Epistemic Logicism

Epistemic logicism is any variant of logicism that seeks to derive a primarily epistemological lesson from the logicist reduction.³³ For an epistemic logicist, the fundamental problem is the justification, or explanation, of mathematical knowledge, and his aim is to account for this by reducing the epistemology of mathematics to that of logic.

On this view, the importance of the logicist reduction stems from the fact that it claims to reveal the true ground of justification for mathematical propositions. Because we are epistemically justified in accepting the basic truth of logic say, because they are self-evident the identification of mathematics with logic would extend this justification from logic to mathematics.³⁴

The principal source for the interpretation which associates Russell’s work on the foundations of mathematics with epistemic logicism is the commonly held conception of him as a philosopher for whom the overriding question of philosophy was one concerning knowledge and certainty.³⁵ And it must be said that it is Russell himself who is chiefly responsible for this interpretation. On the face of it, epistemic logicism is a direct consequence of this “constant preoccupation” of his (Russell 1959, p. 11). The following quotation, in which he speaks of his work leading up to Principia Mathematica, is typical:

The desire to discover some really certain knowledge inspired all my work up to the age of thirty-eight. It seemed clear that mathematics had a better claim to be considered knowledge than anything else; therefore it was to the principles of mathematics that I addressed myself. At the age of thirty-eight I felt that I had done all that lay in my power to do in this field, although I was far from having arrived any absolute certainty.

Indeed the net result of my work was to throw doubt upon arithmetic which had never been thrown before. I was and am persuaded that the method I pursued brings one nearer to knowledge than any other that is

33 The term “epistemic logicism” is taken from Irvine (1989, sec. 2).

34 Irvine (1989, p. 307).

35 See Wood (1959).

available, but the knowledge it brings is only probable, and not so precise as it appears at first sight (Russell 1929, p. 16; for a very similar state-ment, see Russell 1924, pp. 323-4).

The following quotation is even more to the point, since it forges an explicit connection between one important aspect of the concept of rigour, that of rigorous proof, and its potential epistemic virtues:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstra-tions, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new kind of mathematics, with more solid foundations than those that had hitherto been thought secured (1956a, p. 53).³⁶

Imre Lakatos’ paper “Infinite regress and foundations of mathemat-ics” (Lakatos 1962) constitutes what is probably the most striking attempt to follow Russell’s own testimonies and make his logicism a variant of epistemic logicism.³⁷ According to Lakatos, logicism should be looked upon as an attempt to counter the classical sceptical argu-ment from infinite regress. As a sceptic uses it, this arguargu-ment is meant to show that any claim to knowledge can be undermined since no firm foundation can be given either for meaning or for truth;

every effort at establishing either of these can always be met with a sceptical demur, and insofar as one tries to meet this demur by giving the meaning of an expression in terms of other expressions or reduc-ing the truth of a statement to that of other statements, the sceptic can always reply by reasserting his position. Not even the claim that one has reached an absolute end-point – meanings that are perfectly

36 The passage is repeated in Russell (1969), p. 220.

37 According to Lakatos, his interpretation applies to Frege as well, although he uses Russell’s logicism as an illustration. In fact, Lakatos goes so far as to say that he wants to “exhibit modern mathematical philosophy as deeply embedded in general epistemology and as only to be understood in this context” (1962, p. 4).

transparent or truths that are absolutely self-evident – can defeat the sceptic, since the requirement for a warrant applies with no less force to them. The problem, as Lakatos puts it, is that “[m]eaning and truth can be transferred, but not established” (1962, p. 3).

