Diagrams and 

Mathematical Practice

Jean-Yves Beziau and Andrei Rodin (Orgs)

Münster, Germany, September 30, 2024

Part of DIAGRAMS 2024

PROGRAMME

9:00 - 9:30 - First Part

9:00 - 9:30  Presentation of the workshop by the organizers

9:30 - 10:00 Diagrammatic Syntax and Neo-Peircean Calculus of Relations in the Diagrammatization of Mathematical Practices

Nathan Haydon (Tallinn University of Technology, Estonia) and Ahti-Veikko Pietarinen (Hong Kong Baptist University, Hong Kong) 

Following the early work on the logic of relations by Boole, De Morgan, Peirce, and Schröder in the 19th century, the study of relations has largely been represented in mathematical practice by the turn to relation algebras. Following Tarski’s renewed emphasis on the subject in 1941 [12], relation algebra is now recognized as granting a foundation for large portions of mathematics [11, 13], with notable descendants in computer science [1, 4, 10] and cognitive science (such as Formal Concept Analysis [15] and Knowledge Representation [14]). A more recent direction is motivated by category-theoretic allegories [5] and relation-algebraic theories [2], which can be seen as an approach in the tradition of universal algebra with a focus on relations and relational operations.

The works cited suggest logic of relations as a general setting for elements of mathematical foundations, instruction, and practice. We can complement the picture of what Tarski specifically gives the credits for, namely Peirce’s early work on the logic of relations [8, 10, 13], by showing how Peirce’s later logic, his Existential Graphs (EGs, [9]) draws a direct connection to relation algebra, thus superseding the standard comparison with predicate logic by topological features of graphs that preserve composition, much needed for the latter’s direct connection to relation-algebraic operations.

In this vein, recent work in [6, 7] shows that Peirce’s work closely accords with categorical presentations [2, 5]. The diagrammatic, Neo-Peircean calculus of relations [3], contains a sound and complete axiomatization of first-order logic which agrees with algebraic theories and in which additional axioms characterizing an algebraic theory can be presented and reasoned about within the diagrammatic syntax itself.

In this work, we present these recent diagrammatic advances on Peirce’s work in terms of the Neo-Peircean calculus of relations, showing how to translate relation-algebraic expressions and operations into EGs, with examples of diagrammatic algebraic theories and functional and relational properties. The upshot is that Peirce’s work not only inspired relational approaches to mathematics but that his EGs catered for the contemporary settings for diagrammatic practice of mathematics and its philosophy.

References

[1] Bird, R. and De Moor, O. (1996). The algebra of programming. NATO ASI DPD, 152:167–203.

[2] Bonchi, F., Pavlovic, D., and Sobocinski, P. (2017). Functorial semantics for relational theories. https://arxiv.org/abs/1711.08699.

[3] Bonchi, F., Giorgio, A. D., Haydon, N., and Sobocinski, P. (2024). Diagrammatic algebra of first order logic. To appear at LICS.

[4] Codd, E. F. (1970). A relational model of data for large shared data banks. Commun. ACM, 13(6):377–387.

[5] Freyd, P. J. and Scedrov, A. (1990). Categories, Allegories. North-Holland Mathematical Library.

[6] Haydon, N. and Sobocinski, P. (2020). Compositional diagrammatic first-order logic. In Pietarinen, A.-V., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., and Linker, S., eds., Diagrammatic Representation and Inference, pp. 402–418, Cham. Springer.

[7] Haydon, N. and Pietarinen, A.-V. (2021). Residuation in existential graphs. In Basu, A., Stapleton, G., Linker, S., Legg, C., Manalo, E., and Viana, P., eds., Diagrammatic Representation and Inference, pp. 229–237, Cham. Springer.

[8] Peirce, Charles S. Studies in Logic. By Members of the Johns Hopkins University. Little, Brown, and Company, 1883.

[9] Peirce, Charles S. Logic of the Future : Writings on Existential Graphs. Ed. by A.-V. Pietarinen. Vol. 1: History and Applications. Vol. 2/1: The Logical Tracts. Vol. 2/2: The 1903 Lowell Lectures. Vol. 3/1: Pragmaticism. Vol. 3/2: Correspondence. Boston & Berlin: De Gruyter, 2019-2024.

[10] Pratt, V. R. Origins of the Calculus of Binary Relations. In International Symposium on Mathematical Foundations of Computer Science, pp. 142–155. Springer, 1992.

