This volume continues Palgrave’s impressive Handbooks in German Idealism series, already comprising significant collections of essays on the topics of German Idealism in general, Kant, Fichte, and German Romanticism. At the time of writing, volumes on Schelling and on the relation between German Idealism and Existentialism are also on the way.
A book of this kind, collecting up-to-date critical contributions across all of the major areas addressed in Hegel’s systematic philosophy, might be thought to stand in competition with a number of similar recent volumes, perhaps most obviously Baur and Houlgate’s A Companion to Hegel (2011) from Blackwell, de Laurentiis and Edwards’ Bloomsbury Companion to Hegel (2013), and Moyar’s The Oxford Handbook of Hegel (2017). The reality of the situation, however, in the light of the richness of Hegel’s work, is that these collections complement one another. They do so by assembling investigations of Hegel’s work on phenomenology, metaphysics, epistemology, philosophy of mind, ethics, political philosophy, aesthetics, philosophy of religion, philosophy of history, and so on, that are frequently interestingly different in emphasis, evaluation, or interpretation. The essays included in such volumes can be read in isolation, so that somebody interested in, say, topics in Hegel’s philosophy of religion, would benefit from consulting the relevant essays in all four works, without having to engage with each in its entirety. At the same time, due to their scope and size, all four share the virtue of offering readers the opportunity to consider the various topics addressed within them in a systematic context (valuable in the case of Hegel).
The fact that such “competitors” exist also has consequences for how one evaluates the coverage of this volume. The Palgrave Hegel Handbook, to provide one example, has comparatively little in the way of entries which focus upon engagement with Hegel’s work in the twentieth century. Notably, it has no entries which focus upon the reception of Hegel among phenomenologists, critical theorists, or twentieth century French philosophers. This is a particular strength of the Blackwell Companion. It also has comparatively less to say about specific metaphysical topics treated in the Science of Logic; the Oxford Handbook is stronger here. To provide one more example, however, The Palgrave Hegel Handbook clearly offers more than the other collections when it comes to topics in Hegel’s epistemology and philosophy of mind. All four of the volumes that I have mentioned address all of the major aspects of Hegel’s systematic philosophy, but differences in focus such as those in the examples just provided demonstrate another way in which a reader with access to all of them will find that they complement one another. From this point on I will focus this review upon The Palgrave Hegel Handbook alone.
Before discussing the content of the volume, I will make one further remark concerning coverage. It would be unreasonable to expect such a volume to be truly exhaustive in term of the material with which it engages, and the editors make no such claim. Given, however, that the final section of the volume comprises entries on “Hegelianism and Post-Hegelian Thought”, and the editors’ commitment to assessing ‘contemporary controversies concerning his philosophy’ (l), one might think that this would be a good opportunity to include a substantive entry engaging with the already-sizeable and growing body of work concerning Hegel’s colonialism, sexism, and racism, not only in terms of the nature of the implications of his prejudices for the evaluation of his philosophical work, but also in terms of the positive uses made of the resources of Hegel’s thought over the last seventy or so years by those seeking to oppose and overcome such prejudices. Unfortunately, such topics are not treated here. In light of recent collections such as Monahan’s Creolizing Hegel (2017), some engagement with work of this kind would have made a valuable addition to the volume.
The volume opens with a helpful analytical table of contents which roughly indicates the content of each essay. Interesting material is also included in the form of appendices; I particularly enjoyed the schematic presentation of Hegel’s major works as they correspond to the various parts of his mature philosophical system. Importantly, the editors have included Hegel’s various lecture courses in Jena, Heidelberg, and Berlin in this context. Given the richness of many of the transcripts from these lectures, this amounts to a very useful pointer for further reading on the various topics covered in the volume. I was less sure of the editors’ “Agenda for Future Research”. Although the suggestions are certainly valid (and, in my opinion, interesting), and the editors note that this material is ‘suggestive, not exhaustive’ (581), there is no clear criterion according to which some possible projects have been included and others excluded. Why emphasise, for example, the possibility of distinctively Hegelian contributions to contemporary cognitive science (583), but say nothing of the possibility of introducing Hegel to contemporary metaphysical discussions of natural kinds, or of monism, as suggested by Kreines (2015), or of the possible value of Hegelian insights in considering contemporary social pathologies, as explored by Bunyard (2019)? Certainly, there is a multitude of avenues for further research available to those interested in Hegel’s philosophy, but I am not sure how valuable it is for the editors to pick just some and list them.
I also think that the short editors’ introduction to the volume is perhaps not as helpful as it could have been. It aims to provide a sketch of contents and significant goals of the various parts of Hegel’s system, but while this breadth of scope is appropriate here, the brevity of the introduction means that the key claims being singled out are often not explained in sufficient detail. Instead, one gets the impression that the introduction is rather hurriedly emphasizing those elements of Hegel’s philosophy which the editors, perhaps especially Westphal, take to be most significant for contemporary thought (the majority of the references to chapters in the volume are to those by Westphal). I was left feeling that it would have been better simply to indicate the themes of the chapters and let the reader get on with reading them, since there these technical claims receive more adequate and clearer treatment. One valuable addition, however, is a brief run-down of various senses in which Hegel deploys the term “dialectic”. It would have been even better if this account could have been extended with references to analyses of Hegel’s various dialectical arguments, conceptual explications, and the like, as they occur among some of the chapters in the book.
There are twenty-eight essays in this volume. It is impossible within the confines of even a reasonably long review to adequately address even the majority of them. I shall endeavour to say something about seven chapters, composing two of the volume’s eight parts. I will focus my attention on the material concerning Hegel’s engagement with his immediate forbears, his epistemology, and his Science of Logic. This emphasis reflects the interests and expertise of the reviewer and I acknowledge that a case could be made for arguing that some of the most noteworthy essays in the volume are not those which fall into these categories. I will briefly draw attention to what I considered to be some of the most worthwhile essays addressing other topics in the volume at the end of the review.
