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.
This is a good book – and, on the Italian market, a much-needed one. Simone Aurora’s declared aim is to show that Husserl’s Logical Investigations belong to the history and conceptual horizon of structuralism, and in a prominent position at that. The whole book builds up to a defense of the view in the last chapter. Aurora’s case is set up well from the beginning and thoroughly argued at the end. That is why the book is good. The reason why the book is much needed on the Italian market is that it is also an introduction to Husserl’s early philosophy – from On the Concept of Number (1886) to the Investigations (1900-1901) – as it should be written: starting from 19th-century developments in psychology and, importantly, mathematics. To my knowledge, there are no published works in Italian that do so, or do so extensively. Aurora satisfactorily fills the gap.
Chapter 1 is about Husserl’s beginnings – a story Aurora does a good job of telling. A mathematics, physics, and astronomy student in Leipzig in 1876, Husserl would end up, in 1883, writing a doctoral thesis on the calculus of variations with Leo Königsberger in Vienna. He was then briefly Weierstrass’s assistant in Berlin. In 1884 Husserl came across Brentano’s work and lectures; as a result, he steered towards philosophy. By 1887, Husserl’s first philosophical work – his Habilitationsschrift under the supervision of Carl Stumpf in Halle – was complete. Crucial to On the Concept of Number are both the mathematical and the philosophical strands of Husserl’s academic life. The eponymous problem is inherited from Weierstrass, Kronecker, and in general, the whole debate on the foundations of mathematics, which at the time was soaring in Europe. The method with which Husserl tackled it – and this is where the originality of the work lies – was Brentano’s descriptive psychology. Both these backgrounds, their developments and Husserl’s own take on them are well expounded by Aurora.
Chapter 2 is about 1891’s Philosophy of Arithmetic (PA). Overall, Aurora’s presentation is clear and, I believe, effective. The relations with the earlier work are explained and the architecture of the book is clearly laid out. Overall, the main notions (‘collective connection’, ‘something in general’, and so forth) and arguments are satisfactorily presented. Let me mention a couple of worries.
One problem is that Aurora highlights relatively few connections between points discussed in PA on the one hand, and the larger debates and their recent developments on the other. For example, at that stage Husserl, like e.g. Cantor, held a version of the abstraction theory of numbers. That, for example, is where the notion of ‘something in general’ (Etwas überhaupt) comes in. The theory had already been severely criticised by Frege in The Foundations of Arithmetic (1884), a criticism, importantly, that fed into Husserl’s work (as well as into Cantor’s). See ortiz Hill 1997. This might have deserved a few lines. Also, although for most of the twentieth century the abstraction theory was forsaken if not forgotten, in the late 1990s Kit Fine attempted a rescue, sparking some debate (Fine 1998). Again, a quick pointer might have been helpful.
Here is a second worry. Some scholars (A. Altobrando and G. Rang are Aurora’s references) believe they can discern the first traces of the development of Husserl’s notions of eidetic intuition and phenomenological epoché in PA. Aurora is among them, and in particular he reckons abstraction is the place to look: for, according to Husserl, in abstraction one disregards all qualitative (and to some extent relational) aspects of the relevant objects, and is only interested in the latter as empty ‘something in general’. The view is put forward at p. 71. Now, there is no denying that both eidetic intuition and the phenomenological epoché involve some sort of heavy disregarding or bracketing. But surely the philosophical literature is crammed with similar methods and theories – not least the British empiricists’ accounts of abstraction, which is as far as it gets from Husserl’s Ideation or Wesensanschauung. Prima facie similarities, then, are in fact rather thin. Terminology as well as theoretical contexts and functions, Aurora admits, are also very different. We may wonder, at this point, what is left for the interpretation to be based on. I suspect very little if anything.
Chapter 3 is about the transition, in the 1890s, from PA to the Investigations. Two conceptual pairs begin to emerge in this period that will end up being paramount in the later work. The first pair, abstract/concrete, is the subject (or one of the subjects) of the third Investigation; the second, intuition/representation, is one of the main characters of the sixth. Aurora describes well their first appearance in an 1894 essay entitled Psychologischen Studien zur elementaren Logik. Developments in Husserl’s view of intentional objects are also discussed in some detail. The main references in this case are manuscript K I 56 and Husserl’s review of Twardowski’s Zur Lehre vom Inhalt und Gegenstand der Vorstellung, both from 1894.
