The Philosophy of the Kyoto School (2018) is translated into English by Robert Chapeskie and revised by John W. M. Krummel. It introduces the reader to the works of (some of) the members of the Kyoto School. The general structure of the book means that each member is represented by a primary text, which is supplemented by an introductory essay. The general purpose of the latter is to outline the research, life and works of each scholar and to provide the background knowledge necessary to understand how each member relates to the conception of the Kyoto School. In the preface, Fujita Masakatsu, the editor of this book, suggests that readers “read the [introductory] essay first before turning to the original text it discusses” (The Philosophy of the Kyoto School, Ed. Fujita Masakatsu, 2018, vii). In addition to this suggestion, which I strongly recommend that any reader with no prior knowledge of the Kyoto School adhere to, I would recommend reading the two supplementary essays (The Kyoto School and the Issue of “Overcoming Modernity”, and The Identity of the Kyoto School: A Critical Analysis) before tackling any of the chapters, since they answer some of the questions readers with little previous knowledge of the Kyoto School might overlook while reading this book; these questions, nonetheless, do seem important to bear in mind while reading this book. They can be summarised as: Which thinkers do we include in the Kyoto School? and How do we define the Kyoto School?
The answer to the first question is far too complex for a thorough examination in this review, but the Kyoto School is generally considered to have been founded by Kitarō Nishida (1870-1945), a professor at Kyoto University, together with Hajime Tanabe (1885-1965). In relation to this, it seems relevant to answer questions regarding the nature of the Kyoto School. First, it is important to know that it was not a school in the sense of the Frankfurt School. Instead, and as an answer to the second question raised earlier, the Kyoto School is a loose term used to describe philosophers with a direct, or indirect, relationship to Nishida and Tanabe. In practice, this invariably also means to have a relationship with Kyoto University, its Faculty of Letters and/or the Chair of Philosophy at this faculty. The chair which Tanabe held after Nishida. Due to this strong connection with these two philosophers, a thorough outline of their philosophies and disputes seems to be in order, even if the book is structured so that each individual philosopher is given an equal amount of attention.
Nishida graduated from Tokyo Imperial University and later became first an assistant professor (in 1910) and shortly after a full professor (in 1913), both positions held at the Kyoto University Faculty of Letters, where Nishida held the Chair of Philosophy. While Nishida’s philosophical style is described as unsystematic by Masakatsu in the introductory essay, the concept of place is suggested as an important fixture in Nishidian Philosophy. The text included in this volume by Nishida is called Place. Place for Nishida is a concept which is developed in order to describe that which must “[envelope the] opposition between the ‘I’ and the ‘non-I’ and that establishes the so-called phenomena of consciousness” (Ibid. 3). This might be paraphrased as meaning that for Nishida place is a mediator of the I and the non-I, or put differently, of the subject and the object, as we know the discussion from the Western philosophical tradition (see i.e. Plato, Aristotle, Kant, Hegel, and Heidegger). However, place is not platonic, a point which Nishida spells out, writing: “what I refer to as ‘place’ is not the same as what Plato refers to as ‘space’ or ‘receptacle’ [vώqa]” (Ibid. 3). Opposed to Plato’s understanding of space/receptacle, Nishida’s place is “that which permits the relationship between physical space and physical space cannot itself be physical space. What is required is a place wherein physical space is situated” (Ibid. 5, my italics). This means that for Nishida place comes to be the solution to the question of how to understand the relation between I and non-I, subject and object. Critiquing the Kantian notion of the transcendental subject, Nishida posits that consciousness includes meaning and that because of this “we can speak of consciousness as the self-determination of something universal” (Ibid. 6). This led Nishida to the realisation that this cannot be in the case of form and matter; instead, these – to establish knowledge – must be mediated by a different sort of place, concerning which Nishida writes:
“The place that establishes the opposition between form and matter must be different from the place that establishes the opposition between truth and falsity. At the place that establishes knowledge, not only must form and content be distinguishable, but their separation and combination must be free” (Ibid. 6).
This leads to the conclusion that there must be a “place of experience” (Ibid. 6-7). Thus, knowledge and experience are established in the same place, because both knowledge and experience are “phenomena of consciousness” (Ibid. 7). This outline of Western metaphysics, of the subject/object distinction, led Nishida to consider “the idea of self-awareness that reflects the self within itself” (Ibid. 8). Following this revelation, Nishida comes to posit knowing as an act which envelops the opposition between form and matter, or between subject and object. Answering the question of where a self-awareness, which reflects itself within itself, is situated (i.e. placed), Nishida posits the category of true nothing as this place. True nothing is a nothing which has transcended the opposition between being and nothing, between the I and non-I. It has transcended these in such a way that it envelops both – “To speak of subject-object unity, or the disappearance of subject and object, is simply to say that place becomes truly nothing” (Ibid. 9).
