The Ernst Cassirer renaissance is in full order. Since Massimo Ferrari’s anticipation and prediction that the German philosopher would be lifted from the realms of semi-forgottenness in 1994 different lines of reception have swept through the German-, Italian- and English-speaking world. (cf. Ferrari, 1994) It was only a matter of time until this resurgence would carry over to Anglo-American departments, where, along with a renewed interest in Neo-Kantianism, more and more research on Cassirer is being conducted. The newly translated and edited edition of his three volume magnus opum The Philosophy of Symbolic Forms by Routledge is a case in point here. Accordingly, the present work by Samantha Matherne, assistant professor of philosophy at Harvard University, is perhaps only the logical conclusion to a new wave of Cassirer reception in the English-speaking world, appearing in the renowned The Routledge Philosophers series edited by Brian Leitner. Primarily aimed at undergraduate students, the book will surely complement many syllabi on the German philosopher in the English-speaking academy for years to come, especially as the hitherto existing English introduction to Cassirer, John Michael Krois’ Symbolic Forms and History, is by now 34 years old.
In the contemporary reception Cassirer’s philosophy is explicitly advertised as being able to bridge “gaps not only between the so-called ‘analytic-continental divide’ in philosophy, but also between philosophy and other disciplines in the humanities, social sciences, and natural sciences” (p.2) Indeed, apart from purely historical considerations the primary aim of contemporary research on Cassirer seems to be the development of a transcendental philosophy of culture as the investigation of the conditions of possibility that enable cultural artifacts and their world by means of an analysis of the different modalities of symbolization. (cf. Endres et al., 12f.; Luft 2021, 215) Following the influential studies of Peter Gordon (Gordon, 2010) and Michael Friedmann (Friedmann, 2000) the peculiar position of Cassirer in 20th century (German) philosophy is recognized and contextualized and with it a philosophy that seemingly does not outright reject modernity’s proliferation of cultural and life-forms in either a rural conservative individual flight to authenticity (Heidegger) or a detached logic-semantical analysis of scientific propositions (Carnap). Hans Sluga, a reviewer of Gordon’s book, however, expressed his doubts about deriving a reconciliation of culture via Cassirer:
Cassirer was no doubt an accomplished philosopher, an influential teacher, and above all a thoroughly decent and admirable human being, but he does not get close in stature to the much more problematic Heidegger, and he certainly also lacks the philosophical radicalism of a Wittgenstein, Foucault, or Derrida and the incisive scientific acumen of a Russell, Quine, or Rawls. Attempts to revive his fortunes are, I am afraid, doomed to failure. (Sluga, 2011)
However, the contemporary reception of Cassirer wagers that the German philosopher has still a lot to offer for present-day problematics. (cf. Gordon 2021, xiv; cf. Luft/ Ferrari 2021, passim)
How the background of this reception and its repercussions along with the different ‘geophilosophical’ context vis-à-vis existing German introductions (Sandkühler/ Pätzold, 2003; Graeser, 1994; Recki, 2004, 2013; Paetzold, 2014) have shaped the task of presenting a summary and overview of Cassirer’s philosophy will form the frame of this review. The author’s aim to “offer an overview of Cassirer’s philosophical system as a whole that can help the reader navigate his corpus” will determine its immanent threshold of success. (p.2) I will provide a summary of its contents before engaging in a more critical reading.
After setting out from a brief biography of Cassirer, the book unfolds via a historical contextualization of Cassirer within the broader movement of Marburg Neo-Kantianism as the general frame of reference and conceptualization Cassirer worked and philosophized within. “For all the shifts and developments in Cassirer’s body of work, his philosophical system remains, throughout, that of a Neo-Kantian.” (p.18) It is transcendental spontaneity that for Matherne is the central motif of Cassirer’s effort for a philosophy of culture and in connection with the methodological impetus of accounting for the conditions of possibilities of cultural facts the decisive trait of his intellectual lineage. Hence this, after setting the general picture of Marburg Neo-Kantianism as being primarily scientifically oriented right, amounts to a transcendental investigation of the conditions of possibility of meaning-creating/ – making in a shared world. In this sense, (Marburg) Neo-Kantianism tout court had always already been on the way to a philosophy of culture, though it is Cassirer’s merit to conduct this investigation in a way that would do justice to the concept of culture. (cf. p. 31f.)
In practice, this configures the subject’s capability to confer meaning- and form – making processes freely and spontaneously upon the world. Matherne decisively accounts for this by contextualizing Cassirer’s indebtedness to Cohen’s and Natorp’s intellectualist interpretation of Kant’s theory of cognition – the actual conceptuality of what had been forms of intuition, space and time, in Kant. (cf. p. 39ff.) In other words, all forms of cognition and perception remain relative to the transcendental subject’s employment of a range of categories. On this view, Cassirer’s central philosophical innovation consists in invoking the ‘softer’ notions of form and symbol/ function against ‘law’ – the former two permitting a greater range of phenomena attributable to the ‘world of meaning’. (cf. p. 37.)
Accounting for this in more detail, Matherne sets out to retrace the younger Cassirer’s work on epistemology and a theory of concept-formation, largely neglecting the first published monograph Leibniz’ System in its Scientific Foundations and the first volume of Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit. Rather, Cassirer’s first central philosophical innovation is said to have first and foremost occurred within the theory of concepts and the adjacent philosophy of mathematics to form conceptual and scientific basic distinctions, which, insisting on the continuity of Cassirer’s thought, remain invariant up to The Philosophy of Symbolic Forms and beyond. In this way, Cassirer’s elaboration of the distinction between substance-concept and function-concept in the eponymous book are employed to account for the respective processes of objectification (Ver-gegenständlichung) yielding the symbolic forms and their ranges of perception and cognition. This amounts essentially to the primacy of the category of relation over substance from Kant’s transcendental logic to prevent a notion of concepts as being mere copies of pre-existing objects attained by way of abstraction. (cf. Truwant 2015, 291) A spontaneously conceived function – later to be extended as symbolic form – posits a law of succession and orders a series of representations according to it. (cf. 53ff.)
The remaining chapter presents Cassirer’s consequent views in the philosophy of arithmetic and geometry. Matherne summarizes the attained position under the heading of ‘logical structuralism’, “according to which mathematics has its basis in functions of relations that belong to logic and mathematical objects are ideal structures generated on the basis of those functions or relations.” (p. 75) Although introducing Cassirer’s first philosophical innovation in this way diminishes the methodological role of the Neo-Kantian’s historiography of philosophy as a history of problems (Problemgeschichte) in relation to the historization of the a priori and its relevance for the establishment of the function-concept, the presented difference between the two respective views is presented clearly and convincingly.
The historical character of functions comes back in Cassirer’s ‘philosophy of natural science’, which is the topic of the ensuing chapter. In dialogue with the natural scientist, it is the transcendental philosopher’s task to account for the conditions of possibility of the facts of science by means of a reconstruction of the corresponding transcendental functions, which remain relative to the overall scientific context of experience (cf. p. 81 In the context of natural science this task amounts to the elaboration of the fundamental concepts employed by the natural scientist and the positions the yielded concepts occupy within their empirical theories. Hence the elaboration of a taxonomy of the scientific statements of measurement, laws and principles as instantiations of a different order of generality. In turn, the philosopher should, according to Cassirer, make out the invariant relations on a purely conceptual level. (cf. p. 98.) In the last instance, these figure as the transcendental categories, that is, the functions continuously employed in all scientific endeavors such as time, space, or number. Although these may be configured differently over history they serve as the functional a priori building blocks of any scientific theory.
