Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new meta-methodological concept (...) is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web-based proofs. This article claims that Web provings can be studied as a particular type of Goguen's proof-events. Web-based proof-events have a social component, communication medium, prover-interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To demonstrate its claim, the (...) article discusses the Kumo and Polymath projects, both of which employ Web-based communication as part of proving. Web proving is a novel type of proving activity that may have a serious impact on the change in mathematical practices, despite the fact that it is not currently a universally acceptable methodology. (shrink)

This paper outlines a logical representation of certain aspects of the process of mathematical proving that are important from the point of view of Artificial Intelligence. Our starting point is the concept of proof-event or proving, introduced by Goguen, instead of the traditional concept of mathematical proof. The reason behind this choice is that in contrast to the traditional static concept of mathematical proof, proof-events are understood as processes, which enables their use in Artificial Intelligence in such contexts in which (...) problem-solving procedures and strategies are studied. We represent proof-events as problem-centred spatio-temporal processes by means of the language of the calculus of events, which adequately captures certain temporal aspects of proof-events (i.e. that they have history and form sequences of proof-events evolving in time). Further, we suggest a “loose” semantics for the proof events by means of Kolmogorov’s calculus of problems. Finally, we expose the intented interpretations for our logical model from the fields of automated theorem-proving and Web-based collective proving. (shrink)

The concept of proof can be studied from many different perspectives. Many types of proofs have been developed throughout history such as apodictic, dialectical, formal, constructive and non-constructive proofs, proofs by visualisation, assumption-based proofs, computer-generated proofs, etc. In this paper, we develop Goguen’s general concept of proof-events and the methodology of algebraic semiotics, in order to define the concept of mathematical style, which characterizes the proofs produced by different cultures, schools or scholars. In our view, style can be defined as (...) a semiotic meta-code that depends on the underlying mode of signification (semiosis), the selected code and the underlying semiotic space and determines the individual mode of integration (selection, combination, blending) into a narrative structure (proof). Finally, we examine certain historical types of styles of mathematical proofs, to elucidate our viewpoint. (shrink)

This paper is concerned with the construction of theories of software systems yielding adequate predictions of their target systems’ computations. It is first argued that mathematical theories of programs are not able to provide predictions that are consistent with observed executions. Empirical theories of software systems are here introduced semantically, in terms of a hierarchy of computational models that are supplied by formal methods and testing techniques in computer science. Both deductive top-down and inductive bottom-up approaches in the discovery of (...) semantic software theories are refused to argue in favour of the abductive process of hypothesising and refining models at each level in the hierarchy, until they become satisfactorily predictive. Empirical theories of computational systems are required to be modular, as modular are most software verification and testing activities. We argue that logic relations must be thereby defined among models representing different modules in a semantic theory of a modular software system. We exclude that scientific structuralism is able to define module relations needed in software modular theories. The algebraic Theory of Institutions is finally introduced to specify the logic structure of modular semantic theories of computational systems. (shrink)

This paper studies the structure of semantic theories over modular computational systems and applies the algebraic Theory of Institutions to provide a logical representation of such theories. A modular semantic theory is here defined by a cluster of semantic theories, each for a single program's module, and by a set of relations connecting models of different semantic theories. A semantic theory of a single module is provided in terms of the set of ∑‐models mapped from the category Th of ∑‐theories (...) and generating a hierarchy of structures from an abstract model to a concrete model of data. The collection of abstract models representing different modules of a program is formalised as the category of institutions INS, where theory morphisms express refinements, integrations, and compositions between couples of modules. Finally, it is required that a morphism in INS at any level occurs iff the same morphism occurs at the lower level alongside the Th hierarchy. (shrink)

This paper explores the relationship between informal reasoning, creativity in mathematics, and problem solving. It underscores the importance of environments that promote interaction, hypothesis generation, examination, refutation, derivation of new solutions, drawing conclusions, and reasoning with others, as key factors in enhancing mathematical creativity. Drawing on argumentation logic, the paper proposes a novel approach to uncover specific characteristics in the development of formalized proving using “proof-events.” Argumentation logic can offer reasoning mechanisms that facilitate these environments. This paper proposes how argumentation (...) can be implemented to discover certain characteristics in the development of formalized proving with “proof-events”. The concept of a proof-event was introduced by Goguen who described mathematical proof as a multi-agent social event involving not only “classical” formal proofs, but also other informal proving actions such as deficient or alleged proofs. Argumentation is an integral component of the discovery process for a mathematical proof since a proof necessitates a dialogue between provers and interpreters to clarify and resolve gaps or assumptions. By formalizing proof-events through argumentation, this paper demonstrates how informal reasoning and conflicts arising during the proving process can be effectively simulated. The paper presents an extended version of the proof-events calculus, rooted in argumentation theories, and highlights the intricate relationships among proof, human reasoning, cognitive processes, creativity, and mathematical arguments. (shrink)