This is what Nishida calls the logic of nothing, a logic which takes on a new form in the work of Nishida’s successor, Tanabe Hajime (1885-1962). After graduating from the Faculty of Letters at Tokyo Imperial University, he eventually gained a position at Kyoto University in 1919, and later took over the Chair of Philosophy after Nishida’s retirement. The text included in this volume by Tanabe is called Clarifying the Meaning of the Logic of Species. Heavily inspired by historical materialism, Hajime “brought the practical dynamism he had learned from it to the logic of nothing” (Ibid. 43), founding the philosophical notion of the logic of species, a term which is as much a critique of the logic of nothing as it is a development of it. Regarding the internal critique between the members of the Kyoto School, Masakatsu writes:
“We may take this kind of relationship that permits mutual criticism, or of taking critique as a springboard or the criticism received as energy for developing one’s own thought, to be one characteristic feature of the Kyoto School” (Ibid. vii).
This can be assumed to be a direct reference to the fact that Nishida not only accepted Tanabe’s critique, but also used it to further develop the logic of nothing. Leaving this development aside, the following is an outline of Tanabe’s conception of the logic of species. Tanabe states that there are two reasons for writing this essay: “the practical and the logical” (ibid. 25). The practical reason for Tanabe seems to be a wish to understand the rise of ethno-homogenous state ideology in South-East Asia. Tanabe refutes the idea that states are made up of individuals who enter into a contract, as exemplified in the theories of Hobbes’ Leviathan, or Rousseau’s Social Contract. Opposed to such theories as describing at least the Japanese state, Tanabe instead argues that:
“society is not a relationship that simply proceeds from individuals … Rather, unless it possessed a substratum [基体] unbounded by the generational replacement of individuals and to this extent exist as something preceding them, it would be unable to coercively unify them. And since the social substratum is something species-tribal [種族的], wherein individuals are born and included, I thought it should be called a [species]” (ibid. 25)
Tanabe calls this kind of society “communal” (Ibid. 27), which stands in opposition to the “contractual society” (Ibid. 27). Following on from this, Tanabe devotes the remainder of the essay to explaining how an individual comes to accept state coercion, and it is here that the logic of nothing is redeveloped by Tanabe, who argues that: “The true individual becomes individual within the whole only through the mediation of the universal … the affirmation of the subject in absolute negation, is the mutual unification [相即] of the state and the individual as a subjective whole” (Ibid. 27-28). Hence, the mediation between individuality and state is, for Tanabe, that which brings about the true individual (in the same way as the mediation of universal and particular in Nishidian philosophy came to bring about true nothingness). Thus, Tanabe breaks with Nishida in claiming that state coercion is necessary to mediate and, in this way, achieve a subjective whole. With regard to this, in the introductory essay, Nakaoka writes that “To negate the self as an individual is to establish its communal character. Tanabe thus came to believe that ‘the true self is restored by losing itself’ ” (Ibid. 47). The true self for Tanabe is something which envelops both the individual and the species (the universal), but where Nishida claimed an absolutely nothing, Tanabe postulated a true self which needs to lose itself to be found. Thus, Tanabe’s conceptual development of the logic of nothing into the logic of species makes Tanabe’s contribution a much more social/material logic than Nishida’s. Nishida and Tanabe constitute two of the grounding pillars on which the Kyoto School stands, and in their works, we see concepts and topics which are to be taken up, expanded upon or criticised by their direct or indirect heirs.
Kiyoshi Miki (1897-1945) was a direct heir, who entered Kyoto University in 1917 and subsequently studied philosophy under both Nishida and Tanabe. In 1922, Miki went to Germany to attend lectures given by Rickert and Heidegger and in 1924 Miki moved to Paris, “where he spent one year devoting himself to reading [Pascal] while studying French” (Ibid. 66). Miki’s text included here is called The Logic of Imagination, and it represents Miki’s attempt to unify pathos and logos, which eventually led Miki to the logic of imagination conceived of as a “philosophy of action” (Ibid. 59). While paying tribute to Nishidian philosophy, Miki would state clearly that the logic of imagination was to be “considered separately” (Ibid. 59). Miki conceived of action different from the philosophical tradition which conceives it as having an origin in the will, meaning in subjectivism. Opposed to such an understanding, Miki posited that the term should be understood as
“the event of creating things … All acts in the broad sense … have the meaning of production … To act is to make new forms by working upon things and altering their forms (transforming them). Forms, as things that are made, are historical and change through history” (Ibid. 59).