Chapter 4 is about the Prolegomena to Pure Logic, the first part of the Logical Investigations. Aurora does a good job of expounding both Husserl’s arguments against psychologism and his concept of a pure logic and theory of science – the two main themes of the work. As it can and should be expected of an introductory exposition, a few details are at some points glossed over. Yet the main idea, i.e., that there is a basic dimension to science which is called ‘pure logic’ and which is ideal (or, as people tend to say these days, ‘abstract’), objective, and to all appearances, independent of human thought or language, comes across very clearly. There is, however, one distinction that, it seems to me, Aurora fails to recognise (or to report). It is not a major issue for what, after all, is an introductory chapter – but nonetheless a point worth raising. It is the distinction between deduction and grounding.
Between the Prolegomena and the Investigations Husserl defines (or uses) two to four related concepts: on the one hand, deduction or inference (Schluß, or sometimes an unqualified Begründung, in Husserl’s German) and explanatory grounding (the relation between an erklärender Grund and what it is the ground of), both operative in the Prolegomena; on the other hand, foundation (Fundierung), introduced in the third Investigation and operative in the subsequent ones. Now, foundation may (Nenon 1997) or may not have two models, one ontological and one epistemological; and one of these two models, the ontological, may or may not be identical to the explanatory grounding of the Prolegomena – a view for which, I believe, there is something to be said. Your count here will depend on your views on foundation. But whatever these are, there is no doubt at least that deduction and explanatory grounding are distinct in the Prolegomena. That is what does not come across in the book.
Indeed, as far as I can see, in Aurora’s presentation the two concepts from the Prolegomena collapse into one. While explaining what, for Husserl, constitutes the ‘unity of science’, Aurora introduces the concept of Begründung and says that it ‘substantially refers to the notion of inference or logical deduction’ (p. 134). Yet this is something that Husserl explicitly denies. To see this, look at Prolegomena, §63. Here, a distinction is made between explanatory and non-explanatory Begründung, and the former, not the latter, is deemed essential to (the unity of) science. Indeed for Husserl, as for Bolzano (from whom he inherits the notion), what secures the unity of science is an explanatory relation (erklärende Zusammenhang) between true propositions. And while ‘all grounds are premises’ – so that if proposition A grounds proposition B then there is an inference from A to B – ‘not all premises are grounds’. It is not the case, that is, that if there is an inference from A to B then A grounds B. In other words, ‘every explanatory relation is deductive (deduktive), but not every deductive relation is explanatory’.
While Husserl is very explicit in drawing the distinction, he is not so helpful in justifying it. He devotes a few remarks to the task, right after the passage I quoted; but they do not make an argument. Here is how one may be extracted. (Bolzano’s arguments are also available from the Wissenschaftslehre, around §200.)
Let us stipulate deducibility as the modern notion of (classical) logical consequence. If grounding were just logical consequence, the latter would be an explanatory relation (because the former is). But it isn’t: there are cases of valid and sound arguments in which the premises fail to explain the conclusion. For example, p ╞ p, or p & q ╞ p. Indeed, it is hard to see how a proposition, even though it can be inferred from itself, can also ground (explain) itself: it is raining, therefore it is raining – but is it raining because it is raining? Things are even worse with the second case: does the truth of a conjunction ground the truth of one of its conjuncts? It is probably the other way round. To derive a conclusion from a set of premises is not, in and of itself, to explain the former in terms of the latter. But then grounding and deducibility must be distinct.
(I should mention that in an extended footnote at p. 133 Aurora does discuss Husserl’s notion of Begründung vis-à-vis Bolzano’s. So he is definitely aware of the theoretical background, the significance and the facets of the concept. So much so, that the footnote seems to contradict, rather than explain, the main text.)
Chapter 5 is possibly the most felicitous of the whole book, partly because, due to the topic, Aurora’s background in linguistics shines through. We are now past the Prolegomena and into the Investigations proper. Having established in the former that logical and mathematical objects do not, by all appearances, belong to the spatio-temporal world, Husserl is left with the question as to how we can know anything about them – in fact, relate to them at all. Short of an answer, Husserl thinks, the existence of logic and therefore of science in general, as human enterprises, must remain a mystery. And for Husserl the starting point is language, because it is primarily in language – in the meanings of words and sentences – that logical objects make their spatio-temporal appearance. The main result of the first two Investigations are the following: meanings are ideal (non-spatio-temporal) and akin to universals; and universals are genuine objects, irreducible to their instances, to thought, or to language. (It is a substantive question whether this amounts to full-blown Platonism; Aurora believes it doesn’t, and some remarks of Husserl’s certainly point that way.)