Subsequently, the discussion moves on to the philosophy of symbolic forms proper, that is, not just the elaboration of the eponymous trilogy, but also the dispersed articles and texts written between 1920 and 1945. Matherne chooses to frame the philosophy of symbolic form as a philosophy of culture throughout, and, although not outright neglecting its later transformation into a philosophical anthropology, takes her “cue from his early formulations of it in The Philosophy Symbolic Forms and other texts from the 1920s”. (p. 116) While it is conceded that Cassirer’s thought evolved in newer directions at a later stage of his career, the conception of a ‘philosophy of culture’ is by definition a narrower one than that of a philosophical anthropology. And although the reason for this concession is provided for in the continuity of the central status of symbolization as seen in the dictum of the human as animal symbolicum, questions why this should not compel one to conceive of his philosophy of culture as a philosophical anthropology are unanswered. (cf. 116f.) It is perhaps by way of the general relevance of Cassirer for a contemporary philosophy of culture that this conception is motivated. Rather than going the whole way of conceiving of the philosophy of symbolic forms as a philosophical anthropology the more modest task of investigating meaning-making processes fairs equally well with the ascribed position of the German philosopher with regard to the analytic-continental split. Thus, the task of the philosophy of symbolic forms “is ultimately organized around an effort to elucidate the conditions of culture.” (p. 119)
Matherne follows the common distinction between the different forms of culture along the subjective and objective lines. The former is comprised of the different modalities of representation as the triad of expressive, presentative and significative functions, the latter as the continuous progression of objective spirit, that is, culture’s overall context of signification as an “a priori intersubjectively shared structure and activity, which unites human beings […] together.” (p. 120) The different symbolic forms encompass respective “perceptive, intuitive and cognitive” structures and in this way the philosophy of symbolic forms aims to tie an analysis of the transcendental functions of the subject with its objective cultural expressions together (p.125) In contrast to the discussion of the cognitions of mathematics and natural science, the investigation shifts to the broader notion of the various kinds of ‘understanding’ in the human cultural sciences. (p.121) Cassirer posits their specific modality of concept-formation as being aligned with the general model detected in the natural sciences, foreclosing an anticipated discussion of their status as form- or style-concepts. (Form- oder Stilbegriffe)
Matherne then goes on to discuss the methodological requirements to conduct an analysis of the conditions of possibility of culture. The transcendental method is once again evoked, this time in Natorp’s “bi-directional conception”. (p. 124) The correlation of objective and subjective spirit is bifurcated along a reconstructive axis for the subjective side of the equation and constructive axis for its objective side. The latter posits a specific analysis of culture (‘constructive’) and the former accounts for the conditions of possibility of it by reconstructing a corresponding synthesis of transcendental subjectivity. (cf. Freyberg/ Niklas 2019, passim) It would perhaps have been worthwhile to extend and contrast the presented account with the manuscript for a Metaphysics of Symbolic Forms and its thoughts on ‘Basis Phenomena’ for a more rounded account. Matherne’s presentation gives the impression that Cassirer’s seems to privilege the reconstructive side over their correlativity or ‘work-relatedness’ (Werkbezogenheit), though the account remains thus firmly faithful to her overall interpretation of Cassirer. Subsequently the details of symbolization by means of categorial function-concepts, such as cause, time, thing or property, to yield the respective symbolic forms are discussed. (cf. p. 129) These figure as “the concepts that remain constant across all our spiritual activities because they are the a priori conditions that make all spiritual ‘forming’ possible in the first place.” ( p. 129) Matherne takes up Cassirer’s distinction between a category’s quality – its basic logical impetus of ordering series – and its modality, the particular ‘content’ “indexed to ‘regions of culture’ a[s] context” that orders representations. (p.130) With regard to spirit, Cassirer draws attention to ongoing discussions concerning the autonomy of the respective symbolic forms vis-à-vis the others (‘irreducibility thesis’) and whether their consecutiveness is to be conceived of teleologically as progress (‘teleology thesis’), although the latter question is answered affirmatively.
After the determination of the general functional context, Cassirer moves on towards the elaboration of the individual symbolic forms. The triad of expressive, presentative and significative symbolization as different functional modalities of representation provide the guidelines for this elaboration, relating the individual to respective realizations of her own freedom as spontaneity. Accordingly, religion and myth are relegated toward the expressive, language, history, and technology toward the presentative and mathematics, the natural sciences, morality and right toward the significative function of consciousness. (cf. p. 152) Philosophy entertains neither a position of a totalized god-like view of their overall cohesion nor does it count as one symbolic form among the others but figures as a toll to reflect on the symbolic forms. The specific functions and ‘worldviews’ of both myth and religion are presented in clear and minute detail before going over to art as the ‘objective’ demonstration of ‘subjective’ presentation – thereby “revealing to us that we are not passive with respect to our affects and emotions.” (p. 166) Objectification is reflected from the objective side of the dichotomy by the symbolic form of language, which, while still remaining bound to intuition and a substance-based view of categories, fosters the recognition of self-consciousness by the liberated understanding of reality it enables. It is interesting that Matherne specifically mentions that language and technology foster both practical and theoretical recognition of freedom and one wonders to what extent that can be said of the other symbolic forms. While this realization would be imaginable for myth, religion and the latter distinction between specific recognitions of this contention in morality and natural science, respectively, is left unaccounted for.
Both history and technology remain tied to the presentative functions of consciousness and spirit, the former by revealing reality’s distinctively human texture by means of the objective presentation of the past, the latter as the realization of the will’s striving for power toward the free configuration of the world. (cf. p.175f; p.178f.) Lastly, it is, on the side of theoretical reason, mathematics and natural science, that exemplify the significative functions of spirit. Following Cassirer’s views on the philosophy of mathematics, it is the fact that these symbolic forms are devoid of any relation to intuition or perception as to the yielded concepts and ‘things’ that elevates them towards the highest ranks of culture as most grasped realizations of transcendental freedom. It is precisely because these forms remain purely self-referential as expressions of freedom that “spirit truly discovers itself”. (Cassirer in Matherne, p. 184.)
The elaboration of the theoretical accomplishments of subjectivity is followed by their practical counterpart and the question over their position within the overall cohesion of the philosophy of symbolic forms. Recounting Cassirer’s refutation of emotive cognitivism in Axel Högerstrom Matherne insists on the employment of the transcendental method in the realms of morality and right. “ Cassirer endorses a critical approach [to practical philosophy] in which he analyzes morality and right in terms of ‘functions’ that serve as conditions of the possibility of the ‘facts’ of the ‘world of willing and action’. (p. 193, my amendments, N.S.) This deployment of the transcendental method is thereby connected to the demand of a regulative principle, the categorical imperative its claim to a universal, objectifiable moral principle. “Thus, a universal principle is one that enables us to most closely approximate the idea of ‘unity of willing’” thereby conferring objectivity on the ethical progress of consciousness via Sittlichkeit.(p. 194) Right, on the other hand, functions as a symbolic form in the overall context of Cassirer’s philosophy as self-binding to juridical lawfulness. Cassirer’s ‘philosophy of right’ posits a version of natural right that fosters the practical recognition of freedom by means of the postulation of and adherence to collective autonomy via laws. (cf. p. 214) Lastly, the teleological underpinnings of Cassirer’s progressivist understanding of theoretical and practical consciousness are posited as contingent. This is demonstrated in Cassirer’s analysis of National-Socialism in his The Myth of the State. Fascism re-introduced myth in modern consciousness via the symbolic form of technology and the ideas of hero worship, race and the dominance of the state. It is these late analyses that prompted Cassirer to also revise his conception of philosophy late in his career. Against the merely scholastic concept of philosophy, he brought forward its ‘cosmopolitan’ counterpart. Culture’s contingent accomplishments are not to be taken for granted but are to be achieved and upheld by means of struggle. To assign the task of this struggle had been the last innovation of Cassirerean philosophy.
The last chapter aims to reconstruct Cassirer’s influence on the development of not only philosophy but also (art) history, social science, ethnology, and Critical Theory. The presentation is focused on direct engagements with and influences of Cassirer on figures and movements. Accordingly, one learns about, for instance, the German philosopher’s influence on such diverse figures as Langer, Goodmann, Merleau-Ponty, Panofsky, Blumenberg, Habermas et.al. Cassirer’s possible inspiration to contemporary positions in the philosophy of science, such as logical structuralism and ontic scientific realism are addressed. (cf. p. 249f.)
It is puzzling, though, that, given the general narrative of Cassirer, an explicit contextualization of Cassirer within and relation to ‘philosophy of culture’ and its major movements and figures is lacking. This is even more relevant as, despite presenting the philosophy of symbolic forms as a philosophy of culture, Cassirer’s specific concept of culture remains unaccounted for. It appears that, following his Neo-Kantian heritage, the latter can only ever be the constructed empirical totality of culture at a given moment in history. Accordingly, one wonders whether the philosophy of symbolic forms is not prone to becoming ‘sociologized’: an investigation of the constituents and subsequent diversity of culture that would, by means of the quid iuris, be retied to an investigation of the correlative conception of subjective spirit. In the German context, this could be understood along the lines of Luhmann’s project of a ‘system theoretical’ approach to culture and society and its ‘autopoietic’, subjective sources.
Whether one concurs with Matherne’s way of framing Cassirer and his philosophy as being ‘organically’ culturally oriented or not, it is unquestionable that she is an informed and avid reader of the German philosopher. Via the transcendental method, Matherne is able to provide a coherent narrative of Cassirer’s philosophy. The book neatly ties the multi-faceted aspects of the oeuvre together in a rigorous and convincing manner and presents them in a remarkably cohesive way. Indeed, another title for it could have been: Cassirer: A Study on the Unity of his System. It is beyond doubt that the new reception of Cassirer has found a corresponding introduction to its subject.
