The heart of the human experience is suffering. Such, at least, is Arthur Schopenhauer’s abiding thought. For Schopenhauer, in fact, our own personal suffering is just a microcosm of the whole world’s plight, which itself is, as his writings never cease trying to remind us, one characterized by brutality, cruelty, agony, despair, and ultimately death. Our situatedness in the phenomenal world (which is but the illusory shimmer of a primal Wille), he says, is that of a sailor who “sits in a small boat in a boundless raging sea, surrounded on all sides by heaving mountainous waves, trusting to his frail vessel; so does the individual man sit calmly in the middle of a world of torment, trusting the principium individuationis” (49). If the world we experience is in many ways a house of horrors, then, so Schopenhauer argues, it falls to each of us to do what he can to diminish the suffering we find in it—and how else, so he suggests, will accomplishing that be feasible except by having compassion for others? Compassion, hence, it would seem to follow, is the foundation of ethics.
As Schopenhauer says of Mitleid in On the Basis of Morality, when this “compassion is aroused, the weal and woe of another are nearest to my heart in exactly the same way, although not always in exactly the same degree, as otherwise only my own are. Hence the difference between him and me is now no longer absolute” (31). The ethical imperative to die to one’s egoism, and to thereby identify with others rather than only with oneself, however, is for Schopenhauer paradoxically infused with a thoroughgoing fatalism: just as the individual and his ego are themselves illusions, so too is free will. “The person is never free,” claims Schopenhauer, “even though it is an appearance of a free will, because it is the already determined appearance of the free willing of that will” (39). It is the search for life’s justification even in the face of its immense suffering that later drove Friedrich Nietzsche, following Schopenhauer, to reflect on these matters of freedom, value, and meaning. Art consistently proves central to those resulting reflections. For on one plausible interpretation of the matter, Nietzsche’s idea that “it is only as an aesthetic phenomenon” that existence is justified is itself a formulation to be understood as a variation of Schopenhauer’s own pessimistic insight into the purportedly inherent pointlessness of suffering. Existence requires justification precisely because it is not immediately self-justified. As for life, as the Nietzsche of the Birth of Tragedy notes, it cannot be affirmed strictly for what it is—“the truth is terrible,” after all—but rather must be tolerated by way of placative lies. On the view Nietzsche sets out during this period of his thinking, art accordingly presents us with a palatable world, a beautifully transfigured version of what is a reality otherwise too ugly to be embraced unadorned.
Many scholars have examined the numerous, fascinating connections between Schopenhauer and Nietzsche on points of art, ethics, and metaphysics. Many, too, have done so with the aim of locating both figures in their shared intellectual and historical milieu. Paul Downes’s Concentric Space as a Life Principle Beyond Schopenhauer, Nietzsche and Ricœur: Inclusion of the Other, does both of these things, with an eye to philosophical ambitions of its own that make the work remarkably original. Downes is not interested only in telling us what these three key post-Kantian European figures think, but, more vitally, in getting us to identify and think about the important subtleties they themselves have left unthought, or at least unsaid. For Downes, who is interested in our relation to the other in its full ethical and metaphysical complexity and richness, the correct point of departure lies, with Schopenhauer, in seeking “a basic orientation of openness” breaking with egoism (43). The task for thinking, here in turn, demands a form of inquiry that he calls a “spatial phenomenology”: an account of experiential space in all its variegation, including the peculiar spatiality of thought itself. Through a series of close and constructive readings of Schopenhauer and Nietzsche and Ricœur too (among others like Kant, Lévi-Strauss, and Heidegger to name a few), Downes undertakes a comprehensive analysis of the relation between oneself and the other, a spatial alterity ethics, as it were. As he puts it, “There is a spatial system of relations, a primordial spatial discourse pertaining to life that is embedded in the seminal works of Schopenhauer, Nietzsche and Ricœur —and yet overlooked by each of them” (1).
Before turning to Downes’s account of “concentric” and “diametric” spaces, it is valuable to linger over the problem of empathy, or compassion. Nietzsche’s philosophy of will to power, for instance, by rendering the picture of self-consciousness and human flourishing that it does, rejects the deeply humane perspective above for which Schopenhauer is famous. Compassion for the Übermensch is out the window, with cruelty assuming the pride of place. It is somewhat surprising that Nietzsche’s thought, which does begin by acknowledging the role of suffering in existence, nonetheless abandons compassion. How does it end in cruelty? As Downes says, to start with, Nietzsche’s view of tragedy correctly recognizes the deep pain of existence, what the latter himself terms “the eternal wound of life” (55). Dionysianism seeks relief from this primordial suffering in a countermovement of ecstasy, or rapture. And if the world of individuated consciousness is one giving rise to ineradicable suffering, this consciousness itself must be dissolved if suffering is to be vanquished. What results, Downes writes, is “a purported expansion as annihilation,” an “obliteration of boundaries” whose quest consists in the desire “for no boundaries” (54). A Nietzschean pursuit of rapture leads to the monism of the Dionysian, “a collective ritual, the fusion” of one’s diffusion of oneself “with the crowd” (55). In the end, however, this entire economy of desire only spells trouble. Its “intoxication as obliteration of self, as annihilation of boundaries,” says Downes, “whether wine-fueled or through other intoxicatory substances such as hallucinatory drugs,” risks a collapse of self. An assessment of this economy invites the question of what mediating pathways explain the passage from this collapse of identity in the early Nietzsche’s Dionysian, on the one hand, to the subsequent cruelty of the Dionysian in the later Nietzsche, on the other hand, a cruelty including eventually even the lust to inflict it. The desire to expand (or at least this mode of expansion: the dilatio explicated by Jean-Louis Chrétien, for one, is not the same), to be powerful, to ever grow beyond one’s current limits, eventually in Nietzsche’s thinking transforms into the satanic desire to inflict suffering on others, to dominate and control them. As Downes notes, following Károly Kerényi, whoever travels this road ends up psychologically mimicking the cruelty of the sacrificial Dionysian rite in which the victim “was first boiled and then roasted” (95).
According to Nietzsche himself, “cruelty constituted the great joy and delight of ancient man” (109). In turn, as Downes puts it, “joy becomes a celebration of cruelty” (Ibid.). Now as anyone familiar with what Nietzsche says can attest, nothing about such cruelty is said by him to be ethically censurable. After all, the fact that, ordinarily, cruelty is something condemned while compassion is praised is only so, maintains Nietzsche, because of the “slave morality” from which the moral categories of good and evil themselves originated, a moral and psychological economy that his account of will to power presumes itself to have dismantled. But Downes detects a problem waiting to surface. For even if it is possible to overlook the fact that Nietzsche’s characterization of “blissful annihilation” is perhaps just an unwitting variation of the very same pessimistic impulse behind Schopenhauer’s “ascetic negation of the will” that he wishes to reject (43), it still remains the case that, for Nietzsche, “hell is not so much other people as connection with other people” (100). Ultimately, this urge for domineering isolation, which is mobilized in pursuit of self-gratification, undermines itself, for it never finds the satisfaction it is seeking. The Nietzschean impulse to flee from connection (100), to dash the bonds of our ordinary ethical relations and the Apollonian mode of self-consciousness underpinning those relations, culminates in a lust for power that replicates, in the form of a straightforward inversion, the two original terms of the ethical relation it sought to overcome. Where before one was encouraged to have compassion for another’s suffering, now one instead takes delight in inflicting it. At times, Nietzsche appears to attempt to defend this mode of existence by emphasizing that this is simply the way of the world, that “all happening in the organic world consists of overpowering and dominating” (89). As Downes explains, the will to power, which was said to originate as a life principle, instead becomes a will to death, one that accordingly “finds infliction of pain and suffering ‘magical’” (90), so much that the “actual infliction of suffering is elevated into being an end of itself” (Ibid.). The “fundamental monism” recounted in the Birth of Tragedy’s account of an “unmediated life will in music,” a surpassing of the ordinary boundaries of the Apollonian principium individuationis (107), ultimately elevates the psychology of “Dionysian sacrifices” (94) to a “wider cosmological principle” (93), thereby justifying psychological hate and destruction. One here might naturally call to mind Freud, whom Downes does: “Freud’s account of ritual” as an “obsessional neurosis” rooted in the “compulsion to repeat” (97), and which gives expression to a death drive, seems a plausible explanation for the psychology of the one who comes to be dominated by his compulsive desire to dominate others. What began as the attempted liberation from the cultural construction of good and evil leads to little more than an insatiable, self-destructive sadism. “Why,” for example, Nietzsche asks, “is knowledge … linked to pleasure? First and foremost, because by it we gain awareness of our power … any new knowledge … makes us feel superior to everyone” (90). Perhaps Nietzsche really just is more perceptive and honest than the rest of us, when he claims to identify the desire to know as one whose end consists in the satisfying pleasure of feeling superior to others who don’t know what we do. Some people, I suppose, may well desire to know for that reason. But everyone always?