The first two sections of the chapter, on the first Investigation, are nearly flawless. The remaining sections, on the second Investigation, are also effective but, I believe, raise at least one worry. Aurora thinks that, for Husserl, meanings are ‘ideal classes of objects’ (203). Now, he may well not be using ‘class’ in its fully technical sense. But the fact remains that classes, among other things, are (like sets, their close relatives) extensional mathematical constructs. However, in the 1890s, when most of the Investigations were thought out, Husserl was an adamant intensionalist. See for example his review of Schröder’s Vorlesungen as well as The Deductive Calculus and the Logic of Contents, both from 1891. For evidence that Husserl did not change his mind afterwards, see the 1903 review of Palágy’s Der Streit der Psychologisten und Formalisten in der modernen Logik. Aurora’s reading, therefore, if taken literally, is probably incorrect. If we take it charitably, it is misleading.
Despite this, Aurora is completely right in pointing out (204) the indispensability of ideal objects, particularly species (universals), for Husserl’s phenomenological project in the Investigations: if the former go, the latter goes with them.
Chapter 6 is about the third and fourth Investigations. The latter deals with matters of ‘pure grammar’, as Husserl calls it, and here Aurora’s linguistic background is once again both tangible and helpful. Yet it is the first sections, on the third Investigation, that are particularly important. In fact, they are the crux of the whole book. The reason is that the third Investigation is about parts, wholes and the relations between them – and (without going into detail, I will return to it later) the very concept of structure, central to the book for obvious reasons, is defined, in the last chapter, in mereological terms.
To say something of significance on Aurora’s interpretation of the third Investigation I would have to write more than my allowance permits. I will therefore only mention what is at least a presentational flaw. Despite insisting throughout the book and in the chapter on the relevance of the formal sciences in the development of Husserl’s philosophy, Aurora never engages with the several formalizations of the Husserlian theory of parts and wholes. He does mention the first of such contributions, Simons 1982 (334). But we also have Simons 1987, Fine 1995, Casari 2000, and Correia 2004 – which, moreover, all extend Husserl’s theory in many different ways. This, to me, is the only genuinely disappointing feature of, or absence from, the book. All the more so, because the capacity to be mathematized or formalized is one of the definitional traits of structures as set out in the final discussion (310).
Chapter 7 outlines the properly phenomenological parts of the Investigations, namely, the fifth and sixth Investigations. This is where Husserl puts to work all the notions he previously set up and sketches a phenomenological theory of consciousness (especially of intentional consciousness) and knowledge. Aurora’s exposition is careful and effective, with more than one passage I found particularly felicitous.
Chapter 8 is where Aurora lays out and defends his view. These are the main claims:
- Husserl’s philosophy in the Logical Investigations is a structuralist philosophy;
- Some of the aspects of Husserl’s philosophy that make it structuralist are ideally suited to characterise structuralism as such;
- Husserl’s subsequent, transcendental work deals with one of the central problems of structuralism: the origin of structures.
Section 1 is about structuralism in general. The first thing to sort out is, obviously, what a structure is. Borrowing from a number of authors, Aurora characterises structure in terms of two things: part-whole relations, and mathematizability. A structure is ‘a particular type of multiplicity’ whose elements obey laws ‘that confer properties to the whole as such which are distinct from those of the elements’ (309, half-quoting J. Piaget). Moreover, a structure ‘must always be formalizable’ (310). On the basis of this, Aurora characterises structuralism as follows:
Structuralism aims at studying the latent structures within classes of objects…by creating models, i.e. formal descriptions that make the immanent relations between objects of the relevant class predictable and intelligible (311).
It is worth noting that the given definition of structure does not necessitate that of structuralism. It is even more worth noting that this is a good thing. The reason is that, while Aurora wants to argue that the philosophy of the Investigations is structuralist, it is dubious that Husserl’s project in 1900-1901 involved the idea that the phenomenology of the fifth and sixth Investigations should be formalized. True, Husserl did have in mind a formalization of his theory of wholes and parts, and that theory is operative in the phenomenology. But that doesn’t entail that Husserl’s early phenomenology was ever meant to be entirely formalizable – much less that its aim was to ‘make predictions’ about consciousness and knowledge possible. The upshot is that Aurora’s definitions allow for a Husserl who deals in structures but not, strictly speaking, for a structuralist Husserl. This is too underwhelming a conclusion for what is otherwise, as I said at the outset, a well-constructed case. A looser definition of structuralism might perhaps have been suitable.
