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9 karma · joined July 30, 2019
Sure, programs are proofs in the context of Curry-Howard. But not necessarily proofs of what you want them to prove.
One issue that it glosses over is that in mathematics, the object of study is a-priori truths (and mathematicians are usually Platonists). I would express this as saying that math is interested in knowing what's true. So obviously it is applicable to proving things about programs. So far so good.
But programming involves a lot of work which could be described as a-posteriori. As an example on dictionary.com puts it, "an a posteriori argument [...] derives the theory from the evidence". Programmers are designers, wrestling sense out of complex and sometimes poorly expressed specifications, requirements, and realities. This doesn't map onto mathematics: it's neither (in Gowers' terms) theory building nor problem solving, because mathematical theory building is an a-priori business dealing reflexively with mathematical tools, not with theories of the outside reality. A typical large software system is an unwieldy, organic thing, to which mathematically formulated theories apply in the same way as they do to biological organisms. Sometimes math can describe aspects a complex system well, but it can't tell you how to build it, any more than it can tell you how to build the Parthenon.