[11] Schmidt, G. (2010). Relational Mathematics. Encyclopedia of Mathematics and its Applications. Cambridge University Press.

[12] Tarski, A. (1941). On the calculus of relations. The Journal of Symbolic Logic, 6(3):73–89.

[13] Tarski, A. and Givant, S. R. (1988). A formalization of set theory without variables. American Mathematical Society 41.

[14] Wang, Y. (2017). On Relation Algebra: A Denotational Mathematical Structure of Relation Theory for Knowledge Representation and Cognitive Computing. Journal of Advanced Mathematics and Applications, 6, pp. 43-66(24)

[15] Wille, R. (1992). Concept lattices and conceptual knowledge systems. Computer & Mathematics with Applications, 23(6):493–515.

10:00 - 10:30 The Coloured Geometry of 5-Valued Contradiction: the Oppositional Quinque-Segment B52 and its 4D Attractor

Alessio Moretti  (Dipartimento di letteratura francese, Università telematica eCampus, Novedrate, Italy )

The formal study of “opposition” gives rise to a new geometry, of which the “square of opposition” and the “logical hexagon” are famous anticipations. Its first form was a theory of the “bi-simplexes” A2N ([2] and of their “closures” B2N  [5]]), which gave progressively place to the wider concept of “poly-simplex” ([3], [4] and  [1]). The theory is thus based on the concept of “simplex”, seen as a geometrical n-dimensional expression of the concept of natural number. This concept plays at two levels: the “degree of opposition” (“segment” for 2-opposition, “triangle” for 3-opposition, “tetrahedron” for 4-opposition, etc.) and the “number of truth-values used” (bi-simplexes for 2-valued n-oppositions, tri-simplexes for 3-valued n- oppositions, quadri-simplexes for 4-valued n-oppositions, etc.). This results in a discrete “poly- simplicial space” (expressible either as one quadrant of an infinite 2D matrix or as an infinite descending “numerical triangle”), made up of “points” BMN, where each such point is a complex “polytope of opposition” (containing smaller ones) expressing with “m” truth-values a “n- opposition”. The question lasted unanswered as to what place “oppositions” occupy inside mathematics. Since a few years an answer has emerged: oppositions BMN result from a special “projection” of the series of the “Pascalian simplexes” PM (a generalisation of “Pascal’s triangle” P2). Since poly-simplicial “points” BMN can be represented prima facie by the number of the vertices of their polytope of opposition (each BMN has MN-M vertices), for combinatorial reasons the difficulty of studying such structures is proportional to the number of their vertices: it therefore appears that of the infinite number of poly-simplexes BMN only a dozen can be studied relatively easily (having less than 100 vertices). So far six of them have been studied properly (the bi-simplexes B22, B23, B24, and the poly-simplexes B32, B33 and B42, with poly³3). Here we sketch the study of a seventh poly- simplex, the quinque-segment B52, expressing the geometry of 5-valued contradiction. We show it to be a very elegant structure provided with 20 vertices, containing as sub-structures intertwined B32 and B42, and whose “geometrical attractor” seems to be a 4D “runcinated 5-cell”. The quinque- segment is also the intersection of the series of the poly-segments BM2 and the series of the quinque- simplexes B5N. It is informative for several reasons: (1) it is the first quinque-simplex to be studied (on which all higher quinque-simplexes B5N will rely); (2) it sheds more light on the poly-segments BM2; (3) it offers a concrete, although complex view of (linear) 5-valuedness, taking into account conjunctly “truth-approximation” (“almost false” ¼, and “almost true” ¾, which characterise the quadri-simplexes B4N) and “truth-pivotality” (“half way” ½, which characterises the tri-simplexes B3N); (4) it is, with respect to its “geometrical attractor”, the simplest of the four 4-dimensional poly- simplexes of the poly-simplicial space BMN (the other three, not yet fully studied, are the poly- simplexes B25, B34, B43) and as such it could provide precious hints; (5) it confirms, as we will show, the usefulness and, in fact, the necessity of using colours when studying mathematically “oppositions”.

References

[1] R.Angot-Pellissier, "Many-Valued Logical Hexagons in a 3-Oppositional Trisimplex", in J.-Y. Beziau, I. Vandoulakis (eds.), The Exoteric Square of Opposition, Birkhäuser, Cham, 2022, pp.333-345.

