Abstract
The Wolfram Knowledgebase, powered by the entity framework, contains expertly curated data from thousands of diverse domains. We have begun expanding this framework to include mathematical knowledge, making significant strides in the representation of results pertaining to continued fractions, function spaces, and most recently, topology. This paper will focus on our progress in the representation of general topology. We have curated over 700 entities representing concept definitions, theorem statements, and concrete topological spaces, as well as their corresponding properties, including their formal representations as well as references, computed properties, and other metadata. Virtually every formal representation in this project required extensions to the Wolfram Language, mostly for basic set theory. We will outline all of these design choices by way of examples, as well as present additional functionality for querying, usage messages, formatting, and other computations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
You can see that the standard form for an entity is a box containing a string called the entity’s common name. The common name is a property that is available for all entities.
- 2.
There is some interest in introducing a symbolic type language within the Wolfram Language suitable for communicating basic information between proof assistants and Mathematica. This could be used, for example, to use Mathematica’s computer algebra functionalities to assist these proof assistants (see Lewis [7]).
- 3.
We say “essentially” because Wolfram Language syntax is very permissive. Syntactically valid expressions can semantically represent statements in a first-order theory, but are not guaranteed to do so in all cases. For example, ForAll, is intended to be used a first-order quantifier, and functions like Resolve use heuristics (such as assuming variables appearing in inequalities are real numbers) to attempt to remove quantifiers, but ForAll is not axiomitized. The argument structure allows for arbitrary expressions to be passed as arguments, in which cases the expression remains in an unevaluated symbolic form.
- 4.
This is accomplished by defining a default function used to calculate the common name from the canonical name, but which can be manually overwritten.
- 5.
One might ask why these properties even have different names from the corresponding concept properties. After all, under the Curry-Howard correspondence, a universally quantified proposition is essentially a function type. For example, \(\forall _{\texttt {x,x}\in \texttt {Reals}} \texttt {x}^2 \ge \texttt {0}\) can be viewed as a function which takes a real number x and outputs a proof that \(\texttt {x}^2\ge \texttt {0}\). This is natural and powerful in languages such as Lean [9] which are based on dependent type theory, but as the Wolfram Language does not have a type system, this equivalence is not made.
- 6.
The \(\pi \)-Base project [3] uses a similar representation of topological property relations and uses them under the hood to deduce topological properties from the properties asserted of their many example topological spaces, but to our knowledge they do not provide an easy way for users to do their own computations as we do here.
References
Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pa̧k, K., Urban, J.: Mizar: state-of-the-art and beyond. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS, vol. 9150, pp. 261–279. Springer, Cham (2015). doi:10.1007/978-3-319-20615-8_17
Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development - Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-662-07964-5
Dabbs, J.: \(\pi \)-Base. https://topology.jdabbs.com/
Harrison, J.: HOL light: a tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996). doi:10.1007/BFb0031814
Iancu, M.: Towards Flexiformal Mathematics. Ph.D. thesis, Jacobs University (2016)
Kieffer, S., Avigad, J., Friedman, H.: A language for mathematical knowledge management, January 2011. https://arxiv.org/abs/0805.1386
Lewis, R.Y.: An extensible ad hoc interface between Lean and Mathematica. http://www.andrew.cmu.edu/user/rlewis1/leanmm/leanmm_public_draft.pdf
Megill, N.: Metamath: A Computer Language for Pure Mathematics. Lulu Press, Morrisville
de Moura, L., Kong, S., Avigad, J., van Doorn, F., von Raumer, J.: The Lean Theorem Prover (system description). https://leanprover.github.io/papers/system.pdf
Munkres, J.: Topology, 2nd edn. Prentice Hall, Upper Saddle River (2000)
Nipkow, T., Klein, G.: Concrete Semantics - With Isabelle/HOL. Springer, Cham (2014). http://dx.doi.org/10.1007/978-3-319-10542-0
Steen, L., Seebach, J.A.: Counterexamples in Topology, 2nd edn. Springer, New York (1978)
Wolfram Research Inc.: Wolfram Data Repository. https://datarepository.wolframcloud.com/
Wolfram Research Inc.: Wolfram Knowledgebase. https://www.wolfram.com/knowledgebase/
Wolfram Research Inc.: The Wolfram Language. https://www.wolfram.com/language/
Wolfram Research Inc.: Wolfram Language & System Documentation Center. Assumptions and Domains. http://reference.wolfram.com/language/guide/AssumptionsAndDomains.html
Wolfram Research Inc.: Wolfram Language & System Documentation Center. Expressions. https://reference.wolfram.com/language/tutorial/ExpressionsOverview.html
Wolfram Research Inc.: Wolfram Language & System Documentation Center. Knowledge Representation & Access. https://reference.wolfram.com/language/guide/KnowledgeRepresentationAndAccess.html
Wolfram Research Inc.: Wolfram Language & System Documentation. Logic & Boolean Algebra. https://reference.wolfram.com/language/guide/LogicAndBooleanAlgebra.html
Wolfram Research Inc.: Wolfram\(|\)Alpha. https://www.wolframalpha.com/
Acknowledgements
I thank CICM, Michael Trott, and Wolfram Research for this opportunity, as well as Stephen Wolfram, Michael Trott, James Mulnix, Eric Weisstein, Robert Lewis, and José Martín-García for their support and great work on pure mathematics in the Wolfram Language.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Ford, I. (2017). Semantic Representation of General Topology in the Wolfram Language. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds) Intelligent Computer Mathematics. CICM 2017. Lecture Notes in Computer Science(), vol 10383. Springer, Cham. https://doi.org/10.1007/978-3-319-62075-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-62075-6_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62074-9
Online ISBN: 978-3-319-62075-6
eBook Packages: Computer ScienceComputer Science (R0)