Here one clearly sees the influence which historical materialism had on the philosophy of Miki, and this is a definite break with Nishidian philosophy. The acts of creation which Miki attributes to the logic of imagination links this philosophy closely with technology and the arts, both of which Miki conceives of as creative, in the sense that they both create something new. Another figure closely linked to Miki is Jun Tosaka (1900-1945). The connection with Miki is not only in the forming of what has been termed the left-wing of the Kyoto School, but also in the tragic fate they shared, both dying in prison (in Japan) in 1945. Tosaka, another graduate from Kyoto University, was concerned with the notion of the technological spirit, and the text included is What Is the Technological Spirit? Tosaka describes this as “the fundamental spirit of modern culture” (Ibid. 81). Tosaka then goes on to locate this spirit not only in the modern world but also traces it back to ancient philosophy, in effect tracing it back to Plato and Aristotle. Tosaka also postulates a scientific spirit, which is then examined in relation to the technological spirit, concluding that these spirits are like opposite sides of the same coin. The scientific spirit, Tosaka claims, has three characteristics. It is “firstly a positivist spirit … secondly … a rational spirit … [And] I also consider the scientific spirit to the historical spirit … The scientific spirit … must be a spirit of our everyday life and action” (Ibid. 85). Tosaka does not dwell on the question concerning whether the scientific spirit is the technological spirit or the other way around. Instead, the technological spirit is conceived as “another face of the scientific spirit” (Ibid.). This leads Tosaka to argue that even at the level of the laboratory (positivist science) there is a social aspect, thus it is not a “true [absolute] historical understanding” (Ibid. 86). This is a direct critique of Tanabe and the idea that the progress of science will be rolled out deterministically based on the logic of species. Opposed to such an understanding, Tosaka came to claim that even positive science is historically situated and not an absolute.
Differing from Miki and Tosaka’s materialistic concerns, Motomori Kimura’s (1895-1946) philosophy engages with the question of body and spirit and the essay included here is Body and Spirit [Mind]. Kimura graduated from Kyoto University in 1923 and returned in 1933 as an assistant professor. What is of interest regarding Kimura is that from 1939 onwards Kimura oversaw teaching, not in philosophy but in pedagogy and teaching methods. Thus, Ōnishi, in the introductory essay, examines Kimura as “as a scholar (philosopher) of education … Kimura philosophized from the principial depths of praxis = poiesis underlying both the undertaking of the practice called ‘education’ and the act of creating a work of art” (Ibid. 124). For Kimura the body is not the opposite of the spirit. Instead, the body is described as “a principle of expression [表現]. Expression, however, is the manifestation of the inside on the outside” (Ibid. 110). This means for Kimura that the inside is “at the same time outside and vice versa” (Ibid). In this sense, the body becomes a mediator which manifests the inside, or the spirit on the outside (what Kimura calls nature). Hence, in Kimura there is no dualism between body and spirit. Instead, there is a mediation between the spirit and nature through the body. The body comes to act as a point which allows mind and matter to interact with one another. Leaving this point aside, what is important for Kimura in this regard is the concept of expression. Expression, outlined succinctly, is the inside expressed on the outside, as an act of creation, situated on the outside. It is not conceived of as in opposition to the outside (nature) but, instead, as being situated outside of the inside. The conclusion of this line of thought is that:
“[The body]is the self-negation of spirit, and at the same time it is the self-negation of matter. Because the body is thus the self-identity of contradictories [矛盾の自己同] it possesses the capacity of formation, and expressive life is able to express itself in self-awareness through the mediation of the body” (Ibid. 120).