This stated diagnosis of what motivates our desire to know raises a deeper question of whether knowledge itself is even possible at all, a skeptical worry that for its own part leads to further questions concerning the nature of the experienced world and the nature of reality as such. For Schopenhauer, for example, who basically grants the Kantian division between the world in itself and the world of appearance, there is a sense in which any human discursive knowledge is illusory. From this radically Kantian perspective, the spatiotemporal world of individuated entities is itself a deformation of the real—whatever it may be. In a move that will anticipate the work’s subsequent treatment of Ricœur, Downes observes that, in the wake of “the linguistic turn” associated with structuralism, realism goes by the wayside. The structuralist commitment to the “primacy of language” leaves us with conceptual schemes, and that is all. As an heir to Kantianism, such an approach denies that we have access to things as they are in themselves. Here in response to the linguistic idealist, Downes like Claude Romano holds that there is a domain of meaning more fundamental than that which is shaped and structured by language or concepts. It is, he says, “a language of space that is itself prior to language—a spatial protolanguage or discourse” (7). Or more precisely still, “Space is a precondition for language; language is not a precondition for space. Space is itself a system, a system of meaningful relations through the contrasts between diametric and concentric spaces” (156). This space is experientially prior to anything language is able to mediate or structure. To begin bringing into focus what Downes wishes us to see, to see what he means by this spatiality, it is important to recognize that the relevant notion of space is not Cartesian. Descartes’s notion of empty space, a geometrically extended field for scientific abstraction, is not what is at issue (6). Nor is the conception of space at issue Kantian either, for it is not “a transcendental condition as a necessity for thought” (7). Initially, Downes explains what such space is by characterizing both concentric and diametric space in terms of what they are not:
“[Concentric and diametric spaces] are not to be reduced simply to a flawed appeal to the ‘natural’ […] Life is not being treated as a substance, it not ousia as presence but as a relation, a relational space as a directional movement and tension” (9).
“Concentric space is not being postulated as an ancient primordial experience to be re-enacted. Concentric space is not a nostalgia for the premodern or for some period of history lost in the mists of time; it is a current, ongoing experiential possibility [concentric and diametric spaces] are not mere categories, static forms, collections of sterile classifications, schemes or cognitions corresponding to particular ideas, life goals or world views. Rather they are proposed as being prior framing conditions for understanding as projections of primordial experience structured through these spaces—and as such these spaces are not monoperspectival exhortations, but rather conditions for a vast plurality of perspectives to grow and thrive” (11).
“Primordial here is not being invoked in terms of some ancient prehistory. The spatial dimension is proposed as ontologically prior and primordial as a more fundamental truth or experience; a direction of unity for experience; a truth and experience prior to socially constructed realities; a cross-cultural truth; being beyond the limited schema of causal explanations” (27).
Alternatively, sometimes he offers a description of these spaces in positive terms:
“Moving space is the breath of thought” (6).
“Spatial breath is the pulse of thought, not a denial of spatial breath in the inert space of monism. Space is indelibly interactive and immanent in thought” (Ibid.).
“Diametric spatial structure is one where a circle is split in half by a line that is its diameter or where a square or rectangle is similarly dived into two equal halves” (17).
“In a concentric spatial structure, one circle is inscribed in another larger circle; in pure form, the circles share a common central point” (Ibid.).
“A concentric spatial relation is a structure of inclusion compared to diametric spatial structure of exclusion” (20).
“These spaces are being examined as fundamentally directions of movement rather than to be treated as simply static structures” (21).
“They are precognitive framing spaces and are prior to metaphor” (24).
Returning to Schopenhauer’s and Nietzsche’s handling of compassion and cruelty shows how these issues are to be understood in terms of Downes’s spatial phenomenology. From this perspective, Nietzsche is “locked into a diametric spatial understanding” (12). In at least two senses. To begin with, diametric space is oppositional, sometimes exclusionary to the point that one of its binaries cuts off all contact with its opposite term. Taken in an ethical register, to say that Nietzsche’s Dionysian account of self-understanding is underwritten by diametric space is just to say, whatever else it also means, that the self excludes the other. The self who is locked in diametric space lacks compassion and empathy for the other: precisely what Nietzsche himself extols when he valorizes cruelty. Furthermore, there is the second sense to this diametric spatial understanding, one which concerns the thought itself responsible for attempting to conceptualize the ethical relation. Here again, Nietzsche’s thinking is itself diametric: in struggling to subvert the ordinary terms of good and evil, he ends up re-instantiating binaries, either to prioritize one of the original terms over the other, or else to introduce a new dyad in substitution for the original pairing. Thus, according to Downes, “the more [Nietzsche] seeks to break away from the diametric spaces underpinning this, the more he is locked within them in different forms” (86).
Hence, “Inclusion of the Other” requires a “concentric spatial relation” (2). And here again, in both of the two senses established above. Because the “exclusion process in the us/them projection rests on a diametric binary opposition” (2), it will be necessary to overcome this binary in a way that allows the self and other to exist harmoniously, rather than as adversaries. Schopenhauer’s own position can be explained in terms of the spatial terms that frame it: just as compassion “internalises the other as an extension of the self” (33), so here a concentric spatial relation “is not an obliteration of self but a moistening of boundaries between self and other as a governing precondition for compassion” (37). Such a concentric relation depends on an assumed connection rather than an assumed separation (32). Nietzsche’s will to power, as Nietzsche himself says, is a process of expansion, which is seen to be spatial: “Its object thereby is the incorporation of new ‘experiences’ … growth; or more properly, the feeling of growth, the feeling of increased power—is its object” (84). This process, however, as we have seen, is haunted by its inherently egoistic topos, what in turn erects a “thick partition” (37) between oneself and the other. It could be argued that Nietzsche’s account of the Dionysian possesses sufficient conceptual resources to overcome this “diametric oppositional spatial split between self and other, where the individual internalizes the other with stark divisive boundaries of a diametric dualistic spatial relation” (41-42). After all, is not the point about such Dionysian self-consciousness that it entails a sort of orgiastic, monistic fusion, whereby boundaries between oneself and others are dissolved? Downes acknowledges that the Nietzschean position could be initially construed as embodying a form of concentric space, insofar as it articulates a connective notion of “monistic fusion as unity” (43). But, as Downes notes, there is a difference between “destruction and dissolving generally” (37). Nietzsche’s account proposes a dissolution of ordinary self-consciousness so extreme, that, despite overcoming a form of the ordinary oppositional split characteristic of diametric space, it nevertheless fails as an ethical solution to the problem of egoism. It lapses into the annihilation of self. This form of monistic connective space does not make room for the inclusion of the other, then, because there remains no individuated self capable of exercising the recognition necessary to welcome that other. This, it should be noted, is one of the main troubles with crowd psychology and mob mentality, which absorb the individual into a mindless monism. If I am to identify successfully with your pain and so empathize with you, I must so exist as an individual. Nietzsche’s Dionysianism, which refers to “drunkenness and mystical self-abandonment, Dionysian festivities,” and which bring “an effusive transgression of the sexual order,” and with that the “annihilation of the usual limits and borders of existence” (133-34), may escape the diametric separation of Apollonian self-consciousness, but it ushers in an ethical void.