Endres, Tobias/ Favuzzi, Pellegrino/ Klattenhoff, Timo. 2016. “Cassirer, globalized.” In Philosophie der Kultur- und Wissensformen. Ernst Cassirer neu lesen, edited by Endres, Tobias/ Favuzzi, Pellegrino/ Klattenhoff, Timo, Philosophie im Kontext von Gesellschaft und Wissenschaften, vol. 78, Frankfurt am Main: Peter Lang Edition, 9 – 22.
Ferrari, Massimo. 1994. “La ≫Cassirer-Renaissance≪ in Europa“, Studi Kantiani 7: 111–139.
Friedmann, Michael. 2000. A Parting of the Ways. Carnap, Cassirer, and Heidegger. Chicago/ La Salle, IL: Open Court.
Freyberg, Sascha, Niklas, Stefan. 2019. “Rekonstruktive Synthesis. Zur Methodik der Kulturphilosophie bei Ernst Cassirer und John Dewey.” In Ernst Cassirer in seinen systematischen Beziehungen. Zur kritisch-kommunikativen Bedeutung seiner Kulturphilosophie. Deutsche Zeitschrift für Philosophie Sonderbände Vol. 40, edited by Breyer, Thiemo and Niklas, Stefan, 47-68, Berlin/ Boston: De Gruyter.
Graser, Andreas. 1994. Ernst Cassirer. München: Beck.
Gordon, Peter. 2010. Continental Divide. Heidegger, Cassirer, Davos. Cambridge, MA/ London: Harvard University Press.
Gordon, Peter. 2021. “Foreword.” In Cassirer, Ernst. The Philosophy of Symbolic Forms. Volume 3: Phenomenology of Cognition. Trans. by Steve G. Lofts. viii-xv. Oxon/ New York: Routledge.
Luft, Sebastian. 2021. “Cassirer’s Place in Today’s Philosophical Landscape. ‘Synthetic Philosophy,’ Transcendental Idealism, Cultural Pluralism.” In Interpreting Cassirer. Critical Essays, edited by Simon Truwant. Cambridge/ New York: Cambridge University Press, 214-236.
Luft, Sebastian/ Ferrari Massimo. 2021. “Cassirer’s Children”, Special Topics Issue, Journal of Transcendental Philosophy 2(1):1-5.
Paetzold, Heinz. 2002. Ernst Cassirer zur Einführung. Hamburg: Junius.
Recki, Birgit. 2004. Kultur als Praxis: eine Einführung in Ernst Cassirer’s Philosophie der symbolischen Formen. Berlin: Akademie Verlag.
Recki, Birgit. 2013. Cassirer. Stuttgart: Reclam.
Sandkühler, Hans Jörg and Detlev Pätzold (Ed.). 2003. Kultur und Symbol. Ein Handbuch zur Philosophie Ernst Cassirer. Stuttgart/ Weimar: Verlag J.B. Metzler.
Schwemmer, Oswald. 1997. Ernst Cassirer. Ein Philosoph der europäischen Moderne. Berlin: Akademie Verlag.
Sluga, Hans. 2011. “Continental Divide: Heidegger, Cassirer, Davos.” Review of Continental Divide, by Peter Gordon. Accessed November 1, 2021. https://ndpr.nd.edu/reviews/continental-divide-heidegger-cassirer-davos/.
Truwant, Simon. 2015. “The Concept of ‘Function’ in Cassirer’s Historical, Systematic, and Ethical Writings.“ In The Philosophy of Ernst Cassirer: A Novel Assessment, edited by Friedman, J. Tyler and Luft, Sebastian, 289-312, Berlin: De Gruyter.
 See cf. Endres et al, “Cassirer, globalized”, in: Philosophie der Kultur – und Wissensformen. Ernst Cassirer neu lesen. Endres/ Favuzzi/ Klattenhoff (Eds.), pp. 9 -22. Frankfurt am Main: Peter Lang Edition, 2016, for an overview of recent research conducted on Cassirer.
 Where in text citations refer to page numbers only the addressed book is Matherne, Cassirer. Routledge, 2021.
 For cf. Schwemmer 1997, it is precisely the case that Cassirer’s philosophy of culture is always already a philosophical anthropology – “because that which defines the human being – spirit – consists in the configuration and usage of cultural symbolisms. (Ibid., p. 3145, my translation, N.S.)
Why read Jean Cavaillès’ work today? This is the foremost question that we need to address. The history of philosophy pullulates with untimely ideas, obscured innovators, and forgotten precursors: the fact that Cavaillès anticipated decisive developments within French philosophy cannot suffice to revisit his short but dense treatise. Historiographical significance may direct historians to his work, but we need positive, philosophical reasons to convince phenomenologists, philosophers of science, and epistemologists to work their way through a book of which each page demands to be read closely, carefully, and rigorously. To be able to demonstrate the value of Cavaillès’ philosophical pathway, one needs to show how our time calls again for his imagination.
If there ever has been a moment for this untimely treatise. Written while incarcerated by the German authorities in Montpellier and Limoges, Cavaillès did not live to experience its publication, he was executed for his leading role in the French resistance in early April 1944 – just shortly after completing the manuscript. When his friends Georges Canguilhem and Charles Ehresmann published Sur la logique et la théorie de la science in 1947, the moment when one could presuppose acquaintance with Neo-Kantian philosophy, Carnap’s logical syntax, Husserl’s phenomenology, and the debates concerning the foundations of mathematics, had already passed.
The intellectual situation in which the book was conceived had also disintegrated: Cavaillès’ doctoral advisor, the rationalist philosopher Léon Brunschvicg, had died in exile in Southern France; his friend, the philosopher of mathematics Albert Lautman, had been killed for his resistance work; the historian of science Hélène Metzger had become a victim of the Shoah; and the philosopher Gaston Bachelard by then was moving away from philosophy of science to aesthetics. The philosophical positions to which Cavaillès was indebted and the technical debates to which he was responding were reforming: the French epistemology of the 1920’s and 1930’s had lost its compelling force, the fierce disputes regarding the sovereignty of intuitionism, logicism, and formalism in mathematics were tempered, and Kantian, rationalist philosophy had become the target of powerful critiques.
In the 1950’s, the French intellectual scenery underwent profound changes: existentialism came to be fashionable, Marxism became the dominant political ideology, and phenomenology emerged as the dominant philosophical method for studying everything from nausea to wonder. Hegel, Husserl, and Heidegger were the names on the lips and the books of the desks of the agrégés.
Notwithstanding the rather unfortunate circumstances of the book’s publication, Cavaillès’ work still exercised a decisive but ephemeral influence on the young Derrida, the early work of Foucault, and the philosophy of science of Louis Althusser. More lasting was his sway over the work of Suzanne Bachelard, Gilles-Gaston Granger, and Jules Vuillemin, but these philosophers, whose labor was decisive in introducing analytic philosophy in France, never had the same impact on the intellectual scenery as the generation of May ‘68. At the time, there was a particular bias amongst French intellectuals concerning analytic philosophy that leaned too much towards logical positivism: the works associated with the Vienna Circle were considered one-dimensional, politically suspect, and to be lacking in terms of style. In the eyes of philosophers such as Althusser, Cavaillès had adequately portrayed the defects of logical positivism with his critique of Carnap. Consequently, the convergences between the works of the Vienna Circle and that of Cavaillès were overlooked in a climate hostile to scientific philosophy.
Under these conditions, the treatise has come to take up an uncanny place within the history of philosophy. Although Cavaillès’ work is canonized in the French epistemological tradition and now recognized as a critical influence upon the (post-)structuralist generation, the fact that he was ‘thoroughly immersed in the new logical culture’ is often deemphasized or simply ignored (27). The affiliations, conjunctions, and divergences between his work and that of Frege, Hilbert, Brouwer, Carnap, and Reichenbach remain underdeveloped and sometimes even unexplored. While it would be philosophically, but also historically, naive to plead for a restorative rereading of Cavaillès’ work within its autochthonic context, the relevancy of On the Logic and the Theory of Science for this other tradition of doing philosophy certainly deserves further scrutiny.
Indeed, it is precisely the uneasy position of Cavaillès’ work in the history of philosophy that makes rereading the treatise so compelling today. His critique of logical positivism, his elaboration of intuitionism, his conceptualization of Neo-Kantianism, and his divergent reflections on epistemological questions enclose a rich reserve for developing alternatives genealogies of twentieth-century philosophy. At a moment when French philosophers are finally beginning to reconsider Carnap’s contributions to philosophy, analytic epistemologists are exploring the resources of French philosophy for interventions in present-day discussions, and a new generation of French philosophers is reinstalling the fundamental relation between philosophy and mathematics: maybe, the time is finally there to begin to read Cavaillès as a contemporary. Now, more than ever, is his anomalous philosophical trajectory able to generate a conceptual space to assemble estranged thinkers and construct new philosophical itineraries.