Part I considers Hegel’s intellectual background and the nature of his philosophical project. There is a short sketch of Hegel’s intellectual life by Bykova which covers more or less what one would expect it to. Particularly good is Bykova’s treatment of the evolution of Hegel’s philosophical aspirations, from an early enthusiasm for popular philosophy and the moral education of the people to his later, considered belief that the practical benefits of philosophy would be better accomplished on the back of a more thorough-going revision of its more abstract, theoretical underpinnings.
Also featured here is an essay by Baur which carries out the task, crucial in a volume which treats the key themes of Hegel’s epistemology and metaphysics, of reconstructing the major philosophical developments which took place in Germany between the publication of Kant’s first Critique and Hegel’s Phenomenology. This is important because many of the major disputes in the interpretation of Hegel’s work (especially his metaphysics) since at least as far back as the 1970s have turned on how to understand his relation to his forbears, especially Kant. This is a lot to address in one essay and there are elements of the account that could have been made clearer: Baur spends some time, quite properly, explaining Kant’s rejection of the possibility of intellectual intuition for human cognition, but then does not explicitly mention intellectual intuition in his treatment of Fichte or Schelling, despite its crucial importance for their projects. More problematically, he suggests that Schelling and Hegel’s idealisms move away from the more subjective idealisms of Kant and Fichte because the former two come endorse the Spinozistic claim that ‘mind and world are fully co-extensive’ (37), but provides no clear argument as to why they might have been justified in endorsing such a claim. This risks giving the impression that Schelling’s and Hegel’s projects amount to a reversion to pre-critical dogmatism, despite the fact that Baur wants to claim that they do not (23). Still, it is necessary to paint with broad brushstrokes in an essay of this kind, and I think that Baur largely succeeds in characterising the idea of Hegel’s project in the Phenomenology as a series of determinate negations intended to persuade his opponents of the validity of metaphysics which can ‘combine the pre-Kantian thought of Spinoza with the post-Kantian thought of Fichte’ (23).
Westphal also includes an article of his own here which distils some of the key points of the interpretation of Hegel’s epistemology that he has been developing for some decades, most recently and thoroughly presented in his (2018). Westphal’s impressive scholarship is on display here as he relates the project of Hegel’s Phenomenology to key philosophical developments both before and after its publication. I shall not discuss the contents of the essay here: a critical engagement with Westphal’s account of Hegel’s project demands more space than a book review of this kind can accommodate. I will say, however, that although Westphal’s writing is clear, this article is something of a whirlwind of references to various works and topics, and at times demands a not insignificant amount of relevant knowledge on behalf of the reader (although, to be fair, it also provides plenty of references for further reading). While there are articles in this volume that would be useful for students with an interest in Hegel’s work, I would not be quick to direct a student towards this one.
Somewhat oddly also placed here is a piece by Varnier on Hegel’s epistemology. The immediate value of this essay is that it encourages those looking to identify Hegel’s epistemological positions to direct their gaze beyond the 1807 Phenomenology of Spirit, and particularly towards the section of Hegel’s Logic on cognition and the material on theoretical spirit in the Philosophy of Spirit. In fact, Varnier does not go far enough concerning this crucial point; it should be remembered that the primary function of Hegel’s argument in the Phenomenology is to provide (immanent) criticism of the various positions of “natural consciousness” opposed in various ways to Hegel’s own idealist position. Identifying distinctive Hegelian epistemological positions in the arguments of the Phenomenology, therefore, is at the very least a rather murky procedure, as what is identified as an internal problem for a particular shape of consciousness in that work need not automatically imply clear positions that we can take Hegel to endorse. It seems to me that Westphal, in the previous essay in this volume, does not do enough to address concerns of this kind.
Varnier’s article concerns itself principally with two topics. Firstly, he seems to accept something like the metaphysics-first metaphilosophy attributed to Hegel in (Kreines 2015) when he suggests that relevance to epistemology of Hegel’s Logic is that it provides a ‘theory of all ontological structures of science and of common knowledge, which make knowledge possible and certain’ (67). In this context, by asking about how Hegel defends this metaphysical project itself against scepticism Varnier seems to be engaging with important questions about Hegel’s views on the epistemology of metaphysics. Secondly, Varnier also treats Hegel’s views on traditional epistemological matters such as the justification of everyday beliefs and the definition of knowledge. Regarding the first topic – the epistemology of Hegel’s metaphysics – Varnier appeals to various “introductions” Hegel provided to his systematic philosophy, the arguments of which are presumably intended to go some way towards securing the metaphysical claims made in the latter (67). This is not an unusual view, and nor is Varnier’s suggestion that the various determinate negations of natural consciousness carried out in the Phenomenology vindicate thought’s claims about the nature of objective reality (71). Given that this essay is preceded by two others which also address the introductory function of Hegel’s Phenomenology, I would suggest that less time could have been spent on this aspect of the epistemology of Hegel’s metaphysics, in favour of topics that have received less attention, perhaps concerning the matter of how to evaluate the claims to knowledge made in the context of Hegel’s Logic itself, or in his Realphilosophie, for example.
