Sofia Miguens (Ed.): The Logical Alien: Conant and his Critics

The Logical Alien: Conant and His Critics Book Cover The Logical Alien: Conant and His Critics
Sofia Miguens (Ed.)
Harvard University Press
Hardback $59.95 • £47.95 • €54.00

Reviewed by: Nicola Spinelli (King’s College London / Hertswood Academy)

This is the kind of book one hates to review. Not because it is bad; it is an excellent work, rich and profound and relevant at least to: the scholar of half a dozen areas in the history of philosophy (from medieval through early modern, modern, Kant, post-Kantian, to the early analytic philosophy), the philosopher of language, the metaphysician, the philosopher of logic, and the epistemologist. But it is complex – much more complex even than your average 1069-page philosophy collection. Perhaps this is to be expected: one way to think of The Logical Alien is as a commentary (on steroids) of James Conant’s 1991 “The Search for Logically Alien Thought: Descartes, Kant, Frege, and the Tractatus”, itself a long, seminal, profound and – dealing as it does with history and theory and some of the heavyweights of the last five hundred years of philosophy – multitasking paper.[1] The papers collected in the book are written for one third by different authors engaging with Conant’s 1991 paper, and for two thirds by Conant engaging with his former self and with each of the other contributors, occasionally with more than one at the same time. The parts of the book end up being so interconnected at so many levels, that it takes several readings just to find one’s way through it – never mind figuring out what to make of even one of the numerous debates involved or convey it to prospective readers with something resembling accuracy. Yet the book is as difficult to review as it is exhilarating to read. Once you get hooked up (and you do get hooked up), you won’t be finished for a long time.

The central question is taken from Frege and is simple enough: Is there such a thing as thought which is logical but whose logical laws are different from, and incompatible with, ours? Put this way, there would seem to be an equally simple answer: yes. Consider systems with different and incompatible rules of inference: in a classical setting, Excluded Middle and Full Double Negation are laws; in an intuitionistic setting, they aren’t – yet nobody from either camp seriously thinks that the other just isn’t thinking logically. After all, intuitionistic and classical logic are equiconsistent (a proposition is classically provable if and only if its double negation is intuitionistically provable). Of course there is a qualification to make in this case: some logical laws are in common. For example, Non-Contradiction – which in any case seems to be needed for concepts like ‘consistency’, ‘incompatibility’ and ‘disagreement’ to even make sense. What about, then, thought which shares none of our logical laws – not even Non-Contradiction? Conant’s original paper, and much of the discussion in the book, revolve around this insight: that since at least some of what we call logical laws are constitutive of thought as such, thought which does not conform to them is in fact not thought at all. In one form or another is attributed by Conant, past and present, to Frege, Wittgenstein and Putnam (or Putnam at some point of his career).

The insight – which we shall call the Insight – develops in interesting ways. Consider the following way of putting the central question: Are the laws of logic necessary? If the Insight is correct, then, one might say, they are. Not so – at least on the view Conant and his critics are interested in. Since what we call logical laws are constitutive of thought as such, logically alien thought is an impossibility. Discourse about it, then, is what Conant calls philosophical fiction (768).[2] The contrast is with empirical fiction. The latter invites us to contemplate a scenario which happens not to be the case, but which ‘falls within the realm of the possible’. The former invites us to contemplate something which is not even possible. So that in philosophical fiction we ‘only apparently grasp what it would be for [the scenario] to obtain: its possibility can only seemingly be grasped in thought’. But, the view concludes, if logically alien thought is philosophical fiction, then the project of establishing its possibility or impossibility is in fact a non-starter: for in order to affirm or deny that logically alien thought is possible, or even ask whether it is possible, we first need to grasp ‘it’ – the thought with content ‘logically alien thought’ – but that is exactly what we cannot do. Far from being able to answer the question, we seem to have no question to answer. It looked as though we had one; but it turns out we never did. It was a mock-question. Hence, for example and according to Conant (past and present), the austere – non-mystical – Wittgensteinian stance at the end of the Tractatus: the necessity of logic isn’t a question which logic cannot answer; it is a non-question. Hence, too, the Wittgensteinian idea that philosophy should be conceived of not as doctrine, not even as research, but as something called ‘elucidation’: the activity of recognising that some or all of what we take to be profound philosophical problems are in fact simply nonsense.