Lakatos then suggests that Russell’s logicism is an instance of what he calls the “Euclidean programme” (id., p. 4). That is, it is in essence an endeavour to save mathematics from sceptical doubt that is kept alive by the constant threat of the infinite regress. This it attempts to do by showing, in accordance with epistemic logicism, how mathematics can be derived from self-evident logical axioms and abbreviatory definitions by means of self-evidently correct rules of inference. On Lakatos’ reading, the rationale for pursuing the logicist reduction stems from the inner instability of the Euclidean pro-gramme; concerning any putative foundation for (mathematical) knowledge, the sceptic – or the dogmatist at a moment of doubt – can ask questions of the following sort: Are the primitive terms really primitive? Are the primitive truths really primitive? Are the rules of inference really safe? “These questions”, Lakatos explains, “played a decisive role in Frege’s and Russell’s great enterprise to go back to still more fundamental first principles, beyond the Peano axioms of arithmetic” (id., p. 11).³⁸ It is this line of thought which backs up

38 As this quotation shows, Lakatos considers only the case of arithmetic and thus ignores the fact that the early Russell’s logicism was not restricted in this way. It would, indeed, be difficult to reconcile Lakatos’ reading of logicism with Russell’s inclusion of geometry within the scope of logicism.

Whatever else is said of Russell’s treatment of geometry in the Principles, it is at least clear that it was not his intention to give geometrical knowledge an absolutely certain and unshakeable foundation. It follows from his treatment of geometry that a distinction must be drawn between pure and applied geometries. As regards the former, “all geometrical results follow, by the mere rules of logic, from the definitions of various spaces (1903a, §434). The latter in turn has to do with the problem of which of the abstractly definable geometries correctly describes the properties of physical space. If “geometri-cal knowledge” is taken to refer to knowledge of applied geometry, it is (partly) empirical and falls outside the scope of logic and logicism. And if it is taken in the former sense, as what follows from which definitions, it is not

Lakatos’ further claim that the Euclidean programme implies the

“trivialization of [mathematical] knowledge” (id., pp. 4-5); since the logicist can suppress scepticism only by a successful identification of a foundation for which the demand of a further warrant cannot arise, the ultimate foundation can consist only of such truths, principles and terms that must be recognised as “trivialities”.

The weakest point in Lakatos’ rational reconstruction is the claim that Russell’s logicism involves a “logico-trivialization of mathemat-ics”. As I read him, this is implied by Lakatos’ view on the “inner dialectic” of the Euclidean programme: nothing short of a foundation with an epistemically trivial load can be considered a viable founda-tionalist answer to the sceptical argument from an infinite regress.

Hence, since logicism is an instance of this programme, the logicist foundation must be construed accordingly. This inference is where Lakatos’ reconstruction invites an immediate objection. Apart from his own conception of what the Euclidean programme requires, there is no reason to think that the early Russell would have attributed anything like vacuous truth or triviality to logic. And there are very good reasons for thinking otherwise.³⁹ That is, Russell’s acceptance of something like the triviality-thesis was not part of his original logi-cism. In fact, it came only much later and was largely due to Wittgen-stein’s influence.⁴⁰ It was through him that Russell came to the con-viction that logic is in some sense a matter of linguistic rules and hence (again, in some sense) tautologous and “trivial”. Furthermore, Russell says that it was only with great reluctance that he finally accepted the triviality of logical and, therefore, of mathematical truth (1959, pp. 211-12; cf. also Russell 1950-2).

This observation suffices to undermine Lakatos’ reconstruction, which presents Russell’s logicism was directed against the specific target of scepticism arising from the classical infinite regress

argu-clear that self-evidence has any such role to play in this knowledge as Laka-tos finds for it in the case of arithmetic and its reduction to logic.

39 Cf. here Hager (1994, pp. 40-41).

40 As Russell himself says explicitly (1959, pp. 112-119).

ment. There is, however, a more general difficulty besetting any attempt to read Russell’s philosophy of mathematics as a species of epistemic logicism; notwithstanding his later testimonies, issues pertaining to epistemology simply do not surface in the relevant writings (most notably of course in the Principles of Mathematics).⁴¹ Russell’s writings from the early logicist period have very little to say about the epistemology of mathematics. If his reasons for occu-pying himself with the foundations of mathematics had been episte-mological in some relevant sense, then, presumably, he would have addressed various questions about mathematical knowledge. But he does not do so. Instead, he gives the following explanation of the objects of his own work (this is from the Preface to the Principles).