Varnier’s treatment of the second topic is briefer that his treatment of the first, which is a shame. He provides a lengthy passage on the nature of knowledge from the Philosophy of Spirit but decides not to ‘dissect’ it (74). Instead he suggests rather briefly, and in a manner that was not clear to me, that Hegel is arguing both for the strong claim that our use of concepts tracks reality in a manner constitutive of knowledge as a matter of ‘necessity’ (74), and for the ‘irreality… of any and every finite standpoint’ (76). In order for these two claims not to be in tension with one another, it seems that the knowing subject in the former case must not be any individual, finite knower, but somehow the historically developing community of interdependent, human, knowing subjects that might be labelled ‘absolute spirit’ in Hegelian language. Indeed, Varnier suggests that, for the collective subject of absolute spirit, ‘the knower and the known are fully adequate to each other’ (75). Peculiarly, however, he then goes on to suggest that absolute spirit itself is also just another finite perspective, adherence to which invites scepticism (76). It is not obvious to me how to reconcile these two claims. Varnier also suggests that the transition from “essence” to “concept” in the Logic might constitute an argument against the sceptical suggestion that our knowledge might be restricted to appearances, and therefore that we might not know how reality really is, but again he refuses to explore that argument (76). He concludes with some highly interesting remarks on the connection between knowledge and practical reason in Hegel’s work, and suggests a possible connection to be drawn with contemporary virtue epistemology (78), but these promising ideas are, frustratingly, left undeveloped here.
I pass over Part II, which focuses on Hegel’s Phenomenology of Spirit, to consider Part III, which focuses mostly on Hegel’s Science of Logic. As I mentioned earlier, this volume does not include dedicated contributions on Hegel’s specific treatment of various historically significant logical or metaphysical topics. Instead this section includes helpful and interesting essays by Nuzzo and Burbidge, on the method of Hegel’s logical investigations and on his infamous use of Aufhebung, respectively, followed by a rather informal essay by Southgate which attempts to provide a big-picture overview of an account of metaphysical holism developed in the Doctrine of Essence and to connect it to Hegel’s account of freedom.
Southgate’s piece is a curious addition to this volume. Unlike the other chapters, it does not really represent an intervention either into debates in the secondary literature concerning Hegel’s position or argument on some philosophical issue, or into philosophical work on some topic along Hegelian lines. As such, although its principal topics are metaphysical holism, human freedom, and the connection between the former and the latter in Hegel’s thought, there is no real discussion of debates between those who consider Hegel to be a holist in the relevant sense and those who do not, for example. The major focus of the chapter, instead, seems to be to sketch a way in to Hegel’s thought, aimed at illustrating its importance for those uncertain as to whether or not to spend the time working their way through his famously difficult prose. As such, this chapter, more than any other in the volume, appears to be aimed at students approaching Hegel for the first time, or, possibly, academics considering how to introduce Hegel to such students.
With this goal in mind, I think that Southgate does quite a good job here, although some may find his style a little grating. There is a helpful discussion of Hegel’s account of freedom, aimed at defusing the tendency to suppose that Hegel is arguing naively that it is in fact possible to peacefully view all tragic events as merely the outcome of our own development and to assume responsibility for them (187-88). Southgate redirects the reader towards Hegel’s treatment of those relations which seem to actualise and illuminate freedom in the sense of “being with oneself in the other” and provides a helpful discussion of this vital notion (188-89). One might express the concern, however, that, in his attempt to emphasise to the reader the existential significance of Hegel’s notion of freedom, Southgate’s presentation can come across as rather too heavily focused on the individual’s own attitude towards freedom, at the cost of playing down Hegel’s emphasis on the objective social structures required for such attitudes. I should add that I think that few readers will find his attempt to reverse engineer an account of metaphysical holism from this account of freedom, or to try to provide a sense of it by appeal to the experience of running an ultramarathon to be successful (Southgate seems to think that Hegel’s position is in some important sense ineffable, but I do not know why).
Nuzzo’s chapter on the method of Hegel’s logical science is a valuable addition to this volume, drawing as it does on her sustained work on this topic over several years. Nuzzo helpfully situates the discussion of Hegel’s dialectical logic in relation to both Kant’s transcendental logic and to traditional, Aristotelian logic. Of especial value here is her account of Hegel’s critique of the formality of transcendental logic in terms of what he judges to be a ‘failed relation to the object’ (156) because of Kant’s separation of sensibility and understanding. Here Nuzzo’s account helpfully explains that Hegel is not simply ignoring or conflating Kant’s distinction between general and transcendental logic (as it might appear, at times).
Interestingly, on the basis of Hegel’s claim that Kant’s transcendental logic, dependent as it is on the input of sensibility for its objects, is unable to deliver the truth about those objects, Nuzzo moves to suggest that Hegel’s dialectical logic is in fact closer to general logic, precisely because it does not involve transcendental logic’s necessary reliance upon an object given to it from elsewhere (Incidentally, the claim Hegel’s new logical method is prompted by what he sees as the failure of Kant’s transcendental logical method does not seem to fit with Nuzzo’s stated rejection of readings of Hegel which have him construct an ‘path alternative to the generally accepted Kantian one’ (154), but I think that it is the former claim which is more important to her argument). In the case of general logic, of course, this is simply because it can be carried out completely abstractly, without reference to real objects, while Hegel sets for dialectical logic the ambitious task of a thinking that, like general logic, is pure in that it requires no input from externally given real objects, but at the same time delivers the truth about real objects nevertheless.
Nuzzo’s account of the method of such a dialectical logic accurately captures the Hegelian claim that the content of logic should not be separate from its form, and that logical thinking can generate its own content. Of course, even if one accepts Nuzzo’s characterisation of Hegel’s method in terms of the generation of logical content from the dynamic movement of thought itself, the question remains, particularly after the treatment of Hegel’s criticism of Kant, as to why one should suppose that this immanently generated logical content accurately tracks reality. Nuzzo rightly points out that Hegel takes his logical science to amount to an ‘objective thinking’ (161), but the reader may well wonder why this does not amount to anything more than an interesting new style of pre-critical dogmatism. To be fair to Nuzzo though, her chapter is concerned with the method of Hegel’s logic, and not with the question of how that logic also amounts to a metaphysics. This latter question has received plenty of attention in recent work on Hegel, and I think that Nuzzo’s essay succeeds admirably in shedding some light on its chosen subject-matter.