To demonstrate this, let us discuss three examples that disclose how Cavaillès’ philosophical imagination may illuminate our current discourse. Together these examples show how his work can, once again, be utilized by philosophers, how his concepts can be put to work, once more, and why we should begin to reconsider and rethink the significance of On Logic and the Theory of Science. In other words, we will use these three examples to re-assess the purport of the treatise.
The first example relates to the role Cavaillès could play in the reconsiderations of Carnap’s work in French philosophy. In his book Carnap et la question transcendentale (2021), Jean-Baptiste Fournier engages extensively with a problem central to French philosophy: the status of the transcendental. Throughout the book, he attempts to reread Carnap’s early work as an exercise in transcendental philosophy – focusing mainly on Der Raum (1922) and Der Logische Aufbau der Welt (1928). In doing so, Fournier not only broadens the breadth of the discussion of Carnap’s relation to Husserl’s phenomenology, Helmholtz’ epistemology, and Neo-Kantianism, but also – implicitly – integrates Carnap’s philosophy in contemporary discussions of the transcendental in French philosophy.
According to Fournier, the early Carnap was fundamentally concerned with the question of ‘how logic could play a transcendental role’ within philosophy (18). He argues that in Carnap’s work the question of the possibility of objective knowledge is transcendental to the extent that it puts at stake the very conceptualization of the world. In what Fournier designates as Carnap’s transcendental analyses, the world has a double meaning: it is both ‘the horizon and correlate of consciousness’ (298). The world ‘thus functions as a Kantian transcendental object’ (298). The crucial difference with Kant’s conceptualization of the transcendental, however, is that in the work of the German philosopher, there is still the possibility of a complete transcendental deduction. In Carnap’s philosophy, the transcendental question ‘does not arise once and for all but must always be revived for each concept and each science’ (301).
Upon a first reading, it might appear that there is no room for reading such a concept of the transcendental in Cavaillès’ philosophy. His remarks on the transcendental are often dismissive (22-24, 120). Kant’s transcendental philosophy is said to obscure any attempt to account for the normativity of logic since it is ‘fundamentally dependent upon the notions of actions and faculty, which are meaningful only in reference to a concrete consciousness’ (20). In a similar vein, Cavaillès’ ambition to found an absolute logic of science leads him to reject the transcendental logic of Husserl. In his view, Husserl’s elaboration of phenomenology is parasitic upon the acts of consciousness and, therefore, lacks the autonomy needed for logic to ground itself.
While Cavaillès critiques the role given to transcendental subjectivity in Kant and Husserl’s philosophy, he does leave open the possibility of a transcendental analysis without a constitutive role of the subject. Sometimes, it seems that this is precisely what Cavaillès is concerned with: ‘[h]ere lies the role of transcendental analysis: to recognise authentic diversities and to establish the relations between them’ (49). Just like Carnap’s transcendental analyses, the analyses that Cavaillès conducts concern the structure of scientific theories and their conditions. But even though Cavaillès commences with a reflection on scientific theories, in his analyses, there is never any given that is presupposed or left unexamined. Each level of analysis is incorporated and related to a subsequent or antecedent one: in this way, his vision resembles that of Carnap’s Aufbau.
The question Cavaillès continually asks himself is also similar to that of Fournier’s Carnap: what are the conditions under which objective or scientific knowledge is possible? For Cavaillès, this question cannot be answered at one moment in time, it must continually be asked in relation to specific scientific discoveries, and it must always be compared with and integrated in previous analyses or theories. Within his philosophy, the transcendental, therefore, undergoes constant transformation: it is no longer possible to deduce all the conditions and ramifications of a single concept in one place and time. If there is a concept of the transcendental in Cavaillès’ philosophy, it is thus thoroughly dynamic – maybe even plastic.
The second example concerns Cavaillès’ anomalous position within the history of French philosophy and how it may be of use in connecting the work of antagonized thinkers. Catarina Dutilh Novaes, in a recent paper titled “Carnap Meets Foucault: conceptual engineering and genealogical investigations” (2020), discloses the similarities between Carnap and Foucault and argues for a reengagement with their work in current discussions on conceptual engineering. Although she tries to reconstruct the historical relations between these two thinkers, she does not discuss or even mention Cavaillès, when in fact, the philosopher is one of the missing links: Foucault probably first encountered Carnap’s work in On Logic and the Theory of Science. More importantly, his philosophy could have brought the concerns and considerations of these two philosophers together – since Cavaillès’ philosophy harbors a plane common to both: a more extensive conceptual space, where the dialectics of history and the complexities of (formal) epistemology come together, a central place to revisit the ramifications of what it means to tinker and ameliorate concepts.
In Cavaillès’ work, there are extensive genealogical analyses of concepts and a methodology of explication. Concepts are contemplated regarding their history and are clarified by rendering explicit the inferences that they imply. Furthermore, the French philosopher does not shy away from ameliorating concepts in order to enlarge their function. Yet throughout these efforts, there is never any reference to an essential constituent of a concept. As in Dutilh Novaes’ account of marriage as well the political aspects of Carnap’s concept of explication, pragmatics plays a crucial role, the function of a concept co-determines its value. At the same time, Cavaillès would never go so far to argue that the value of a concept lies in its use – a danger that seems apparent in Dutilh Novaes’ alliance of Foucault and Carnap. In his philosophy, truth remains the last measure when determining the function of a concept.
The last example bears on the philosophical project of two thinkers: Alain Badiou and Quentin Meillassoux. Both reclaim a Platonic theory of truth, bring renewed attention to the discourse of science in French philosophy, and place mathematics, once more, at the heart of philosophy. Taken together, their work reinvigorates a hybrid of rationalism and materialism in French philosophy. To further delineate and deepen their itinerary, one needs conceptual spaces to perceive the limits of and think through the problems particular to their philosophical project. For this, encounters with precursors such as Cavaillès, who have dealt extensively with questions concerning the nature of mathematics and the dialectics of history, are indispensable.
First, something about Badiou. Even though in his book Figures of Post-War French Philosophy (2009), Badiou discusses the relation between Cavaillès’ philosophy and his life as a resistance fighter, he has never engaged extensively with On the Logic and the Theory of Science. In the third chapter of the book, “Georges Canguilhem (1904—1995) Jean Cavaillès (1903-1944)”, Badiou gives us only a brief summary of Cavaillès’ position, which regardless of its fragmentary nature still reveals a concurrence between their projects: ‘the philosophy of mathematics must rid itself of all reference to a constituent mathematical subject, and should examine the internal necessity of mathematical notions’ (10). The absence of engagement with Cavaillès’ philosophy in Badiou’s work is rather surprising, given the importance that both ascribe to set-theory, their collective embracement of ruptures, and their communal passion for a revised Platonic concept of mathematics.
Needless to say, there are also critical differences between their conceptualization of these three points. While it is only in a complicated encounter between their thought that one could be able to explicate the ramifications of this divergence, a brief and somewhat cursory sketch may already reveal the contours of the paths taken and the advancements that can be obtained from such an analysis.
In their work, Badiou and Cavaillès utilize Cantor’s set-theoretical discoveries to conceptualize an ontology that is open, an ontology that resists totality. For them, set theory represents a break within thought, an event that ruptures the intelligible. Yet whereas both affirm the irreducibility of time within the generative movement of thought, Cavaillès is the only one who has thoroughly studied the fundamental historicity of the scientific event: this constituent of Cantor’s discovery is left entirely unexplored by Badiou. This singular difference characterizes their divergent notion of events: while Badiou’s concept of events leaves little room for describing the origination and construction of a scientific theory, Cavaillès aims to rationally reconstruct the very movement of science. As he puts it in the second part of On Logic and the Theory of Science: ‘by defining a structure of science that is nothing but science manifesting to itself what it is, we specify and justify the preceding characteristics’ (30).
In Badiou’s philosophy, there is no such rational reconstruction of scientific discoveries, there is no effort to define the determinate conditions of scientific progress, as there is no account of the labor involved in the progress in mathematics or science in general. His perspective on the history of science is shaped by the great moments of science, he views science through the eyes and work of aristocratic revolutionaries – not the communal work involved in thinking through theorems, experiments, and demonstrations. For Badiou, it seems that events take place within a vacuum; they are indeterminate, contingent, and original. In this regard, Badiou aligns himself with a century old tradition of French philosophy: the importance he attaches to contingency has been present in French thought ever since Émile Boutroux’s De la Contingence des Lois de la Nature (1874).