In fact, the Apollonian and Dionysian, Downes writes, “is a response to experience of an existential void” (74), what Nietzsche himself identifies in On the Genealogy of Morals as the suffering incurred “‘from the problem of [finding one’s] own meaning’” (74). On the subject of the void—or the “nothing,” one might say—it is Heidegger whom Downes discusses most extensively. But it is a different aspect of Downes’s engagement with Heidegger that I would like to highlight instead. In the spirit of Being and Time’s existential analytic, might not one claim, with Heidegger, that the entire problem of the relation between self and other has heretofore been misconstrued? While the spatial phenomenology on offer does well to have highlighted how experiential space is not Cartesian (or even Kantian), has it not, so the argument continues, failed to eliminate perhaps another classic residue of Cartesianism, the vision of intersubjectivity that envisions self and the other as essentially disconnected? Put differently, is to characterize compassion as an achievement, as Schopenhauer does, to overlook the deeper ontological bond between oneself and another? As Heidegger famously claims when explaining why traditional skepticism about the “external world” and other minds is misplaced, Dasein is Mitsein, a being for whom its mode of being always already includes others being with and alongside oneself. The other, thus, “is in some way a dimension of the primordial structures of self” (174). This Heideggerian line of objection will resurface down the page, when turning to Downes’s treatment of Ricœur’s use of metaphor.
As for the failure of Nietzsche’s account of the relation with the other to reach any satisfying ethical solution, this is so, as Downes says, in part due to the fact that, trapped in one’s egoism, the individual who rejects compassion in the name of will to power thereby confines himself to the “dungeon of diametric space [operating] as a sealed compartment” (87). Understandably, we want to say this is bad, and for many reasons. But does a spatial phenomenology allow us to reject such egoism on genuinely ethical grounds? In considering this question, I want to mention a potential tension that emerges in Downes’s view. When, for instance, it is written in the book’s introduction that a spatial phenomenology “is paving the way for a question of wellbeing for all people” (11), the thought naturally arises: precisely what view of man is at issue here? How are we to understand the notion of wellbeing, what is our measure of good in doing so? On the one hand, the work’s language appears to endorse a full-blown realism about human nature when remarking that space is “a truth and experience prior to socially constructed realities; a cross-cultural truth” (27). On the other hand, sometimes the language suggests something less universal but pluralistic or even relativistic: “these spaces are not monoperspectival exhortations, but rather conditions for a vast plurality of perspectives to grow and thrive” (11). The tension between these statements is most evident, as just mentioned, I think, when focusing specifically on Nietzsche’s own metaethical critique of good and evil. Sometimes, Downes appears to endorse Nietzsche’s rejection of the claim that there is any absolute notion of good and evil worth preserving: “Nietzsche challenges the good-evil diametric opposition and is aware of the need to seek a more fundamental understanding” (108). A similar point seems to be made when criticizing oversimplistic binaries, including, evidently, that again of the one between good and evil: “Nietzsche perceives the limitations of the diametric oppositions, good-evil, pleasure-pain and seeks to overcome them for a purportedly more fundamental level of reality and experience” (111). And the apparent historicity of values—including morals—is stated explicitly two pages later: “Morals derive from emotions and experience, thus a shift in experiential habits can bring a transformation of values” (113). On one reading, it appears that Downes wants to reject Nietzsche’s diametric spatial account of the relation between self and other in order to retain the ethical importance of compassion, but I wonder whether compassion’s value can be truly affirmed without also endorsing the kind of metaethical realism from which Downes’s spatial phenomenology seems to decouple itself. When Nietzsche says, “Who is really evil according to the meaning of the morality of resentment? … just the good man of the other morality, just the aristocrat, the powerful one, the one who rules, but who is distorted by the venomous eye of resentfulness” (85), I think those of us who see the value of compassion are inclined to condemn Nietzsche’s exaltation of power and cruelty as not simply being mistaken in just any sense, but as wrong morally. I am not sure how an analysis of compassion in strictly spatial terms will accommodate that judgment. It is also worth mentioning in this same context that despite “the death of God” being a central lodestar for Schopenhauer, Nietzsche, and Ricœur, Downes does not explicitly say anything about the issue. Addressing this theological horizon of their thinking would perhaps be one way of also addressing the metaethical questions to which spatial phenomenology’s account of compassion and alterity ethics gives rise.
It is, I think, to this task that the the work’s third figure, Ricœur, is meant to answer. Just as with Ricœur himself who takes Husserl’s side against structuralism by denying the primacy of language, so too Downes holds that there are “structures of relation prior to metaphor” (146). Because concentric space operates “at a preconceptual, precognitive, prerepresentational framing level” (58), there is room for the establishment of an ethical relation prior to the interference of power-relations or any of the other potential complications attending language and tradition. A truly empathetic face-to-face encounter with the other is possible. Against the “monistic tendency” (63) of Nietzsche and the pessimism of Schopenhauer, Ricœur sets out a conception of alterity ethics that is capable of grounding a constructive dialogical exchange, a true sharing of views and perspectives in good faith, one that is in principle oriented by the ideals of truth, reason, and justice, rather than manipulation, power, and oppression. The trick, in short, says Downes, is to appreciate with Ricœur that a “concentric relation allows for distinction and difference that does not have to lead to opposition” (147). Here, Ricœureaen metaphor is to be understood as a spatial “precondition or prior system of relations to language interacting with language” (145), opening onto, as Husserl and Romano each say, “an autonomous system of meaning and relations” before language (146), and that accordingly invites us to “seek structures of being” (Ibid.). To return to Heidegger again (and the language of his phenomenological ontology), what is at stake in a work such as Basic Problems of Phenomenology, for example, is an “interrogation of the copula” (146), an inquiry entailing the recognition of a “domain of truth prior to [the] apophantic judgment of Aristotle” (163). Spatial phenomenology is an heir to that sort of effort.
In a theme which recurs throughout the work, Downes claims that whatever the analysis of experience happens to be on offer (whether from Descartes, Kant, Schopenhauer, Nietzsche, or Heidegger), it fails to prioritize space in favor of something else. The most obvious example of this tendency to downplay space would be in Heidegger’s own prioritization of temporality. One response to this subordination of space, it might be suggested, would be instead to adopt a hybrid approach: one that, for example, synthesizes the role of space and time in structuring experience and thought. In a word, why not see place—rather than just space or time—as the fundamental structure of experience? Downes acknowledges this position explicitly (Jeff Malpas’s work on place is discussed) but ultimately rejects the claim that place is prior to space—such an approach, he says, “[collapses] the subtlety of space into mere place” (162). I cannot adjudicate this important debate, nor the related question of how concentric and diametric spaces relate to one another, adequately here, so I must simply note it in passing.
To conclude, it should be said that some readers may be frustrated with Downes’s dense formulations that require multiple readings. What he means is not always immediately clear. But that, I don’t think, is because what Downes is saying is inherently muddled, but rather because we are today too often superficial, inattentive, and distracted readers, so we are less and less accustomed to authentically thinking as we’re reading. It can be disorienting and unsettling to encounter a text that expects serious effort from us. This, maybe, is I think part of what makes Downes’s book enjoyably challenging. In a time when the space of discourse is increasingly less a space of reasons, a work as this, sensitive and subtle and deeply humane, is a thoughtful refuge from the shrill and shallow, one that repays the attention we provide it.