I should point out that there is a slightly misleading slip in the language of this paper. Nuzzo describes the relation between the question of the relation between Hegel’s dialectical logic and his attitude to Kant’s transcendental logic and the question of the relation between the conclusion of the Phenomenology of Spirit and the idea of logic present in the Science of Logic as ‘all but self-explanatory’ (155), where she means “anything but self-explanatory” (She does go on to provide an excellent explanation later in the chapter). Unfortunately, although no one paper in the volume exhibits a high volume of typos, mistakes, or awkward phrasings, there is quite a number of such things spread across the book as a whole. In general, the Palgrave Hegel Handbook would have benefitted from more careful editing on this score.
Burbidge, in his contribution, provides an illuminating discussion of Hegel’s use of the term Aufhebung, which describes the kind of transitions or inferences key to every part of Hegel’s mature philosophy. Burbidge’s chapter compliments Nuzzo’s. Whereas she focused on Hegel’s attempt to present a logical science that generates its own content, his attention is on the nature of the development of that content. In particular, he is concerned to explain how it is that Hegel is able to argue that more complex thought determinations emerge out of simpler and less determinate ones, without surreptitiously assuming those more complex determinations as a goal in advance. Of course, this has always been a common complaint made against Hegel’s procedure, and Burbidge makes quite a good case for thinking that it is unfounded. He shows, particularly by appealing to remarks Hegel makes towards the end of his treatment of quantitative concepts, and in his account of the absolute idea, that the basic parts of a move that can be described as an Aufhebung, the movement, that is, through which more complex determinations are generated from simpler ones, are firstly the immanent negation of some determination, followed in turn by the negation of the determination to which the first negation gave rise. This ‘doubled transition’ (171) amounts to a slippage between the two determinations in question, with each implying but excluding the other. Finally, this slippage between determinations can be grasped as a single unity, in which the one-sided conceptions of the previous determinations have been replaced with a conception that grasps them as belonging to this new determination in such a way that they have been both “annulled” and “preserved”, as the verb aufheben can suggest.
Burbidge’s account of the dialectical transitions which make up the argument of Hegel’s Logic does not require Hegel to assume in advance the outcome of those transitions, but there are other worries that might be expressed about it. It is not obvious what it is that gives rise to the moment at which the continual slippage between opposing thought-determinations is grasped as a whole. Burbidge invokes the unifying function that Kant attributes to the categories in the transcendental deduction, but it is not clear how helpful this is. Burbidge himself acknowledges that Kant’s discussion of how the categories unify sensible intuitions that have been synthesised by the imagination is somewhat removed from Hegel’s focus on the relations between concepts alone. What Burbidge seems to want from Kant is a discussion of the understanding, since it is the unifying activity that Kant attributes to the understanding that Burbidge sees in Hegel’s Aufhebung. But even then it is not clear exactly what or who is responsible for this unifying activity in Hegel’s case. What is more, Burbidge seems to slip rather too quickly between Kant’s account of the understanding as a faculty for unifying the deliverances of sensibility under concepts and Hegel’s insistence that the understanding is (primarily) a kind of thinking that separates and statically opposes thought determinations, risking giving the impression that Kant and Hegel are talking about the same thing. This cannot be right, but Burbidge’s suggestion that it is ‘understanding’s “power of the negative”, which collapses the double transition with its inherent contradiction and infinite progress into a simple, unified concept’ (172) is mystifying because it gives precisely this impression. What this account seems to require is a discussion of the kind of thinking Hegel describes as “reason”, but this is strangely absent.
Burbidge goes on to provide a useful overview of some of dialectical transitions which occur early in the first book of Hegel’s Logic. In fact, I think that it would have been better still if he had spent a little more time spelling some of these out, rather than focusing on the account of the concepts of being, nothing, and becoming as his most detailed example of an instance of a logical Aufhebung. In the case of this first dialectical transition it is uniquely difficult to see how the original, one-sided determinations of being and nothing are opposed at all, and thus to get the transition on the move. Things become much clearer in the case of, for example, the treatment of the (qualitative) finite and infinite, which Burbidge treats only briefly.
Turning from the Logic to the metaphysics of nature, Burbidge proceeds to provide a whistle-stop tour through some notable features of contemporary particle physics, biochemisty, and biology (which I am not competent to assess), suggesting that these provide evidence for thinking that the kind of dialectical transitions that Hegel explores in his Logic may appear in the activity and development of nature in ways that Hegel did not anticipate. Burbidge thinks that this amounts to a challenge to Hegel’s view that ‘the universe is grounded in a rational structure that is prior to, and independent of nature’ (180). It is not clear to me that such developments need push Hegel to abandon the idea that logic can be treated as an a priori science, independent of the study of nature, but one whose metaphysical implications might be expected to govern nature. Certainly, we might concede to Burbidge that if nature does indeed appear to run on Aufhebung-like processes, then a thinking which takes its cue from the presence of such dialectical transitions in nature and reflects upon them in an abstract context might come to resemble Hegel’s Logic, but this does not guarantee his conclusion that ‘there is no a priori logical structure, but human thought is affected by what it discovers in the changes and transformations of nature’ (181). Aside from the challenges mounted by Kant, Hegel, and others to conceiving of logic on such an empirical basis, it remains the case that if (and admittedly it is a big “if”) Hegel can make a case for the development of such a logic a priori and show that it has metaphysical implications, then he should not be too troubled by the discovery of natural processes which conform to the structures of thought. Burbidge is right to draw attention to outdated claims and failings in Hegel’s own philosophy of nature, but I do not think that these need to cause problems for Hegel’s big picture concerning the relation between thought and reality and the way that Burbidge seems to think that they might.
