Russell - A Very Short Indroduction (6 page)

Russell saw idealism and its concomitant monism as arising from a view about
relations
which, once refuted, opens the way to pluralist realism. Relations are expressed by such sentences as ‘A is to the left of B’, ‘A is earlier than B’, ‘A loves B’. On the idealist view, Russell claimed, all relations are ‘internal’, that is, they are properties of the terms they relate, and, in a full description, appear as properties of the whole which they form with their relata. This is sometimes plausible; in ‘A loves B’ A’s loving B is a property of A – that is, is a fact about the nature of A – and the complex fact denoted by ‘A loves B’ has the property of being a
loving-of-B-by-A
. But if all relations are internal it immediately follows that the universe constitutes what the idealist philosopher Harold Joachim calls ‘a significant whole’, for it means that it is part of the nature of anything to be related to everything else, and that therefore a full description of any one thing will tell us everything about the whole universe, and vice versa. Bradley puts the point like this: ‘Reality is one. It must be single because plurality, taken as real, contradicts itself. Plurality implies relations, and, through its relations it unwillingly asserts always a superior unity’ (
Appearance and Reality
, 519).

In opposition to this view Russell argued that the idealists commit a fundamental mistake. This is that they take all propositions to be of subject-predicate form. Consider the sentence ‘The ball is round’. This can be used to express a proposition in which the property of roundness is predicated of a given ball (‘predicated’ means: applied to, said of). In Russell’s view, the idealists wrongly took it that all propositions, even relational ones, are ultimately of subject-predicate form; which means that every proposition must, in the final analysis, constitute a predication on reality as a whole, and that relations as such are unreal. For example: on the idealist view the proposition ‘A is to the left of B’ should properly be understood as saying, ‘Reality has the property of Aappearing-to-be-to-the-left-of-B’ (or something like this). But if one sees that many propositions are irreducibly relational in form, one thereby sees that monism is false. To say that many propositions are irreducibly relational is to say that relations are real or ‘external’ – they are not grounded in the terms they relate; the relation ‘to the left of’ does not belong intrinsically to any spatial object, which is to say that no spatial object must of necessity be to the left of other things. For it to be true that ‘A is to the left of B’, Russell argued, there therefore has to be an A
and separately
a B for the former to stand to the latter in the relation ‘left of’. And of course to say that there are more things than one is to reject monism.

Rejection of monism constitutes a rejection of idealism for Russell because it is crucial to idealism that the relation of experience to its objects should be internal; which is in effect to say that there is no such relation; which is again in effect to say that relations are unreal. But on Russell’s opposed view that relations are real, experience cannot be conflated with its objects; which is to say that those objects exist independently of being experienced. And this is central to what Russell and Moore meant by realism.

It is disputable whether Russell is right in thinking that all the idealists (including Leibniz), and before them the Schoolmen with their metaphysics of substance and attribute, were committed to the view that all propositions are subject-predicate in form. But he certainly took himself to have discovered a highly important flaw in previous philosophy. With the rejection of idealism he went for a time to the other extreme, that of being a realist about everything. By his own account he was a ‘naïve realist’ in the sense of one who believes that all the perceived properties of material objects are genuine properties of them, a physical realist in believing that all the theoretical entities of physics are ‘actually existing entities’ (
MPD
48–9), and a Platonic realist in believing also in the existence, or at least in the ‘being’ (where this is a qualified and perhaps lesser kind of existence), of ‘numbers, the Homeric gods, relations, chimeras, and four-dimensional spaces’ (
The Principles of Mathematics
(
PoM
) 449). Russell later trimmed this luxuriant universe by applying ‘Ockham’s razor’, the principle that entities should not be multiplied unnecessarily. For example, if physical objects can be exhaustively explained in terms of subatomic entities, then a basic inventory of the universe should not contain
both
trees
and
the quarks, leptons, and gauge particles of which trees are made. This, later, was how he applied the technique of analysis. But he still believed in an inclusive realism in
PoM
, to which he turned after encountering the work of Giuseppe Peano in Paris in 1900.

The foundations of mathematics

Leibniz had dreamed of a
characteristica universalis
, a universal and completely precise language, use of which will solve all philosophical problems. Russell recognized, in his book on Leibniz, that this was a desire for a symbolic logic, by which Russell then meant the ‘Boolean algebra’ developed by George Boole in the mid-nineteenth century. But at that juncture he did not think Leibniz was right to suppose that philosophical problems can be solved by employing the technicalities of a deductive logical system, for the reason that the truly important questions of philosophy are about matters that are ‘anterior to deduction’, namely, the concepts or facts referred to in the premisses from which inference starts. Whatever these are, Russell argued, they are not supplied to us by logic; logic can only help us in reasoning about them.

But Russell changed his mind when he encountered Peano’s work. Peano’s advances in logical technique (they had been anticipated by Gottlob Frege, but neither Peano nor Russell then realized this) immediately suggested to Russell ways of stating the fundamental principles of logic, and of showing two centrally important things: first, how all the concepts of mathematics can be defined in terms of them, and secondly, how all mathematical truths can be proved from them. In short, it suggested to Russell how to show that logic and mathematics are identical. This is the aim of both
PoM
, and its more fully worked out version,
Principia Mathematica
(
PM
).

The project of deriving mathematics from logic is known as ‘logicism’. In
PoM
Russell did not attempt a rigorous assault on this part of the programme, limiting himself instead to an informal sketch. The rigorous assault was left to
PM
. Chief among Russell’s reasons for delaying the task until
PM
was his discovery of a paradox which threatened the whole enterprise.

