Michèle Friend's Pluralism in Mathematics: A New Position in Philosophy of Mathematics (henceforth simply ‘Pluralism’) advances toleration between competing theories in the Philosophy of Mathematics. The area is ripe for such a treatment; there are many different competing views concerning the nature of mathematical enquiry, which individually provide responses to philosophical challenges and motivate their own elegant mathematical programmes. Moreover, there have been no book-length discussions of these issues – so Friend's book represents an attempt to fill a salient hole in the literature.

There is much in Pluralism that Philosophers of Mathematics will want to debate, and indeed there is much material, often quite technical in nature, that calls for further scrutiny. Here, however, I want to bring to the fore some reasons for all philosophers to be interested in the work, and possible directions for future research.

Friend's book is (by and large), clearly written. Especially helpful is the practice of displaying key definitions, propositions, and viewpoints to add clarity and aid exposition. However the work also has a pleasant narrative, simplifying assumptions are (explicitly) made early on, and then slowly the position is refined through analysis and criticism. The final result is that, by the end of the piece, one feels as though one has been on something of a journey of intellectual development, and even the staunchest opponent is likely to see the merits of Friend's position.

The overall methodology is fairly simple; to provide reasons to adopt a spirit of tolerance in the Philosophy of Mathematics, and then elucidate some specifics of one such account. This is attempted through four main parts. Part I motivates the advocacy of tolerance in the Philosophy of Mathematics from different, widely accepted, philosophical views. Part II then provides the basic characterisation of the Pluralist viewpoint, and also philosophically characterises some logico-mathematical notions central to her position. Part III then examines some objections and possible problems of paradox that might be raised for the Pluralist. Finally Part IV puts Pluralism to work in interpreting various kinds of disagreement about the nature of Mathematics.

Part I is especially interesting for those involved in classic debates, certainly there are many different controversial moves made in motivating Pluralism from each position, and there is work to be done in analysing whether Friend is successful here. Part II will appeal most to specialists in (Mathematical) Logic and its Philosophy; for here we find characterisation of logic as dialectical tool for communication (as opposed to the existence of a formal logical proof setting the standard for correctness). This has interesting consequences, and comes packaged with an associated notion of proof (which more closely unifies informal and formal mathematical reasoning) and rigour to accommodate the new conception of logic. The same is true for Part IV which is largely concerned with bringing Pluralism to bear on some classic debates in The Foundations of Mathematics and Pluralist reconstructions of mathematical reasoning.

It is Part III however that, once the view is on the table, contains material of most interest to philosophers from diverse areas. For here we find arguments, problems, and solutions that crop up across philosophical disciplines. It will, therefore, be useful to say a few words about the position and content of Part III before continuing.

Firstly, what is Pluralism? We may characterise it (in broad terms) as follows:

This contrasts strongly with the following view:

Clearly Monism implies the negation of Pluralism, but it should be noted just how strong a view Pluralism is. Indeed, there are positions between Pluralism and Monism; we might, for example, insist that out of the many and varied attitudes towards discipline X, only two are satisfactory. Such a Dualist position would be intermediate. Further, the relativisation to discipline X means that really Pluralism is a family of positions. One can easily be a Pluralist about a discipline X whilst being a Monist about a different kind of discourse Y. Indeed, within Mathematics alone, Friend covers a vast array of Pluralisms of different stripes.

There is much to say about the specifics of exposition and application to mathematical cases. For the general philosophical audience, however, Pluralism gains its highest degree of interest when considered in its most general form. We wish to say that any mathematical discipline should be actively cultivated and regarded as legitimate. This leads to some classic problems for Pluralism and similar philosophies.

The essential rub of a very general Pluralist Philosophy is the following; like many theories of this broad structure, it facilitates the generation of paradoxes. Friend identifies three:

1. (I) The External Paradox.

2. (II) The Internal Paradox.

3. (III) The Transcendental Paradox.

Paradoxes (I) and (II) will be familiar in nature and admit of a unified solution, whereas (III) requires a little more exposition. Nevertheless, all three are instructive for considering the behaviour of Pluralism and how its study might prove to be informative.

For the External Paradox we begin with the following observation. Pluralism, by its very nature, advocates a spirit of tolerance between philosophical views. Given this, one might argue that the Pluralist should be tolerant of Philosophical views which advocate an intolerance of Pluralism! Does the Pluralist then have to admit that she is in fact wrong (by her opponent's lights) and thus give up on her position?

