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Formal Methods and Mathematical Intuition

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Protocols, Strands, and Logic

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13066))

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Abstract

The paper offers a retrospective on earlier developments and work in formal methods at The MITRE Corporation, emphasizing the leading work of Joshua Guttman and some of his colleagues. It then provides a short introduction to dimension theory and its history as a methodologically contrasting development in mathematics in which counterintuitive examples and counterexamples play a role in motivating mathematical growth by means of conjectures and refutations or proofs and refutations. The paper ends with a broad methodological comparison between developments in formal methods and developments in domains of mathematics like dimension theory in which mathematical intuition and proofs and refutations play a significant role.

Approved for Public Release; Distribution Unlimited. Public Release Case Number 21–2577. The author’s affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE’s concurrence with, or support for, the positions, opinions, or viewpoints expressed by the author. ©2021 The MITRE Corporation. ALL RIGHTS RESERVED.

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Notes

  1. 1.

    See my papers [19,20,21]. I am writing a book on dimension theories and their history deriving from these papers and including much new material.

  2. 2.

    In German: Lässt sich eine Fläche (etwa ein Quadrat mit Einschluss der Begrenzung) eindeutig auf eine Linie (etwa eine gerade Strecke mit Einschluss der Endpunkte) eindeutig beziehen, so dass zu jedem Puncte der Fläche ein Punct der Linie und umgekehrt zu jedem Puncte der Linie ein Punct der Fläche gehört?

  3. 3.

    In German: ... dass Flächen, Körper, ja selbst stetige Gebilde von \(\rho \) Dimensionen sich eindeutig zuordnen lassen stetigen Linien, also Gebilden von nur einer Dimension, dass also Flächen, Körper, ja sogar Gebilde von \(\rho \) Dimensionen, dieselbe Mächtigkeit haben, wie Curven ... .

  4. 4.

    In French: ...je le vois, mais je ne le crois pas.

  5. 5.

    See Kennedy’s statement [22, page 7]: ‘Peano was so proud of this discovery that he had one of the curves in the sequence put on the terrace of his home, in black tiles on white.’.

  6. 6.

    Compare [22] for English translations of some of Peano’s works and notes on and appraisals of his mathematics.

  7. 7.

    In German: Wenn in einer q-dimensionalen Mannigfaltigkeit bei einer eindeutigen und stetigen Abbildung eines q-dimensionalen Kubus das Maximum der Verriückungen kleiner ist als die halbe Kantenlänge, so existiert ein konzentrischer und homothetischer Kubus, der ganz in der Bildmenge enthalten ist.

  8. 8.

    For an additional case of the drive for improved results through counterexamples, one may look to the celebrated case of Brouwer’s work of his 1910 paper, ‘Zur Analysis Situs’ (‘On Analysis Situs’) [2], in which, through a set of counterintuitive examples, he demolished the previous topological work of Arthur Schoenflies (1853–1928).

  9. 9.

    However, it must be pointed out that to build full mathematical theories of topology and dimension at the level of IMPS formalization would take considerable effort in formal theory construction. Algebraic simplification rules could be a part of that construction.

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Author information

Authors and Affiliations

  1. The MITRE Corporation, McLean, VA, 22102, USA

    Dale M. Johnson

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  1. Dale M. Johnson
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Editors and Affiliations

  1. Department of Computer Science, Worcester Polytechnic Institute, Worcester, MA, USA

    Daniel Dougherty

  2. Th.M. Siebel Center for Computer Science, University of Illinois Urbana-Champaign, Urbana, IL, USA

    José Meseguer

  3. Institute of Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark

    Sebastian Alexander Mödersheim

  4. J83K, The MITRE Corporation, Bedford, MA, USA

    Paul Rowe

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Dedicated to Joshua Guttman, Colleague and Friend.

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Johnson, D.M. (2021). Formal Methods and Mathematical Intuition. In: Dougherty, D., Meseguer, J., Mödersheim, S.A., Rowe, P. (eds) Protocols, Strands, and Logic. Lecture Notes in Computer Science(), vol 13066. Springer, Cham. https://doi.org/10.1007/978-3-030-91631-2_12

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