Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake.

The myth of associative discrimination and the Court of Justice’s great vanishing act: part 2

This article complements an article (part 1) recently published in this journal (72(1) NILQ 29–60) contending that the notion of associative discrimination as a term of art renders it so vulnerable to manipulation that it can be used to narrow the scope of the legislation. That argument was rooted in the UK Supreme Court’s reasoning in Lee v Ashers Bakery [2018] UKSC 49. Part 2 continues the theme, but this time to show that the vulnerability can work the other way, producing, first, an ‘extended’ notion of associative discrimination and, second, radically broad notions of direct and indirect discrimination. This limb of the thesis also argues that a case heralded as one of associative discrimination, CHEZ [2016] CMLR 14, was no such thing. It concludes that the ambitious approach of the European Court of Justice and its Advocates General will blur the traditional form-based distinction between direct and indirect discrimination.

A Systematic Approach to Consistent Truncations of Supergravity Theories

Exceptional generalised geometry is a reformulation of eleven/ten-dimensional supergravity that unifies ordinary diffeomorphisms and gauge transformations of the higher-rank potentials of the theory in an extended notion of diffeormorphisms. These features make exceptional generalised geometry a very powerful tool to study consistent truncations of eleven/ten-dimensional supergravities. In this article, we review how the notion of generalised G-structure allows us to derive consistent truncations to supergravity theories in various dimensions and with different amounts of supersymmetry. We discuss in detail the truncations of eleven-dimensional supergravity to N=4 and N=2 supergravity in five dimensions.

Near Approximations in Modules

Rough set theory is a mathematical approach to imperfect knowledge. The near set approach leads to partitions of ensembles of sample objects with measurable information content and an approach to feature selection. In this paper, we apply the previous results of Bagirmaz [Appl. Algebra Engrg. Comm. Comput., 30(4) (2019) 285-29] and [Davvaz et al., Near approximations in rings. AAECC (2020). https://doi.org/10.1007/s00200-020-00421-3] to module theory. We introduce the notion of near approximations in a module over a ring, which is an extended notion of a rough approximations in a module presented in [B. Davvaz and M. Mahdavipour, Roughness in modules, Information Sciences, 176 (2006) 3658-3674]. Then we define the lower and upper near submodules and investigate their properties.

An Holistic Extension for Classical Logic via Quantum Fredkin Gate

An holistic extension for classical propositional logic is introduced in the framework of quantum computation with mixed states. The mentioned extension is obtained by applying the quantum Fredkin gate to non-factorizable bipartite states. In particular, an extended notion of classical contradiction is studied in this holistic framework.

Diagnosis of Active Systems with Abstract Observations and Compiled Knowledge

An active system (AS) is a discrete-event system (DES) with asynchronous behavior, which is represented by a network of components that are modeled as communicating automata. When being operated, an AS performs a trajectory within its behavior space, while generating a sequence of observations, namely a temporal observation. The model of the AS and a temporal observation are the two key ingredients of the diagnosis task, which aims to find out possible faulty behavior via abductive reasoning. Among other knowledge, such reasoning requires knowing what is observable and what is not. This essential distinction constitutes the observability of the AS. In the literature, the observability of a DES boils down to qualifying each state transition either as observable or unobservable, which contrasts with the way humans observe reality, typically by mapping a collection of observations to a single, abstract perception. Moreover, the occurrence of single state transitions is not necessarily what we can observe or what we want to observe for diagnosis purposes. This paper presents an extended notion of observability, where each observation is associated with a behavioral scenario rather than a single state transition, where a scenario is defined as a regular language on state transitions. To speed up the online diagnosis engine, specific diagnosis-oriented knowledge is compiled offline. Eventually, the diagnosis technique based on abstract observability is extended to cope with temporal uncertainty.

Varieties of Deliberation: Framing Plurality in Political CSR

This article argues that the concept of deliberation is construed too narrowly in political corporate social responsibility (CSR) and that a concept of deliberation for political CSR should err toward useful speech acts rather than reciprocity and charity. It draws from the political philosophy, labor relations, and business ethics literatures to outline a framework for an extended notion of deliberative engagement. The characters of deliberative behavior and deliberative environment are held to generate four modes of engagement: strategic deliberation, unitarist deliberation, pluralist deliberation, and deliberative activism. The article concludes by arguing that political CSR will be better positioned to realize its potential by moving away from primarily consensus-centered objectives to a more responsive range of deliberative goals and practice.

The myth of associative discrimination and the Court of Justice’s great vanishing act: part 2

This article complements an article (part 1) recently published in this journal (72(1) NILQ 29–60) contending that the notion of associative discrimination as a term of art renders it so vulnerable to manipulation that it can be used to narrow the scope of the legislation. That argument was rooted in the UK Supreme Court’s reasoning in Lee v Ashers Bakery [2018] UKSC 49. Part 2 continues the theme, but this time to show that the vulnerability can work the other way, producing, first, an ‘extended’ notion of associative discrimination and, second, radically broad notions of direct and indirect discrimination. This limb of the thesis also argues that a case heralded as one of associative discrimination, CHEZ [2016] CMLR 14, was no such thing. It concludes that the ambitious approach of the European Court of Justice and its Advocates General will blur the traditional form-based distinction between direct and indirect discrimination.

Japanese -tati and generalized associative plurals

This chapter presents descriptive generalizations of plural marking in Japanese with the morpheme -tati and proposes an account for its distributional and interpretive properties that are puzzling in many ways. The semantic peculiarities of -tati plurals, such as their tendency to be definite and the lack of generic and kind interpretations, result from the use of -tati as an associative plural marker. When -tati attaches to an individual-denoting expression, it denotes a plurality that consists of the referent of the expression and entities associated with. It is argued that -tati maintains this associative meaning even when it combines with a common noun. The extended notion of associativity allows X-tati, where X is a common noun, to include non-Xs in its denotation as long as such entities are closely associated with X, yielding similative plurals. This potential heterogeneity can solve most, if not all, of the puzzles posed by -tati plurals.

$$ \mathcal{N} $$ = 2 extended MacDowell-Mansouri supergravity

We construct a gauge theory based in the supergroup G = SU(2, 2|2) that generalizes MacDowell-Mansouri supergravity. This is done introducing an extended notion of Hodge operator in the form of an outer automorphism of su(2, 2|2)-valued 2-form tensors. The model closely resembles a Yang-Mills theory — including the action principle, equations of motion and gauge transformations — which avoids the use of the otherwise complicated component formalism. The theory enjoys H = SO(3, 1) × ℝ × U(1) × SU(2) off-shell symmetry whilst the broken symmetries G/H, translation-type symmetries and supersymmetry, can be recovered on surface of integrability conditions of the equations of motion, for which it suffices the Rarita-Schwinger equation and torsion-like constraints to hold. Using the matter ansatz —projecting the 1 ⊗ 1/2 reducible representation into the spin-1/2 irreducible sector — we obtain (chiral) fermion models with gauge and gravity interactions.