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# Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration

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Christoph Lange's Ph.D. defense (2011-03-11)

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### Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration

1. 1. Overview Service Integration Knowledge Representation Conclusion & Future Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration Christoph Lange Jacobs University, Bremen, Germany KWARC – Knowledge Adaptation and Reasoning for Content 2011-03-11 Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 1
2. 2. Overview Service Integration Knowledge Representation Conclusion & FutureWhy Mathematics? Mathematics ubiquitous foundation of science, technology, and engineering these have in common: rigorous style of argumentation symbolic formula language similar process of understanding results Mathematical Knowledge complex structures . . . that have been well studied and understood Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 2
3. 3. Overviewtime. Note factoring a quartic into two real quadratics is different than sign each Service Integration Knowledge Representation Conclusion & Future trying to ﬁnd four complex roots. Deﬁnition: A function f is analytic on an open subset R ⊂ C if f is complexSemiformal Mathematical Knowledge differentiable everywhere on R; f is entire if it is analytic on all of C.2 Proof of the Fundamental Theorem via Liouville Theorem 2.1 (Liouville). If f (z) is analytic and bounded in the complex plane, then f (z) is constant.Informal We now prove Theorem 2.2 (Fundamental Theorem of Algebra). Let p(z) be a polynomial Formalized = Computerized with complex coefﬁcients of degree n. Then p(z) has n roots. Proof. It is sufﬁcient to show any p(z) has one root, for by division we can then write p(z) = (z − z0 )g(z), with g of lower degree. Note that if p(z) = an z n + an−1 z n−1 + · · · + a0 , (2) then as |z| → ∞, |p(z)| → ∞. This follows as an−1 a0 p(z) = z n · an + + ··· + n . (3) z z 1 Assume p(z) is non-zero everywhere. Then p(z) is bounded when |z| ≥ R. 1 1 Also, p(z) = 0, so p(z) is bounded for |z| ≤ R by continuity. Thus, p(z) is a bounded, entire function, which must be constant. Thus, p(z) is constant, a contradiction which implies p(z) must have a zero (our assumption). [Lev] Semiformal – a pragmatic and practical compromise 2 anything informal that is intended to or could in principle be formalized combinations of informal and formal for both human and machine audience Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 3
4. 4. Overview Service Integration Knowledge Representation Conclusion & FutureCollaboration in Mathematics History of collaboration in the small: Hardy/Littlewood in the large: hundreds of mathematicians classifying the finite simple groups “industrialization” of research Utilizing the Social Web research blogs: Baez, Gowers, Tao Polymath: collaborative proofs Collaboration = creation, formalization, organization, understanding, reuse, application Polymath wiki/blog: P ≠ NP proof Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 4
5. 5. Overview Service Integration Knowledge Representation Conclusion & FutureAn Integrated View on a Collaboration WorkflowThe author(s): The reader(s): The reviewer(s): 0 original idea (in one’s “What does that 1 read paper (← ) mind) mean?”: missing 2 verify claims background, 1 formalize into used to different 3 point out problems structured document notation with the paper and 2 search existing its formal concepts “How does that knowledge to build work?” on “What is that good 3 validate formal for?” structure look up background 4 present in a information in cited comprehensible way publications 5 submit for review Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 5
6. 6. Overview Service Integration Knowledge Representation Conclusion & FutureLooking up Background Knowledge “What does that mean?” Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 6
7. 7. Overview Service Integration Knowledge Representation Conclusion & FutureAdapting the Presentation to FamiliarTerminology “What does that mean?” – here: unfamiliar unit system (imperial vs. metric) Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 7
9. 9. Overview Service Integration Knowledge Representation Conclusion & FuturePointing out and Discussing Problems Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 9
10. 10. Overview Service Integration Knowledge Representation Conclusion & FutureCollaboration Still has to be Enabled! Many collaboration tasks not currently well supported by machines For other tasks there is (limited) support creating and formalizing documents – semiformal!? search existing knowledge to build on – semiformal!? computation (recall unit conversion) – but not inside documents publishing in textbook style – could it be more comprehensible? adapting notation (e.g. ⋅ ×, n k Cn ) – not quite on demand k Existing machine services only focus on primitive tasks Can’t simply be put together, as they . . . . . . speak different languages . . . take different perspectives on knowledge Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 10
11. 11. Overview Service Integration Knowledge Representation Conclusion & FutureDocument Perspective: XML Markup XHTML+MathML(+OpenMath) ... is [itex] <mn>9144</mn> <mo>&InvisibleTimes;</mo> <mo>m</mo> [/itex] from city ... Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 11