[ 2] A.Moretti, "Geometry for Modalities? Yes: Through n-Opposition Theory." In: J-Y. Beziau, A.Costa-Leite,  A.Facchini A. (eds.): Aspects of Universal Logic, N.17 of Travaux de logique, University of Neuchâtel, 2004).

[3] A.Moretti,  The Geometry of Logical Opposition. PhD Thesis, University of Neuchâtel, Neuchâtel, 2009.

[4] A.Moretti,  "Tri-simplicial Contradiction: The Pascalian 3D Simplex for the Oppositional Tri-segment."  In  .J.-Y. Beziau, I. Vandoulakis (eds.), The Exoteric Square of Opposition, Birkhäuser, Cham, 2022, pp.347-479.

[5] R.Pélissier,   “Setting” n-opposition", Logica Universalis, 2, 2008, pp.235-263.

11:00 - 12:30 Second Part 

11:00 - 11:30 Empirical studies of mathematical diagrams: Lessons from five years of investigations

Mikkel Willum Johansen  (University of Copenhagen, Denmark)

In this talk I will report on a series of empirical investigations of mathematicians’ use of mathematical diagrams I have been part of conducting. The basic approach of the investigations has been to combine large-scale quantitative investigations with closed readings and qualitative analysis. As a core result, we investigated the prevalence of diagrams in mathematics journals from 1815 to 1915 (Johansen & Pallavicini, 2022). The investigation showed that diagrams were relatively common in mathematical research publication around year 1900, then they disappear for half a century and reappear during the 1960’s. In 2015 about 65% of the papers included in the survey included at least one diagram. Guided by this overall map of the use of diagrams in mathematical practice we conducted several focused investigations. Especially, comparison of the types of diagrams used in the period around 1900 and in periods after 1960 shows marked differences; in short, to reenter mathematical publications diagrams had to (partly) comply to formalist norms of rigor (Johansen & Pallavicini 2021). However, in recent years the attitude seems to have been relaxed and an in-depth analysis of the use of diagrams in published proofs showed that diagrams are frequently used not only heuristic but also epistemic purposes (Sørensen & Johansen, 2022).

In this talk I will give an overview of the main results outlined above, I will discuss the different roles diagrams play in mathematical practice and how this practice may inform our typology and definition of (mathematical) diagrams. Finally, I will discuss how mathematicians’ use of diagrams challenges what the (loosely be called) ‘formalist epistemology’ of mathematics, and calls for a more pragmatic understanding of the ways mathematicians obtain conviction.

References

Johansen, M.W., Misfeldt, M. and Pallavicini, J.L. (2018): A Typology of Mathematical Diagrams. In: Gem Stapleton, Francesco Bellucci, Amirouche Moktefi, Peter Chapman and Sarah Perez-Kriz (eds): Diagrammatic Representation and Inference - 10th International Conference. Springer, pp. 105-119 (LNCS). https://doi.org/10.1007/978-3-319-91376-6_13

Johansen, M.W. and Misfeldt, M. (2018): Material representations in mathematical research practice, Synthese 197:3721–374. https://doi.org/10.1007/s11229-018-02033-4.

Johansen, M.W. & Pallavicini, J. L. (2021). The Fall and Rise of Resemblance Diagrams. In: Basu, A., Stapleton, G., Linker, S., Legg, C., Manalo, E., Viana, P. (eds): Diagrammatic Representation and Inference. 12th International Conference. Springer, pp. 331–338 (LNCS). https://doi.org/10.1007/978-3-030-86062-2_33

Sørensen, H.K. and Johansen, M.W. (2022). Epistemic roles of diagrams in short proofs. In: Valeria Giardino, Sven Linker, Richard Burns, Francesco Bellucci, Jean-Michel Boucheix and Petrucio Viana (Eds.): Diagrammatic Representation and Inference: 13th International Conference. Springer, p. 235–242 (LNCS). https://doi.org/10.1007/978-3-031-15146-0_20.

Johansen, M.W. & Pallavicini, J. L. (2022). Entering the valley of formalism: trends and changes in mathematicians’ publication practice—1885 to 2015. Synthese, 200(239). https://doi.org/10.1007/s11229-022-03741-8

12:00 - 12:30 Linear Proof versus Diagramnatic Proof -  Study of an Example:  There is  no Cube of Opposition 

Caroline Pires Ting   (Federal University of Rio de Janeiro, Brazil, and  Macau International Instititute) and  Jean-Yves Beziau (Federal University of Rio de Janeiro, Brazil, Brazilian Research Council and Brazilian Academy of Philosophy) 

 The theory of opposition is based on of one of  the most famous diagrams of the history of philosophy, logic and mathematics, the square of oppositon.