Another (minor) unclarity is Aurora’s appeal to mereology throughout the book. In and of itself, this appeal is perfectly fine. Yet not all mereologies admit of the sort of relations between parts that structuralists require. For example, and in stark contrast with the structuralist’s mantra, in classical mereology there is a sense in which the whole is just the sum (fusion) of its parts! Yet Aurora never engages with the distinction between classical and non-classical mereologies in any significant way. Moreover, it is unclear why formalizations of structures should be mereological rather than, say, algebraic (like most of Aurora’s examples of formal structures) or order-theoretic.
Be that as it may, Aurora is entirely correct when he points out that, if part-whole discourse is crucial to structuralism, then Husserl’s theory is ideally suited to form the core of any structuralist system: it is (or can be made) robust, it is philosophically profound, and, importantly, being a non-classical mereology, it is strong enough to describe the right sort of relations the structuralist needs.
At the very end, Aurora points out that one of the distinctive features of Husserl’s structuralism is its engagement with the problem of the origin of structures. In particular, Husserl is interested in understanding the relations between the subjects who come to be aware of structures and the structures themselves. This is indeed what the Investigations are all about. It is also one of the threads of Husserl’s whole philosophical career. As Aurora puts it (effectively, I believe), ‘this attempt at conciliating genesis and structure, first carried out in the Logical Investigations, is peculiar to Husserlian structuralism, and it is the question that Husserl will try to answer – through an ever more complex philosophical elaboration – in all his subsequent works.’
Casari, E. 2000. “On Husserl’s Theory of Wholes and Parts.” History and Philosophy of Logic 21 (1): 1-43.
Correia, F. 2004. “Husserl on Foundation.” Dialectica 58 (3): 349-367.
Fine, K. 1995. “Part-whole”. In Smith, B. and Woodruff Smith, D. (eds.). The Cambridge Companion to Husserl (Cambridge: CUP), pp. 463-486.
Fine, K. 1998. “Cantorian Abstraction: A Reconstruction and Defense.” Journal of Philosophy 95 (12): 599-634.
Nenon, T. 1997. “Two Models of Foundation in the Logical Investigations.” In Hopkins, B. (ed). Husserl in the Contemporary Context: Prospects and Projects for Transcendental Phenomenology (Dodrecht: Kluwer), pp. 159-177.
Ortiz Hill, C. 1997. “Did Georg Cantor Influence Edmund Husserl?” Synthese 113 (1): 145-170.
Simons, P. 1982. “Three Essays in Formal Ontology.” In B. Smith (ed.). Parts and Moments. Studies in Logic and Formal Ontology (Philosophia Verlag: München-Wien), pp. 111-260.
Simons, P. 1987. Parts. A Study in Ontology (Oxford: OUP).
Thinking Thinking: Practicing Radical Reflection, edited by Donata Schoeller and Vera Saller, is a collection of essays that reflect on the very process of reflection. The topics revolve around the activity and the experience of thinking. As such, the nine authors address questions related to language-use, the body as source of meaning, and subjective experience. They offer a broad picture of contemporary discussions and debates in phenomenology, philosophy of language, and psychotherapy.
In a way, each essay points toward theoretical constructions and attempts to define their epistemological blind spots. The phenomenological postulate stating that what is described is tightly linked to the way it is given and to the experience of the subject for whom it is given lies at the root of every contribution. The „logical, syntactical and semantic structures of propositions (11)” are insufficient to account for the complexity of thought-in-process. Furthermore, it is the contributors’ conviction that these habitual conditions of thinking leave aside the vitality of the process. This implies that we should consider the embodied processes and the preconscious dimension accompanying thinking.
