Transmathematica 2021

Transmathematica 2021: The 3rd International Conference on Total Systems was held online, using Zoom, on 5th July 2021 from 12.45 - 18.00 London Time. Edited video recordings were uploaded to a new Transmathematica channel on YouTube. We now present the conference proceedings and announce Transmathematica 2022: The 4th International Conference on Total Systems.

Accusation Theory

Accusations play a pivotal role in human communication. There seems be be some changes in the contemporary political situation. However, there is a lack of a philosophical conceptualisation that is necessary for appropriate descriptions of accusations and further philosophical scrutiny. Accustation Theory (AT) is proposed as such a theoretical framework. This paper aims (1) to present some general tools for the description of accusations as speech acts, (2) will try to understand fundamental features of accusations, (3) understand some elements of the practice of accusing, (4) will present a short overview of relevant literature and (5) outline some possible lines of further research.

Construction of the Transreal Numbers from Hyperreal Numbers

We construct the transreal numbers and arithmetic from subsets of hyperreal numbers. In possession of this construction, we propose a contextual interpretation of the transreal arithmetical operations as vector transformations.

Opinion on Patrick Suppes 1957 Introduction to Logic

We give an opinion on the merits of Patrick Suppes' 1957 book, "Introduction to Logic."

Review of Suppes 1957 Proposals For Division by Zero

We review the exposition of division by zero and the definition of total arithmetical functions in ``Introduction to Logic" by Patrick Suppes, 1957, and provide a hyperlink to the archived text. This book is a pedagogical introduction to first-order predicate calculus with logical, mathematical, physical and philosophical examples, some presented in exercises. It is notable for (i) presenting division by zero as a problem worthy of contemplation, (ii) considering five totalisations of real arithmetic, and (iii) making the observation that each of these solutions to ``the problem of division by zero" has both advantages and disadvantages -- none of the proposals being fully satisfactory. We classify totalisations by the number of non-real symbols they introduce, called their Extension Type. We compare Suppes' proposals for division by zero to more recent proposals. We find that all totalisations of Extension Type 0 are arbitrary, hence all non-arbitrary totalisations are of Extension Type at least 1. Totalisations of the differential and integral calculus have Extension Type at least 2. In particular, Meadows have Extension Type 1, Wheels have Extension Type 2, and Transreal numbers have Extension Type 3. It appears that Suppes was the modern originator of the idea that all real numbers divided by zero are equal to zero. This has Extension Type 0 and is, therefore, arbitrary.

Promise Theory and the Alignment of Context, Processes, Types, and Transforms

Promise Theory concerns the 'alignment', i.e. the degree of functional compatibility and the 'scaling' properties of process outcomes in agent-based models, with causality and intentional semantics. It serves as an umbrella for other theories of interaction, from physics to socio-economics, integrating dynamical and semantic concerns into a single framework. It derives its measures from sets, and can therefore incorporate a wide range of descriptive techniques, giving additional structure with predictive constraints. We review some structural details of Promise Theory, applied to Promises of the First Kind, to assist in the comparison of Promise Theory with other forms of physical and mathematical modelling, including Category Theory and Dynamical Systems. We explain how Promise Theory is distinct from other kinds of model, but has a natural structural similarity to statistical mechanics and quantum theory, albeit with different goals; it respects and clarifies the bounds of locality, while incorporating non-local communication. We derive the relationship between promises and morphisms to the extent that this would be a useful comparison.

Book Review: The Cult of Pythagoras

We review ``"The Cult of Pythagoras: Math and Myths" by Alberto A. Martinez, 2012. The book sets out a number of mathematical myths and dissolves them by a combination of checking historical sources and calculating results using the mathematics of the time. We pay particular attention to a chapter on division by zero. We record the earliest dates for particular solutions to the problem of division by zero and the ambition to have total mathematical systems.

The Transrational Numbers as an Abstract Data Type

In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to its opposite, and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.

Arithmetical Datatypes, Fracterms, and the Fraction Definition Problem

Datatypes and abstract datatypes are positioned as results of importing aspects of universal algebra into computer science and software engineering. It is suggested that 50 years later, by way of a transfer in the opposite direction, outcomes of research on datatypes can be made available via elementary arithmetic. This idea leads to the notions of an arithmetical signature, an arithmetical datatype and an arithmetical abstract datatype and to algebraic specifications for such entities. The area of fractions in elementary arithmetic is chosen as an application area and while taking a common meadow of rational numbers as the basis, an arithmetical datatype equipped with an absorptive element. The use of datatypes and signatures makes syntax available for giving precise definitions in cases where lack of precision is common place. Fracterm is coined as the name for a fraction when primarily understood as a syntactic entity. The main contribution of the paper is to provide a detailed terminology of fracterms. Subsequently the fraction definition problem is stated, a distinction between explicit definitions of fractions and implicit definitions of fractions is made, and an outline of a survey of both forms of definitions of the notion of a fraction is given.

Promise Theory as a Tool for Informaticians

Promise theory was designed and developed from 2005 onwards by Mark Burgess and his coworkers. It totalises the notion of a promise so that it applies to both animate and inanimate promisers. The focus of promise theory is on applications in informatics and systems design. This paper extends the account of promises by providing more detailed requirements on promises. In particular, the requirement of determinacy, the requirement of decomposition of aggregate agents, and the feature of a promise bias are introduced. The paper further includes an account of threats, as well as of risks, both viewed as an extension of promise theory. It is finally indicated by means of a series of informal examples how and where various kinds of promises and threats may occur in informatics.