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LangMathKnow08 - A language for mathematical knowledge...

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arXiv:0805.1386v2 [cs.LO] 26 Aug 2008 A language for mathematical knowledge management Steven Kieffer 1 , Jeremy Avigad 2 , and Harvey Friedman 3 1 Simon Fraser University 2 Carnegie Mellon University 3 Ohio State University Abstract. We argue that the language of Zermelo Fraenkel set the- ory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowl- edge. We then describe a syntactic sugaring of that language that pro- vides a way of writing remarkably readable assertions without straying far from the set-theoretic semantics. We illustrate with some examples of formalized textbook definitions from elementary set theory and point-set topology. We also present statistics concerning the complexity of these definitions, under various complexity measures. 1 Introduction With the growing use of digital means of storing, communicating, accessing, and manipulating mathematical knowledge, it becomes important to develop appro- priate formal languages for the representation of such knowledge. But the scope of “mathematical knowledge” is broad, and the meaning of the word “appropri- ate” will vary according to the application. At the extremes, there are competing desiderata: At the foundational level , one wants a small and simple syntax, and a precise specification of its semantics. In particular, one wants a specification as to which inferences are valid. At the human level , one wants to have mathematical languages that are as easy to read and understand as ordinary mathematical texts, yet also admit a precise interpretation to the foundational level. For ordinary working mathematicians, the foundational interpretation is largely irrelevant, but some sort of formal semantics is necessary if the information encoded in mathematical texts is to be used and manipulated at the formal level. Of course, one solution is simply to pair each informal mathematical assertion with a formal translation, but then there is the problem of obtaining the formal translations and ensuring that they match the intention of the informal text. As a result, it is more promising to use semi-structured languages that integrate Work by Avigad and Friedman partially supported by NSF grant DMS-0700174.
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features of both the foundational and human levels. This results in a smooth spectrum of languages in between the two extremes. At intermediate “expert user” levels, one may want a language whose structure is close to that of the underlying foundational framework, yet is as humanly readable as possible. To complicate matters, there are features of mathematical knowledge that are not captured at the level of assertions: mathematical language is used to communicate definitions, theorems, proofs, algorithms, and problems, among other things. At the level of a mathematical theory, language is also used to communicate relationships between these different types of data. The formal information that is relevant will vary depending on the application one has in mind, be it database access and search, theorem proving, formal verification, etc.
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