In the original 1991 paper, Conant follows the development of this line of thought – call it elucidativism about logic – from Descartes through Aquinas,  Leibniz, Kant, Frege, to Wittgenstein and Putnam. He does not defend elucidativism, but he clearly favours it. In the first part of The Logical Alien, his critics either follow up on 1991-Conant’s historical claims in the paper (which is included in the book), or take issue with theoretical claims, or both. The following is an overview of the contributions. A.W. Moore’s is about Descartes and what he ought to have thought about modality. In particular, whereas 1991-Conant claims that Descartes’ official view was that necessary truths (amongst which are the laws of logic) are contingently necessary, Moore argues that statements to that effect to be found in Descartes are aberrations rather than expressions of the official view. Matthew Boyle’s chapter is about Kant’s and Frege’s conceptions of logic and of the formal. Arata Hamawaki’s paper is about a distinction between Cartesian and Kantian skepticism. I have to say that, while the former contributions are excellent reads, I found this one rather difficult to follow and, despite the theme, somewhat underwhelming. Barry Stroud’s paper is the skeptical contribution: historically, doubts are cast on 1991-Conant’s reading of Frege; theoretically, issue is taken with the notion that necessary truths are apt to being explained. Peter Sullivan objects to 1991-Conant’s view of Frege, and argues that the latter is more Kantian than is usually thought. The contribution also contains a very good summary of the dialectic of the 1991 article (in case you struggle to follow it). Along with Moore’s, perhaps the best of the (mainly) historical contributions (to my taste). Martin Gustafsson and Jocelyn Benoist concentrate on post-tractarian Wittgenstein: the former to examine the relations between language use and rule-following, the latter to show how Wittgenstein’s treatment of private languge is an exercise in elucidation. Finally, Charles Travis’ chapter, the longest, discusses Frege, Wittgenstein and the heart of the elucidative enterprise. Undoubtedly the most important of the critical essays. I agree with many points he makes, and I will be saying something similar in the remainder of this review – but from a very different perspective. The second part of The Logical Alien consists of present-day Conant discussing both his 1991 paper and the critics’ contributions. I see no point in saying anything here, except that he (and probably the editor, Sophia Miguens) did an excellent job of making the Conant’s own chapters a single narrative rather than a collection of discrete replies.

Now, upon my first reading of the 1991 paper, and on every subsequent reread, and indeed as I was ploughing through the book, I thought it a shame that there was (virtually) no reference to the phenomenological tradition at all. This is not to say that there should have been: as far as I can tell, phenomenology has never been among Conant’s interests, and that this should be reflected in a book about his work is, after all, only natural. On the other hand, at least some of the debates in The Logical Alien might have benefited from a phenomenological voice; and others are relevant to discussions within the phenomenological tradition. And since I am writing this review for a journal called Phenomenological Reviews, I will allow myself to expand on the above and bring phenomenology into the melee.

I have already said what the central view at stake in the book is: that the question as to whether there can be logically alien thought is a non-question, because its formulation involves something akin to a cognitive illusion. The further question, however, is: Why is grasping a thought about an impossibility itself impossible? Why, in other words, should we buy the claim that in philosophical fiction, as Conant says, we only seem to grasp a thought but we really do not? Why is the thought that there may be logically alien thought, despite appearances, no thought at all?