 For one excellent statement to this effect, see Julian Young, Nietzsche’s Philosophy of Art (CUP, 1992).
 This question will not be foreign to those familiar with analytic phenomenology, for which the phenomenality of thought has been a topic of attention for some time. See Charles Siewert, The Significance of Consciousness (Princeton, 1998) and David Pitt, The Quality of Thought (OUP, in progress).
 Downes, quite accurately, summarizes the nonconformist psychology typifying the one who sincerely rejects the morality of “the herd,” thereby hovering above and beyond good and evil. Such an individual’s concern, says Downes, “is to challenge flattened notions of comfort and security underpinning some conceptions of wellbeing, while he quests for an experiential intensity connected with risk and danger of destruction of self” (110). As a sociological observation, it bears noting that recent events surrounding Covid-19 have revealed something of an irony, that many of those in our universities who self-avowedly claim to live as Nietzschean „free spirits“ have themselves instead embraced the logic of “safety and security,” and quickly succumbed to mass panic and hypochondria.
 It should be noted Downes anticipates that somebody might object to his phenomenological interpretation of the Dionysian in terms of concentric and diametric spaces by claiming that such an account is itself a quasi-Apollonian attempt to impose form on what Nietzsche treats as a formless expansive sea (76). His reply, which strikes me as correct, is that this is not an imposition of form, but rather the uncovering of “structural regularities in Nietzsche’s own habits of thought and experience” (Ibid.). The fact that the work identifies many other instances where Nietzsche’s thought exemplifies the same diametric opposition it notes in this context strengthens its position.
 As Downes observes, some of Nietzsche’s readers such as Walter Kaufmann have recognized the implicit binary structure of Nietzsche’s thought, noting in this context that monistic fusion and diametric closure are themselves inverted diametric images identified as power and impotence, respectively.
This volume is the latest edition in Palgrave Macmillan’s History of Analytic Philosophy series, and it deals exclusively with the philosophical thought of Anton Marty, a student of Franz Brentano at Würzburg and Hermann Lotze at Göttingen. The reason for such a volume is that Marty is often overlooked and underestimated. In both the analytic and phenomenological traditions, Brentano, Alexius Meinong and Edmund Husserl receive most of the attention and Marty is often seen as merely a defender of Brentano – not a philosopher in his own right. This book seeks to disrupt these preconceived notions about Marty, in a way that clearly demonstrates the promise of his ideas for contemporary research (for both the analytic and phenomenological traditions and beyond) while breathing “new life into his thought”. (vii) For example, pieces by François Recanati and Mark Textor highlight Marty’s original contributions while engaging in fresh critical discussion of his work alongside that of Paul Grice and Brentano. Kevin Mulligan does something similar with Ludwig Wittgenstein. Other authors, like Ingvar Johansson, showcase Marty’s contributions (for example, with space) that have been excluded from the history of philosophy. This volume feels less like a simple overview of a forgotten thinker and more like a critical introduction that simultaneously launches the reader into fruitful dialogue with both contemporary and longstanding issues in analytic philosophy. This book is organized into three parts: Issues pertaining to philosophy of language; philosophy of space and time; and the metaphilosophical aspects of existence and being in his thought.
In the first part, focusing on philosophy of language, Textor’s chapter stands out as particularly well executed, and which would appeal to a broader audience than just the analytic tradition. What is said here will be of great value to scholars in the phenomenological tradition who study the early work of Edmund Husserl or the Munich Circle students who studied with him before the outbreak of WWI. Issues surrounding the nature of language and signification, statements expressing wishing, commanding and questioning, and especially judgment are central to the works of Johannes Daubert and Adolf Reinach, who both read Marty, and then later students such as Roman Ingarden. Textor identifies Marty’s theory of language as ‘intentionalist semantics’ – Marty defined the word language as synonymous with intentional indication of the inner life of the person – and this metaphysical view of meaning comes with two commitments: first, mental facts concerning desire and belief are the most fundamental to what signs mean; and second, the speaker means something if and only if she does it with the purpose of producing an attitude for or in an audience. (34) This is where we see Marty and Grice roughly align. Textor focuses his essay on this second commitment – communicative intention –, but while he does so, he explores an alternative view of meaning put forth by Brentano. That is the idea that some utterances have meaning independently of whether they were made with the purpose of influencing others; therefore, with regard to the primary source of meaning, the utterance meaning takes priority over the speaker’s intended meaning for it. (35) Textor engages with Brentano’s position to remedy problems that both Marty and Grice fall prey to, specifically occurring with non-communicative utterances. Textor, however, isn’t painting Brentano as the answer to all of our problems, but rather delves into the shortcomings his view faces and then demonstrates how it can be rescued and developed to achieve greater insight about speaker meaning. He takes Brentano’s work on the meaning of utterances expressed in judgments and extends it, to create a model that will connect judgment and non-natural meaning, looking to the mechanism of belief acquisition. For example, if we believe a speaker to be trustworthy, we are more likely to make a rational judgment based on the information they share. Textor ends with: “There are further details to be filled in to complete Brentano’s picture, but I hope that I gave the reader some reasons to take Brentano’s proposal to be the basis for an alternative to Grice’s and Marty’s that is worth completing further.”(64) Textor primarily uses, as source material, Brentano’s logic lectures (EL 80), taken from the Würzburg course of the winter semester 1869/70 entitled Deduktive und Induktive Logik. Brentano lectured for many years on logic, while at Würzburg and later Vienna, and it is great to see these lectures being highlighted and utilized. Here, we see their value communicated, and Textor provides his own (excellent) translations – this is more than simply a passing mention of Brentano’s academic teaching history.
This piece by Textor is a real gem, because the reader gets a thorough journey into theories of language that were happening just prior to the activities of Gottlob Frege and Edmund Husserl. For the latter, in particular, it was setting the stage for what he would write in the Logical Investigations (1900-1901). For an early phenomenology scholar like myself, this chapter is great for the discussion of Brentano logic lectures and the Marty writings that rarely receive any attention and yet have such a central role to play in the ideas of the early movement. Also, it is wonderful to read Brentano’s logical insights about language, and see them given serious consideration alongside someone like Grice, and in fact used to help Grice, as this work often takes a backseat to his intentionality thesis contained in the Psychology From an Empirical Standpoint and his later reism.
In the second part, the chapters on the consciousness of space and time are some of my favorite. This section was my reason for wanting a copy of this book, if I am honest. Once again, these chapters will appeal and prove very helpful to those in both the analytic and phenomenological traditions who wish to understand the discussions of the consciousness of time and space that informed major figures in the 20th century, and for me this means Husserl. This topic is yet another that Husserl lectured on early in the 20th century, and this theme continued to be a popular one with both the Munich and Göttingen Circles, for example in the works of Max Scheler, Moritz Geiger, Hedwig Conrad-Martius, and Roman Ingarden. It was also one that Henri Bergson wrote about, and it informed the position expressed during the famous debate with Einstein in Paris. In this second part of the volume the first essay by Johansson, “A Presentation and Defense of Anton Marty’s Conception of Space” goes beyond a defense of Marty. Johansson clearly demonstrates how Marty’s ideas on the topic are relevant and important not just to history of philosophy but also to the field of physics. As he points out at the beginning, there are two kinds of space: the perceptual and the physical. He focuses on the physical expression of space, bringing together ideas from Marty with elements of Immanuel Kant, Graham Nerlich, and himself to defend a container conception of space and space-time and to show why contemporary physics should give it serious consideration. Marty’s theory holds that space has a mind-independent existence, where all bodies, properties, events, spatial points and relations are all contained within this ontologically preexisting space. (100 – 102) He also leaves open the possibility that space could be empty, which goes against not only Kant but also Brentano. This position is opposed to the relational theory of space, which was held by Leibniz. While Marty’s theory most likely falls under the modern label “substantivalism” (i.e., the theory that space exists in itself in addition to the material objects within it), it doesn’t fit squarely: while it can conform to the general definition of substantivalism, Marty’s conception of space is ontologically more basic or, rather, primary to what is contained within it, and this makes him distinct from other “substantivalists”, like Barry Dainton. By the way, there is a great discussion of Dainton here, too. This chapter offers a wonderful historical run down, along with comparison of Marty’s conception, and in such an accessible way. If you are rusty on the topic or new to it, this chapter is a great primer and will also leave you with some points to think about.