With that, I draw my discussion of just some of the essays assembled in this volume to a close. As additional highlights not addressed here, I would direct the reader’s attention to an essay by Collins which considers the role of Hegel’s account of religion in the context of the argument of the Phenomenology (85-108), an essay by Testa on Hegel’s treatment of embodied cognition and agency (269-95), an essay by Yeomans on the relation between Hegel’s logic and his political thought (373-88), and Motroshilova’s account of the development of Hegel’s treatment of the history of philosophy (485-517). By way of conclusion I shall simply state that there is a great deal in this volume that will be of interest to Hegel scholars and students, and that the Palgrave Hegel Handbook provides a valuable addition to the resources available to anyone engaging seriously with almost any facet of Hegel’s work.
Baur, M. and Houlgate, S. 2011. A Companion to Hegel. Oxford: Blackwell.
Bunyard, T. 2019. “Demagogy and Social Pathology: Wendy Brown and Robert Pippin on the Pathologies of Neoliberal Subjectivity.” Araucaria, Vol 21 Issue 42: 505-527
de Laurentiis, A. and Edwards, J. 2013. The Bloomsbury Companion to Hegel. London: Bloomsbury.
Kreines, J. 2015. Reason in the World: Hegel’s Metaphysics and its Philosophical Appeal. Oxford: Oxford University Press.
Moyar, D. 2017. The Oxford Handbook of Hegel. Oxford: Oxford University Press.
Monahan, M. 2017. Creolizing Hegel (London: Rowman and Littlefield International.
Westphal, K. 2018. Grounds of Pragmatic Realism: Hegel’s Internal Critique and Reconstruction of Kant’s Critical Philosophy. Leiden: Brill.
This is a book long overdue. Other authors have made more or less recent phenomenological and transcendental-idealist contributions to the philosophy of mathematics: Dieter Lohmar (1989), Richard Tieszen (2005) and Mark van Atten (2007) are perhaps the most important ones. Ten years is a sufficiently wide gap to welcome any new work. Yet da Silva’s contribution stands out for one reason: it is unique in the emphasis it puts, not so much, or not only, on the traditional problems of the philosophy of mathematics (ontological status of mathematical objects, mathematical knowledge, and so on), but on the problem of the application of mathematics. The author’s chief aim – all the other issues dealt with in the book are subordinated to it – is to give a transcendental phenomenological and idealist solution to the evergreen problem of how it is that we can apply mathematics to the world and actually get things right – particularly mathematics developed in complete isolation from mundane, scientific or technological efforts.
Chapter 1 is an introduction. In Chapters 2 and 3, da Silva sets up his tools. Chapters 4 to 6 are about particular aspects of mathematics: numbers, sets and space. The bulk of the overall case is then developed in Chapters 7 and 8. Chapter 9, “Final Conclusions”, is in fact a critique of positions common in the analytic philosophy of mathematics.
Chapter 2, “Phenomenology”, is where da Silva prepares the notions he will then deploy throughout the book. Concepts like intentionality, intuition, empty intending, transcendental (as opposed to psychological) ego, and so on, are presented. They are all familiar from the phenomenological literature, but da Silva does a good job explaining their motivation and highlighting their interconnections. The occasional (or perhaps not so occasional) polemic access may be excused. The reader expecting arguments for views or distinctions, however, will be disappointed: da Silva borrows liberally from Husserl, carefully distinguishing his own positions from the orthodoxy but stating, rather than defending, them. This creates the impression that, at least to an extent, he is preaching to the converted. As a result, if you are looking for reasons to endorse idealism, or to steer clear of it, this may not be the book for you.
Be that as it may, the main result of the chapter is, unsurprisingly, transcendental idealism. This is the claim that, barring the metaphysical presuppositions unwelcome to the phenomenologist, there is nothing more to the reality of objects than their being “objective”, i.e., public. ‘Objectivation’, as da Silva puts it, ‘is an intentional experience performed by a community of egos operating cooperatively as intentional subjects. … Presentifying to oneself the number 2 as an objective entity is presentifying it and simultaneously conceiving it as a possible object of intentional experience to alter egos (the whole community of intentional egos)’ (26-27). This is true of ideal objects, as in the author’s example, but also of physical objects (the primary type of intentional experience will then be perception).
There are two other important views stated and espoused in the chapter. One is the Husserlian idea that a necessary condition for objective existence is the lack of cancellation, due to intentional conflict, of the relevant object. Given the subject matter of the book, the most important corollary of this idea is that ideal objects, if they are to be objective, at the very least must not give rise to inconsistencies. For example, the set of all ordinals does not objectively exist, because it gives rise to the Burali-Forti paradox. The other view, paramount to the overall case of the book (I will return to it later), is that for a language to be material (or materially determined) is for its non-logical constants to denote materially determined entities (59). If a language is not material, it is formal.
Chapter 3 is about logic. Da Silva attempts a transcendental clarification of what he views as the trademark principles of classical logic: identity, contradiction and bivalence. The most relevant to the book is the third, and the problem with it is: how can we hold bivalence – for every sentence p, either p or not-p – and a phenomenological-idealist outlook on reality? For bivalence seems to require a world that is, as da Silva puts it, ‘objectively complete’: such that any well-formed sentence is in principle verifiable against it. Yet how can the idealist’s world be objectively complete? Surely if a sentence is about a state of affairs we currently have no epistemic access to (e.g., the continuous being immediately after the discrete) there just is no fact of the matter as to whether the sentence is true or false: for there is nothing beyond what we, as transcendental intersubjectivity, have epistemic access to.
