Ordinals by Abstraction

The neo-Fregean programme in the philosophy of mathematics seeks to provide foundations for fundamental mathematical theories in abstraction principles. Ian Rumfitt (2018) proposes to introduce ordinal numbers by means of an abstraction principle, (ORD), which says, roughly, that ‘the ordinal number attaching to one well-ordered series is identical with that attaching to another if, and only if, the two series are isomorphic’. Rumfitt’s proposal poses a sharp and serious challenge to those seeking to advance the neo-Fregean programme, for Rumfitt proposes to save (ORD) from threatening paradox by avoiding dependence on an impredicative comprehension principle. However, such a principle is usually taken to be required by the neo-Fregean account of the cardinal numbers. Thus if neo-Fregean foundations for elementary arithmetic are to be saved, we must explain how we can avoid paradox for (ORD) in another way. In this chapter, the prospects for doing so are explored.

Solving the Caesar Problem—with Metaphysics

According to neo-Fregean Platonism, abstraction principles—such as the principle that the direction of line a is identical to the direction of line b iff a and b are parallel—may in some cases be regarded as introducing new singular terms (e.g., “the direction of line a”) and as fixing the truth-conditions of genuine identity statements featuring them. If neo-Fregeanism is to vindicate Frege’s idea that a plausible philosophy of arithmetic can and should treat the natural numbers as a species of object, it must address the so-called “Caesar Problem”: the problem of explaining in general terms which objects given in other terms they are to be distinguished from. This chapter pilots a novel solution to the Caesar Problem via the notion of a real definition: a definition whose purpose is not to explain a meaning, but to characterize the essential nature of the thing introduced.

A General Overview of RESTful Web Services

This chapter aims to present a general overview of today's dominant software architectural style for developing web services, namely REST, by comparing the core elements of this paradigm with the big web service model. The study evaluates the HTTP requests, responses, and thus, the SOAP/JSON payloads involved in consuming a big web service and a RESTful service that is developed in the ASP.NET Core Web API framework. After summarizing the REST constraints, the chapter elucidates how the example RESTful web service satisfies these constraints and lists some scenarios suited to each paradigm. The study notes the object-oriented elements that are inherent in RESTful services, specifically how polymorphism and abstraction principles can be applied to RESTful services.

Frege’s Relation to Dedekind: Basic Laws and Beyond

Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in terms of their general backgrounds and core projects. This essay provides a reexamination of Frege’s critical reactions to Dedekind, in Grundgesetze and some related texts. The reexamination includes documenting their interaction in some detail and putting it into a broader context, both philosophically and systematically. It also involves separating Frege’s less compelling criticisms of Dedekind from those that have deeper, more lasting significance. The essay ends with a suggestion for how to reconcile Fregean and Dedekindian forms of logicism, based on distinguishing two distinct but complementary kinds of abstraction principles.

A Framework for Halal Knowledge Metadata Representations

End users and consumers of halal industry are facing difficulties in finding verified halal information. This occurred due to information that is stored in silos at every point of activity for every process chain, employing different structures and models, creating an issue of information verification. Integration of multiple information systems generally aims at combining selected systems so that information can be easily retrieved and manage by users. A proposed five components metadata representation development methodology is presented in this paper so that they form a unified new whole and give users the illusion of interacting with one single information system, therefore, data can be represented using the same abstraction principles (unified global data model and unified semantics) without any physical restructuring.

Structuralist Neologicism†

ABSTRACT Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all (known) properties of the numbers, achieving a key neofregean advantage, while preserving the key structuralist advantage, which objects play the number role does not matter.

Dynamic Abstraction

Any abstractionist approach to thin objects faces the threat of paradox, as illustrated by Frege’s inconsistent Basic Law V. The neo-Fregeans Hale and Wright respond by severely restricting the class of acceptable abstraction principles. Their approach is static in the sense that they hold the domain fixed. This approach to abstraction is criticized, and an alternative approach is developed which permits abstraction on a vast class of equivalence relations. This alternative approach is dynamic in the sense that abstraction on an extensionally specified domain (i.e. a domain specified by means of a plurality of objects) may result in a larger such domain. A form of absolute generality is nevertheless possible, provided that the associated domain is understood in an intensional sense (i.e. it cannot be specified by means of a plurality).