The next essay by Clare Mac Cumhaill “Raum and ‘Room’: Comments on Anton Marty on Space Perception” is the perfect follow up to Johansson. Cumhaill’s piece elaborates and extends what Johansson discussed, in particular on perception, and then in the comparisons of Marty to others who write on space and time, and again in a very approachable and engaging way. The essay contains an informative outline of Marty’s conception of the ontology of space, a section on Marty’s critiques of Kant and Brentano on the topic of space and time, and an inquiry into whether any contemporary theory of perception can handle Marty’s notion of space and time. The most promising for Cumhaill is Naïve Realism, but this comes with its own difficulties. A highlight for me was the section comparing Husserl and Marty; it was full of insights. I actually wanted more Husserl and comparison talk of him, because of what I stated earlier, but what is there is great (in particular on 137, the sections of the letters Marty wrote to Husserl are a fun read).
Thomas Sattig closes out this part of the volume with a bang, with his chapter: “Experiencing Change: Extensionalism, Retentionalism, and Marty’s Hybrid Account.” Sattig builds on the previous two chapters to discussing contemporary ideas concerning our experience of change: after some helpful encapsulations of extentionalism and retentionalism, there is a wonderful summary of Marty’s account, and at the close there are some challenges raised against Marty’s view. Marty’s position is called a “hybrid account” because, as pointed out in section three, the notion of how we experience change combines elements from both the extensionalist and retentionalist views, and in a presentist framework (i.e., only the present is actual, the past and future are not). (163) This chapter, like the others, is well organized, accessible and has an engaging style; it even has some lovely diagrams with leaves to help illustrate (great diagrams are necessary for discussions of time). The challenges to Marty’s view are excellent, and the suggested fixes for the holes or omissions in Marty’s theory offered are thorough, but Sattig also leaves room for the reader to think and form their own insights about these shortcomings.
While I only discussed chapters from the first two sections of this book, this should not in anyway convey to anyone reading this review that the third section is subpar or weak – it isn’t. The reader will get more fantastic pieces that really turn the spotlight on Marty’s work, which is much needed and deserved.
I really enjoyed what this volume had to offer and it reminded me of why I found Marty invaluable and fascinating during my graduate and postgraduate work. He’s an amazing talent and brilliant scholar in his own right, not simply a defender of Brentano and fellow priest who left the cloth with convictions about the infallibility of the pope. I really appreciated how this book was organized, and enjoyed how the chapters in each section relate but thoughtfully expand in various directions. The discussion of Marty is always balanced; the presentation of Marty feels very well rounded, and the contributors are always willing to talk about the errors as much as the successes. Furthermore, the fact that much of this book contains his lesser-known works is fantastic and asset to any collection or library. This volume also offers some great excursions into the history of philosophy, and this not only provides the context for Marty’s ideas but also what made him such a great philosopher.
If I have anything critical to say, besides wanting more Husserl, it is that some might come to the idea that Marty is an analytic philosopher or more of a forefather to the analytic tradition than to phenomenology or any other discipline. This can be gathered by the title of the book series and then the index of authors cited in the chapters. The introduction to this volume tries to convey that this is not what is being argued; it attempts to show that Marty’s work had significant influence on the analytic tradition, more influence than we currently feel he had, given that so much of his work is overlooked. But once you get into chapters, it is easy to forget what was said in the introduction and jump to conclusions, because sometimes the feel or approach is itself very analytic. However, I will say, it would be shortsighted to jump to such conclusions and/or to not to read this book. This volume offers a wonderful picture of Marty that is insightful, thought provoking, and inspirational. As I said many times (proportionally to how many times I noticed this in my reading), it is also an approachable and engaging to read. As a scholar of Husserl and Reinach, I see a lot of potential ties to my own work. Marty is one of many forefathers that both the analytic and phenomenological traditions share, and we should celebrate this man and his mind rather than divide ourselves into camps. Hey, we both share great taste in Austrians of the 19th century! Brentano and his students were immensely productive, interdisciplinary and incredibly brilliant; they changed the 20th century dialogue for philosophy – period. That being said, I highly recommend this book for both scholars of analytic philosophy and phenomenology, as well as those interested in the topics discussed between its covers.
Die Rolle des Raumes, der bislang in Heideggers Denken neben jener der Zeit bzw. der Zeitlichkeit kaum wahrgenommen wurde, ist in den letzten Jahren immer häufiger in den Fokus der Forschung gerückt worden. Es wird dabei betont, dass vor allem die kleinen Schriften über die Kunst, die im Laufe der 1960er Jahre anlässlich von Heideggers Begegnung mit einigen zeitgenössischen Künstlern entstanden sind, von einem starken Interesse Heideggers am Phänomen des Raumes zeugen. Denn diesen Texten lässt sich eine Raumauffassung entnehmen, die im Vergleich zur Räumlichkeit des Daseins in Sein und Zeit oder auch zur Konzeption des Raumes als Wohnen in den 1940er und 1950er Jahren neue Verhältnisse zwischen Raum und Zeit, Raum und Dasein, Raum und Körper und nicht zuletzt zwischen Raum und Welt entstehen lässt. In diesem Forschungskontext, der der Spur des späten Heidegger auf der Suche nach seiner revidierten Raumauffassung folgt, verortet sich auch Andrew J. Mitchells Heidegger unter Bildhauern. Körper, Raum und die Kunst des Wohnens. Wie der Titel bereits verrät, stellt Mitchell Heideggers Konzeption des Raumes in seinem Verhältnis zum Körper und zur Kunst – insbesondere zur plastischen Kunst – dar. Zu diesem Zweck untersucht und interpretiert er in Anlehnung an Heideggers Denken die Werke der Bildhauer Ernst Barlach, Bernhard Heiliger und Eduardo Chillida, denen er jeweils ein Kapitel widmet.
Der erste Satz des Buches fasst implizit seinen Ausgangspunkt und sein Ziel zusammen: „Die Bildhauerei lehrt uns, was es heißt, in der Welt zu sein.“ (9) Eine fragwürdige, sehr allgemeine und sogar tendenziöse Annahme – könnte man denken. Auch die Erklärung, die der Autor kurz darauf vorschlägt – „In dieser Welt zu sein heißt stets, einen materiellen Raum von Strahlung zu betreten.“ (9) –, bleibt erklärungsbedürftig. Wenn man aber die Ungenauigkeit dieser Annahme vorläufig akzeptiert und sich von ihr durch den Text leiten lässt, wird im Laufe der Lektüre verständlich, dass dieser vermeintlich unverständliche Ansatz das Programm des gesamten Werkes Mitchells zum Ausdruck bringt. Denn dem Schlüsselbegriff ‚Grenze‘ folgend, will der Autor in seinem Buch zeigen, dass Heidegger durch eine Auseinandersetzung mit der Bildhauerei eine Raumkonzeption entwickelt, auf Basis derer der Unterschied zwischen Raum und Kunst aufgehoben wird. Mitchell zeigt darüber hinaus, dass, indem Raum zur Kunst und Kunst zum Raum wird, Heidegger ein neues Verständnis des Verhältnisses des Daseins zu seinem Wohnend-Sein bzw. zu seinem In-der-Welt-Sein entwirft.