Along the years, many geometric  generalizations have been proposed: polygons and polyhedra. Among polygons, the most famous one is Blanché's hexagon of opposition,  among polyhedra, the cube of opposition. But if, on the one hand the hexagon of opposition is a figure of opposition following the original pattern of the  square of opposition and improving it by reconstructing it together with two other squares included in the hexagon, on the other hand a cube of opposition cannot have  its six faces as standard squares of opposition.

The fact that there is no cube of opposition in this sense can be seen in a few seconds using  a colored three-dimensional diagram. This contrasts with a linear proof which is rather long and tedious. 

Based on that, we will examine the following questions which are of general interest for the value and usefulness of diagrammatic proofs: Do these two proofs have the same strength and validity? Are they equivalent? And we will compare the situation with other cases, especially the two versions of  Cantor's proof about the non-enumerability of the reals.

References

[1] J.-Y.Beziau, “There is no cube of opposition”, in J.-Y.Beziau and G.Basti, The Square of Opposition: A Cornerstone of Thought, Birkhäuser, Basel, 2017, pp.179-193

[2].R.Blanché, Structures intellectuelles - Essai sur l'organisation systématique des concepts, Vrin, Paris, 1966.

[3] M.Correia,  "Boethius on the Square of Opposition", in J.-Y.Beziau and D.Jacquette, Around and Beyond the Square of Opposition, Birkhäuser, Basel, 2012, , pp.41-52. 

[4] M.Correia,  "Aristotle’s Squares of Opposition", South American Journal of Logic, Vol. 3, n. 2, pp. 313–326, 2017.

14:00 - 15:30 Third Part 

14:00 - 14:30 A Diagrammatical Perspective on the Algebras of Logic

Jasmin Özel (University of Siegen, Germany)

The use of diagrams as a tool for the evaluation of proofs experienced a period of growth in the 18th and especially 19th century (Euler, Venn, Carroll, Peirce). Yet, the simultaneous Algebra of Logic tradition steered clear of the use of diagrams. Even Ladd-Franklin, as a student of Peirce, and thus familiar with his Existential Graphs, refrained from using diagrams in her writings. The goal of this talk will be to explain why the Algebra of Logic tradition largely avoided diagrams as a tool of reasoning—and to show how the tradition could have benefited from including diagrams. 

Focusing on the changes in notation that Ladd-Franklin introduced in contrast to previous Algebras of Logic, I will first ask whether a diagrammatic expression was perhaps simply not conducive to a more convenient visual grasp of the axioms and postulates we generally find in the Algebra of Logic tradition. Following Ambrose & Lazerowitz (1960), I will argue that this is not the case; the diagrammatic expression of the antilogism, for instance in Venn-diagrams, is just as straightforward as it is in the case of syllogisms. The second part of my talk will be focusing on the different function that diagrams can play in the contexts of syllogisms, proof theory, and the early beginnings of model theory— and why diagrams occupied different roles for the logicist tradition in the early 20th century as opposed to the psychologist tradition we find on the European continent at the time. My hypothesis will be that while diagrams can serve an important purpose when it comes to the description of the general structure of logical arguments, their “universal algebra”, the Algebra of Logic tradition’s pronounced focus on questions concerning the choice of notation prevented the algebrarians of logic from fully taking advantage of diagrams. I will close with a discussion of the potential advantages of a diagrammatically expressed “universal algebra”. In particular, I will argue that the expression of Ladd-Franklin’s Algebra of Logic in terms of diagrams demonstrates that her antilogism made it possible to express claims including modal operators.

[1] F. Abeles. Christine Ladd-Franklin’s antilogism. In Verburgt, L. & Cosci, M. (eds.) Aristotle’s syllogism and the creation of modern logic: Between tradition, and innovation, 1820s-1930s. Bloomsbury Publishing Plc., 2023.

[2] A. Ambrose and M. Lazerowitz. Logic: The Theory of Formal Inference. Holt, Rinehart Winston, New York, 1961.

[3] M.R. Cohen, E. Nagel. An Introduction to Logic. Harcourt, 1937.