Russell’s first task was to define the concepts of mathematics using as small a number as possible of purely logical notions. (Here follow three paragraphs of informal technicality, which need not detain the reader.) Letting ‘p’ and ‘q’ stand for propositions, these notions are: negation (not-p), disjunction (p or q), conjunction (p and q), and implication (if p then q). To these operations are added symbols for representing the inner structure of propositions: ‘Fx’ is a functional expression in which ‘x’ is a variable standing for any individual, and ‘F’ is a predicate letter standing for any property. Thus ‘Fx’ says that x is F (an instance of what it symbolizes is: ‘the tree is tall’). One of the important technical advances that Russell was able to use is a way of
quantifying
such functions. Using notation which is now standard in logic, quantification is expressed like this: (x) expresses ‘all xs’, so (x)Fx says that
all
xs are F, (∃x) expresses ‘at least one x’, so (∃x)Fx says that
at least one
x is F. And finally there is the notion of identity: ‘a = b’ says that a and b are not two objects but one and the same object. With this simple language it is possible to define the concepts of mathematics.

Earlier mathematicians had investigated the relations among mathematical notions and had recognized that they are all reducible to the natural numbers (the counting numbers 1, 2, 3 . . . ), although no one had so far demonstrated this precisely. The first step in the programme was therefore to define the natural numbers in logical terms. This is what Frege had already done, although Russell did not at the time realize this.

The definition exploits the notion of classes: 2 is defined as the class of all couples, 3 as the class of all trios, and so on. In turn, a ‘couple’ is defined as a class having members x and y where x and y are not identical and where, if there is any other member z of the class, z is identical with either x or y. The general definition of number is stated in terms of sets of similar classes, where ‘similarity’ is a precise notion denoting a one–one relation: two classes are similar if a one–one relation is specifiable as holding between their members.

With these notions in place, a raft of problems is solved, among them: how to define 0 and 1 (Russell pointed out that these are two of the most difficult notions in mathematics), how to overcome ‘one and many’ puzzles (how many things is a chair: is it one, or – if you count its parts and constituents – many?), and how to understand infinite numbers. Once the whole numbers are defined, the others (positive and negative numbers, fractions, real numbers, complex numbers) present relatively little difficulty.

The first part of the programme – defining mathematical concepts in terms of logical ones – therefore seems largely unproblematic, once the right technicalities are available. The second – the distinctively logicist part of showing that mathematical truths can be proved from the fundamental principles of logic – turns out to be vastly more difficult.

The main reason for this, from Russell’s point of view at the time, was his discovery of paradox. The paradox relates to a notion which, as the foregoing sketch shows, is central to the project: the notion of classes. In the course of his work Russell was led to ponder the fact that some classes are, and some are not, members of themselves. For example, the class of teaspoons is not a teaspoon, and therefore is not a member of itself; but the class of things which are not teaspoons is a member of itself because it is not a teaspoon. What, then, of the class of all those classes which are not members of themselves? If this class is not a member of itself, then by definition it is a member of itself; and if it is a member of itself, then by definition it is not a member of itself. So it is both a member of itself and not a member of itself. Thus paradox.

At first Russell thought that some trivial mistake was to blame, but after much effort to put things right, and after consulting Frege and Whitehead, it became clear to him that disaster had struck. Russell published
PoM
without having found a remedy. But by the time he and Whitehead came to write
PM
he had, he thought, found a way out – but his strategy proved highly controversial. Matters can be described as follows.

The attempt to deduce the theorems of mathematics from purely logical axioms cannot proceed, Russell found, without supplementary axioms to make possible the task of proving certain theorems in arithmetic and set theory. Two of these supplementary axioms (their details do not matter; I mention them for completeness) are the ‘axiom of infinity’, which states that there are infinite collections in the world, and the ‘axiom of choice’ (sometimes called the ‘multiplicative axiom’) which states that for every set of disjoint non-empty sets there is a set which shares exactly one member with each of the member sets. The axioms are needed so that numbers can be defined in terms of classes, as sketched above. But they both appear to involve a difficulty, which is that they are existential in character, that is, they say ‘
there is
such-and-such’ – in the first case, a number, in the second, a set – and this is a problem because logic should not be concerned with what does or does not exist, but only with purely formal matters. But Russell found a solution: it is to treat mathematical sentences as conditionals, that is, as sentences of ‘if – then –’ form, with the axioms occupying the ‘if’ gap: as such they say, ‘if you premiss this axiom, then –’. Because these conditionals are derivable from the axioms of logic, the apparent importation of existential considerations does not matter.

But much greater difficulty arose with a third supplementary axiom, the ‘axiom of reducibility’. This is the axiom Russell adopted to overcome the paradox problem, but which other logicians find hard to accept.

The axiom of reducibility is tied to Russell’s ‘theory of types’. An informal way of understanding this theory is to note that the paradox discovered by Russell arises because the property of not being-amember-of-itself is applied to the class of all classes having that property. If a restriction could be introduced which ruled that this property is applicable only to the member classes and not to the class of those classes, the paradox would not arise. This suggests that there should be something like a distinction of levels among properties, such that those attributed at one level are not attributable at a higher level.

There is a version of type theory – it is a simpler version than Russell’s – which captures this intuition and seems plausible to some logicians. It was suggested by the mathematician-philosopher Frank Ramsey and is called the ‘simple theory of types’. It puts matters like this: the language which applies to a given domain has level 1 expressions – names – which refer to objects in the domain, and it has level 2 expressions – predicates – which refer only to properties of those objects, and it has level 3 expressions – predicates of predicates – which refer only to properties of those properties . . . and so on. The rule is that every expression belongs to a particular type and can only be applied to expressions of the next type below it in the hierarchy. In line with the informal sketch just given, one sees how this strategy suggests a solution to the paradox problem.