Um die Entwicklung und zugleich die Ergebnisse der Heideggerschen Auseinandersetzung mit dem Raum-Begriff von den 1920er bis zu den 1960er Jahren darstellen zu können, gliedert Mitchell sein Werk in fünf chronologisch aufeinanderfolgende Teile. Auf eine lange Einleitung, die von Sein und Zeit (1927) über die Kunstwerksabhandlung (1935) bis zu den späten 1960er Jahren durch die bedeutendsten Etappen das Verhältnis von Dasein, Kunst und Raum im Denken Heideggers rekonstruiert, folgen drei aufeinander aufbauende Kapiteln, die die Zusammenhänge zwischen dem Denken Heideggers und der Kunst Ernst Barlachs (1.Kapitel), Bernhard Heiligers (2. Kapitel) und Eduardo Chillidas (4. Kapitel) untersuchen. Das dritte Kapitel hingegen ist einen Exkurs über Heideggers Vortrag Die Herkunft der Kunst und die Bestimmung des Denkens. Eine Darstellung dieser Abschnitte wird im Folgenden jene Aspekte fokussieren, die Mitchel zufolge für die Entwicklung des Denkens Heideggers in Bezug auf das Verhältnis von Raum, Kunst und Mensch eine besonders wichtige Rolle spielen.
Statt den Leser in das Thema des Buches einzuführen oder einen systematischen bzw. historischen Hintergrund zur Orientierung zu umreißen, versetzt die Einleitung ihn sofort ins Zentrum der Betrachtung. Durch eine Sprache, die deutlich eine starke Beeinflussung durch Heideggers Stil erkennen lässt, gewinnt der Leser einen unmittelbaren Zugang zur Thematik des Werkes: das neue Verhältnis von Körper und Raum, das sich deutlich in den Vorträgen und kleineren Schriften Heideggers der 1960er Jahre zeigt. Schon die ersten Seiten des Werkes entwerfen eine innovative Interpretation der Entwicklung der Raumauffassung im Denken Heideggers. Denn Mitchell stellt keinen Bruch zwischen der Raumauffassung von Sein und Zeit und jener der späten 1960er Jahre fest. Er vertritt vielmehr eine Kontinuitätsthese: Die in den 1960er Jahren von Heidegger entwickelte Auffassung des Raumes und seines Verhältnisses zum Körper „schreitet“ laut Mitchell „auf einem Denkweg durch Sein und Zeit zur Abhandlung über ‚den Ursprung des Kunstwerks‘“. (10) Damit bestreitet Mitchell jedoch nicht, dass sich die Raumkonzeption Heideggers im Laufe seines Denkens deutlich verändert hat. Er plädiert aber für die These, dass Heideggers Werke der 1920er und 1930er Jahre den Kern seiner späteren Raumauffassung bereits in sich tragen. Eben diese kontinuierliche Entwicklung des Heideggerschen Raumverständnisses wird von Mitchell in der Einleitung auf kurze und prägnante Weise dargestellt. Er zeigt zuerst, dass die Auffassung des Raumes in Sein und Zeit Grenzen aufweist, die seiner Analyse zufolge dadurch entstehen, dass Heidegger die Räumlichkeit des Daseins „vom daseinsmäßigen Nutzen des Zeugs (des ‚Zuhandenen‘) her“ (13) denkt. (Vgl. 11–17) Aufgrund dessen bleibe der Raum in Sein und Zeit ausschließlich ein funktionaler Raum, dessen Existenz vom handelnden Menschen abhängig ist. (Vgl. 17) In einem zweiten Schritt zeigt Mitchell, wie Heidegger die Auffassung eines funktionalen Raumes überwindet und im Kunstwerksaufsatz eine Konzeption entwickelt, die auf einem vom Dasein unabhängigen Raum basiert. (Vgl. 17-24) Diese neue Idee eines autonomen, „anti-utilitaristischen“ (21) Raumes wird Mitchell zufolge im Kunstwerksaufsatz im Schlüsselbegriff ‚Erde‘ expliziert: „Erde nennt eine exzessive und abgründige Phänomenalität, eine Erscheinung, die auf keiner unterliegenden Substanz beruht.“ (19) Auf dieser veränderten Auffassung des Raumes, die nun von Heidegger als Erscheinung bzw. als Lichtung der Wahrheit (vgl. 21) verstanden wird, basieren Mitchell zufolge die Veränderungen in Bezug auf das Verhältnis von Körper und Raum, die sich in Heideggers Denken in den 1960er Jahren anlässlich seiner Auseinandersetzung mit den Plastiken verschiedener Künstler äußern.
Vor dem Hintergrund der dargestellten Entwicklung untersucht Mitchell im ersten Kapitel seines Buches (vgl. 31-48) den Zusammenhang zwischen dem Spätdenken Heideggers und der Kunst Ernst Barlachs. Der Begriff der Seinsverlassenheit bildet dem Autor zufolge das Bindeglied zwischen Heideggers Denken und Barlachs Kunstwerken. In diesem Zusammenhang deutet Mitchell Verlassenheit als „Weg, Sein als weder völlig präsent (es hat Seiendes verlassen) noch als völlig absent zu verstehen“ (33) und somit das Seiende als „etwas Offenes, das in die Welt ausgeschüttet ist“, (34) zu erfahren. Die stark metaphorischen, fast poetischen Züge der Sprache Mitchells beeinträchtigen bisweilen ein systematisches, eindeutiges Verständnis des Textes. Dennoch lässt sich Mitchells Interpretation der Werke Barlachs in Bezug auf Heideggers Denken erkennen: Indem die formlosen Körper-Skulpturen Barlachs ein Seiendes ohne bestimmte Grenze bzw. ein offenes, nicht abgeschlossenes Objekt verkörpern, stellen sie laut Mitchell die Spannung zwischen Präsenz und Absenz dar, die der Seinsverlassenheit eigen ist, und werden somit als Ausdruck der „Unbestimmtheit des irdischen Lebens“ (43) gedeutet. Außerdem betont Mitchell, dass eine implizite Kritik an der Welt der Technik und am Formideal des Nationalsozialismus als deren Konsequenz vorgenommen wird: „Barlachs Skulpturen sind mehr geformt als jeder Nazi-Körper es sein könnte, gerade durch ihre Weigerung, Form zu verdinglichen oder zu kristallisieren und sie von ihren sie ermöglichenden Bedingungen abzuziehen.“ (47)
Dieses Verhältnis von Raum und Körper, das die formlosen, offenen Skulpturen Barlachs bereits implizit thematisieren, wird zum Hauptthema in Heideggers Rede Bemerkungen zu Kunst-Plastik-Raum, die er 1964 anlässlich seiner Auseinandersetzung mit den Kunstwerken Bernhard Heiligers gehalten hat. Auf Basis dieses Textes zeigt Mitchell im zweiten Kapitel seines Buches (vgl. 49–72), dass Heidegger das Verhältnis von Kunst und Raum eindringlich untersucht, dass er grundlegende Fragen über die Möglichkeit einer Auseinandersetzung mit dem Raum für den Künstler aufwirft und dass dabei auch das Verhältnis von Körper und Raum zunehmend an Bedeutung gewinnt. Bei dem Versuch, dieses Geflecht von Verhältnissen, Bezügen, Verweisen und Zusammenhängen zwischen Kunst, Raum und Körper zu entwirren, entwirft Heidegger laut Mitchell eine neue Auffassung des Raumes, die dazu zwingt, auch seinen Bezug zur Kunst und zum Dasein neu zu denken. Gegen die klassische Raumauffassung, die die Definition des Raumes mit den Körpern verbindet, zeigt Mitchell, dass Heidegger den Raum vom Raum und nicht vom Körper her denkt. Auf dieser Weise