The reason lies in the following view, endorsed at lest to some extent by Frege, embraced by tractarian Wittgenstein and assumed in Conant and his critics’ discussions: To grasp a thought is to grasp what the world must be like for the thought to be true and what the world must be like for the thought to be false.[3] A thought for which either of these things cannot be done is a thought for which, as Frege would put it, the question of truth does not genuinely arise. It is then not a thought but a mock-thought. This is the basis of Wittgenstein’s notion that tautologies and contradictions have no content: for we just cannot imagine what the world what have to be like for tautologies to be false or contradictions true. For all the depth and complexity of the debates which Conant’s 1991 paper has sparked, and which are well represented in The Logical Alien, if what we may call the Assumption falls it is hard to see how the rest might stand. For if grasping the content of a thought is decoupled from grasping its truth-(and-falsity-)conditions, or from even bringing truth into the picture, then even if philosophically-fictitious scenarios are impossible we can still grasp them – if only to deem them impossible. Thoughts about them are not mock-thoughts; or, if they are, they are so in a weaker sense than Conant seems to envisage – too weak for the work he wants mock-thoughts to do.

Conant is aware of this. In his reply to Stroud he highlights how the 1991 paper pinpoints a tension in Frege between 1) his elucidative treatment of the logical alien in the foreword to Grundgesetze, and 2) his commitment to the idea that tautologies and axioms are true.[4] If the Insight and the Assumption are true, then 1) and 2) are (or very much seem to be) incompatible. Conant suggests that the ‘deeper wisdom’ to be found in Frege, which is also the strand of Frege’s thought which Wittgenstein develops, is 1). The claim that axioms and tautologies, despite having negations which are absurd, are true is treated by Conant as stemming from Frege’s conception of content (thought) as ‘explanatorily prior’ to judgement. So that it is only if we think that the content of a judgement pre-exists the judgement that we can take judgements about impossible scenarios to have a content. Otherwise we would have to say: there is no judgement to be made here, and therefore there is no content.

I will not go into the minutiae – or even the nitty-gritty – of Fregean scholarship. But surely the move only pushes the problem a step further. Grant that judgeable content should not be thought of as explanatorily prior (whatever that means exactly) to judgement, the question is: Why buy the claim that we cannot judge about impossibilia – not even to say that they are impossibilia? If we can, there is judgement; and therefore there is content. Are there views on the market which do not take judgeable content as explanatorily prior to judgement, and according to which we can and do judge about impossibilia?

Husserl held just such a view throughout his career. There are several ways to see this. Begin with the Investigations. There, meanings are ideal objects (universals) instantiated by the act-matter of classes of meaning-intentions. The latter are intentional acts through which a subject intends, or refers to, an object. Their matter is, with some oversimplification, their content.[5] Notice that the content of a meaning-intention is not the meaning: without an act there is no content – though there is a (perhaps uninstantiated) meaning. So even in the early Husserl, despite his ostensible Platonism, it is not obvious that judgeable content is prior to, or even independent of, judgement. In the fourth Logical Investigation, a distinction is made between nonsensical (Unsinnig) and absurd (Widersinnig) meanings. A nonsensical meaning is a non-meaning: an illegal combination of simpler meanings (illegal, that is, with respect to a certain set of a priori laws). A syntactical analogue would be a non-well-formed string of symbols: ‘But or home’. So, when it comes to nonsensical meanings, there just is no content (no act-matter). An absurd meaning, by contrast, is a (formally or materially) contradictory one: ‘Round square’. In this case there are both a meaning and an act matter; it’s just that to intentional acts whose matter or content instantiates the absurd meaning there cannot correspond an intuition – intuition being the sort of experience which acquaint us with objects: perception, memory, imagination. So we cannot see or remember or imagine round squares, but we can think about them, wonder whether they exist, explain why they cannot exist, and so on. Moreover, the very impossibility of intuitively fulfilling an absurd meaning-intention is, in Husserl, itself intuitively constituted and attested: attempting to intuit the absurd meaning leads to what Husserl calls a synthesis of conflict.

