The Analytic Turn in Early Twentieth-century Philosophy
1- Frege and Russell: Decompositional and Transformative Analysis
The papers in Part One explore the work of Frege and Russell, the two main instigators of the analytic turn that gave rise to analytic philosophy. As indicated above, both Frege and Russell came to philosophy through concern with the foundations of mathematics, and both sought to demonstrate the logicist thesis that arithmetic (and geometry as well, in the case of Russell) could be reduced to logic by offering transformative analyses utilizing the new quantificational logic. It was in their philosophical attempts to justify their logicist projects that analytic philosophy was born.
In ‘Frege-Russell Numbers: Analysis or Explication?’, Erich Reck takes as his starting-point the logicist definition of the natural numbers as equivalence classes of equinumerous classes which both Frege and Russell gave, and considers the status of this definition, focusing primarily on Frege’s views. Was it intended as an ‘analysis’, in the sense of revealing what the natural numbers ‘really’ are, or as an ‘explication’, in the sense of offering a reconstruction that does essentially the same job but in a more powerful and rigorous theoretical system? The Platonism that many have attributed to Frege would seem to suggest the first, while the second is compatible with a more conventionalist reading that brings Frege closer to Russell and Carnap. Reck does not attempt the difficult task of deciding the issue on textual grounds, but he does elucidate the conceptions of analysis involved in asking the question and discuss the constraints on such definitions that might narrow down the possibilities.
As far as Frege’s Platonism is concerned, Reck argues that this should not be interpreted as invoking a ‘Platonic heaven’ of abstract objects such as numbers, which we apprehend by some quasi-perceptual ‘intuition’. The most charitable and sophisticated reading, he suggests, is that developed by Tyler Burge,[^12] according to which getting at ‘the facts of the matter’ is taken to involve reasoning and theory construction rather than (quasi-)empiricist observation. Nevertheless, even this sophisticated reading seems to conflict with a more conventionalist reading, and as Reck notes, there are certainly passages where Frege offers something very close to Carnap’s notion of explication (in lectures that Carnap actually attended).[^13]
One way of approaching the issue is by comparing the Frege-Russell definition with alternative definitions such as those subsequently provided by John von Neumann and, more recently, by Crispin Wright and Bob Hale. Taking these three cases, how do we decide whether to identify the natural numbers with theFrege-Russell numbers , thevon Neumann numbers or theWright-Hale numbers , as Reck calls them? Like the Frege-Russell numbers, the von Neumann numbers are classes (set-theoretic objects), which satisfy the Dedekind-Peano axioms, but they arguably do not do justice to the role of numbers in ‘bringing together’ equinumerous collections. The Wright-Hale numbers, on the other hand, seem to do justice to the application of numbers, but do they really count as logical objects? Would Frege have been happy with Wright’s and Hale’s ‘neo-logicism’?
Clearly, there are different constraints in different theoretical contexts, and the question of what the numbers ‘really’ are can only be answered in a particular conceptual framework. As Reck suggests, this might help us in reconciling the Platonist and conventionalist strands in Frege’s thought, even if Frege himself may not have seen it in this way. Indeed, for any interpretation of Frege’s thought that might be offered, we might well be tempted to ask an analogous question. Does the interpretation offered count as an ‘analysis’ or an ‘explication’? Are there ‘facts of the matter’ as to what Frege really meant? The question Reck addresses in his paper clearly has implications beyond the specific case of the natural numbers.
Frege’s and Russell’s logicist definition of the natural numbers as equivalence classes of equinumerous classes is also the starting-point of James Levine’s paper, ‘Analysis and Abstraction Principles in Russell and Frege’. Although they offered the same definition, however, Levine argues that they used that definition in quite different ways (providing a further illustration of the Carnapian message of Reck’s paper, we might add). For Frege, it played a role in his claim that numbers are ‘self-subsistent objects’, whereas for Russell, it was taken as showing that numbers can be dispensed with in giving an inventory of the world. Underlying these two different philosophical approaches were two different conceptions of analysis and propositional contents. Central to Russell’s philosophy from the time of his rejection of idealism, Levine argues, was the principle that every propositional content can be uniquely analyzed into ultimate simple constituents, a claim that Frege did not endorse. This meant that, for Russell, every proposition had a privileged representation (even if no one had yet been able to give it), which mirrored its content at the ultimate level of analysis. If two sentences of different forms could be used to assert the same propositional content, therefore, then they could not both be privileged representations. Frege, on the other hand, insisted throughout his life that one and the same content (‘thought’, in his later terminology) could be analyzed in indefinitely many ways, without assuming that there was some one way that was uniquely privileged.