definiert Heidegger den Raum als Räumen. Dies ermöglicht, „Raum nicht länger abstrakt und homogen, sondern selbst schon sich versammelnd und furchend und ausstreckend und zurückschnappend in Gebiete, Fernen, Richtungen und Schranken“ (58) zu denken. Diese neue Raumauffassung fordert, dass auch das Verhältnis von Dasein und räumendem Raum vom Raum her gedacht wird – und nicht mehr wie in Sein und Zeit vom Dasein her. Aus dieser Perspektive neu gedacht, lässt sich Mitchell zufolge das Verhältnis von Dasein und Raum als ein sich gegenseitiges Durchdringen und Prägen verdeutlichen. (Vgl. 60) Entsprechend heißt In-der-Welt-Sein, dass das Dasein durch die Welt geprägt ist und dass sich die Welt konsequenterweise, wenn auch verdeckt, in jedem Dasein zeigt. Eben dieses unsichtbare Verhältnis des Menschen zur Welt und zugleich die unsichtbare Präsenz der Welt in jedem Menschen werden laut Mitchell von Heidegger in Heiligers Kopf-Werken zum Ausdruck gebracht: „Wenn der Künstler einen Kopf modelliert, so scheint er nur die sichtbaren Oberflächen nachzubilden; in Wahrheit bildet er das eigentlich Unsichtbare, nämlich die Weise, wie dieser Kopf in die Welt blickt, wie er im Offenen des Raumes sich aufhält, darin von Menschen und Dingen angegangen wird.“ (61) In diesem Verhältnis von Welt und Mensch kommt den Begriffen des Zwischen, der Bewegung und der Relationalität in der Argumentation Mitchells besondere Relevanz zu. (Vgl. 63–67) In Anlehnung an den kurzen Dankesbrief, den Heidegger nach einem Besuch des Heiligers Ateliers schrieb, (vgl. 63) und auf Basis einiger Bemerkungen Heiligers, der selbst seine Skulpturen als Kunstwerke in Bewegung bzw. als etwas Offenes, in dem Offenheit waltet und Welt erscheint (vgl. 63), beschreibt, deutet Mitchell die Welt als Zusammengehörigkeit von Menschen und Dingen bzw. als ein geheimnisvolles Dazwischen. (Vgl. 65–66) Dadurch will Mitchell an den Werken Heiligers zeigen, welche Deutung von Welt und Mensch sich aus der Heideggerschen Auffassung des Raumes als Räumen ergibt. Der Versuch Mitchells, diese Idee der Welt als Zwischen und ihre Bedeutung für den Menschen zu verdeutlichen, wird jedoch durch seine literarische Sprache, die das Verständnis erschwert, ausgedrückt: Mitchell schreitet an dieser Stelle seiner Betrachtung durch intuitive Verbindungen zwischen den Sätzen, er bedient sich metaphorischer Bilder, die schnell aufeinanderfolgen und die intuitiv aufeinander verweisen. Der Diskurs scheint existenziell poetische Gedanke hervorrufen und das Terrain des philosophischen Argumentierens bzw. der Kunstkritik verlassen zu wollen. Diese existenzielle Richtung verstärkt sich im nachfolgenden Paragraph ‚Artikulation 2: Verfall und Erosion‘. (Vgl. 67–72) Mitchell betont, dass die Kunstwerke Heiligers, die die Relationalität zwischen Mensch und Welt ausdrücken, „die Tatsache [attestieren], dass Bewegung ein Abnutzen ist“. (67) In diesem Sinne expliziert der Autor weiter, dass „ein Werden hin zu etwas […] ein Werden weg von etwas“ (67) ist. Eben dieses Thema der ‚Distanzierung von etwas‘ wird von Mitchell in seiner Deutung der Werke Heiligers betont, weil er darin den Ausdruck einer grundlegenden Weise des In-der-Welt-Seins sieht. Ausgehend von dieser Deutung der Werke Heiligers bringt Mitchell einen anderen Wesenszug des Verhältnisses von Mensch und Welt zum Ausdruck. Denn die Welt wird nun nicht als etwas verstanden, das den Menschen prägt, sondern als etwas, das uns verbraucht bzw. „erodiert“: (68) Insofern Mensch und Welt sich gegenseitig durchdringen und prägen und sich daher in einer ständigen Bewegung bzw. einem ständigem Werden befinden, das nicht nur ein Werden zu etwas, sondern auch ein ‚Weg von etwas‘ ist, verbraucht die Welt den Menschen. Mit den folgenden Worten drückt Mitchell diesen Gedanken in all seiner Radikalität aus: „Wir sind durch Welt verwittert, erodiert im Zwischen. Unsere Absprache besteht darin, gemeinsam zu erodieren.“ (68) Indem die Skulptur den Menschen in diesem Zwischen hält – so Mitchell weiter – und Verbindung zwischen Mensch und Welt stiftet und daher Mensch und Welt verändert, erweist sich die Skulptur für diesen Erosionsprozess des Menschen als mitverantwortlich. (Vgl. 71)
Bevor Mitchell auf das Verhältnis des Heideggerschen Denken und der Kunst Eduardo Chillidas eingeht – ein Verhältnis, das dem Autor zufolge eine weitere Entwicklung des Verhältnisses von Raum, Körper und Kunst im Denken Heideggers darstellt –, setzt sich Mitchell in einem kurzen Exkurs mit Heideggers Die Herkunft der Kunst und die Bestimmung des Denkens auseinander. (Vgl. 73–81) Mit der Interpretation Mitchells, die ausgehend vom Blick Athenas auf die Steingrenzen (vgl. 77) darauf zielt, die Zusammengehörigkeit von τέχνη und ϕύσις im Denken Heideggers zu begründen, ist die Heidegger-Forschung längst vertraut. „Der Ruf der ϕύσις ist“, schreibt Mitchell, „für die menschlichen Werke also eine Einladung die Welt zu prägen, doch zugleich auch sich selbst von der Welt prägen zu lassen.“ (80) Besonders interessant und originell ist dagegen der Gedanke, dass das Bas-Relief in einer ausgezeichneten Weise diese Zusammengehörigkeit von ϕύσις und τέχνη bzw. von Natürlichem und Künstlichem zum Ausdruck bringt. (Vgl. 80) Diesbezüglich weist Mitchell darauf hin, dass es vielleicht kein Zufall ist, dass die drei Bildhauer, mit denen Heidegger sich auseinandergesetzt hat, im Relief arbeiten. (Vgl. 80)
Im vierten Kapitel seines Werkes stellt Mitchell den letzten Schritt und daher das endgültige Ergebnis der Auseinandersetzung Heideggers mit dem Raum und dem Körper dar, das Heidegger laut Mitchell 1968 anlässlich der Begegnung mit den Kunstwerken Chillidas entwickelt hat. (Vgl. 83–109) Der grundlegende Gedanke dieses Schritts und der Wandel im Verhältnis zur vorherigen Raumkonzeption Heideggers besteht Mitchell zufolge darin, dass, indem Heidegger eine physikalische bzw. metaphysische Auffassung von Raum explizit ablehnt, jeder Unterschied zwischen Kunst und Raum aufgehoben wird. Wenn daher die Werke Barlachs und Heiligers noch von einer Trennung von Raum und Kunst zeugen, die auf unterschiedliche Art und Weise überbrückt wird, konstatiert Heidegger anlässlich der Begegnung mit den Werken Chillidas, dass eine solche Trennung und konsequenterweise eine Überbrückung der Lücke zwischen Kunst und Raum überhaupt nicht denkbar ist. (Vgl. 84–86) Denn Kunst ist keine „Besitzergreifung des Raumes“ (84), sondern sie ist schon immer ein räumender Raum, ein Ort gewordenen