Say, then, that whilst engaging in philosophical fiction we try to make sense of logically alien thought, and we fail. This failure consists, in Husserlian phenomenology, in the arising of a conflict in our intuition, as a consequence of which we deem the scenario impossible. In the Husserlian framework this failure does not entail that there was never any thinking taking place with the content ‘logically alien thought’: it was ‘merely signitive’ thinking – thinking to which, a priori, no intuition can correspond – but contentful thinking nonetheless. We cannot intuit the impossible, but we can think about it.

So in Husserl the impossibility – the philosophical-fictitiousness – of logically alien thought does not entail that, when we think of logically alien thought, we only seem to do so. When we think of logically alien thought, we actually do think about logically alien thought; and one of the things we reckon when we think about logically alien thought is that it is impossible. All of this, notice, without appealing to the explanatory priority of judgeable content over judgement – which is what Conant finds disagreeable in Frege. Husserl, then, seems to be in a position to agree with Conant that judgeable content doesn’t come before judgement, and yet disagree with Conant that there is any wisdom whatever in Frege’s elucidative treatment of the logical alien.

All this is reflected in Husserl’s view of logic. From the Investigations throughout his career, Husserl maintained that logic comes in layers. In the official systematisation (Formal and Transcendental Logic, §§ 12-20) these are: 1) the theory of the pure form of judgements; 2) the logic of non-contradiction; 3) truth-logic. The first of the three is what in the fourth Investigation was called ‘grammar of pure logic’, and its job is to sort the meaningless – combinations of meaning which do not yield a new meaning – from the meaningful. It is the job of the logic of non-contradiction to sort, within the realm of the meaningful, the absurd meanings from the non-absurd. It is debatable whether truth is operative in this second layer of logic; I understand Husserl as denying that it is. But in any case, truth is not operative in the first layer. When Conant and his critics discuss the laws of logic, they take them to be such that, first, they are constitutive of thought, and second, truth plays a crucial role in them; and they take thoughts which misbehave with respect to truth, such as tautologies and contradictions, not to be thoughts at all (giving rise to tension in Frege). From a Husserlian perspective, what makes a thought a thought is not the laws of truth, but the laws of the grammar of meanings. Truth has nothing to do with it – nor, as a consequence, with what it is to be a thought.

The second part of Conant’s reply to Stroud (roughly, from p. 819 onwards) connects the above to another phenomenologically relevant strand of The Logical Alien: Kant and the project of a transcendental philosophy. The starting point is the difference between Frege’s approach on the one hand, and Kant’s and Wittgenstein’s on the other. The issue is, again, the central one of the relations between thoughts and judgements. Conant’s aim is to show that Frege can conceive of thought as separate from judgement – of content as distinct from the recognition of the truth of content – only by committing himself to the following conjunctive account: whenever an agent S judges that p, a) S thinks that p, and b) S recognises that p is true. These are two distinct acts on the part of S. This is contrasted with Kant’s (and, later on, Wittgenstein’s) disjunctive approach: there is a fundamental case of judgement in which S simply judges that p; and there are derived cases, different in kind from the former, in which S entertains the thought that p without recognising its truth – for example, in what Kant calls problematic judgements (‘Possibly, p’). Conant does not seem to provide a reason why we should be disjunctivists rather than conjunctivists – other than the claim that conjunctivism is at odds with the wider Kantian transcendental project. The implication being that if one buys into the latter at all, then one ought to be a Kantian rather than a Fregean when it comes to the relations between content and judgement.

What is, for Conant, Kant’s transcendental project? This is spelled out in the excellent reply to Hamawaki and Stroud.[6] To be a Kantian is first of all to put forward transcendental arguments. According to Conant, a transcendental argument is something close to an elucidative treatment of what he calls Kantian Skepticism: the worry, not that the external world may not be as experienced or not exist all, but that we may not be able to ‘make sense of the idea that our experience is so much as able to afford us with the sort of content that is able to present the world as seeming to be a certain way’ (762). Kant’s way to resolve the worry is to show that the scenario in which our experience is not able to present the world at all is philosophical fiction: if we probe the Kantian-skeptical worry enough, we find it unintelligible.