Cavaillès, with his commitment to studying the history of science and grounding the necessity of scientific progress, takes a radically different route. From the very beginning of his career, he is concerned with mapping the developments that lead to Cantor’s discoveries and examining the role of other scientists, who assisted in clarifying the ramifications of Cantor’s work. In this way, the event is contextualized, its constituents explained, and its conditions determined. For Cavaillès, this labor is critical. In his view, demonstration is at the heart of science, his aim is always to explicate the progress of science, to apprehend the event: ‘in its generative movement … to recover this structure not via description but apodictically as it unfolds and demonstrates itself’ (31). Ultimately, Cavaillès wants to incorporate the event within science, he aims to ground the development of science within science: ‘a science of science, and hence a part of itself’ (30).
By contrast, Badiou has not shown the least interest in science as a communal enterprise, as a self-illuminating development, or as an ‘Riemannian volume, closed and yet without any exterior’, his view of science is restricted to mysterious breaks within the movement of science that simply shock thought (30). This becomes poignantly clear when he speaks about the difference between truth and knowledge or when he reveals his disdain for normal science. In general, he attaches no importance to the observation, experimentation, and theorizing of scientists working in a settled paradigm. The realm of truth in Badiou’s work is thus restricted to ineffable instants in science, politics, aesthetics, and love. Although these events must be incorporated and related to a world, this work remains one of fidelity to the event – not one of a continuous critical examination of the consequences and limits of a theorem, conviction, style, or commitment.
Crudely put, for Badiou, when the event comes to you, you are immediately enlightened about its significance. For Cavaillès, truths are irreducibly genetic, they inherently involve the labor of explicating its inferences. This labor is, at least, twofold: there is paradigmization, where the demonstration is ‘longitudinal, coextensive with the demonstrative sequence’ and there is thematization, where one inaugurates ‘a new system of interconnection on the basis of the old one, understood no longer as a particular phase within a larger movement, but as an object of reflection in its current configuration’ (71). In Badiou’s philosophy, there is no such distinction, he does not seem to recognize the constructive, laborious activities of scientific communities – only the grandiose seems to deserve being subsumed under the idea. Yet with this gesture, Badiou risks splitting science into the profane and the profound, instead of safeguarding the significance of incorporating events within more expansive webs of knowledge.
Now, something about Meillassoux. In his book After Finitude (2006), Meillassoux aims to reconstruct philosophy through a philosopheme called correlationism: ‘the idea according to which we only ever have access to the correlation between thinking and being, and never to either term considered apart from the other’ (5). For Meillassoux, the modern line of thought that dissolved Locke’s distinction between primary and secondary qualities has obscured the possibility of thinking the thing-in-itself. His aim is to reactivate this distinction by arguing that ‘all those aspects of the object that can be formulated in mathematical terms can be meaningfully conceived as properties of the object in itself’ (3).
In On Logic and the Theory of Science, Cavaillès takes on a similar task: from the very beginning of his book, he is concerned with critiquing the philosophy of consciousness, his worry being that philosophers, such as Kant and Husserl, make science and mathematics dependent upon the acts of consciousness. In his view, science progresses, not consciousness, it might drag consciousness along or happen within consciousness, but certainly does not depend upon it for progress. By distinguishing radically between subject and object, he wants to reinstate mathematics’ autonomy and necessity. As Hourya Benis Sinaceur remarks in her paper “From Kant to Hilbert: French philosophy of concepts in the beginning of the XXth century” (2006): by ‘determining the objective structures of objectivity’, Cavaillès brings about ‘[a] truly Ptolemaic revolution’ (330). It is no longer a question of explicating the constitutive role of transcendental subjectivity, but of explicating the role of conceptual development within the objectivity of mathematics.
For Meillassoux, mathematics plays an equally important role in regaining access to the absolute, but this absolute is one of this world, it is attained through a reflection on the time before or after the existence of human beings. It is at this point that Meillassoux and Cavaillès part ways: Meillassoux wants to make use of the rigorousness of mathematics to say something about the world independent of the correlation between thinking and being and, consequently, about the thing-in-itself, while Cavaillès wants to dissociate subjectivity and objectivity, right now, by founding an absolute logic and science of science, he is determined to never subordinate the autonomy and necessity of mathematics to the ways of the world. For Cavaillès, ‘[t]o know the world, to understand the world – this is a programme that already represents the abandonment of complete creative autonomy, the renunciation of a necessity beholden to nothing other than itself’ (28).
According to Cavaillès, mathematics is never a mere tool for measuring or a medium that helps us access the worldly thing-in-itself, it cannot be subordinated to physics. Although the relation between physics and mathematics remains underdeveloped in Cavaillès’ work, he makes a few profound remarks on the problem of the relation between mathematics and physics. In his view, ‘the concatenation of physics has no absolute beginning, any more than that of mathematics does … experience itself as a system of acts is internally organised in such a way that it is impossible to interrupt its continual unfolding’ (88). It is at the singular intersection of the two sequences that theoretical physics is born. Yet this intersection itself can never be formalized, the correspondence can never be presupposed, and the one can never be incorporated in or reduced to the other. Mathematical physics is hence the name of a problem – not a state of science that we can simply presuppose.
Even if both thinkers endorse a particularly strong concept of logic and mathematics, Cavaillès is the only one who tries to account for this. Throughout Meillassoux’ undertaking, much is put up for grasp: the necessity of the laws of nature, the primacy of consciousness, and the very idea of the transcendental. Nevertheless, the validity of the logic that he uses to construct his philosophy is never questioned, conceptualized, or grounded. Like Badiou, Meillassoux rarely reflects upon the role of logic in his undertaking; he never seems to wonder whether other conceptualizations of logic or the relation between mathematics and the world are possible. Consequently, Meillassoux’ account of the relation of the a priori of logic and the a posteriori of the empirical within science remains somewhat obscure. Crucially, there seems to be no possibility that an event might change the logic he uses to construct his philosophy and thus transform his own conclusive propositions. In other words, Meillassoux’ logic is thoroughly static – whereas Cavaillès’ rendering is thoroughly dynamic.
This difference is also inherent to the conception of their own philosophical projects. In Meillassoux’ work, there is no account of the movement of thought itself. This feature is clearly visible in his philosophical style, which distances itself in the writing from the writing, it considers arguments without considering the dialectics of deliberation. Cavaillès’, on the other hand, never distances himself from the theories or concepts that he is discussing, he constantly closes the gap between his own thought and the thought of the philosopher, logician, or scientist that is discussing. There is a decisive need within his philosophy to give an account of the movement of thought itself. Naturally, this raises the question whether his philosophical project does not end in an infinite regression. His critique of Husserl’s logic using Gödel’s incompleteness theorems as well as his reflections on Spinoza’s idea of the idea suggest that Cavaillès was aware of this problem within his philosophy. Still, the question remains unresolved. A biographical fact may illuminate the two possible pathways that Cavaillès envisioned. In his final days, Cavaillès asked a priest for two books: a copy of the New Testament and a copy of Hegel’s Science of Logic.
Let us end with a few remarks on the new translation. Robin MacKay has, once more, proved to be a great translator. His exceptional ability to render unusually difficult and heterogenous French philosophical texts into stylistically gratifying and elegant English prose has been a more than welcome gift to those unable to read the original texts. His translations of the philosophy and literature of Gilles Châtelet, François J. Bonnet, and Gabriel Catren, not to mention a few short texts by Cavaillès, all have an unprecedented quality. The publishing house, Urbanomic, which MacKay founded in 2006, has been an invaluable source for those trying to renegotiate the limits of philosophical creativity. The series of books published together with Sequence Press is the perfect place for Cavaillès’ treatise to re-appear, his work resonates in a strange but enrapturing harmony with that of Land, Laruelle and Zalameo.
The other translator, Knox Peden, has done much to make Cavaillès’ known in the Anglo-Saxon world. His work Spinoza Contra Phenomenology: French Rationalism from Cavaillès to Deleuze (2014) is an excellent introduction not only to Cavaillès’ work, but also to the French rationalist tradition of which Cavaillès is part. He convincingly argues how we should not only read figures, such as Jean-Toussiant Desanti and Althusser, for historiographical reasons, but for philosophical reasons as well: their philosophical attitude – regardless of its conspicuous shortcomings – remains persuasive. Peden’s introduction adequately narrates Cavaillès’ biography while demonstrating the importance of the concept of necessity in Cavaillès’ work and elaborating upon his position vis-à-vis phenomenology.
