The Concept of Abstraction

Philosophers deal with abstractions. Being reflective, they also have come up with theories about what these abstractions are. Aristotle is no exception, and indeed gave what came to be a canonical account of abstraction. Here I shall investigate what Aristotle thinks abstraction is. I shall conclude that Aristotle views abstraction as selective attention. As its very name suggests, abstracting (ἀφαιρέω) consists in taking away something from an object. The root verb, αἱρέω, suggests additionally a sense of grasping or of choosing, of taking for oneself something of what lies ready to hand. These lexical meanings leave open a wide range of conceptions of ‘abstraction’. Does the abstraction consist in taking out something and discarding the rest? Or does it consist in taking away something and keeping what is left? We can call the first one the selection view, and the latter the subtraction view. The Greek gives an ambiguity between the two because ‘ἀφαίρεσις’, being a verbal noun, could be derived from the active form ‘ἀφαιρεῖν’, which generally does have the sense of ‘removal’, or from the middle form ‘ἀφαιρείσθαι’, which generally has the sense of ‘take away for oneself’ or ‘steal’. On linguistic grounds of common usage, the selective reading of ‘ἀφαίρεσις’ has the advantage, as the middle voice forms are far more common than the active voice forms. Yet, as Aristotle is a philosopher, and philosophy stretches or distorts the ordinary usage of language, the philological evidence does not settle the issue. For that, we must turn to Aristotle’s texts

Aristotle\u27s Abstract Ontology

Aristotle has a metaphysics of individual substances, substrata persisting through time that are neither in nor said of a subject. That I do not dispute. However, when we move from the individual to the universal, from perception to knowledge, Aristotle has a metaphysics of relations. This I will try to sketch out here. Aristotle appeals to abstraction at key places in his philosophy. Somehow abstraction gets us to the first principles and to the objects of the most fundamental sciences. Somehow universals are abstracted from singulars and have no transcendent existence. Aristotle never states his theory of abstraction formally or too explicitly. Yet he surely uses it. Perception is the abstraction of forms from substances; knowledge is the abstraction of the universal from singulars; the objects of the mathematical sciences are “the things said from abstraction”. [An. III.4; Metaph. I.1; Phys. I.1] Likewise, he speaks of “cutting off a part of being” and making a science about it. [Metaph. 1003a24-5] Physics concerns substances qua movable; geometry considers substances qua figure. [Metaph. 1026a7-10; 1061a28-1062b11; 1077b22-1078a21] We start with the individual substances given in sense perception and then isolate aspects of them, the abstracta, for study in particular sciences. Above all, on account being abstractions, the essences of substances can be real without existing in their own right. Aristotle tends to express abstractions in the language of what later came to be called “reduplicatives”: something may be considered “qua this” or “qua that”. Yet he has not left us a treatise on abstraction. This lack of explicit doctrines I think accounts for many of the disputes over his ontology. Abstractions have the formal features of relations. Aristotle appeals to these features in addressing the problem of universals

A Case Study in Capacity Planning for PEPA Models with the PEPA Eclipse Plug-in

AbstractWe report on the addition of Capacity Planning facilities to the PEPA Eclipse Plug-in, a software tool for analysing performance models written in the PEPA language. The PEPA language allows the compositional description of complex systems consisting of different kinds of processes. The capacity planning addition allows modellers to automatically search for the populations of processes that allows for an optimal trade-off between the performance of the system and the cost of acquiring or operating the components of the system under the modeller's control