Da Silva’s first move is to put the following condition on the meaningfulness of sentences: a sentence is meaningful if and only if it represents a possible fact (75). The question, then, becomes whether possible facts can always be checked against the sentences representing them, at least in principle. The answer, for da Silva, turns on the idea, familiar from Husserl, that intentional performances constitute not merely objects, but objects with meanings. This is also true of more structured objectivities, such as states of affairs and complexes thereof – a point da Silva makes in Chapter 2. The world (reality) is such a complex: it is ‘a maximally consistent domain of facts’ (81). The world, then, is intentionally posited (by transcendental intersubjectivity) with a meaning. To hold bivalence as a logical principle means, transcendentally, to include ‘objective completeness’ in the intentional meaning (posited by the community of transcendental egos) of the world. In other words, to believe that sentences have a truth value independent of our epistemic access to the state of affairs they represent is to believe that every possible state of affairs is in principle verifiable, in intuition or in non-intuitive forms of intentionality. This, of course, does not justify the logical principle: it merely gives it a transcendental sense. Yet this is exactly what da Silva is interested in, and all he thinks we can do. Once we refuse to assume the objective completeness of the world in a metaphysical sense, what we do is to assume it as a ‘transcendental presupposition’ or ‘hypothesis’. In the author’s words:
How can we be sure that any proposition can be confronted with the facts without endorsing metaphysical presuppositions about reality and our power to access reality in intuitive experiences? … By a transcendental hypothesis. By respecting the rules of syntactic and semantic meaning, the ego determines completely a priori the scope of the domain of possible situations – precisely those expressed by meaningful propositions – which are, then, hypothesized to be ideally verifiable. (83)
Logical principles express transcendental hypotheses; transcendental hypotheses spell out intentional meaning. … The a priori justification of logical principles depends on which experiences are meant to be possible in principle, which depends on how the domain of experience is intentionally meant to be. (73)
There is, I believe, a worry regarding da Silva’s definition of meaningfulness in terms of possible situations: it seems to be in tension with the apparent inability of modality to capture fine-grained (or hyper-) intensional distinction and therefore, ultimately, meaning (for a non-comprehensive overview of the field of intensional semantics, see Fox and Lappin 2005). True, since possible situations are invoked to define the meaningfulness, not the meaning, of sentences, there is no overt incompatibility; yet it would be odd to define meaningfulness in terms of possible situations, and meaning in a completely different way.
Chapter 4, “Numbers”, has two strands. The first deals with another evergreen of philosophy: the ontological status of numbers and mathematical objects in general. Da Silva’s treatment is interesting and his results, as far as I can see, entirely Husserlian: numbers and other mathematical objects behave like platonist entities except that they do not exist independently of the intentional performances that constitute them. One consequence is that mathematical objects have a transcendental history which can and should be unearthed to fully understand their nature. The phenomenological approach is unique in its attention to this interplay between history and intentional constitution, and it is to da Silva’s credit, I believe, that it should figure so prominently in the book. Ian Hacking was right when he wrote, a few years back, that ‘probably phenomenology has offered more than analytic philosophy’ to understand ‘how mathematics became possible for a species like ours in a world like this one’ (Hacking 2014). Da Silva’s work fits the pattern.
And yet I have a few reservations, at least about the treatment (I will leave the results to readers). For one thing, there is no mention of unorthodox items such as choice sequences. Given da Silva’s rejection of intuitionism in Chapter 3, perhaps this is unsurprising. Yet not endorsing is one thing, not even mentioning is quite another. I cannot help but think the author missed an opportunity to contribute to one of the most engaging debates in the phenomenology of mathematics of the last decade (van Atten’s Brouwer Meets Husserl is from 2007). Da Silva’s seemingly difficult relationship with intuitionism is also connected with another conspicuous absence from the book. At p. 118 da Silva looks into the relations between our intuition of the continuum and its mathematical construction in terms of ‘tightly packed punctual moments’, and argues that the former does not support the latter (which should then be motivated on different grounds). He cites Weyl as the main purveyor of an alternative model – which he might well be. But complete silence about intuitionist analysis seems frankly excessive.
A final problem with da Silva’s presentation is his dismissal of logicism as a philosophy of, and a foundational approach to, mathematics. ‘Of course,’ he writes, ‘Frege’s project of providing arithmetic with logical foundations collapsed completely in face of logical contradiction (Russell’s paradox)’ (103). The point is not merely historical: ‘Frege’s reduction of numbers to classes of equinumerous concepts is an unnecessary artifice devised exclusively to satisfy logicist parti-pris … That this caused the doom of his projects indicates the error of the choice’. I would have expected at least some mention of either Russell’s own brand of logicism (designed, with type theory, to overcome the paradox), or more recent revivals, such as Bob Hale’s and Crispin Wright’s Neo-Fregeanism (starting with Wright 1983) or George Bealer’s less Fregean work in Quality and Concept (1982). None of these has suffered the car crash Frege’s original programme did, and all of them are still, at least in principle, on the market. True, da Silva attacks logicism on other grounds, too, and may argue that, in those respects, the new brands are just as vulnerable as the old. Yet, that is not what he does; he just does not say anything.
The second strand of the chapter, more relevant to the overall case of the book, develops the idea that numbers may be regarded in two ways: materially and formally. The two lines of investigation are not totally unrelated, and indeed some of da Silva’s arguments for the latter claim are historical. The claim itself is as follow. According to da Silva, numbers are essentially related to quantity: ‘A number is the ideal form that each member of a class of equinumerous quantitative forms indifferently instantiates’, and ‘two numbers are the same if they are instantiable as equinumerical quantitative forms’ (104). Yet some types of numbers are more or less detached from quantity: if in the case of the negative integers, for example, the link with quantity is thin, when it comes to the complex numbers it is gone altogether. Complex numbers are numbers only in the sense that they behave operationally like ones – but they are not the real (no pun intended) thing. Da Silva is completely right in saying that it was this problem that moved the focus of Husserl’s reflections in the 1890s from arithmetic to general problems of semiotic, logic and knowledge. The way he cashes out the distinction is in terms of a material and a formal way to consider numbers. Genuine, ‘quantitative’ numbers are material numbers. Numbers in a wider sense, and thus including the negative and the complex, are numbers in a formal sense. Since, typically, the mathematician is interested in numbers either to calculate or because they want to study their relations (with one another or with something else), they will view numbers formally – i.e., at bottom, from the point of view of operations and structure – rather than materially.