This article is concerned with the concept of ambiguity in argumentation. Ambiguity in linguistics lies on the coexistence of two possibly interpretations of an utterance, while the role of contextual factors and background/encyclopedic knowledge within a specific society seems to be crucial. From a systemic point of view, Halliday has proposed three main language functions (meta-functions): a) ideational function, b) interpersonal function, c) textual function. Language could reflect speaker’s experience of his external and internal world, interpersonal relationships and organization of (...) text, respectively. Lexico-grammatical choices under a micro-level perspective and context (the environment of language) may lead to inconsistent interpretations through semantic or syntactic ambiguities. In philosophy and argumentation logic, strategies of ambiguity have been investigated by Aristotle, since the first sophistic movement. In his Topics, Metaphysics and Rhetoric, has pointed out the notion of “τὸ διττῶς / διχῶς λεγόμενον”, meaning that a term can have different senses and double interpretation. In this paper we discuss how we reconstruct the meaning of an utterance in dialogue through the mechanism of interpretation and how we analyze and construe ambiguities, combining the insights of argumentation theory and text linguistics. Research results show that in case of misunderstanding, the “best interpretation” is the less defeasible one according to contextual presumptions. (shrink)

In recent decades, research in the square of opposition has increased. New interpretations, extensions, and generalizations have been suggested, both Aristotelian and non-Aristotelian ones. The paper attempts to compare different versions of the square of opposition. For this reason, we appeal to the wider categorical model-theoretic framework of the theory of institutions.

While collaboration has always played an important role in many cases of discovery and creation, recent developments such as the web facilitate and encourage collaboration at scales never seen before, even in areas such as mathematics, where contributions by single individuals have historically been the norm. This new scenario poses a challenge at the theoretical level, as it brings out the importance of various issues which, as of yet, have not been sufficiently central to the study of problem-solving, discovery, and (...) creativity. We analyze the case of collective and web-based proof events in mathematics, which share their temporal and social nature with every case of collective problem-solving. We propose that some ideas from cognitive architectures, in particular, the notion of codelet—understood as an agent engaged in one of a multitude of available tasks—can illuminate our understanding of collective problem-solving and act as a natural bridge from some of the theoretical aspects of collective, web-based discovery to the practical concern of designing cognitively inspired systems to support collective problem-solving. We use the Pythagorean Theorem and its many proofs as a case study to illustrate our approach. (shrink)

In this paper, we suggest the broader concept of proof-event, introduced by Joseph Goguen, as a fundamental methodological tool for studying proofs in history of mathematics. In this framework, proof is understood not as a purely syntactic object, but as a social process that involves at least two agents; this highlights the communicational aspect of proving. We claim that historians of mathematics essentially study proof-events in their research, since the mathematical proofs they face in the extant sources involve many informal (...) components, often not completely formalizable, and convey some kind of semantic content calling for understanding and verification. We illustrate the application of this methodological approach in some outstanding historical cases, paying particular attention to the process of proof interpretation that makes a proof-event alive. Finally, we suggest a classification of proof-events, according to the conditions imposed upon problem-solving. This enables us to speak about broad classes of proof-events in history of mathematics that share a common characteristic. (shrink)

As automation in artificial intelligence is increasing, we will need to automate a growing amount of ethical decision making. However, ethical decision- making raises novel challenges for engineers, ethicists and policymakers, who will have to explore new ways to realize this task. The presented work focuses on the development and formalization of models that aim at ensuring a correct ethical behaviour of artificial intelligent agents, in a provable way, extending and implementing a logic-based proving calculus that is based on argumentation (...) reasoning with support and attack arguments. This leads to a formal theoretical framework of ethical competence that could be implemented in artificial intelligent systems in order to best formalize certain parameters of ethical decision-making to ensure safety and justified trust. (shrink)

The senses convey impressions, perceptions and ultimately feelings, from which meaning emerges, revealing a platform where perception and thinking are actually very closely involved. From this perspective, the distance between descriptive, mental (cognitive) meaning and emotive, affective (non-cognitive) meaning shrinks, creating a platform to investigate how meaning is generated by posing the question ‘how does meaning actually make sense?’. This project investigates meaning as derived from the physical nature of our brains, our bodies and our physical experiences. We construct a (...) collaborative dialogue between art and the field of logic, specifically studies of embodied mathematics and computational theory as related to the construction of meaning. Here ideas from modern symbolic logic, such as abstract model theory (i.e. theory of institutions), can provide a suitable framework to explore a theory of meaning. (shrink)