[4] H. Curry. A Mathematical Treatment of the Rules of the Syllogism. In Mind 45(178), 209–216.

[5] R. H. Dotterer 1943. A Supplementary Note on the Rules of the Antilogism. In The Journal of Symbolic Logic. Vol. 8 , No. 1. (1943) , pp. 24-24.

[6] C. Dutilh Noves. Reassessing logical hylomorphism and the demarcation of logical constants. In Synthese. Vol. 185 (2012), pp. 387–410.

[7] R. W. Holmes Exercises in Reasoning, with an Outline of Logic. 1939, 1940.

[8] F. Janssen-Lauret. Grandmothers of Analytic Philosophy: The Formal and Philosophical Logic of Christine Ladd-Franklin and Constance Jones. In Minnesota Studies in Philosophy of Science. Minnesota, 2021.

[9] C. F. Ladd-Franklin. The Antilogism. In Mind. New Series, Vol. 37, No. 148 (Oct., 1928), pp. 532-534.

[10] C. Ladd On the algebra of logic. In Studies in Logic, By Members of the Johns Hopkins University edited by C. S. Peirce Boston: Little, Brown, and Company; Cambridge: University Press, John Wilson and Son, 1883.

[11] C. I. Lewis. Interesting Theorems in Symbolic Logic. In The Journal of Philosophy, Psychology and Scientific Methods. 10 (9) (1913), pp. 239-242.

[12] C. A. Mace. The Principles of Logic: An Introductory Survey.

[13] J. R. Norman & J. R. Sylvan. Directions in Relevant Logic. 2012.

[14] J. Parker. An Explication of the Antilogism in Christine Ladd- Franklin’s “Algebra of Logic” – Symbolic Notation in “Algebra of Logic”. Convergence. https://maa.org/press/periodicals/convergence/anexplication- of-the-antilogism-in-christine-ladd-franklins-algebra-of-logicfrom- elimination-toOnline resource

[15] A. Pietarinen. Christine Ladd-Franklin’s and Victoria Welby’s correspondence with Charles Peirce. In Semiotica. (2013)

[16] A. Reichenberger. Gender Equality and Diversity. Challenges and Perspectives for the Historiography of Science. forthcoming.

[17] R. Rogers. The Single Rule of the Antilogism and Syllogism. In Hermathena Vol. 19, No. 42 (1920), pp. 96-99.

[18] S. Russinoff. The Syllogism’s Final Solution. In The Bulletin of Symbolic Logic. Vol. 5, No. 4 (Dec., 1999), pp. 451-469.

[19] S. Uckelman. What Problem Did Ladd-Franklin (The She) Solve(d)? In Notre Dame Journal of Formal Logic. 2021, 62 (3). pp. 527-552.

[20] R. M. Sabre. Extending the Antilogism. In Logique et Analyse. Vol. 30 , No. 117/118., pp. 103-111.

[21] E. Shen. The Ladd-Franklin Formula in Logic: The Antilogism. In Mind. 36 (1927): 54–60.

[22] S. Stebbing. A Modern Introduction to Logic. Methuen: London (1930).

[23] A. P. Ushenko. The Theory of Logic: An Introductory Text. 1936.

[24] W. C. Wilcox. The Antilogism Extended. In Mind. 1969 , Vol. 78 , No. 310. , pp. 266-269.

14:30 - 15:00 Achieving Mathematical Understanding via Natural Language, Symbols and Diagrams

Andrei Rodin (Archives  Poincaré, University of Lorraine, France) and George Shabat (Moscow Center for Continuous Mathematical Education, Russia) , 

A presentation of  mathematical reasoning typically combines three types of expressive means:  a natural language, elements of symbolic syntax and diagrams. Using some historical and some recent examples we show how these expressive means work together in mathematical discourses, and analyse their specific epistemic roles. Using O.B. Bassler’s distinction between local and global surveyability of mathematical  proofs we show how mathematical diagrams mediate between informal linguistic explanations of mathematical arguments, on the one hand, and formal syntactic computations, on the other hand.  Along with some historical examples we consider the recent case of formalised computer-assisted mathematics and show how using dynamic diagrams in the case helps to achieve a better  understanding. 

Reference

[1] O.B.Bassler,  "The Surveyability of Mathematical Proof: A Historical Perspective", Synthese. Vol. 148 (2006), pp. 99–133.

15:00 - 15:30  General Discussion