From Logical Calculus to Logical Formality—What Kant Did with Euler’s Circles

John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations

Linear lambda terms as invariants of rooted trivalent maps

The main aim of the article is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between $\alpha$-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the "easy" direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure. Building on recent work in applied category theory, we show how deterministic dynamical systems, discrete and continuous, can be composed in a hierarchical style. In mathematical terms, we reformulate some existing operads of wiring diagrams and introduce new ones, using the general formalism of C-sets (copresheaves). We then establish dynamical systems as algebras of these operads. In a computational vein, we show that Euler's method is functorial for undirected systems, extending a previous result for directed systems. All of the ideas in this paper are implemented as practical software using Catlab and the AlgebraicJulia ecosystem, written in the Julia programming language for scientific computing.Comment: In Proceedings ACT 2021, arXiv:2211.0110

The abstraction effect on logic rules application

The aim of this study is to analyze the relationship between training on abstraction and the comprehension of logic rules. In order to evaluate the possibility of improvement on logic performance we have selected the particular case of the DeMorgan’s laws. The dispute between the natural logic approach and the mental models theory is analyzed from the perspective of such abstraction effect. Two experiments are reported. The first one suggests that the presentation of a formal proof promotes a better comprehension of DeMorgan´s laws than the use of visual resources or colloquial examples. The second one offers a stronger test for the same abstraction effect. Some limitations concerned with the syntactic meaning of negation and the differences between constructive and evaluative conditions are discussed. Since the meaning of abstraction for the psychology of reasoning is pointed out as critical some suggestions for further research and possible educational applications are mentioned.Fil: Macbeth, Guillermo Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Entre Ríos. Facultad de Ciencias de la Educación; ArgentinaFil: Razumiejczyk, Eugenia. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Entre Ríos. Facultad de Ciencias de la Educación; ArgentinaFil: Campitelli, Guillermo Jorge. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Edith Cowan University; Australi

Understanding Student Computational Thinking with Computational Modeling

Recently, the National Research Council's framework for next generation science standards highlighted "computational thinking" as one of its "fundamental practices". 9th Grade students taking a physics course that employed the Modeling Instruction curriculum were taught to construct computational models of physical systems. Student computational thinking was assessed using a proctored programming assignment, written essay, and a series of think-aloud interviews, where the students produced and discussed a computational model of a baseball in motion via a high-level programming environment (VPython). Roughly a third of the students in the study were successful in completing the programming assignment. Student success on this assessment was tied to how students synthesized their knowledge of physics and computation. On the essay and interview assessments, students displayed unique views of the relationship between force and motion; those who spoke of this relationship in causal (rather than observational) terms tended to have more success in the programming exercise.Comment: preprint to submit to PERC proceedings 201

Harmony and Technology Enhanced Learning

New technologies offer rich opportunities to support education in harmony. In this chapter we consider theoretical perspectives and underlying principles behind technologies for learning and teaching harmony. Such perspectives help in matching existing and future technologies to educational purposes, and to inspire the creative re-appropriation of technologies

Recent studies on signs: Commentary and perspectives

In this commentary, I reply to the fourteen papers published in the Sign Systems Studies special issue on Peirce’s Theory of Signs, with a view on connecting some of their central themes and theses and in putting some of the key points in those papers into a wider perspective of Peirce’s logic and philosophy

“I just do not understand the logic of this”:intervention study aimed at secondary school students’ development of logical reasoning skills

Logical reasoning is key for the development of critical thinking as a 21st century skill. Logical reasoning has been part of one of the Mathematics courses in the Netherlands since 2015. One of the objectives of this domain is to support pre-university students’ reasoning skills in a variety of societally relevant topics. Because courses in formal logic often did not result in the desired outcomes, we developed an intervention in which societally relevant contexts, such as newspaper articles, were central. The focus of the lessons during our intervention at eight schools was on developing and learning to use appropriate formalisations, visualisations, and schematisations, which were intended to support students’ reasoning. Important was the specific attention to the links between the different representations (based on the model of concreteness fading). Other literature-based design characteristics were: students exchanging ideas in small groups, formative feedback and class discussions on the strategies students used. We showed that the reasoning skills of students from the experimental group improved significantly and that those students also used significantly more formalisations. We provided evidence that our approach stimulated and supported the learning of logical reasoning and recommend to include this domain in all mathematics subjects. Preferably, it should be taught cross-curricular. In the light of the ongoing curriculum reform in the Netherlands (curriculum.nu), this offers great opportunities to seek collaboration with other subjects where reasoning and analysing arguments play an important role

The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems

Since its inception as a student project in 2001, initially just for the handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library has been continuously improved and extended by joining scrupulous research on the theoretical foundations of (possibly non-convex) numerical abstractions to a total adherence to the best available practices in software development. Even though it is still not fully mature and functionally complete, the Parma Polyhedra Library already offers a combination of functionality, reliability, usability and performance that is not matched by similar, freely available libraries. In this paper, we present the main features of the current version of the library, emphasizing those that distinguish it from other similar libraries and those that are important for applications in the field of analysis and verification of hardware and software systems.Comment: 38 pages, 2 figures, 3 listings, 3 table