Consider, then, the case of the Cantor-Hume principle,[^14] asserting the equivalence between (Na) and (Nb):[^15]
(Na) The conceptF is equinumerous to (i.e., can be correlated one-one with) the conceptG .
(Nb) The number ofF s is equal to the number ofG s.
On Russell’s view, if (Na) and (Nb) have the same propositional content, then at most only one of them can offer a privileged representation of that content, since they are of different forms. So their equivalence suggests that talk of numbers can be ‘reduced’ to talk of the one-one correlation of concepts, so that we do need to suppose the existence of numbers in addition to that of concepts. For Frege, on the other hand, the possibility of contextually defining numbers in this way does not imply that numbers are not objects. On the contrary, the fact that number statements can be true and that constituent number terms such as ‘the number ofF s’ are proper names is enough to show that numbersare objects. The issue is how we can apprehend such objects, given (as Frege himself stressed) that they are notactual objects, i.e., spatio-temporal objects that have causal effects. It was here that he appealed to the equivalence between (Na) and (Nb). According to Frege, we apprehend numbers by understanding the sense of sentences in which number terms appear, an understanding that is grounded (and hence shown to have a logical source) by our grasp of sentences such as (Na) together with our recognition of the equivalence captured in the Cantor-Hume Principle.[^16]
What we have in the case of the Cantor-Hume Principle is what is often called an ‘abstraction principle’, and Frege’s and Russell’s different conceptions of analysis clearly lead to different views of the use of such principles. In fact, it is significant in this respect that Frege himself never called it an ‘abstraction principle’, a phrase which itself suggests that one of the two sentences involved is on a different and ‘higher’ (i.e., more abstract) level to the other - numbers being ‘abstracted’ from the relation of one-one correlation obtaining between concepts. Indeed, from Russell’s diametrically opposed perspective, the use of the phrase is also misleading, since it seems to grant that numbers are objects, just ‘higher’ or more abstract objects. As Levine notes (p. [16] below), Russell at one point remarks that the principle of abstraction should really be called ‘the principle which dispenses with abstraction’, since it “clears away incredible accumulations of metaphysical lumber” (1914, p. 51). In Russell’s case, the reductionism made possible by abstraction principles takes the form of eliminativism - ‘analyzing away’ the supposed abstract objects. Not only the use of abstraction principles but also the very name they are given, then, reflects the underlying conceptions of analysis.
What led Russell to this eliminativist view of abstraction principles? He may have shared Frege’s concern to demonstrate logicism, but he adopted a diametrically opposed approach to the use of abstraction principles. As Levine shows, at the root of this disagreement lies their different conceptions of analysis, and in particular, their different attitudes to the principle that every propositional content can be uniquely analyzed into ultimate simple constituents, which Russell endorsed but Frege did not. This principle was adopted by Russell in his initial rejection of idealism. But adopting this principle does not in itself determine which of the two sentences involved in an abstraction principle is to be seen as the more fundamental (as the more privileged representation, in Levine’s terminology), nor whether eliminativism is to be preferred to a more moderate reductionism. Why should (Na) be seen as more fundamental than (Nb), for example, and why, if we do this, should we think of numbers being ‘analyzed away’ rather than just being shown to be ‘higher’ objects?
Levine identifies the source of Russell’s concern with abstraction principles in his interest in theories of serial order, which arose in his engagement with Hegelian idealism. Take the case of events, considered as ordered by the temporal relations ofbefore ,after andsimultaneous with . On an absolute theory, to say that two events are simultaneous with one another is to say that they both occur at one time, moments of time being treated as just as real as events, and the relation ofoccurring at being treated as just as basic as the ordering relations. On a relative theory, on the other hand, events and the ordering relations are taken as basic, and moments of time are then defined in terms of these. (There is no absolute framework of temporal moments in which events are located.) Immediately after his rejection of idealism, Russell adopted absolute theories of order, but he soon came to endorse relative theories. In the case of number, for example, he moved from regarding numbers as just as real as (and distinct from) classes to treating them as definable in terms of (and hence reducible to) classes.
