Aristotle's Theory of Abstraction by Allan Bäck

Author:Allan Bäck
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

6.6 The Physicist and the Mathematician

The details presented above of Aristotle’s theory of mental processes help to explain why Aristotle says that the physicist uses “addition”, i.e., synthesis , and her experience in acquiring the objects for the study of physics, while the mathematician uses “subtraction ”, i.e., abstraction, in acquiring the mathematical objects (Modrak 2001: 116). [Eth. Nic. 1142a16–9] I have claimed that the objects of both physics, bodies in motion , and mathematics, shapes and numbers, require both abstraction and synthesis in their formation. [Metaph. 1077b34–1078a5; (ps.) Alexander , in Metaph. 735, 37–736, 9] Even the perception of individual bodies, shapes, and [instances of] numbers requires both abstraction and synthesis . How then do physics and mathematics differ? Why then does Aristotle say that the objects of mathematics come from abstraction, while those of physics come from synthesis ?

One difference may lie in the types of abstraction and synthesis being required. Both do require synthesis and abstraction in order to have perceptions of individual things, both substantial and accidental, as well as thoughts of universals. But, once we get the universal objects, the two differ. Aristotle says that physics considers “objects having the principle of motion within themselves”, whereas mathematics deals with “things that are at rest, although its subjects cannot exist apart.” [Metaph. 1064a30–3; cf.1025b19–21; 1026a12–5] That is, the objects of mathematics do not exist in re separately from matter but are considered as being separate from matter and from their being movable. Again, natural objects have final causes ; the abstractions of mathematics do not—they are considered apart from those causes . [Part. An. 641b10–2; cf. Cael. 299a11–6]

In order for the physicist to consider individual things qua movable, she must abstract away from the color, texture etc. of those objects and consider them only qua moving. [Phys. 202a7–8] Then she must abstract away from universal properties necessarily belonging to those objects: being colored; being animate (or: being inanimate). The physicist considers, say, the individual bronze spheres universally, as moving bodies and just as moving bodies. So like mathematics natural science demands abstraction.

On the other hand, being movable is an accident, an inseparable or proper one, of physical objects.54 The physicist then adds on to or synthesizes the essential accident of movable with the definition of the physical object (the perceptible substance) so as to have an object to study. [Cf. Metaph. 1029b31; 1030b15; 1031a4–5; (ps.) Alexander , in Metaph. 733, 23–38] In terrestrial and in celestial physics yet other accidents are added on: everlasting or perishable; moving rectilinearly or moving circularly. [Cael. 269b1–2; Metaph. 1069a30–1069b1]

Like the physicist , the mathematician starts by thinking about individual things, both substances and accidents. The very perception of these individual things involves both abstraction and synthesis . The geometer then focuses on the accidents of shape, while the arithmetician on those of number. Such foci follow processes of abstraction. The geometer abstracts also from the individual features of geometrical objects, including the perceptible ones (if there be any55), like having a particular length or diameter.

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