LangMathKnow08 - A language for mathematical knowledge...

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.

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.