What led Russell to endorse relative theories of order? In his paper Levine is more concerned with the differences between Frege and Russell than with the details of the evolution of Russell’s ideas, but he does note that the change coincides with Russell’s acceptance of logicism in 1901/1902.[^17] Russell was able to endorse the logicist definition of numbers as classes without subscribing to Frege’s realism, however, because of his different conception of analysis. This is Levine’s main point, and it illustrates not only the dependence of metaphysical views on conceptions of analysis but also, in the case of Russell, the significance of the period between 1900 and 1905. This period has long been recognised as crucial in the development of Russell’s thought, and much light has been shed on it by the authors of the next two papers, Nicholas Griffin and Peter Hylton.[^18] Griffin looks in more detail at Russell’s early conception of analysis, and Hylton discusses the transformative conception of analysis that was introduced by the theory of descriptions in 1905.
In ‘Some Remarks on Russell’s Early Decompositional Style of Analysis’, Griffin shows how fundamental Russell’s early conception of analysis was in his thinking after his break with idealism, a conception that was essentially decompositional, that is, that treated analysis as a process of identifying the constituents of something. Russell initially conceded to idealism that a complete analysis was only possible where the complexes to be analysed were mere collections rather than unities, unities involving relations that could not be separated out. But he nevertheless rejected the key doctrine of the British idealists that all relations are internal. What exactly did this doctrine mean, however, and why did Russell reject it? In answering these questions, Griffin focuses on the debate that Russell had with Harold Joachim (1868-1938) in 1905-7, a debate in which the question of the nature of relations was central. Russell glossed what he called the ‘axiom of internal relations’ as the view that all relations are grounded in the natures of their terms. But according to Russell, ‘the nature of a term’ could mean either ‘all the propositions that are true of the thing’ or ‘the adequate analysis of the thing’, and he accused the idealists (Hegelians) of failing to recognise this distinction, a failure that follows, he claimed, from their principle that every proposition attributes a predicate to a subject (cf. pp. [11-13] below).
Understanding this principle to be restricted to the case of atomic propositions, however, Griffin points out that Russell’s claim is only correct on the assumption thatall the properties of a thing are included in an adequate analysis of it. For only then is it true that if every (atomic) proposition attributes a property to a thing, then the set of all (atomic) propositions that are true of a thing is the same as the set of propositions that give its analysis. But such an assumption, Griffin goes on to argue, makes all such propositions come out as ‘analytic’ - at least, on the traditional definition of an ‘analytic’ proposition as one in which the predicate is contained in the subject - and this cannot have been Russell’s view. Indeed, Russell had himself criticized this view in his book on Leibniz. So how can he have maintained the assumption? Griffin’s answer is that Russell did not, in fact, accept that all - or even most - propositions that are apparently of subject-predicate form are actually of that form; many should be construed instead as relational. Russell rejected, in other words, what he saw as the Hegelian principle that every proposition attributes a predicate to a subject.
As Griffin notes, however, such a defence of Russell’s early decompositional conception of analysis is not completely successful, for it does not solve the problem of simple terms (things). By definition, simple terms have no parts, and so cannot be analysed; in which case, it would seem, they cannot have properties. Griffin states the options for Russell here, but does not attempt to resolve the problem. He concludes his paper by highlighting the importance that the question of relational propositions had in the development of Russell’s early philosophy and the extent to which Russell’s break with Hegelianism was gradual: it took him several years to think through the implications of his rejection of the doctrine of internal relations in the context of his decompositional conception of analysis. That conception was not new; what was new was the use he made of it.
