Modelling the way mathematics is actually done

Abstract

Whereas formal mathematical theories are well studied, computers cannot yet adequately represent and reason about mathematical dialogues and other informal texts. To address this gap, we have developed a representation and reasoning strategy that draws on contemporary argumentation theory and classic AI techniques for representing and querying narratives and dialogues. In order to make the structures that these modelling tools produce accessible to computational reasoning, we encode representations in a higherorder nested semantic network. This system, for which we have developed a preliminary prototype in LISP, can represent both the content of what people say, and the dynamic reasoning steps that move from one step to the next. CCS Concepts• Computing methodologies → Philosophical/theoretical foundations of artificial intelligence;

Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups

“…Researchers have taken an external interest in this too, using the Polymath project (and the related mini-Polymath) to study the social features of collaboration, such as in the work of Martin and Pease (2013) and Martin (2015). They also study and model the social conventions that arose (Barany 2010), the conversational patterns used in the online mathematics (Corneli et. al.…”

“…They also study and model the social conventions that arose (Barany 2010), the conversational patterns used in the online mathematics (Corneli et al . 2017), the use of explanation in mathematics (Pease et al . 2019), and the kind of knowledge that is produced by Polymath projects (Allo et al 2013).…”

Big Math and the One-Brain Barrier: The Tetrapod Model of Mathematical Knowledge

“…Nevertheless, the authors believe that a confederated, well- We acknowledge the fact that our tetrapod proposal does not incorporate the fact that mathematics is a social process and that for Big Math problems we will need mathematical social machines, i.e., ''an environment comprising humans and technology interacting and producing outputs or action which would not be possible without both parties present'' [5]. We conjecture that the social machine aspect is one that will live quite comfortably on top of tetrapod-shaped mathematical software systems and can indeed not fully function without all of its five aspects interacting well; see [13,30] for a discussion and further pointers to the literature.…”

Argumentation Theory for Mathematical Argument

“…The reasoning was communicated by a combination of speech and marks on a chalkboard, and is reproduced in Figure 5. This example has been modelled in IATC by Corneli et al (2017b). The problem initially appears difficult to solve without computer algebra system, but a simple algebraic solution is available once the correct strategy is found.…”

“…As a limited proof of concept showing the plausibility of adding a computational deduction and verification layer on top of IATC representations, Corneli et al (2017b) give a detailed expansion of one step of a mathematical proof using simple rules for transforming the underlying graph structures. It is worth emphasising that the representations of reasoning afforded by language elements in Tables 1 and 2 do not themselves encode the meta-level reasoning associated with such graph transformations.…”