Thus, the main theoretical result of the chapter is that, inasmuch as mathematics is concerned with numbers, it is ‘essentially a formal science’ (120). In Chapter 7, da Silva will put forward an argument to the effect that mathematics as a whole is essentially a formal science. This, together with the idea, also anticipated in Chapter 4, that the formal nature of mathematics ‘explains its methodological flexibility and wide applicability’, is the core insight of the whole book. But more about it later.
Chapter 5 is about sets. In particular, da Silva wants to transcendentally justify the ZFC axioms. This includes a (somewhat hurried) genealogy, roughly in the style of Experience and Judgement, of ‘mathematical sets’ from empirical collections and ‘empirical sets’. The intentional operations involved are collecting and several levels of formalisation. The details of the account have no discernible bearing on the overarching argument, so I will leave them to one side. It all hinges, however, on the idea that sets are constituted by the transcendental subject through the collecting operation, and this is what does the main work in the justification. This makes da Silva’s view very close to the iterative conception (as presented for example in Boolos 1971); yet he only mentions it once and in passing (146). Be that as it may, it is an interesting feature of da Silva’s story that it turns controversial axioms such as Choice into sugar, while tame ones such as Empty Set and Extensionality become contentious.
Empty Set, for example, is justified with an account, which da Silva attributes to Husserl, of the constitution of empty sets that I found fascinating but incomplete. Empty sets are clearly a hard case for the phenomenological account: because, as one might say, since collections are empty by definition, no collecting is in fact involved. Or is it? Consider, da Silva says, the collection of the proper divisors of 17:
Any attempt at actually collecting [them] ends up in collecting nothing, the collecting-intention is frustrated. Now, … Husserl sees the frustration in collecting the divisors of 17 as the intuitive presentation of the empty collection of the divisors of 17. So empty collections exist. (148)
It is a further question, and da Silva does not consider it, whether this story accounts for the uniqueness of the empty set (assuming he thinks the empty set is indeed unique, which, as will appear, is not obvious to me). Are collecting-frustration experiences all equal? Or is there a frustration experience for the divisors of 17, one for the divisors of 23, one for the round squares, and so on? If they are all equal, does that warrant the conclusion that the empty sets they constitute are in fact identical? If they are different, what warrants that conclusion? Of course, an option would be: it follows from Extensionality. Yet, I venture, that solution would let the phenomenologist down somewhat. More seriously, da Silva even seems to reject Extensionality (and thus perhaps the notion that there is just one empty set). At least: he claims that there is ‘no a priori reason for preferring’ an extensional to an intensional approach to set theory, but that if we take ‘the ego and its set-constituting experiences’ seriously we ought to be intensionalists (150).
Chapters 6 is about space and its mathematical representations – ‘a paradigmatic case of the relation between mathematics and empirical reality’ (181). It is where da Silva deals the most with perception and the way it relates with mathematical objects. For the idealist, there are at least four sorts of space: perceptual, physical, mathematical-physical and purely formal. The intentional action required to constitute them is increasingly complex, objectivising, idealising and formalising. Perceptual space is subjective, i.e., private as opposed to public. It is also ‘continuous, non-homogeneous, simply connected, tridimensional, unbounded and approximately Euclidean’ (163). Physical space is the result of the intersubjective constitution of a shared spatial framework by harmonization of subjective spatial experiences. This constitution is a ‘non-verbal, mostly tacit compromise among cooperating egos implicit in common practices’ (167). Unlike its perceptual counterpart, physical space has no centre. It also admits of metric, rather than merely proto-metric, relations. It is also ‘everywhere locally’, but not globally, Euclidean (168). The reason is that physical space is public, measurable but based merely on experience (and more or less crude methods of measurement) – not on models.
We start to see models of physical space when we get to mathematical-physical space. In the spirit of Husserl’s Krisis, da Silva is very keen on pointing out that mathematical-physical space, although it does indeed represent physical space, does not reveal what physical space really is. That it should do so, is a naturalistic misunderstanding. In the author’s words:
At best, physical space is proto-mathematical and can only become properly mathematical by idealization, i.e., an intentional process of exactification. However, and this is an important remark, idealization is not a way of uncovering the “true” mathematical skeleton of physical space, which is not at its inner core mathematical. (169)
Mathematical-physical space is what is left of the space we live in – the space of the Lebenswelt, if you will – in a representation designed to make it exact (for theoretical or practical purposes). Importantly, physical space ‘sub-determines’ mathematical-physical space: the latter is richer than the former, and to some extent falsifies what it seeks to represent. Euclidean geometry is paradigmatic:
The Euclidean representation of physical space, despite its intuitive foundations, is an ideal construct. It falsifies to non-negligible extent perceptual features of physical space and often attributes to it features that are not perceptually discernible. (178)
The next step is purely formal representations of space. These begin by representing physical space, but soon focus on its formal features alone. We are then able to do analytic geometry, for example, and claim that, ‘mathematically, nothing is lost’ (180). This connects with da Silva’s view that mathematics is a formal science and, in a way, provides both evidence for and a privileged example of it. If you are prepared to agree that doing geometry synthetically or analytically is, at bottom, the same thing, then you are committed to explain why that is so. And da Silva’s story is, I believe, a plausible candidate.
