**Organizers:** Jean-Yves Beziau, Rio de Janeiro, Brazil (jyb.unilog@gmail.com) and Andrei Rodin, Nancy, France (andrei.rodin@univ-lorraine.fr)

**Workshop description**

Mathematical practice is classically articulated around three notions: definitions, axioms, and proofs. Blaise Pascal gave a short and clear masterful description of this articulation in his short booklet *De l’esprit géométrique et de l’art de persuader* (1657), giving some rules for these three notions. Later on, Alfred Tarski was strongly influenced by this booklet, as shown by his 1937 Paris talk “Sur la méthode deductive”. But, if nowadays if it is common to think of proofs with diagrams, in particular after the three volume book *Proofs without Words *(1993, 2000, 2015) by Roger B.Nelsen, the question of the uses of diagrams for stating and representing axioms and definitions has not yet been systematically studied. This workshop aims at promoting these uses (considering the interaction between the three notions).

Moreover, diagrams are convenient as illustration and source of inspiration, for conceptualization and understanding, also important features of mathematical practice.

This workshop is open to reflections on the functioning of diagrams as well as inquiries about the philosophy of mathematics, the nature of reasoning and proving, through a diagrammatic methodology.

**Keywords:** Definition, Axiom, Proof, Postulate, Truth, Reasoning, Symbol, Category Theory, Model Theory, Proof Theory, Set Theory

#### Accepted talks

- Nathan Haydon and Ahti-Veikko Pietarinen: Diagrammatic Syntax and Neo-Peircean Calculus of Relations in the Diagrammatization of Mathematical Practices
- Alessio Moretti: The Coloured Geometry of 5-Valued Contradiction: the Oppositional Quinque-Segment B52 and its 4D Qttractor
- Mikkel Willum Johansen: Empirical studies of mathematical diagrams: Lessons from five years of investigations
- Mario Román Garcia and Ahti-Veikko Pietarinen: Linear Mathematics and the Rule of Duplication in Graphical Logic
- Caroline Pires Ting and Jean-Yves Beziau: Linear Proof versus Diagrammatic Proof – Study of an Example: There is no Cube of Opposition
- Jasmin Özel: A Diagrammatical Perspective on the Algebras of Logic
- Andrei Rodin and George Shabat: Achieving Mathematical Understanding via Natural Language, Symbols and Diagrams

#### Abstracts

##### 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.

**The Coloured Geometry of 5-Valued Contradiction: the Oppositional Quinque-Segment **B_{5}2** and its 4D Qttractor**

*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” A_{2}N ([2] and of their “closures” B_{2}N [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” B_{M}N, 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 B_{M}N result from a special “projection” of the series of the “Pascalian simplexes” P_{M} (a generalisation of “Pascal’s triangle” P_{2}). Since poly-simplicial “points” B_{M}N can be represented prima facie by the number of the vertices of their polytope of opposition (each B_{M}N has M^{N}-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 B_{M}N 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 B_{2}2, B_{2}3, B_{2}4, and the poly-simplexes B_{3}2, B_{3}3 and B_{4}2, with poly³3). Here we sketch the study of a seventh poly- simplex, the quinque-segment B_{5}2, 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 B_{3}2 and B_{4}2, 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 B_{M}2 and the series of the quinque- simplexes B_{5}N. It is informative for several reasons: (1) it is the first quinque-simplex to be studied (on which all higher quinque-simplexes B_{5}N will rely); (2) it sheds more light on the poly-segments B_{M}2; (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 B_{4}N) and “truth-pivotality” (“half way” ½, which characterises the tri-simplexes B_{3}N); (4) it is, with respect to its “geometrical attractor”, the simplest of the four 4-dimensional poly- simplexes of the poly-simplicial space B_{M}N (the other three, not yet fully studied, are the poly- simplexes B_{2}5, B_{3}4, B_{4}3) 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.

**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

**Linear Mathematics and the Rule of Duplication in Graphical Logic**

*Mario Román Garcia (Oxford University, UK) and Ahti-Veikko Pietarinen (Hong Kong Baptist University, Hong Kong) *

The common view in the foundations of mathematics has long been that the only point where we care about truth in mathematics is the moment in which we state the axioms. In practice, far from this, we find that axioms become justified by their theorems. A proof of an old theorem grounded in new axioms is indeed an important achievement, because what we were looking for about that proof was not a new truth but instead the bidirectional justification of the right axioms with the right theorem(s).