Another thinker who continues this line of examination into the spirit is Shinichi Hisamatsu (1889-1980), who became a professor at Kyoto University in 1946. The text included is called The Metaphysical Element of the East. In this text Hisamatsu elaborates pivotal concept in Hamamatsu’s philosophy of the Eastern nothing. Hamamatsu’s life and works are perhaps those which dwell mostly on the topic of religion, and Nishida once had to write a letter reprimanding Hisamatsu for “[trying] to drop out of university just before graduation in order to practice Zen” (Ibid. 150). Hence, the practice of Zen is an important factor in the development of Hisamatsu’s thoughts, a practice which can be said to have been inspired by a direct suggestion from Nishida, who was also a Zen practitioner. The Eastern nothing is an integral part of Hisamatsu’s religion of awakening. The latter is a metaphysical thought or system which Hisamatsu claims cannot be found in the West, while the former is described as a concept different to, but not in opposition to, Western thought. Hisamatsu stipulates that Western thought, since the Greeks, has revolved around the concept of Being, positing that in the East a different line of thought concerning this developed. Hisamatsu explains that:
“This ‘Eastern nothing’ is something that cannot be fit into the category of what exists in actuality. Without being something metaphysical from the standpoint of all beings or “being”, it is something metaphysical that negates and transcends being itself” (Ibid. 143).
This is thus a concept which draws heavily on the concept of absolute nothingness in Nishidian philosophy, and for this reason Hisamatsu’s philosophy falls within the frame of the Kyoto School, as it directly deals with one of the pivotal concepts of the Kyoto School.
Toratarō Shimomura (1902-1995) is described by Takeda in the introductory essay as the man who brought the Kyoto School to a close, and while the book does, in fact, contain an additional philosopher, this is not an overestimation on Takeda’s part, considering that Shimomura was the last of the philosophers included in this book to pass away. Shimomura’s work included in this volume is The Position of Mathematics in Intellectual History. In this text Shimomura tries to discern the difference between Eastern and Western culture, specifically regarding scientific/academic inquiry (science, for Shimomura, becomes academic inquiry as natural sciences stem from the mathematics of the ancient Greek philosophers). Shimomura asserts that academic inquiry is a Western term which originates from the West and points out that:
“ ‘academic inquiry’ [gakumon 学問] in our mother tongue, if we follow its classical usage, meant something close to that which takes ‘statecraft’ [治国平天下] or ‘moral conduct for living’ [修身処生]—ultimately things of a religious or political-moral, generally practical nature—or ‘practical inquiry’ [実学] as its subject matter” (Ibid. 164).
This means that the subject matter of these inquiries differs in one very important sense; namely, one is theoretical, and the other is practical. Following this insight, Shimomura argues that each culture, or what Shimomura and the Kyoto School call ethnic spirit, has its own kind of “Religion, academic inquiry, and art, too …[which] thereby form a system of culture, and, through the mediation of the ethnic spirit, express the world; the world thus realizes itself in them” (Ibid. 165). Therefore, it is through an inquiry into European academic inquiry (understood as a moment) that Shimomura comes to regard history, and academic inquiry itself, as being mediated through the spirit and experienced by that spirit in its historical moment.
Closing this volume, but not the Kyoto School, is Keiji Nishitani (1900-1990), whose included work is Nihility and Emptiness. This was the only work known to me prior to reading this book, though my knowledge is superficial. In this work by Nishitani, we again see the notions of nothingness (nihility) and emptiness coming into play as pivotal concepts for the Kyoto School. Keta, in the introductory essay on Nishitani, writes that Nishitani’s relationship to Zen is important if one is to understand the philosophy of Nishitani. Like Hisamatsu, Zen Buddhism became a practice for Nishitani which would resolve the crisis of not feeling that any of the philosophers studied up until that point (primarily Western philosophers, as this was Nishitani’s speciality) had been able to fill a growing internal void. Keta writes that: “at the age of thirty-three he began practicing Zen at the Meditation Hall of Shōkoku Temple in Kyoto. He would later state that through this practice he somehow managed to extricate himself from this crisis” (Ibid. 219-220). The basic premise of Nishitani’s philosophy is that science (the scientific method) overlooks both religious and philosophical questions, by mechanizing or rationalising humans, society and nature. This, Nishitani argues, leads to the fact that “contemporary nihilism arises … from an awakening to the meaninglessness at the root of this world and of human beings” (Ibid. 207). This meaninglessness, nihility, is for Nishitani overcome by the concept of Buddhist emptiness [空], which Nishitani equates with Eckhart’s notion of detachment: “What Eckhart called ‘detachment’ [離脱], … a transcendence that is a freeing not only from the self and the world but even from God …This point emerges with greater clarity in the standpoint of what is referred to in Buddhism as ‘emptiness’ [空]” (Ibid. 209). The concept of emptiness is described as “the completion of an orientation toward negation. As a standpoint that has negated nihility as the negation of being” (Ibid. 2014). Such a standpoint seems in alignment with the development of Nishidian philosophy as outlined in this book, and while Nishitani’s concept of emptiness differs from Nishida’s absolute nothingness, it still follows in a line of critiques, redevelopments and new articulations that seem to be the hallmark of the Kyoto School.
