Mathematical Objects as Abstractions

Aristotle’s philosophy of mathematics is part of his general philosophy, and consequently he has to relate concepts of mathematics to his distinction between formand matter, genus and differentia specifica, essential and non-essential attributes, etc., and this may make it difficult to extract just what is relevant in the context of philosophy of mathematics. There is, furthermore, no (known) treatise on the philosophy of mathematics by Aristotle. His remarks on mathematics and philosophy of mathematics are scattered throughout all of his texts. Concerning the first distinction, mathematical objects are not pure forms, and they are not sensible objects, but they are separable from sensible objects in thought. This process of separation is described as a process of abstraction. In this activity the mathematician, or metaphysician, eliminates non-essential attributes, or attributes not to be taken into consideration.

The mathematician

investigates abstractions (for in his investigation he eliminates all the sensible qualities, e.g. weight and lightness, hardness and its contrary, and also heat and cold and the other sensible contraries, and leaves only the quantitative and continuous [...] and the attributes of things qua quantitative and continuous, and does not consider them in any other respect ...)

Thomas Heath illustrates the process of abstraction vs. the process of adding elements or conditions by the contrast between a unit, a substance without position, and a point, a substance having position. The process of abstracting in Aristotle is not a process of finding common properties among individuals, but rather a process of subtracting. According to John J. Cleary it is not an epistemological theory, but a logical theory with ontological consequences. He furthermore maintains that clarifying the ‘qua’ locution in the quotation above is the crucial point in understanding Aristotle’s mathematical ontology. This is exactly the strategy of Jonathan Lear, who introduces a “qua-operator” to analyse the abstraction process in Aristotle’s philosophy

of mathematics. My focus will be on Lear’s analysis.

Since mathematical objects are not pure forms they must inhere in some kind of matter, called intelligible matter, as distinct from sensible matter. Even the straight line ... may be analysed into its matter, continuity

(more precisely continuity in space, extension, or length), and its form. ‘Though the geometer’s line is length without breadth or thickness, and therefore abstract, yet extension is a sort of geometrical matter which enables the conception of mathematics to be after all concrete’.

If mathematical objects are not separable from sensible objects, and if they, in some way, are inherent in sensible objects, how are they related to the objects of physics and metaphysics? Physical objects have attributes in addition to mathematical ones. They can be moving, for example, but mathematics abstracts from movement. Physical objects, like mathematical, contain planes, etc., but the mathematician does not treat planes and points qua attributes of physical bodies, and he does not study them qua limits or boundaries

of physical bodies, as the physicist does. The relation between the objects of mathematics, physics, and metaphysics are described in the following way by Aristotle.

The physicist is he who concerns himself with all the properties active and passive of bodies or materials thus or thus defined; attributes not considered as being of this character he leaves to others, in certain cases it may be to a specialist, e.g. a carpenter or a physician, in others (a) where they are inseparable in fact, but are separable from any particular kind of body by an effort of abstraction, to the mathematician, (b) where they are separate, to the First Philosopher.

Note that Aristotle’s process of abstraction does not give rise to abstract ideas, and it is in that way not affected by e.g. Berkeley’s attack on abstract ideas, or Frege’s attack on psychologism.