The Internal Paradox is somewhat similar in nature, but can be generated without reference to other positions. We note that Pluralism is not an ‘anything goes’ philosophical view; despite her tolerance of diverse positions she still insists on certain standards of protocol (such as standards of rigour within the mathematical discipline in question). However, we might think that in doing so we insist on certain forms of dogmatism.

Both paradoxes admit of a unified solution by being precise about the sense in which the Pluralist is tolerant. We can (following Friend), stratify claims about mathematical discourse into different levels.

At the first level we have claims about particular mathematical theorems and particular philosophical claims. An example of the former is the statement ‘Fermat's Last Theorem is true in the standard model of Peano Arithmetic’ and a sentence of the latter kind is ‘The Continuum Hypothesis has a determinate truth-value’. Second level sentences concern full and comprehensive theories (both philosophical and mathematical) such as ‘ZFC is capable of representing any mathematical theorem’ and ‘All mathematical objects are mental constructions’. Finally, third level discourse involves comparing these theories and making claims about mathematics ‘in general’. Pluralism, Friend argues, is a position that appears at the third level as it makes claims concerning the relationship between statements occurring at the previous two levels.

This provides Friend with the resources to block the first two paradoxes. The Pluralist, Friend maintains, is a Pluralist about the first two levels, which does not preclude insisting (on the basis of good reasons) on issues of protocol concerning the third level. Further, she is a principled agnostic and so does does not have to accept that her opponents are correct in their intolerance (though, given a good argument, she is also open to being convinced otherwise!). This provides us with the resources to dodge the external paradox.

Similarly, the internal paradox is avoided; the claims of protocol are of third level (concerning mathematical enquiry generally). The Pluralist can thus maintain that while she advocates a complete Pluralism concerning second level discourse, she can nonetheless keep certain ‘goal posts steady’ (181) during third level enquiry (though she is, in principle, open to revision of third level claims). These two responses are indicative of a standard response that can be made by views subject to these sorts of paradoxes; make a distinction between the kind of claims one is making and the views between which tolerance is advocated. The reader may wish to dispute the details and application of Friend's taxonomy. However, if her characterisation of the issues is accepted, the point is well taken; being a Pluralist about certain forms of discourse does not necessarily entail pulling the rug from under one's own feet.

The Transcendental Paradox is a little more involved and arises from trying to formulate a global version of Pluralism that attempts to cover more than just the first two levels. Friend notes, drawing on the work of Priest, that certain contradictions conform to the following ‘inclusion schema’. For conditions φ and ψ:

1. (1.) Ω = {x: φ(x)} exists and ψ(Ω).

2. (2.) For all x ⊆ Ω such that ψ(x):

3. (2.)(i) δ(x) ∉ x

4. (2.)(ii) δ(x) ∈ Ω

We then let φ(x) hold iff x is a characteristic or combination of characteristics found in philosophies of mathematics. We define ψ(x) to hold iff x could be made into a philosophical position. We are now in a position to run the following paradoxical reasoning:

The set Ω in this case clearly exists (it is simply the finite set of all characteristics of philosophies of mathematics generated to date). This is effectively the globally pluralistic position; at any one time the view countenances tolerance of all combinations of characteristics of philosophical views. Indeed ψ(Ω) also holds; Ω itself is a philosophical theory. Then, for any x ⊆ Ω, we let δ(x) be the result of applying an arbitrary change in underlying logic to the combination of characteristics represented by x. An application of δ to any particular x thus results in a new philosophy such that δ(x) ∉ x (satisfying (2.)(i)). However, as Ω ⊆ Ω we can then apply δ to Ω so that δ(Ω) ∉ Ω. But by assumption, Ω is meant to contain all possible combinations of philosophical theories (satisfying (2.)(ii)), contradiction.

We can put the argument more informally as follows. By attempting to characterise a ‘global’ form of Pluralism, we try to hold global Pluralism extensionally fixed. However, Pluralism can (by its very nature as tolerant) grow, say by changing it's underlying logic. Any attempt to extensionally ‘fix’ a global Pluralism is thus doomed to contradiction.

Friend's ‘resolution’ of the paradox is striking; she simply accepts the paradox as a dialethia (id est true contradiction) and adopts Priest's Logic of Paradox (LP) as her underlying logic. LP is a logic that does not validate the rule of ex falso quodlibet, and so avoids her Pluralism lapsing into triviality. The advocacy of dialethism and paraconsistent logics is by far the most controversial aspect of Pluralism, and the one I think can be put under the most scrutiny. Further, such scrutiny could well prove to be instructive for other areas of Philosophy.