I don’t believe Conant reads Kant as endorsing elucidativism – that is, I don’t believe Conant reads Kant as making the final step: if the scenario in which experience does not present us with a world is unintelligible, then so is the scenario in which it does. But he does say that this ‘is arguably the closest Kant ever comes to an extended philosophical engagement with something approximating the question of the intelligibility of the idea of a form of cognition that is logically alien to ours’ (772). If one is a transcendentalist, in any case, one has to put forward transcendental arguments; and if Conant is right in his reading of what a (Kantian) transcendental argument is, then a transcendentalist needs to be in a position to reason from the unintelligibility of a scenario to the unintelligibility of the question as to whether the scenario is possible. But to do so – recall the (alleged) tension between Fregean conjunctivism and the Kantian project – a transcendentalist ought to avoid seriously distinguishing between content and judgement.

Another strand of Conant’s discussion of Kant, and at some level a consequence of the nature of transcendental arguments as described above, is the recognition that any account of our cognitive capacity must be given from within the exercise of our cognitive capacity – so that no account of the latter can be given in non-cognitive terms. Conant calls this ‘the truth in idealism’ (776). And this is what, for Conant, ultimately is to be a Kantian: to pursuse a philosophical project in the light of the truth in idealism. Needless to say, Wittgenstein counts as a Kantian par excellence; and so does the elucidativist half of Frege.

The phenomenologically alert reader will not have missed the fact that the truth in idealism is in fact a central tenet of Husserl’s post-Investigations philosophy. Suffice it to quote the title of Section 104 of Formal and Transcendental Logic: “Transcendental phenomenology as self-explication on the part of transcendental subjectivity”. I am less sure about Conant’s reading of transcendental arguments: granted that they do involve the recognition of the unintelligibility of skeptical scenarios, it is unclear why that should not simply be thought of as some sort of reductio ad absurdum, or perhaps of a quasi-aristotelian elenchos, rather than as something pointing to elucidation. Be that as it may, Husserl’s mature philosophy is a view in which the truth in idealism is preserved and in which, however, elucidativism is avoided – because even in the mature Husserl absurd thoughts are contentful.

Consider the relation between content and judgement. In the mature Husserl the interdependence of content and the mental is reasserted and strengthened with the notion of meaning as noema, introduced alongside the old Platonistic one in the 1908 Lectures on the Theory of Meaning, and center-stage in the first volume of Ideas in 1913. The main difference here is that the noema, one of whose component is intentional content, exists only insofar as the relevant mental act – in our case, the relevant thinking episode – does. As to the relations of noema and judgement, Husserl does think that it is possible to thematise a propositional content without judging that it is true. Yet this is claimed within a broader story – genetic phenomenology – of how more sophisticated intentional performances, together with their productions (including propositional contents), arise from more fundamental ones. The chief text here is Experience and Judgement. So Husserl could be said to hold something like what Conant calls the disjunctive account: the act of merely entertaining a thought is derivative of the act of straightforwardly judging. But this is not to say that one cannot merely entertain a thought! It simply means that we would not be able to mereley entertain thoughts if we were not able to straightforwardly judge. Indeed, for Husserl the existence of a noema such as, say, ‘ABCD is a round square’, while dependent on the relevant meaning-intention, is independent of the possibility of there being round squares at all. We can and do entertain the thought whether round squares exist, ask ourselves whether they do, and judge that they don’t. (The simplicity of the example might lead to error: it might appear as though, in this case, phenomenologically or introspectively, there were no distinction between entertaining and judging, for it is immediately clear that there are no round squares. All you have to do is try with more covert absurdities; to take a pertinent example, Frege’s very own Basic Law V.)

