Mill, John Stuart (1806–73)

3. Mathematics

The strategy which Mill applies to mathematics is broadly similar to his approach to logic. If it was merely verbal, mathematical reasoning would be a petitio principii, but semantic analysis shows that it contains real propositions.

Mill provides brief but insightful empiricist sketches of geometry and arithmetic. The theorems of geometry are deduced from premises which are real propositions inductively established. (Deduction is itself largely a process of real inference.) These premises, where they are not straightforwardly true of physical space, are true in the limit. Geometrical objects – points, lines, planes – are ideal or ‘fictional’ limits of ideally constructible material entities. Thus the real empirical assertion underlying an axiom such as ‘Two straight lines cannot enclose a space’ is something like ‘The more closely two lines approach absolute breadthlessness and straightness, the smaller the space they enclose’.

Applying his distinction between denotation and connotation, Mill argues that arithmetical identities such as ‘Two plus one equals three’ are real propositions. Number terms denote ‘aggregates’ and connote certain attributes of aggregates. (He does not say that they denote those attributes of the aggregates, though perhaps he should have done.) ‘Aggregates’ are natural, not abstract, entities – ‘collections’ or ‘agglomerations’ individuated by a principle of aggregation. This theory escapes some of the influential criticisms Frege later made of it, but its viability none the less remains extremely doubtful. The respects in which aggregates have to differ from sets if they are to be credibly natural, and not abstract, entities are precisely those in which they seem to fail to produce a fully adequate ontology for arithmetic. (One can, for example, number numbers, but can there be aggregates of aggregates, or of attributes of aggregates, if aggregates are natural entities?)

However this may be, Mill’s philosophical programme is clear. Arithmetic, like logic and geometry, is a natural science, concerning a category of the laws of nature – those concerning aggregation. The fundamental principles of arithmetic and geometry, as well as of logic itself, are real. Mill provides the first thoroughly empiricist analysis of meaning and of deductive reasoning itself.

He distinguishes his view from three others – ‘Conceptualism’, ‘Nominalism’ and ‘Realism’. ‘Conceptualism’ is his name for the view which takes the objects studied by logic to be psychological states or acts. It holds that names stand for ‘ideas’ which make up judgments and that ‘a proposition is the expression of a relation between two ideas’. It confuses logic and psychology by assimilating propositions to judgments and attributes of objects to ideas. Against this doctrine Mill insists that:

All language recognizes a difference between doctrine or opinion, and the fact of entertaining the opinion; between assent, and what is assented to…. Logic, according to the conception here formed of it, has no concern with the nature of the act of judging or believing; the consideration of that act, as a phenomenon of the mind, belongs to another science.

(1843: 87)

The Nominalists – Mill cites Hobbes – hold that logic and mathematics are entirely verbal. Mill takes this position much more seriously than Conceptualism and seeks to refute it in detail. His main point is that Nominalists are only able to maintain their view because they fail to distinguish between the denotation and the connotation of names, ‘seeking for their meaning exclusively in what they denote’ (1843: 91) (see Nominalism §3).

Nominalists and Conceptualists hold that logic and mathematics can be known non-empirically, while yet retaining the view that no real proposition about the world can be so known. Realists hold that logical and mathematical knowledge is knowledge of universals which exist in an abstract Platonic domain; the terms that make up sentences being signs that stand for such universals. Versions of this view were destined to stage a major revival in philosophy, and semantic analysis would be their main source, but it is the view Mill takes least seriously.

In the contemporary use of the term, Mill is himself a nominalist – he rejects abstract entities (see Abstract objects §4). However, just as severe difficulties lie in the way of treating the ontology of arithmetic in terms of aggregates rather than sets, so there are difficulties in treating the ontology of general semantics without appealing to universals and sets, as well as to natural properties and objects. We can have no clear view of how Mill would have responded to these difficulties had they been made evident to him. But we can be fairly sure that he would have sought to maintain his nominalism.

However, his main target is the doctrine that there are real a priori propositions (see A priori). What, he asks, goes on in practice when we hold a real proposition to be true a priori? We find its negation inconceivable, or that it is derived, by principles whose unsoundness we find inconceivable, from premises whose negation we find inconceivable. Mill is not offering a definition of what is meant by such terms as ‘a priori’, or ‘self-evident’; his point is that facts about what we find inconceivable are all that lends colour to the use of these terms.

They are facts about the limits, felt by us from the inside, on what we can imagine perceiving. Mill thought he could explain these facts about unthinkability, or imaginative unrepresentability, in associationist terms, and much of his work claims to do so. This associationist psychology is unlikely nowadays to convince, but that does not affect his essential point: the step from our inability to represent to ourselves the negation of a proposition, to acceptance of its truth, calls for justification. Moreover, the justification itself must be a priori if it is to show that the proposition is known a priori.