Abstract
Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL. Harrison’s technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL’s axiomatic type classes. However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism.
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Italics in original.
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Italics in original.
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In fact the relevant proposition, ∗ 54 ⋅ 43, is a statement about sets. Many of the propositions laboriously worked out here are elementary identities that are trivial to prove with modern automation.
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Punctured means that one point is removed.
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Malicious code is another matter. In HOL Light, one can use OCaml’s String.set primitive to replace T (true) by F. Given the variety of loopholes in programming languages and systems, not to mention notational trickery, we must be content with defences against mere incompetence.
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Because the HOL Light libraries were ported en masse, corresponding theorems generally have similar names and forms.
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Acknowledgements
Dedicated to Michael J C Gordon FRS, 1948–2017. The development of HOL and Isabelle has been supported by numerous EPSRC grants. The ERC project ALEXANDRIA supports continued work on the topic of this paper. Many thanks to Jeremy Avigad, Johannes Hölzl, Neel Krishnaswami, Andrew Pitts, Andrei Popescu and the anonymous referee for their comments.
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Paulson, L.C. (2019). Formalising Mathematics in Simple Type Theory. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_20
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