Firstly, he attempts to establish the thesis that “all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles” (p. xv). The various parts of this thesis are defended

“against such adverse theories as appeared most widely held or most difficult to disprove” (ibid.), a task that involves the presentation of

“the more important stages in the deductions by which the thesis is established” (ibid.) Secondly, he discusses the fundamental logical concepts and principles which underlie the purported deduction. In his often quoted words, this “discussion of the indefinables [...] is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple” (ibid.) This description of the aims of the book makes no mention of anything that has to do with our knowledge.⁴² The only apparent

41 This means of course that Russell’s later descriptions of his own logicism tend to obscure its real import.

42 It is also worth contrasting two descriptions that Russell himself gave of the development which lead him to write the Principles. In My Philosophical Development he tells that “as a boy” his interest in mathematics was fuelled by practical goals and ambitions (1959, p. 208). Later, however, this gave way to an interest “in the principles upon which mathematics is based” (id., p. 209).

exception is the somewhat casual reference to acquaintance that occurs in one of the passages quoted above. Exactly how this term should be understood in this particular context, however, is a less than straightforward matter. It could be taken as an indication of the epistemology which could with some plausibility be read into the early Russell and which surfaces in his later writings; this epistemol-ogy builds, in one way or another, on the immediate cognitive rela-tion of acquaintance.⁴³ From this reading there is but a short step to epistemic logicism. That is to say, this interpretation of Russell’s position, if spelled out, would be likely to yield the familiar founda-tionalist picture of justification, combined with platonism about mathematical objects. The latter portraits mathematical truths as

According to his testimony, the change arose from a wish to combat

“mathematical scepticism” (ibid.) Its possibility was suggested to him by the obviously fallacious proofs which his mathematics teachers at Cambridge wanted him to accept and which could be overcome only by finding “a firmer foundation for mathematical belief” (ibid.) It was this kind of re-search, he writes, which eventually lead him to mathematical logic. This recollection should be compared to how the origin of the Principles was described in that very work: “About six years ago [sc. around 1896], I began an investigation into the philosophy of Dynamics. I was met by the difficulty that, when a particle is subject to several forces, no one of the component accelerations actually occurs, but only the resultant acceleration, of which they are not parts; this fact rendered illusory such causation of particulars by particulars as is affirmed, at first sight, by the law of gravitation. It appeared also that the difficulty in regard to absolute motion is insoluble on a rela-tional theory of space. From these two questions I was led to a re-examination of the Principles of Geometry, thence to the continuity and infinity, and thence, with a view to discovering the meaning of the word any, to Symbolic Logic” (1903a, p. xvi-xvii). Coffa’s distinction between the two senses of rigour is a very apt way to bring out the contrast between these two accounts; Russell, it seems, had largely forgotten about the semantic sense of rigour by the time the series of autobiographical recollections began to flow from his pen.

43 The term “Platonic Atomism” is used by Hylton (1990a) to refer to that set of doctrines which Russell and Moore developed after their rejection of idealism.

being about mind-independent, abstract objects. The former exhibits mathematical knowledge as possessing a foundational structure, which divides mathematical truths into two epistemological kinds:

those deriving their justification from other mathematical truths, and those the justification of which is immediate or non-inferential (on this view, there is an intimate connection between an axiomatic organisation of a branch of mathematics and our knowledge of that particular branch, and the axiomatisation itself receives its motivation from epistemological considerations, in accordance with epistemic logicism). It is in connection with the second epistemological kind that acquaintance enters the picture; its role is to deliver a crucial part of the foundationalist picture of knowledge, to wit, to render a chain of truths genuinely justificatory by supplying non-arbitrary end-points for the chain. To back up its claim to that role, it is pointed out, firstly, that acquaintance is a direct cognitive relation between the knowing subject and the relevant subject-matter. In this sense it is analogous to ordinary sense-perception; if the justificatory chain is regarded as consisting of beliefs appropriately related to one another, the role of acquaintance can be said to be that of explaining how the justification of mathematical beliefs can terminate with something non-belief-like (mathematical objects and (primitive) truths about these objects). Secondly, to make them suitable for serving as end-points of justification, acquaintance-based beliefs must be invested with some further property which makes them self-justifying, or justified independently of their relation to other truths. As we have already seen, the standard move here is to resort to self-evidence, which is a sort of propositional correlate of the notion of acquaintance.

Since Russell does very little in the Principles to elaborate on the putative epistemological reading of acquaintance, most of the details would remain to be filled in. The use of the word “acquaintance”

naturally suggests ideas that are familiar enough; in particular, the justification of mathematical knowledge, according to this view, is in the last instance a matter of a quasi-perceptual connection between

the knowing subject and certain objects.⁴⁴ The issues surrounding this notion need not detain us, since the present point is just to indicate one way to understand Russell’s reference to acquaintance in the passage from the Principles. The suggestion, then, is that this notion has a definite epistemological role to play, one that at least roughly accords with what is suggested by epistemic logicism.

I have two objections to the interpretation which presents the early Russell’s philosophy of mathematics as a species of epistemic logicism. Firstly, the epistemic reading of “acquaintance” is not forced upon us. Even though it may be tempting to associate Rus-sell’s talk of acquaintance with epistemology and, in particular, a foundationalist account of mathematical knowledge, other statements that he makes in the Principles about the status of the logical founda-tions undermine this association.⁴⁵ To begin with, consider what he writes immediately after the passage about acquaintance:

Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them; there is a

44 Michael Resnik (1980, p. 162) gives the name “epistemic platonism” to this doctrine, according to which “our knowledge of mathematical objects is at least in part based upon direct acquaintance with them, which is analo-gous to our perception of physical objects”.

45 It is worth pointing out that only a few years after the Principles Russell explicitly rejected the idea or requirement that the fundamental logical premises should possess self-evidence or intrinsic obviousness. This point is made briefly in (1906a, pp. 193-4) and elaborated in his (1907b), where it is argued that the reasons for accepting an axiom are largely “inductive” in character. That is, the reasons for accepting a primitive proposition is not (or is not simply) its self-evidence: “[t]he primitive propositions with which the deductions of logistic begin should, if possible, be self-evident to intui-tion; but that is not indispensable, nor is it, in any case, the whole reason for their acceptance. This reason is inductive, namely that, among their known consequences (including themselves), many appear to intuition to be true, none appear to intuition to be false, and those that appear to intuition to be true are not, so far as can be seen, deducible from any system of indemon-strable propositions inconsistent with the system in question” (id., p. 194).

process analogous to that which resulted in the discovery of Neptune, with the difference that the final stage the search with a mental tele-scope for the entity which has been inferred is often the most difficult part of the undertaking (1903a, p. xv).

What this passage suggests is that knowledge of the indefinables is not simply a matter of immediate perception, or perception-like process, through which the relevant entities force themselves upon us. The indefinables should rather be described as “inferred entities”

in something like the sense that is familiar from Russell’s later writ-ings: they are hypothetical entities postulated for the purposes of fulfilling a certain role.

If this is on the right track, it might well be that acquaintance, as Russell uses it in the passage under discussion, has no epistemological connotations at all; it may be that all he is saying is that the early parts of the book are meant to make readers sufficiently familiar with the

If this is on the right track, it might well be that acquaintance, as Russell uses it in the passage under discussion, has no epistemological connotations at all; it may be that all he is saying is that the early parts of the book are meant to make readers sufficiently familiar with the