Räumens. Diese radikal neue Konzeption des Raumes und seines Verhältnisses zur Kunst bewirkt – so Mitchell – Veränderungen in der Auffassung des Verhältnisses von Raum, Werkzeug und Kunstwerk, von Raum und Menschen, von Raum und Sprache und von Raum und Körper. In Bezug auf das Werkzeug behauptet Mitchell, dass die Funktion des Werkzeugs als Medium zwischen Künstler und Materie in Frage gestellt wird. (Vgl. 91) Denn es gibt keine Leere mehr zwischen den beiden, die durch Werkzeuge gefüllt bzw. überbrückt werden muss. Mitchell verdeutlicht des Weiteren, inwiefern sich auch der Bezug des Daseins zum Raum ändert: Das Dasein verliert sein Privileg als Handelnder, der Räume bildet, stiftet, eröffnet oder ermöglicht. Vielmehr wird das Dasein vom Räumen des Raumes gedacht und ist daher schon dem All des Seienden zugehörig. (Vgl. 100-104) Inwiefern sich auch das Wesen der Sprache in Bezug auf diese neue Raumkonzeption verändert, wird von Mitchell nicht ausführlich erklärt. Er stellt in Heideggers Versuch, den Raum etymologisch zu erhellen, lediglich eine „Betonung der Sprache“ (105) fest. Diesbezüglich sagt er sogar: „‚Kunst und Raum‘ bringt uns dazu, eine Zwiefalt zu denken: dass Raum sprachlich und Sprache räumlich sei.“ (105) Leider erklärt Mitchell nicht, wie genau diese von ihm behauptete Zusammengehörigkeit oder sogar Identität von Raum und Sprache zu verstehen ist. Erklärungsbedürftig bleibt bedauerlicherweise auch die Verbindung, die Mitchell in den letzten Sätzen dieses Abschnittes zwischen Körper, Raum und Wahrheit herstellt. (Vgl. 108–109) Außerdem ist auf eine Irritation zu verweisen, mit der sich der Leser bei der Lektüre dieses Kapitels konfrontiert sieht. Im dritten Teil dieses Kapitels mit der Überschrift ‚Setzen Bringen Zusammenarbeiten‘ (94–99) setzt sich Mitchell mit dem Unterschied zwischen dem ‚sich-ins-Werk-Setzen‘ und dem ‚ins-Werk-Bringen‘ der Wahrheit in der Kunst auseinander. Der Autor macht darauf aufmerksam, dass – wie Heidegger selbst im ‚Zusatz‘ zu Der Ursprung des Kunstwerks bemerkt – in der Entwicklung des Heideggerschen Denkens ein Wandel vom Setzen zum Bringen stattfindet. (Vgl. 94) Dieser Wandel wird jedoch von Mitchell darin identifiziert, dass ‚Setzen‘ ein Moment von Gewalt mit sich bringe, während ‚Bringen‘ etwas Weicheres darstellt, indem es eine Begleitung und nicht eine Gewalt betone. (Vgl. 97) Aus diesem Grund erklärt der Autor: „Die Wahrheit des Werkes erscheint daher in ‚Kunst und Raum‘ weniger insistent als in ‚Der Ursprung des Kunstwerkes‘.“ (97) Dabei übersieht Mitchell aber den bedeutendsten Unterschied zwischen den beiden Ausdrücken, der darin besteht, dass der erste (sich-ins-Werk-Setzen) reflexiv ist und der zweite (ins-Werk-Bringen) eben nicht. Und dies bewirkt eine grundlegende Veränderung des Verhältnisses von Wahrheit und Kunst und konsequenterweise auch eine Veränderung der Rolle des Künstlers. Denn während die Wahrheit im Kunstwerksaufsatz als die ‚sich-Setzende‘ aktiv im Kunstwerk erscheint bzw. geschieht, gewinnt der Künstler in den späteren Auffassung Heideggers eine viel stärkere Rolle, indem er die Wahrheit ins Werk bringt.
Das abschließende Kapitel fasst die Ergebnisse der vorhergehenden Kapitel zusammen und zeichnet dadurch den Weg, auf welchem Heidegger ausgehend von der Begegnung mit den formlosen Körpern Barlachs über jene mit den Köpfen Heiligers bis zu der Auseinandersetzung mit den Vögeln Chillidas seine Raumauffassung in den 1960er Jahren entworfen hat. Vor dem Hintergrund dieser neuen Raumkonzeption versucht Mitchell auf den letzten zwei Seiten, den Menschen in den Mittelpunkt der Betrachtung zu stellen und sein Verhältnis zu sich selbst, zu den anderen, zu seinem In-der-Welt-Sein und zur Wahrheit neu zu konturieren. Leider zeichnet sich auch dieser Abschnitt durch eine sehr kryptische Sprachverwendung aus. Aufgrund dessen bleibt es schwer nachvollziehbar, inwiefern Mitchell das aus der neuen Raumsauffassung entstandene Verhältnis von Mensch, Plastik und Raum als eine Aufforderung für den Menschen, sein Leben zu ändern, versteht. (Vgl. 114)
Abgesehen von diesen Unklarheiten der Darstellung trägt das Buch zweifellos zur Klärung der in der Heidegger-Forschung tendenziell vernachlässigten Thematik des Raumes bei und ergänzt diese um interessante Überlegungen und Denkanstößen. Denn Mitchell unternimmt in seinem Buch den gewagten Versuch, auf Basis sehr kurzer und zuweilen unsystematischer Texte des späten Heidegger eine systematische Raumkonzeption darzustellen. Es gelingt Mitchell jedoch nicht immer, die Schwierigkeiten zu umgehen, die ein solches Vorhaben unvermeidlich mit sich bringt. An einigen Stellen erweckt der Text den Eindruck, als ob der Autor, indem er in Anlehnung an die Texte Heideggers und mithilfe seiner Begrifflichkeit die Werke der drei Bildhauer deutet, ihnen Inhalte, Bedeutungen oder Verweise zuspricht, die diesen Kunstwerken andernfalls nicht zukommen. Eine andere Schwierigkeit, auf die bereits hingewiesen wurde, ist die Sprachverwendung. Oft wird eine sehr poetische Sprache verwendet: Einige Zusammenhänge und Verweise werden intuitiv aufgebaut und daher bleiben einige Gedanke erklärungsbedürftig. Auf Grund dessen entsteht der Eindruck, als habe sich der Autor nicht immer bemüht, seine Überlegungen zu erklären, und es stattdessen vorgezogen, á la Heidegger mit der Etymologie der Worte zu spielen und seinen Diskurs durch intuitive Verbindungen aufzubauen. Dies macht einige Textpassagen auch für den Heidegger-Kenner sehr schwer verständlich. Ob und inwiefern die Übersetzung Trawnys zu diesen Schwierigkeiten beiträgt, bleibt unklar. Außerdem lassen sich einige Ungenauigkeiten in der Auslegung der Texte Heideggers feststellen.
Trotz dieser kritischen Anmerkungen ist der Versuch Mitchells lesenswert. Denn der Leser erhält durch das Werk nicht nur einen Überblick über die kontinuierliche Entwicklung des Denken Heideggers über den Raum von Sein und Zeit bis zu den späten 1960er Jahren, sondern dem Leser werden darüber hinaus zahlreiche interessante Deutungsperspektiven des Heideggerschen Denkens angeboten, die sich als originell erweisen und über die Betrachtung Mitchells hinaus für eine Auseinandersetzung mit den Themen Raum, Dasein, Welt und selbstverständlich auch Kunst im Rahmen des Spätdenkens Heideggers fruchtbar gemacht werden können.
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.
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 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.