Succeeding in drawing a red line through the main topics, interests and fields which comprise the works of the members of the Kyoto School, this book is an important contribution to scholars in the West with an interest in the appropriation of Western metaphysics in the East (Japan/Zen Buddhism), to scholars of the Kyoto School in particular, or to those interested in the specific topics dealt with by individual members of the Kyoto School. The primary texts, with their introductory essays, elicit a development of the thought(s) of the Kyoto School which would be hard to elicit for an individual scholar with limited knowledge of Japanese philosophical tradition, Zen Buddhism, or the history of the Japan (ca. 1850-2000), and without access to the translated works. For such scholars, this book is of vital importance as an introduction to this school of philosophy, and the introductory texts and supplementary essays help the reader obtain an outline of each member’s philosophy, their project and the historically important events surrounding their lives, even if it is accomplished from a bird’s eye view. Therefore, I recommend readers with no knowledge of the history of either the Kyoto School or Japan to read the supplementary essays at the end of the book before engaging with the primary texts or their introductory essays. In particular, I found The Kyoto School and the Issue of “Overcoming Modernity” by Kunitsugu Kosaka to be an essay which is very informative for the novice scholar. In this essay Kosaka elaborates not only on the development of the general project of the Kyoto School as an attempt to overcome modernity, but also on the claim that some of the members of the Kyoto School “beginning with Nishida Kitarō, have been stamped with the label of having been collaborators in Japan’s activities during World War II” (Ibid. 233). This is not unlike similar claims levelled against the philosophy of Heidegger or even Nietzsche, both of whom are philosophers who can be said to have had an influence, directly or indirectly, on the members of the Kyoto School. While the book is an introduction to the Kyoto School, it does, however, assume knowledge of philosophical concepts, particularly of metaphysical and ontological concepts. This is not a criticism of the book but a note for any potential reader. Moreover, while it might seem daunting for some readers to immerse themselves in the depths of philosophical inquiry, the task of reading these texts is not insurmountable for anyone willing to spend some time brushing up on key concepts.
A key aspect, or method, of the Kyoto School seems to be that of mutual criticism, and while this does not make the general project of the Kyoto School compatible with the Frankfurt School (e.g. with Adorno and Horkheimer’s critique of modernity/enlightenment), I would point out that this is an aspect where that these two schools converge. In addition to this, both schools also seem to have been engaged with the question of the relationship between Being and Nothing, subject and object, though they differ enormously in their conclusions. Leaving this point aside, as the book does not dwell too much on this question, it seems important to mention finally that while the book introduces the Kyoto School as endeavouring to present an Eastern philosophy which differs from Western philosophy, these two terms are ambiguous for several reasons. Firstly, because the Kyoto School is firmly anchored in Japanese Zen Buddhism or a critique of it, as opposed to an Eastern philosophy that spans other Buddhist ways of thinking, or even other countries. Secondly, because of their engagement with a certain kind of Western philosophy, mainly Heidegger and Nietzsche. In addition to these two points, some members also engage with historical materialism (i.e. Miki and Tosaka). All in all, this is a serious book worth attention from any scholar interested in metaphysical or ontological questions answered from a position different from the normative Western perspective. Though different from the western perspective, Nishida’s general claim is that Japanese culture is well-versed in both the Eastern and Western perspectives, and thus exceptionally suited to provide a bridge between them.
“The original characters of Eastern culture and Western culture are such that they ought to be mutually complementary, not such that one is superior to the other or one must be integrated into the other. What is important is instead to uncover the broader and deeper roots that run through both Eastern culture and Western culture, and from there to shine a new light on both cultures. Nishida argued that this is precisely the world-historical role Japan (being well versed in both cultures) bears today” (Ibid. 240).
In paraphrasing this rather lengthy quote, one might say that the goal of Nishidian philosophy was to bridge the gap between two cultures, or metaphysical systems and that the subsequent members of the Kyoto School should be thought of as engaging with this project either affirmatively, critically or descriptively. Thus, what makes up the Kyoto School, and what merits its name, is a sense of dealing with common themes centred around the idea of shining a light on these two cultures by uncovering their common roots.
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.