Transmathematica

Aggregation Dynamics

Bas van Vlijmen — 2023-11-18

Aggregation Dynamics is a generalisation of ecology where organisms are replaced by aggregations. This totalises the notion of the ecology of living things to the ecology of all things. Aggregations are defined recursively: a thing is an aggregation if an already recognised aggregation acts on it accordingly.

The argument begins by problematising human notions of problem and solution. Reflection on this results in hypotheses such as the Innovation Illusion and the Control Illusion. Aggregation Dynamics suggests that behaviour and organisation of aggregations is emergent and largely information-driven. From that basis, the illusions seem plausible.

Special Accusation Types

Jan Aldert Bergstra — 2023-06-13

Straight accusations are considered a normal case for accusations with special accusation types referring to other forms of accusations. Three special accusation types wil be considered: anonymous accusations, non-evidential accusations, and self-accusations. Anonymous accusations (AA's) are accusations with an anonymous accuser. We describe the remarkable effects which anonymous accusations may have, and we propose various key properties of anonymous accusations: (i) the viral character of AA's, (ii) the potentially explosive effect of AA's, and (iii) the forensic challenge creation characteristic of AA's. These characteristics suggest, and in may contexts impose, rather restrictive rules of engagement for dealing with AA's. Secondly we describe non-evidential accusations (NEA's). Such accusations do not allow any meaningful form of validation of the body of the accusation. Nevertheless NEA's play a significant role nowadays. Finally we provide some remarks on self-accusations. A self-accusation may also be non-evidential.

Transreal Explicit Construction of Universal Possible Worlds

Tiago dos Reis — 2023-02-03

In an earlier paper we supplied an indirect proof of the existence of universal possible worlds that have the topological property of being hypercyclic, which means they can access every world in sequences of worlds that approach arbitrarily closely to every possible world. That proof states that there are universal worlds but it does not exhibit such a world explicitly. We now explicitly construct two such universal worlds. A continuous universal world constructs possible worlds with transreal co-ordinates directly. A discrete world provides a binary hypercyclic vector which can be used to create transfloating-point co-ordinates that approximate transreal co-ordinates. We also discuss the philosophical implications of universal worlds for an omniscient observer and human science.

Transreal Foundation for Floating-Point Arithmetic

James Anderson — 2023-07-23

Software Engineering relies, to a large extent, on formal software standards and logical means for specifying and verifying computer programs. Among these the IEEE 754 standard for floating-point arithmetic is widely used. We criticise the standard from the standpoint of transreal arithmetic. Transreal arithmetic was derived from projective geometry using a double cover to provide signed infinities on the horizon and nullity at the point of projection. These infinities and nullity have some similarities with IEEE 754 floating-point infinities and NaNs but there are important differences. We explore the differences by analysing the standard at three levels: commentary within the standard, abstract datatypes, and bit patterns. We find that all of the differences are coincident with faults in the standard. Obviously a correct standard would better support the specification, development and testing of numerical software. We discuss how the standard can be corrected, in its own terms, or by adopting transreal arithmetic as its theoretical foundation. We also discuss emulation of transreal arithmetic in IEEE 754 processing systems and address accusations that transreal arithmetic plagiarised the standard.