Naturally, this is not a perfect translation, and there are choices made by the two translators that are questionable. For example, Peden and MacKay have chosen concatenation as a translation of enchaînement. They choose this word to stress the conceptual continuity between Cavaillès’ philosophy and that of Spinoza and Descartes, but Cavaillès probably picked up this word from Brouwer’s work on the concept of rijen, which is usually translated into English by the word sequences. Footnotes clarifying the obscure references would also have been helpful – especially given the exceptional difficulty and sometimes even obscurity of Cavaillès’ thought.
Still, this new translation is a great opportunity to re-engage with Cavaillès’ treatise. It opens up the possibility of creating a moment that adheres to his vision. A philosopher, whose work breathed cosmopolitanism, and who already at the age of 26 argued in a review of The Second Davos University Conference (2021) that ‘any limitation involves a privation: like those fulgurations with which Leibniz’s God engendered the monads, it is the same spiritual universe that is expressed by French rationalist reflection and German phenomenology’ is one that deserves to be read outside of a small circle of specialists in French epistemology (10). Cavaillès always thought that ‘rapprochement’ between particularisms within philosophy is part of the road to progress (9). For him, it was ‘obvious that they will benefit from coming out of their splendid isolation (like two inland seas), to open up between them channels of communication that will procure for both of them greater movement and fecundity’ (10).
Phenomenological Approaches to Physics is a welcome attempt to bridge the gap between two areas of philosophy not often mentioned in the same career, let alone the same breath. The collection provides fertile ground for further work on phenomenological approaches to physics—and science more generally—however, as much as the collection is promising, it is also disappointing in the preparatory nature of much of the material. While this is a general vice of the phenomenological tradition—consider how many of Husserl’s published works are introductions to phenomenology—in order to appeal to one of the primary audiences of the collection, phenomenology-curious philosophers of physics, further developments with clear consequences are needed. Many of the papers stop just as they’ve really started. This collection is of value for many purposes: as a general introduction to phenomenology, as a guide to the consequences of phenomenology for science and physics, as a pointer to areas of application for the budding phenomenologist, but it also provides some indications of particular lines of further development.
The editor’s introduction is relatively long, but deservedly so, as it does a lot, providing expositions of ten themes from Husserl’s oeuvre: anti-psychologism, intentionality, descriptions and eidetics, the epistemic significance of experience, phenomenology as first philosophy, anti-naturalism, the life-world, historicity and genetic phenomenology, embodiment and intersubjectivity, the epochē, transcendental reduction, and transcendental idealism. The sketch of Husserl produced is that of an epistemological internalist who develops a theory of the objective from fundamental subjectivity, who denies empiricism about logic and mathematics, and who holds that phenomenology is a first philosophy which comprises analyses of the essential structures of subjectivity, the ground of all knowledge, therefore legitimizing all other forms of knowledge, sciences. Any reader interested in a first pass at the role of these themes in Husserl’s work could probably do so no more efficiently than looking through the first half of this introduction. A highlight of the introduction is a sketch of the relevance of other phenomenologists, Heidegger and Merleau-Ponty, to the philosophy of physics. The themes brought up in the introduction and elsewhere are suggestive: Heidegger’s pluralism regarding scientific standards and the difference in the concepts of time in physics and history; his preemption of the theory-ladenness of observation; his praise of Weyl; his primacy of practical understanding over theoretical knowledge; Merleau-Ponty’s participatory realism; his analysis of measurement and rejection of instrumentalism, realism, and idealism, in favor of structuralism.
Part 1: On the Origins and Systematic Value of Phenomenological Approaches to Physics
Robert Crease’s “Explaining Phenomenology to Physicists” is a response to philosophy-phobic physicists, like Hawking, and aims to show how the projects of phenomenological philosophy and physics differ. This amounts to a sort of introduction to the Husserlian distinction between the natural, or naturalistic, attitude of the physicist in her workshop and the more skeptical attitude of the epochē adopted by the phenomenologist. Note that Crease makes the same point that Maudlin and other metaphysically oriented philosophers of physics often emphasize, that mathematical formulae do not comprise a theory but require an interpretation, an ontology (57). How this interpretation is established and justified is the common project of the phenomenologist and the analytic metaphysician. But herein lies a problem with the Crease essay, which is that it while it distinguishes analytic (narrowly focused on the logical analysts of science of the early 20th century), pragmatic, and phenomenological approaches to the sciences, Crease does not say enough to distinguish a defense of phenomenological approaches to physics from a defense of a philosophical approach to physics whatsoever. Now Crease may make the point that phenomenology preempted concerns with the metaphysics of physics or concerns regarding the applicability of mathematical idealization to nature that have more recently become central to the philosophy of physics. Further, it is not clear that this is a fair reading of the aims of the logical empiricists. What is the logical empiricist project of establishing how scientific, “theoretical” terms get their meaning if not a concern with the “framing” of scientific theories and “the reciprocal impact of that frame and what appears in it on their way of being” (55)? This is not to say there is no distinction to be drawn, but the discussion here is not fully convincing as an argument for the value of phenomenology in studies of physics in particular.
Mirja Hartimo’s contribution, “Husserl’s Phenomenology of Scientific Practice,” fills out Crease’s sketch of the phenomenological approach and specifies how Husserl preempts the naturalistic, practice-oriented turn in contemporary philosophy of science. This “naturalism” is to be opposed with ontological or methodological naturalism, both of which Husserl rejected. Hartimo recapitulates the difference between the natural and phenomenological attitudes and its production by the epochē, in which existence is “bracketed.” The case is made that the phenomenological attitude is not inconsistent with the natural attitude (indeed Husserl had, for the most part, the same natural understanding of the sciences as did his contemporaries in Göttingen). The Göttingen view comprises a pre-established harmony between mathematics and physics, “the axiomatic ideal of mathematics served for Husserl, as well as for his colleagues, as an ideal of scientific rationality, as a device that was taken to guide empirical physical investigations ‘regulatively’.” (67) This influences the focus on Galileo in Crisis: physics is fundamentally mathematical in nature (68). Harmony amounts to an isomorphism of the axioms and the laws, with the axioms of physics being a formal ontology, a formal definite manifold (69). Husserl’s two differences with the Göttingen consensus are: (1) scientists should also develop material ontologies, which provide specific normative ideals for the mathematization of nature and its connection to intuition; (2) the normativity of the exact sciences does not extend to all scientific domains, a normative pluralism. (2) is particularly important because phenomenology itself falls short of the axiomatic ideal, due to the inexactness of the relevant essences.
Pablo Palmieri’s contribution, “Physics as a Form of Life,” is an odd fish. It presents itself not as a presentation of Husserl’s account of the lifeworld and its relevance to physics but rather as focusing on a foundational question raised by Husserl: “why is it that the axioms of mathematical physics are not self-evident despite the evidence and clarity that is gained through the deductive processes that flow from them?” (80) To answer this question Palmieri embarks on an analysis of physics as a form of “Life” in the sense of some historical development. The three epochs of physics which characterize its form of life are (1) the youth of Galileo’s axiomatic physics, (2) the senescence of Helmholtz’s work on the anharmonic oscillator and the combination of tones, and (3) the “posthumous maturity” of physics following quantum physics. These historical studies are interesting and valuable in themselves, especially the Galileo study, particularly regarding the influence of Galileo’s aesthethics on his mathematization of nature (84). Unfortunately, how these studies relate to the overall aim of the essay is unclear and is shrouded by the sort of allegorical and flowery prose that turns away many from “continental” approaches more generally. Palmieri’s description of the third stage of physics’ life as “posthumous maturity” describes a “disarticulation” in physics that comes to a head for Palmieri in Heisenberg’s use of (an)harmonic oscillator framework for quantum mechanics. The result of such a “translation” is not a direct analog to the classical treatment of spectra, due to the lack of rules for “composition of the multiplicity into the unity of an individual, by the interpretation of which we might generate the individual utterance that once performed will elicit in our consciousness a corresponding perception in any of the sensory modalities whatever” (100). The obscurity of such bridge principles to observation is, again, exactly the crisis of which Husserl was concerned. The upshot seems not to be, as it was for Husserl, a call to action for phenomenological analysis, but rather the essential mystery of nature as “[i]t is nature herself that precludes herself from knowing reflexively her own totality of laws” (83). While this is supposed to have the status of an explanation it is only buttressed with metaphor:
This being hidden of nature as a totality, or her desire or necessity to hide herself from further scrutiny, which I would be tempted to qualify as nature’s vow of virginity, explains why the axioms of mathematical physics must appear to our intuitions as obscure (84).
This pessimistic conclusion conflicts with phenomenology’s self-conception as a progressive research programme, leaving Palmieri’s own position mysterious, and one suspects that is how he wants it.