At the core of Griffin’s account of the defensibility of Russell’s early decompositional conception of analysis is the claim that many apparently subject-predicate propositions are implicitly relational. This is not a claim that Russell would have made at the beginning of the 1900s. InThe Principles of Mathematics , for example, he wrote: “On the whole, grammar seems to me to bring us much nearer to a correct logic than the current opinions of philosophers; and in what follows, grammar, though not our master, will yet be taken as our guide” (1903, p. 42). Russell’s debate with Joachim, however, occurs around the time of ‘On Denoting’, when Russell was developing the theory of descriptions, and the claim is certainly characteristic of his views then. Central to the theory of descriptions is the idea that a sentence may need to be transformed - and indeed, radically transformed - to adequately represent the relevant thought or proposition. This idea of transformation is discussed by Peter Hylton in ‘“On Denoting” and the Idea of a Logically Perfect Language’.
Hylton begins by clarifying Russell’s idea of a logically perfect language, a language which mirrors the structure of both the world and the thoughts that represent that world, and in which each ultimate element (simple object) of the world is denoted by one and only one word. Given that our ordinary language is not such a language, associated with the idea is a certain conception of analysis, the aim of which is to transform our ordinary sentences into sentences of the logically perfect language. But what constraints are there on such transformations? Hylton identifies what he calls Russell’s ‘Principle of Acquaintance’ as the key principle, which Russell himself formulates at the end of ‘On Denoting’ as follows: “in every proposition that we can apprehend … all the constituents are really entities with which we have immediate acquaintance.” Although this principle was not new in 1905, Hylton argues, it did not impose any significant constraint on analysis up to that point. In the immediate aftermath of his break with idealism, Russell allowed acquaintance with all sorts of entities; and during the period in which he held his theory of denoting concepts (from 1900/1901 to early 1905), any constraint that such a principle might have imposed was negated, since that theory allowed propositions to have constituents, namely, denoting concepts, that could denote things with which we were not acquainted. It was only when that theory was rejected in favour of the theory of descriptions that the principle finally came to impose a real constraint on analysis.
As far as Russell was concerned, what was crucial about the theory of descriptions was that it enabled him to maintain, in an unqualified form, the view that he had first adopted in rejecting idealism - that a proposition quite literallycontains the objects which it is about. That view had been restricted by the theory of denoting concepts, which had provided a way of dealing with what were accepted as counterexamples. But that theory had also left mysterious the relation of denoting itself - the relation that was taken to obtain between denoting concepts and the things denoted. Russell’s theory of descriptions dispensed with this relation (except, perhaps, in the one case of the variable), but its development came at a cost: the cost of admitting that ordinary sentences need to be radically transformed to yield their ‘real’ logical form, a form that can only be fully revealed in the logically perfect language. In other words, the theory of descriptions allowed Russell to retain his early decompositional conception of analysis, in all its original simplicity, but only by supplementing it with adifferent conception of analysis - the idea of analysis as transformation.
Hylton goes on to consider the further development of this idea in Russell’s later conception of a logical construction and in the work of W. V. O. Quine (1908-2000). In the case of the former (which I will just say something about here), this was reflected in Russell’s ‘supreme maxim in scientific philosophizing’: “Wherever possible, logical constructions are to be substituted for inferred entities” (1917, p. 115; quoted on p. [6] above, in discussing Carnap’sAufbau ). The role that the Principle of Acquaintance plays in Russell’s philosophy might seem to make the need for inferred entities particularly acute. For if we are (apparently) able to talk about a lot of things with which we are not acquainted, then must we notinfer their existence to explain how our talk can beabout such things? Russell denies, however, that such talk is indeed about such things (even if they do exist), and has no way of making sense of entities that are different in kind from those with which we are acquainted. Instead, he suggests, we have to constructanalogues of those entities out of the entities with which we are acquainted (i.e., out of our sense data). But this only reinforces Hylton’s central point - that “Russell is committed to the possibility, in principle, of an extremely far-reaching programme of philosophical analysis” (p. [19] below]. Virtually nothing is what it seems, on Russell’s philosophy after 1905, and it requires extensive analysis to show what the sentences we use are really about.
Russell’s conception of logical construction forms the topic of the final paper in Part One, ‘Logical Analysis and Logical Construction’, in which Bernard Linsky sheds light on the source of this conception in Russell’s philosophy of mathematics, and argues against two influential interpretations of it. Linsky takes as his starting-point Russell’s famous remark inIntroduction to Mathematical Philosophy : “The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil.” (1919, p. 71) Russell had in mind here the ‘postulation’ by Richard Dedekind (1831-1916) of the irrational numbers as limits of a series of ratios, whereas Russell saw himself as actually ‘constructing’ them by defining them as classes. The Dedekind-Peano axioms in the theory of the natural numbers also count as ‘postulates’ which in Russell’s (and Frege’s) logicist project are derived as (supposed) theorems of logic. The logicist definitions of the numbers thus provide the model of logical construction.