It really does seem to be a phenomenological fact that content and judgement are distinct. As the Husserlian case shows, one can maintain that that is so while still allowing the distinction to be derived rather than fundamental. Not only this: one can maintain the distinction, thereby blocking elucidativism, and still subscribe to the truth in idealism and be counted as a Kantian by Conant’s own standards. Or so, at any rate, it seems.

So being a Husserlian may be one way of being a Kantian without being an elucidativist. I hope it is and I hope there are others. Elucidativism usually divides people into three categories: those who buy it, those who don’t, and those who dismiss it as empty gobbledegook. I don’t dismiss it – but I don’t buy it either. For example, the argument for it discussed, and indeed put forward, by Conant seems to me to prove too much. This is a point Stroud makes in his contribution.[7]  In the reply, Conant is, I think, too concerned to show Stroud’s (alleged) misunderstandings to take his commonsense worries seriously. Regardless of that dialectic, consider any proof by contradiction in mathematics: we set up a proposition, we show that the proposition is inconsistent (either with itself or with other assumptions), we conclude that the negation of the proposition is true. If the elucidativist is right, the latter step is unwarranted: if a proposition turns out to be nonsense (which it does, being a proposition about an impossible scenario) then its denial is also nonsensical. So, if the view is correct, a large part of mathematics either is merely a cognitive illusion or, at best, is an exercise in elucidation. And yet the proposition, say, that there are infinitely many primes – whose negation is absurd in the same sense in which logically alien thought is – seems to be a perfectly legitimate proposition. So does the question whether there is a greatest prime, even though, it turns out, it makes no sense to suppose that there is. For some of us, intuitions in this respect are just too strong. In comparison, the elucidativist manoeuvre really seems sleight of hand of sorts.

Of course, even we must bow to argument. And in any case, since the stakes could not be higher, high-quality discussion is always welcome. The Logical Alien provides plenty – as I said, enough to go on for a long time. That is one reason to recommend the book – eve if, like me, you are not in the elucidativist camp. Another reason, relevant to the phenomenologically-minded reader, is that there seems to me to be a family resemblance, however faint, between elucidativism and certain strands of the phenomenological tradition broadly construed: Deleuze’s operation in Logic of Sense, Derrida with his différance, Sartre’s manoeuvres in Critique of Dialectical Reason. The Logical Alien might add something meaningful to those discussions, too.

[1]     J. Conant. 1991. “The Search for Logically Alien Thought: Descartes, Kant, Frege, and the TractatusPhilosophical Topics 20 (1): 115-180.

[2]     Part II, Section X, “Reply to Hamawaki and Stroud on Transcendental Arguments, Idealism, and the Kantian Solution of the Problem of Philosophy”: 758-782. Arabic numerals in parentheses in the main text refer to pages in The Logical Alien.

[3]     I say ‘assumed’, but it is in fact at the heart of Travis’ piece. Sullivan discusses it, too.

[4]     Part II, Section XI, “Reply to Stroud on Kant and Frege”: 783-829.

[5]     For an excellent overview of Husserl’s philosphy of language and its development, see Simons 1995.

[6]     Part II, Section X: “Reply to Hamawaki and Stroud on Transcendental Arguments, Idealism, and the Kantian Solution to the Problem of Philosophy”: 758-782.

[7]     Part I, “Logical Aliens and the ‘Ground’ of Logical Necessity”: 170-182.

Jairo José da Silva: Mathematics and Its Applications: A Transcendental-Idealist Perspective

Mathematics and Its Applications: A Transcendental-Idealist Perspective Book Cover Mathematics and Its Applications: A Transcendental-Idealist Perspective
Synthese Library, Volume 385
Jairo José da Silva
Hardcover 93,59 €
VII, 275

Reviewed by: Nicola Spinelli (King’s College London / Hertswood Academy)

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-pand 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).[1] 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).[2] 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.

[1]     Unless impossible worlds are brought in – but as far as I can see that option is foreign to da Silva’s outlook.

[2]     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.