Norman Sieroka’s “Unities of Knowledge and Being—Weyl’s Late ’Existentialism’ and Heideggerian Phenomenology” is a fascinating exposition of Weyl’s latter existentialist turn and his engagement with Heidegger’s work. Weyl claims that physics is dominated by “symbolic construction”, of which axiomatic mathematics stands as paradigm, which are empirically evaluated holistically. Weyl’s account of symbolic construction is dependent on the understanding that these symbolic systems are constructed out of particular concrete tokens. Similarly it is essential to the symbolic construction that it is intersubjective and the practitioners of a symbolic system are peers embedded in a wider public. The core of mathematics and the sciences is not logic, but rule-bound “practical management” of symbols (109). This practical level must be fundamental or else we fall into a circle of physical reduction and symbolic representation.
Weyl’s 1949 paper “Science as Symbolic Construction of Man,” explicitly invokes Heidegger’s concept of the existential basicness of being-in-the-world as a point of agreement. Weyl does not, however, accept Heidegger’s anti-scientific attitude that concludes from this, that science is “inauthentic”. Weyl held that scientific practice and philosophical reflection were mutually enriching — particularly moral reflection in the shadow of the bomb. Heidegger’s rejection of science is due to symbols being merely present-at-hand, as they do not figure in the “care-taking encounter of daily life” (114). The weight of evidence and experience clearly sides with Weyl here. Sieroka raises examples of bridge-building and experimental physics. More simply, even the manipulation of symbols in themselves is care-taking in that they are to be interpreted and not only by oneself, in a dubious “private language”, but by some community. Here is a missed opportunity to engage with Heidegger’s later work, though it cannot be said to have influenced Weyl. Something like “The Question Concerning Technology” shows that Heidegger did not think that modern science and technology were independent of daily life, but rather have a radical and destabilizing effect that inhibits Dasein from encountering its own essence. Though, it is not clear how much this is a rejection of the verdicts of Being and Time, or should correct Sieroka and Weyl’s intepretations. The extension of the critique by way of Fritz Medicus, Weyl’s colleague, to a critique of “thrownness” and the general receptivity or passivity of Dasein to Being seems beside the point and reliant on a misunderstanding of Heidegger. Medicus’ “piglets” complaint about the thrownness of Dasein can only rest on a misunderstanding of the role of historicity in Dasein’s being (see Division 2, Chapter 5). Intersubjectivity is fundamental to Dasein. Being-with is “equiprimordial” with Dasein’s Being-in-the-World and is an existential characteristic of Dasein, even when it is alone (149-169). Being-with defines Dasein’s inherent historicity. Dasein is thrown into a culture, into a way of life.
Sieroka’s comparison of Weyl and Cassirer, that Cassirer’s theory of symbolic forms provides a unity of knowledge, while Weyl’s provides a unity of being, owing to his existentialist inflection, is interesting but perfunctory. It makes one wonder what such a distinction could tell us about the difference of method between phenomenology and neo-Kantianism, how this might relate to the interpretational dispute at the center of the Davos debate, and how Weyl’s conception of physics and mathematics could have played a role in such rifts.
Part 2: Phenomenological Contributions to (Philosophy of) Physics
“A Revealing Parallel Between Husserl’s Philosophy of Science and Today’s Scientific Metaphysics” by Matthias Egg aims to show how the crises that Husserl saw as central to the contemporary sciences and his solution are echoed in the scientific metaphysics of Ladyman and Ross (2007). The crisis is rooted in the substitution of the lifeworld for mathematical idealities, which amounts to a forgetting of the “meaning-fundament” of the sciences, undermining their own epistemological standing. Egg frames his comparison of Husserl and the scientific metaphysicians with Habermas’ critique of Husserl’s project of making science presuppositionless, providing a basis for absolute practical responsibility. The supposed failure is that it is left unexplained how a more perfect theoretical knowledge is to have practical upshot. The lacuna is Platonic mimesis, wherein the philosopher “having grasped the cosmic order through theorizing, the philosopher brings himself into accord with it, whereby theory enters the conduct of life,” (129), which is in direct ontological opposition with Husserl’s transcendental idealism, as Habermas sees it. (Does Habermas commit the naturalistic fallacy?) Husserl’s model claims only that the procedure or methodology of theoretical knowledge provides normative force on our practical affairs, in Egg’s example, our doing of physics. Egg presents Ladyman and Ross as agreeing with Husserl’s science-cum-Enlightment project, particularly, that science must be central to our worldview as it allows for a unified, intersubjectively valid approach to world even beyond theoretical practice. This too, falls short of Habermas’ mimetic ideal —their project could only be preserved in the “ruins of ontology” (130). Ladyman and Ross share some skepticism about strong metaphysics but accept weak metaphysics. Unfortunately, Egg stops just before saying anything more substantive than an observation of convergent philosophical evolution. There is more to be said particularly regarding the link between this sort of communicative conception of the scientific project and structural realism which puts Ladyman and Ross and Husserl in the same camp. The metaphysical essays to follow cover some of what I would like to say, but let me gesture at a possible development. In Ideas II and the fifth Cartesian Meditation, Husserl develops an account of scientific objectivity such that it is constituted by intersubjective agreement via “appresentation.” What is intersubjectively available are the appearances of objects, but what is agreed upon are the invariant structures supposed to explain the experiences of the community. Heelan’s (1978) hermeneutic interpretation of Husserl provides a picture in which the infinite tasks of mathematization and measurement link together the lifeworld and the scientific image which is constituted by it. There is a structural realist position to be examined here which could provide a unified account of everyday and scientific perception.
Lee Hardy’s “Physical Things, Ideal Objects, and Theoretical Entities: The Prospects of a Husserlian Phenomenology of Physics” attempts to square Husserl’s phenomenology with scientific realism. Husserl’s seeming positivism is especially problematic given that Husserl argues “that the objective correlates of the mathematical laws of the physical sciences simply do not exist in the physical sense. They are ideal mathematical objects, not real physical things” (137). Hardy restricts Husserl’s instrumentalism to scientific laws rather than scientific theories tout court. Husserl’s view is that knowledge of physical objects is gained by mathematical approximation, leaving room open for the positing of actual physical entities. Hardy’s argument, a rational reconstruction of a path not (explicitly) taken by Husserl, depends on a distinction that seems both interesting and suspicious. Hardy wishes to distinguish instrumentalism about the laws from instrumentalism regarding theories, the difference between the two lies in the fact that laws specify functional interdependencies of physical quantities which state how empirical objects behave, but theories explain why physical quantities behave as they do. So then, the instrumentalist holds that the semantic value of theories is limited to that of the laws, which predict observable behavior. The realist holds that scientific theories have as semantic values the behavior of unobservables. Husserl’s radical empiricism is in apparent tension with the realist’s explanation, Hardy reconstructs the received view:
(1) A obtains if and only if p is true.
(2) p is true is and only if p is evident.
(3) p is true if and only if A is intuitively given in an act of consciousness.
Ergo, (4) A obtains if and only if A is intuitively given in an act of consciousness.
Theoretical entities cannot be so given, so statements about them can never be true, so we ought not be committed to them. This interpretation Hardy rejects in favor of one which changes the role of experience from semantic-metaphysical to epistemic:
S is justified in believing p if and only if the correlative states of affairs A is given to S in an intuitive act of consciousness (143).
Hardy specifies that the perceivability condition on existence was meant to be dependent on an ideal possibility, not an actual possibility (dependent on sensory apparatuses). This point goes some way towards specifying the meaning of transcendental idealism, though this seems to go astray in attempting to recover realism. Transcendental idealism requires that possible perception by a transcendental subjectivity constitutes (the preconditions for) existence. Hardy picks up the thread in the Crisis regarding the essential approximative nature of the sciences as their conclusions are mediated by ideal, mathematical constructions:
Exact, objective knowledge is possible only by way of a passage through the ideal; and for that very reason will never be more than approximative knowledge of the real (146).
In Crisis, Hardy claims, Husserl distinguishes the ideal, physical object and the perceived object ontologically: the objects of ordinary life are not “physical” objects. It is these limit-idealized objects that Husserl is anti-realist with respect to. The trouble with Hardy’s distinction between theories and laws and between real objects and idealized objects is that the approximation relation is left unexplained. There remains an explanatory gap as to why physical objects should be subject to laws that properly only have idealities as their subjects.