In this paper, we explore an example of these axiom shifts. Mathematics is usually content with the lack of negative evidence, but we might want to determine the more scientific criterion of positive (interactive, back-and-forth, game-theoretic, [8]) evidence before asserting the truth of a theorem. Scientific consequences of the proposed shifts in the foundations of mathematics that follow this direction are many, among them constructivism [1, 2] and linear mathematics [3, 6, 9]. From the vantage point of the latter, which we take to be the more exciting development, we analyze diagonal arguments for barber-style paradoxes, noticing that the diagonals need not be invariably supported: the culprit is the duplicating diagonal.

This leads to the point of the preferred means and methods of analysis. Once we have at our disposal a diagrammatic syntax by which to build up one’s constructions for logical analysis, we are prone to enter the linear realm. This was shown first by Charles Peirce in 1883, who pointed out not only the indispensable usefulness of game-theoretic semantics for such analyses [4, 7] but also that the diagrammatic syntax of logical graphs raises the important issue of whether we unvaryingly may use the rule of duplication (iteration) [5]. Today we would ask whether logical inferences are resource conscious, as avowed by linear logic. Here we gather evidence from Peirce’s hitherto unexplored writings both for and against the linear rendering of the graphical method of logic and its rule of duplication. Such changes in the perspective towards the interactive, diagrammatic, and linear foundations of mathematics have profound implications for what we mean by “proof” and “truth” in mathematics.

**References**

[1] Bauer, Andrej. Five stages of accepting constructive mathematics. Bulletin of the American Mathematical Society, 54(3):481–498, 2017.

[2] Bishop, Errett & Bridges, Douglas. Constructive Analysis 279. Springer, 2012.

[3] Girard, Jean-Yves. Linear logic: its syntax and semantics. 1995.

[4] Hilpinen, Risto. On C. S. Peirce’s Theory of the Proposition: Peirce as a Precursor of Game-Theoretical Semantics. The Monist 65(2), pp. 182–188, 1982.

[5] 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.

[6] Pietarinen, Ahti-Veikko. Logic, Language-Games and Ludics. Acta Analytica 18(30/31), pp. 89–123, 2003.

[7] Pietarinen, Ahti-Veikko. Peirce’s Game-Theoretic Ideas in Logic. Semiotica 144(14), pp. 33–47, 2003.

[8] Pietarinen, Ahti-Veikko. Games as Formal Tools versus Games as Explanations in Logic and Science. Foundations of Science 8(1), 317–364.

[9] Shulman, Michael. Linear logic for constructive mathematics. arXiv preprint arXiv:1805.07518, 2018.

**Linear Proof versus Diagrammatic 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 opposition.

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 concept*s, 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.

**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.

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

[13] J. R. Norman & J. R. Sylvan. *Directions in Relevant Logi*c. 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.

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[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.

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**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.

#### Background of organizers

Jean-Yves Beziau is a Swiss Logician, Philosopher and Mathematician, PhD in mathematics and PhD in Philosophy. He has been living and working in different places: France, Brazil, Poland, Corsica, California (UCLA, Stanford, UCSD), Switzerland. He is currently Professor at the University of Brazil in Rio de Janeiro, former Director of Graduate Studies in Philosophy and former President of the Brazilian Academy of Philosophy. He is the creator of the *World Logic Day*, yearly celebrated on January 14 (UNESCO international days), the *World Logic Prizes Contest*, the founder and Editor-in-Chief of the journal *Logica Universalis* and *South American Journal of Logic*, the book series Logic PhDs, *Studies in Universal Logic* and area logic editor of the *Internet Encyclopedia of Philosophy*. He has published about 200 research papers and 30 edited books and Special Issues of Journals.

Andrei Rodin is a lecturer in Poincaré Archives, University of Lorraine (France) working in the history and philosophy of mathematics and philosophical logic. He is the author of *Axiomatic Method and Category Theory* (Springer 2014) and a number of articles. Andrei Rodin’s Ph.D. thesis (1995) is on the axiomatic and conceptual structure of the first four Books of Euclid’s *Elements*; in 2020 he defended the Habilitation thesis on *The Axiomatic Structure of Scientific Theories*.