errors, and so on. Yet, mathematics is surely not immune to error, and for all the richness of the subject, mathematicians can usually agree at least informally on what has turned out to be right or wrong1, or good by other criteria like fruitfulness or beauty. Gaifman 2004 claims that errors are common but any such analysis would be unedifying:

An agent might even have beliefs that logically contradict each other. Mersenne believed that 267-1 is a prime number, which was proved false in 1903121ya , cf. Bell (195173ya ). [The factorization, discovered by Cole, is: 193,707,721 197945ya , 269–270). The explosion in the number of mathematical publications and research reports has been accompanied by a similar explosion in erroneous claims; on the whole, errors are noted by small groups of experts in the area, and many go unheeded. There is nothing philosophically interesting that can be said about such failures.2