Let us first acknowledge the fact that the acceptance of a genuinely true contradiction is about as controversial as possible for philosophical positions that have actually been held. We should be mindful, however, that controversy does not imply incoherency. Indeed Friend takes great pains to make the conclusion more palatable; arguments that merit further analysis.

However, there are two closely related choices she makes in Pluralism that I think can (and should) be questioned. The outcome for Friend is, I think, positive; they do not undermine her project, but rather present new and possibly fruitful avenues for inquiry.

The first is a somewhat technical point that masks a wider phenomenon underlying any logically non-classical Philosophy. As noted earlier, the underlying logic Friend adopts for her Pluralism (and hence, one assumes, underlies philosophical inquiry) is Priest's LP. She is at pains to point out how elegant the system is; it has the same class of logical truths as classical logic, it has a rich and interesting structure, et cetera. However, for all its merits LP has one staggering drawback, especially in the context of philosophical enquiry; it invalidates modus ponens.

To see this, observe that LP allows three truth values; true, false, and both true and false. Given then the sentences φ and ψ, we can have a model of LP on which φ is both true and false, ψ is false only, and φ→ψ is both true and false. To see this, assume that φ is both true and false and ψ is false only. Then φ→ψ is true in virtue of the falsity of the antecedent (which is also true!), and also false in virtue of the truth (but also falsity!) of the antecedent and falsity of the consequent. Thus we have a situation in which φ is true (but also false!), φ→ψ is true (but also false!), yet ψ is false only. Hence modus ponens is invalid.

Why is this important? It is simply that we consistently use rules such as modus ponens in philosophical enquiry. Friend is no exception to this rule (for instance, picking fairly arbitrarily a few pages from the middle of the book, we find an application of modus ponens on p194). Without further argument as to why these applications of modus ponens are legitimate, the reasoning is not adequately justified. This is especially so given that these applications occur in very broad philosophical contexts (as opposed to a ‘narrow’ discussion of classical mathematics), precisely when LP is going to be the background logic.

This is indicative of a wider problem for any philosophy that rejects classical logic and modes of inference. Intuitionists, for example, occasionally lapse into the following mode of reasoning in non-mathematical contexts:

1. (1.) Assume that ¬P.

2. (2.) But ¬P implies some absurdity ⟘.

3. (3.) Thus, P.

Such an argument depends crucially (in the move to (3.)) on the Principle of Indirect Proof, which is illegitimate in an intuitionistic system. Thus, the argument does not go through without further reasons for the acceptability of the rule of inference in this case.

The lesson is clear; if one rejects classical logic, one has to be very careful with one's subsequent argumentative structure. We should note that Friend is not wedded to the use of LP, she is explicitly clear on this in Chapter 11. However, there are other logics that could be used, many of which have a conditional that satisfies modus ponens (so called ‘relevance logics’). Investigation of these issues for the Pluralist could well prove to be fruitful.

Of course, it is philosophically preferable given a change of underlying logic (or, even better, prevention of the counterintuitive dialethia) to have motivation for this choice.

This brings us on to a slightly more troubling aspect of Friend's work. Throughout Pluralism, traditional metaphysical enquiry and associated epistemology is generally regarded as an activity that, while it has lead to some interesting mathematics, has mostly lead to dogmatism and the stifling of interesting views.

There certainly is some truth to this claim. However, it pushes Friend to regard Pluralism as the view of the ‘principled agnostic’. While she is open to new developments in metaphysical and epistemological enquiry informing future incarnations of the Pluralist viewpoint (even to the point where one philosophical view triumphs!), by and large she eschews any enquiry of this form in Pluralism.

We might question whether such an approach is well-placed. For, it seems like we might obtain significant insight of a pluralistic nature precisely from such an enquiry. Indeed, in other pluralistic works (such as Nancy Cartwright's The Dappled World), the Pluralism is justified in terms of a particular conception of reality as multifaceted and highly complex. Why not explore the prospects for such a metaphysical and epistemological underpinning of Pluralism in the mathematical case (and beyond)? Such a project is one that would require significant work in itself, but is one largely ignored by Friend in favour of an agnostic justification. Indeed, it would be fitting to be pluralist about the justification of Pluralism!

Pluralism represents a significant contribution to a neglected area of the Philosophy of Mathematics, and provides insight into a broad area of problems. One would like more developments of certain parts of the project, but of course there is only so much that can be accomplished in one book. Friend recognises this, and the reviewer is both heartened and in agreement with her that ‘the important point is to start the vogage’ (258). There is much to disagree with in Friend's work. However, in this modest yet progressive aim, she is fairly successful.