In his essay ‘Logical Atomism’, Russell offers a further formulation of the maxim quoted above: “Wherever possible, substitute constructions out of known entities for inference to unknown entities”. He then immediately suggests that an instance of this maxim is what he has called ‘the principle of abstraction’ or ‘the principle which dispenses with abstraction’ (1924, p. 326). As we have seen in considering Levine’s paper, this is the principle that Russell saw as governing his treatment of abstract objects such as numbers. So the message would seem to be that the appeal to abstract objects as inferred entities is to be replaced by the logical construction of analogues that have the same (or at least analogous) formal properties. This message lies at the heart of Linsky’s criticisms of two particular interpretations of Russellian logical construction. On the first, developed during the early 1930s in the work of the Cambridge School of Analysis, logical constructions provide metaphysical reductions, showing how entities of one kind (such as numbers) can be ‘reduced’ to, entities of another kind (such as classes). On the second, based on the more recent work of William Demopoulos and Michael Friedman, logical constructions exhibit the mathematical structures that can be taken as applicable to the empirical world (with the help of appropriate representation theorems). I will focus here on the first interpretation, since (as indicated above) the Cambridge School of Analysis itself forms part of the early history of analytic philosophy.
A paradigm example of logical construction, on the first interpretation, is the ‘reduction’ of committees to their members: a committee is nothing over and above the individual people that make up that committee and their relevant activities. The idea was extended to the case of material objects (which Russell had himself considered inThe Analysis of Matter of 1927): tables and chairs, for example, were seen as logical constructions out of sense data. On such an interpretation, Russell’s position comes out as similar to traditional phenomenalism. But on Linsky’s account, Russell is not claiming that material objects ‘really are’ bundles of sense data. Rather, he is attempting to define entities that have the same (or analogous) formal properties as material objects, by means of which all the fundamental claims about the material world, such as that no two material objects can be in the same place at the same time, can be proved as theorems.
On Linsky’s view, then, logical construction is not a form of reductive analysis but exemplifies what Carnap came to call ‘explication’; and it is significant in this respect that Carnap did indeed have Russellian logical construction in mind here (cf. pp. [6-7] above). Linsky is reluctant to call it ‘analysis’ at all, or at least ‘analysis proper’, which he characterizes as “the process of finding those ultimate constituents of reality out of which the world in so far as we directly know it through acquaintance is constructed” (pp. [9-10] below). But this is just decompositional analysis, and there are many other uses of the term ‘analysis’, not least in Russell’s own writings, as Linsky recognizes. When Russell talks of ‘the analysis of matter’, for example, he is indeed referring to logical construction and not just decompositional analysis. The important point, though, is that the conceptions (whatever they are called) are distinguished and their relationships clarified; and Linsky is right to suggest that the interpretation of logical construction within the Cambridge School of Analysis was distorted by the influence of Wittgenstein’sTractatus . Russell did not take himself to be analyzing ordinary language, and saw no methodological difficulty in offering ‘analogues’ or ‘substitutes’ or ‘explications’ of our ordinary notions. For him, the type of analysis exemplified in logical construction did not involve reducing entities of one kind to entities of another kind, but rather, replacing postulated entities by constructed entities that do analogous work within the relevant theoretical system.
Although I have suggested that the appearance of the theory of descriptions in 1905 is the single most important event in the development of analytic philosophy, then, the analytic turn itself was a far more complex event. Even in the particular case of Russell’s philosophy, there were several key stages. Russell’s and Moore’s rebellion against idealism may have accorded pride of place to decompositional analysis, but this became supplemented by transformative analysis, made possible by the quantificational logic that Frege invented and utilized in offering his own analyses. But Russell’s use of transformative analysis was different from Frege’s, and has itself given rise to different interpretations and developments. All this is part of the complex methodological inheritance that continues to shape analytic philosophy today.