Claire Petitmengin’s chapter invites us to take account the corporeal experience of the scientist at work. The author collected a series of scientists’ descriptions of their ideational processes to clarify a source of pre-reflective meaning. In so doing, she provides interesting epistemological considerations regarding the relation between lived experience and the genesis of new ideas. In this perspective, “non-rational” tasks such as walking and drawing prove to be decisive in many researchers’ methods of investigation. Albeit underexamined, this “shifting of the center of attention from the head to the body (34)” should be considered. By turning away from discursive modes of thinking, the scientist can open to a “‘felt’ dimension of experience which seems to be…the very dimension of meaning (37).” The skeptic is tempted to question the probity of a so-called felt meaning. But we should keep in mind that such questioning weakens the moment we give up rigid distinctions between body and mind.
In her chapter, Susan A. J. Stuart similarly situates bodily experience at the center of her investigation, this time in explicit relation to language. Discussing Thomas Reid’s theses on artificial language and natural language, she argues for a priority of kinaesthetic, perceptual, and especially “enkinaesthetic” (i.e., the affects we have of our neuro-muscular processes) determinants at the origin of language and considers them as “artificial.”
Eugene Gendlin also approaches an implicit dimension of cognition. He argues that this “background” is not as vague as we might suppose. From the outset, it has a certain “precision” and is decisive, for example, in the formation of concepts. The author suggests various analytic possibilities for this “thinking with the implicit.” Referring to a similar notion of “background,” the prime concern of Donata Schoeller’s chapter is the “thoughtful process of articulation.” She argues that “what is said clarifies aspects of a background that functions in the meaning of what is said (112).” In other words, she examines the cultural and biographical ‘contexts’ that come into play when we formulate and articulate any experience through language. The essay is particularly interesting for its discussion of new possibilities in the methodology of scientific inquiries. These possibilities extend to the theory and practice of psychotherapies.
Both Terrence W. Deacon and Vincent Colapietro’s chapters examine the role of language in relation to the process of thinking. The former offers a neurologically-oriented account of language as “a variation on the emergent dynamics of mental processes in general (157).” He argues that this best fits with our experience of language (i.e., not as a construction and analysis following rules). As for Colapietro, he discusses Peirce’s fallibilism and the experience of ignorance and error as constituent of self-knowledge.
Language is also central in the last chapter, that of Steven C. Hayes’. He examines the relation between knowing and its verbal-symbolic correlate. His thesis is that “human language and cognition…fundamentally alters and shapes our subjective experience and the perspective from which we view it (209).” However, despite the ostensible simplicity of this statement, the author shows clearly that this commonplace appearance arises from our failing to question the meaning of the very terms and concepts used in its formulation. The concern of Hayes’ contribution is the specific meaning of language, cognition, symbols, and perspective-taking, as well as our use of them. This is especially relevant if we are to manipulate them in theoretical investigations concerning their role in our subjective life.
Vera Saller’s contribution addresses the notion of abduction, understood as the “creativity within the framework of rational thinking (182).” Peirce argued that abduction was the “only logical operation which introduces [a] new idea (183),” and should be distinguished from hypothesis. The author also points out that Peirce compared abduction and perception (201). As to the possibility of new ideas, Saller stresses the importance of everyday thinking in their formation: “it is in the problem solving of everyday life, that new thoughts arise (186).”
Remember the story of a Buddhist monk whose disciple urgently asks him about a serious spiritual issue. The master answers with a question: “Have you finished eating?” “Yes,” answers the impatient disciple. The master then replies: “Then go wash your bowl!” In short, there is no better way of solving a preoccupying reflective problem than going about our everyday tasks hoping for the “momentary but significant flash (190),” i.e., the abductive moment. This moment of understanding is clearly not of an exclusively cognitive nature, the more so that it “comes along with a pleasant bodily emotion (202).”
But Saller’s contribution is also interesting as she compares this process with the detective metaphor in psychoanalysis. The literature examining the detective leitmotif in psychoanalysis sometimes commits a crucial error: neglecting the patient’s own work. This is often in favor of a misguided image of the analyst as the sole investigating instance in the cure. The analyst is thus constantly chasing the truth of the subject lying on the couch. Moreover, in this conception of the analyst, he or she attributes to the patient a knowledge that can only come from him/herself. The author points out this mistake and brings forth the “abductive inferences” that the method of free association is supposed to facilitate.
Thinking Thinking: Practicing Radical Reflection serves multiple purposes, including theoretical inquiry (scientific and philosophical), as well as practical concerns (psychotherapy and other social practices). Combining numerous perspectives, notably phenomenology and the philosophy of language, it is relevant to a broad range of researchers and practitioners.