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.