Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

350 jeremy avigadexpression z/n, we are thinking of n as a complex number as well. But theabsolute value function converts a complex value to a real value, so expressionslike |z/n| denote real numbers, and many of the products and sums in thatproof denote real multiplication and addition. In the proof of the lemma, it iscrucial that we keep track of this last fact: the ≤ ordering only makes sense onthe real numbers, so to invoke properties of the ordering we need to knowthat the relevant expressions are real-valued.The elaborate structure of implicit inferences comes to the fore when we tryto formalize such reasoning. With a formal verification system, these inferencesneed to be spelled out in detail, and the result can be painstaking and tedious(see Avigad et al., 2007). In Isabelle, for example, one has to use a functionreal(n) to cast a natural number, n, as a real. Coq has mechanisms that apply such‘coercions’ automatically, but, in both cases, the appropriate simplificationsand verifications are rarely carried out as automatically as one would like.The fact that methods of reasoning that we are barely conscious of when weread a mathematical proof requires so much effort to formalize is one of thescandals of formal verification, and a clear sign that more thought is needed asto how we understand such inferences. I suspect that the structural perspectivedescribed in McLarty’s article, combined with the locale mechanisms describedin Section 12.7, holds the germ of a solution. When we prove theorems aboutthe natural numbers, that is the structure of interest; but when we identifythe natural numbers as a substructure of the reals, we are working in anexpanded locale, where both structures are present and interact. To start with,everything we know in the context of the natural numbers and the real numbersindividually is imported to the combined locale, and is therefore available tous. But there are also new facts and procedures that govern the combinationof the two domains. Figuring out how to model such an understanding so thatproof assistants can verify the proof of Lemma 1, as it stands, will go a longway in explaining how we understand proofs that make use of mixed domains.12.10 ConclusionsI have described four types of inference that are found in ordinary mathematicalproofs, and considered some of the logical and computational methods thathave been developed to verify them. I have argued that these efforts are notjust pragmatic solutions to problems of engineering; rather, they address coreissues in the epistemology of mathematics, and should be supported by broaderphilosophical reflection.