Chapter 7 is where it all happens. First, and crucially, da Silva defends the view that mathematics is formal rather than material in character. I should mention straight away that his argument, a three-liner, is somewhat underdeveloped. Yet it is very clear. To say that mathematics is essentially formal is, for da Silva, to say that mathematics can only capture the formal aspects of reality (as the treatment of space is meant to show). The reason is as follows. Theories are made up of symbols, which can be logical or non-logical. The non-logical symbols may, in principle, be variously interpreted. A theory whose non-logical symbols are interpreted is, recall, material rather than formal. Therefore, one could argue, number theory should count as material. Yet, so da Silva’s reasoning goes, ‘fixing the reference of the terms of an interpreted theory is not a task for the theory itself’ (186). The theory, in other words, cannot capture the interpretation of its non-logical constant: that is a meta-theoretical operation. But then mathematical theories cannot capture the nature, the specificity of its objects even when these are material.
That is the master argument, as well as the crux of the whole book. For it follows from it that mathematics is essentially about structure: objects in general and relations in which they stand. This, for da Silva, does not mean that mathematics is simply not about material objects. That would be implausible. Rather, the claim is that even when a mathematical theory is interpreted, or has a privileged interpretation, and is therefore about a specific (‘materially filled’) structure, it does not itself capture the interpretation (the fixing of it) – and thus it is really formal. Some mathematical theories are, however, formal in a stricter sense: they are concerned with structures that are kept uninterpreted. These are purely formal structures. Regarding space, Hilbert’s geometry is a good example.
Da Silva’s solution to the problem of the applicability of mathematics is thus the following. Mathematics is an intentional construction capable of representing the formal aspects of other intentional constructions – mathematics itself and reality. Moreover, it is capable of representing only the formal aspects of mathematics and reality. It should then be no surprise, much less a problem, that any non-mathematical domain can be represented mathematically: every domain, insofar as it is an intentional construction, has formal aspects – which are the only ones that count from an operational and structural standpoint.
This has implications for the philosophy of mathematics. On the ground of his main result, da Silva defends a phenomenological-idealist sort of structuralism, according to which structures are the privileged objects of mathematics. Yet his structuralism is neither in re nor ante rem. Not in re, because structures, even when formal, are objects in their own right. Not ante rem, because structures are intentional constructs, and thus not ontologically independent. They depend on intentionality, but also on the material structures on whose basis they are constituted through formalisation. This middle-ground stance is typical of phenomenology and transcendental idealism.
I have already said what the last two chapters – 8 and 9 – are about. The latter is a collection of exchanges with views in the analytic philosophy of mathematics. They do not contribute to the general case of the book, so I leave them to prospective readers. The former is an extension of the results of Chapter 7 to science in general. A couple of remarks will be enough here. Indeed, when the reader gets to the chapter, all bets are off: by then, da Silva has put in place everything he needs, and the feeling is that Chapter 8, while required, is after all mere execution. This is not to understate da Silva’s work. It is a consequence of his claim (217) that the problem of the applicability of mathematics to objective reality, resulting in science, just is, at bottom, the problem of the applicability of mathematics to itself – which the author has already treated in Chapter 7. Under transcendental idealism, objective, physical reality, just like mathematical reality, is an intersubjective intentional construct. This construct, being structured, and thus having formal aspects to it, ‘is already proto-mathematical’ and, ‘by being mathematically represented, becomes fully mathematical’ (226). The story is essentially the same.
Yet it is only fair to mention that, while in this connection it would have been easy merely to repeat Husserl (the approach is after all pure Krisis), that is not what da Silva does. He rather distances himself from Husserl in at least two respects. First of all, he rejects what we may call the primacy of intuition in Husserl’s epistemology of mathematics and science. Second, he devotes quite a bit of space to the heuristic role of mathematics in science – made possible, so the author argues, by the formal nature of mathematical representation (234).
As a final remark, I want to stress again what seems to me the chief problem of the book. Da Silva’s aim is to give a transcendental-idealist solution to the problem of the applicability of mathematics. Throughout the chapters, he does a good job spelling out the details of the project. Yet there is no extensive discussion of why one should endorse transcendental idealism in the first place. True, a claim the author repeatedly makes is that idealism is the only approach that does not turn the problem into a quagmire. While the reader may be sympathetic with that view (as I am), da Silva offers no full-blown argument for it. As a result, the book is unlikely to build bridges between phenomenologists and philosophers of mathematics of a more analytic stripe. Perhaps that was never one of da Silva’s aims. Still, I believe, it is something of a shame.
Boolos, G. 1971. “The Iterative Conception of Set”. Journal of Philosophy 68 (8): 215-231.
Bealer, G. 1982. Quality and Concept. Oxford: OUP.
Fox, C. and Lappin, S. 2005. Foundations of Intensional Semantics. Oxford: Blackwell.
Hacking, I. 2014. Why is there Philosophy of Mathematics at all? Cambridge: CUP.
Lohmar, D. 1989. Phänomenologie der Mathematik: Elemente enier phänomenologischen Aufklärung der mathematischen Erkenntnis nach Husserl. Dodrecht: Kluwer.
Tieszen, R. 2005. Phenomenology, Logic, and the Philosophy of Mathematics. Cambridge: CUP.
Van Atten, M. 2007. Brouwer Meets Husserl: On the Phenomenology of Choice Sequences. Dodrecht: Springer.
Wright, C. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: AUP.
 Unless impossible worlds are brought in – but as far as I can see that option is foreign to da Silva’s outlook.
 The notion of quantitative form is at the heart of Husserl’s own account of numbers in Philosophy of Arithmetic – and it is to da Silva’s credit that he takes Husserl’s old work seriously and accommodates into an up-to-date phenomenological-idealist framework.