Arezoo Islami and H. A. Wiltsche’s “A Match Made on Earth: On the Applicability of Mathematics in Physics” shows how phenomenology can provide a response to Wigner’s puzzle, “the unreasonable effectiveness of mathematics,” by moving on from why-questions to how-questions. The puzzle arises from a rejection of Pythagorean mathematical monism towards which the phenomenologist is officially neutral, due to the epochē, setting aside why-questions altogether. To answer the how-questions, the phenomenologist must also provide both synchronic and diachronic accounts of how we apply mathematics. The authors explicate constitution and replacement. They show what is meant by the horizon of experience, all the non-actual aspects of some experience which frame one’s interpretation of it, one’s anticipations. From this constitution is explicated:
It is this process of intending objects through specific noemata and then constantly projecting new sensory data against horizons of possible further experiences that phenomenologists call constitution. Of particular importance in this context are those aspects of experience that remain invariant… (169)
From these invariances of the noemata, lawlike relations are found and suitably objective properties can be described of the noema. This structure generalizes to scientific constitution from the example of perceptual constitution. Aiming to intend all of reality through mathematical noemata is Galileo’s great leap forward. Doing so is to replace the lifeworld with the scientific image. Nature is mathematical because we have made it so. While I am largely sympathetic with this approach, and hold that it contributes to a structuralist view that is worth developing, to satisfy mysterions like Wigner specific accounts of such constitution is needed.
Thomas Ryckman’s essay, “The Gauge Principle, Hermann Weyl, and Symbolic Constructions from the ‘Purely Infinitesimal’,” provides a mini-history of Weyl’s development of the gauge principle (a fuller history in Ryckman 2005), in which Weyl is motivated to investigate Lie groups and algebras by phenomenology on the one hand and Naturwirkungphysik on the other. Naturwirkungphysik is a standard explanation, “that all finite changes are to be comprehended as arising through infinitesimal increments” (182). In practice this is to take locally defined tangent spaces to be explanatorily fundamental. For Weyl, this standard of locality is justified by appeal to not just phenomenological epistemology, that direct givenness to the ego is the ground of all essential insight into the structure of things, and this givenness is attenuated at spatial distance, but to full blown transcendental idealism:
insofar as symbolic construction of the “objective reality” of the purportedly mind-independent objects of physics is, per Husserl, a constitution of the sense of such objects as having “the sense of existing in themselves” (184-5).
Just as the previous essay establishes, the objects of mathematical physics are constructions which intend transcendent objects. However these objects are only fixed up to an isomorphism, any further “essence” is beyond cognitive grasp and therefore unreal (188). Ryckmann provides an able and clear derivation of the gauge principle in QED and a quick rundown of how this generalizes in the Standard Model. While this is a valuable contribution to the collection, those familiar with Ryckman’s past work will wish that the closing remarks regarding the standard model and the Weyl-Nozickean (2001) slogan, “objectivity is invariance,” were expanded upon. I look forward to further development of the alternative view implied by Ryckman’s interpretational challenge this slogan, which centers locality as the source of gauge transformations (199).
Part 3: Phenomenological Approaches to the Measurement Problem
Steven French’s “From a Lost History to a New Future: Is a Phenomenological Approach to Quantum Physics Viable?” does well to show that the phenomenological background of Fritz London was deeply influential on his approach to the measurement problem (with Bauer) and that this influence has been covered over by misinterpretation. The measurement problem is essentially the apparent inconsistency of deterministic dynamics of quantum mechanics and the collapse of the wave function. London and Bauer have been taken to merely restate von Neumann’s notorious solution, that the uniqueness of the interaction of the system with a conscious observer explains how and when the “collapse” occurs. French shows this picture presented by Wigner, which fell to the criticism of Shimony and Putnam, to be a straw man. French argues that London and Bauer’s phenomenological account of quantum measurement can stand up to such criticisms and for London. Quantum mechanics presupposes a theory of knowledge, a relation between observer and object “quite different from that implicit in naive realism” (211). Measurement, considered subjectively, is distinguishable from the unitary evolution of the quantum state by introspection giving the observer the “right to create his own objectivity” (212). This is not some (pseudo-)causal mind-world interaction that creates a collapse but rather a precondition for the quantum system to be treated objectively and by a different mathematical function, the precondition being a reflective act of consciousness in which the ego-pole and object-pole of experience are distinguished, not a substantial dualism, “thereby cutting the ‘chain of statistical correlations’” (212-3). The discussion that follows, while suggestive, shows that it is not clear how this general phenomenological view about the nature of objectivity is supposed to remove the particular quantum measurement problem. Whether this is the fault of French or of London and Bauer is unclear; the most direct quotation from London and Bauer suggests that this distinction of the ego and the object somehow licenses the transition from representing the measurement situation by the wave function, ψ, to representing the system as in a particular eigenstate. This is much too oblique, given that the nature of such fundamental acts of consciousness is, even to the phenomenological initiate, obscure, and requires some substantive claims about the determinate nature of consciousness. French too must find the explanation as given by London and Bauer incomplete as he invokes decoherence, decision theory, and the “relational” interpretation as elements of a fuller story, presenting something, protestations aside, very close to Everettianism indeed. If such a distinctive and useful interpretation can be fleshed out on phenomenological grounds, it would be the most direct and substantive proof of the progressive nature of a phenomenological programme.
Michel Bitbol’s “A Phenomenological Ontology for Physics: Merleau-Ponty and QBism” is another breath of fresh air in the collection, exploring a phenomenological approach other than Husserl’s. Taking the primacy of lifeworld and Bohr’s challenge to traditional scientific epistemology as starting points, the essay sets up correspondence between Fuch’s participatory realism and Merleau-Ponty’s endo-ontology. More generally Bitbol takes recent developments in the philosophy of quantum mechanics, like Peres’ no-interpretation and Zeillinger’s information-theoretic approach, to “all seem to be pointing in the same direction,” in line with the phenomenological approach to the sciences as tools for navigation in the world. These are the pragmatists, as distinguished from the interpreters. Bitbol goes on to describe how the anti-interpretational approach is phenomenological by establishing an epochē for quantum physics. Rather than understand the states of quantum systems in a Hilbert space as properly predicative, we bracket any ontological posit and treat these states functionally as informational bridges between the preparation and outcome of experiments. Bitbol then considers a question a level up:
[W]hat should the world be like in order to display such resistance to being represented as an object of thought? Answering this question would be tantamount to formulating a new kind of ontology, a non-object-based ontology, an ontology of what cannot be represented as an object external to the representation itself (233).
For Merleau-Ponty (and Michel Henry), the non-objectual ontology is provided by the priority of the body and raw, original experience.
This is an ontology of radical situatedness: an ontology in which we are not onlookers of a nature given out there, but rather intimately intermingled with nature, somewhere in the midst of it… we cannot be construed as point-like spectators of what is manifest; instead, we are a field of experiences that merges with what appears in a certain region of it. This endo-ontology is therefore an ontology of the participant in Being, rather than an ontology of the observer of beings (236).
Here the central self-consciousness of transcendental idealism becomes self-perception of the body. In physics, this is translated into a participatory realism, wherein the observer is involved in the creation of Being. Merleau-Ponty’s own statement of the relationship between his phenomenology of embodiment and physics starts from the observation that physics always attempts to take in the subjective as a part of or a special case of the objective. This is something of a category error, and in quantum mechanics it seems that there is a concrete proof of the impossibility of eliminating the subjective, or better yet shows that the objective-subjective distinction is not well formed. These are interesting points and one wishes that Bitbol (and Merleau-Ponty himself) would have spelled out this metaphysical picture in more detail. While the correspondence with QBism seems somewhat plausible, it is not shown that either view commits one to the other or that this endo-ontology provides an advance on the anti-metaphysical orientation of the QBist. The remarks regarding probability are paltry and given the significance of probabilities in quantum mechanics, a full account of it is necessary if there is to be much uptake—the primary limitation here seems to be that Merleau-Ponty did not get to consider this matter much prior to his death.
In contrast, “QBism from a Phenomenomenological Point of View: Husserl and QBism” by Laura de La Tremblaye is one of the fullest contributions in the collection. This essay serves as an able introduction to non-denomenational QBism, presented as a generalization of probability theory and cataloged as a participatory realist, -epistemic “interpretation” of quantum mechanics. QBism “stands out as an exception” (246) in this category because it focuses on belief, adding the Born Rule as an extra, normative rule in Bayesianism (the axiomatization is not explicitly shown). QBism removes the ontological significance of the collapse of the wave function, the state description and reality are decoupled, the collapse is a statement of some (ideal) agent’s belief state. Accordingly, “knowledge” yielded by measurements is redefined as information about the system that is accepted via measurement (250). While the probabilities assigned are subjective, the updating rules are objective.
It is no trivial task to draw a clear line between the subjective and the objective aspects of the Born rule… Fuchs and Schack invoke a completely new form of intersubjectivity. It is through the use of Bayesian probabilities that the multiplicity of subjectivities elaborates a reasoning that can be shared by everyone, and that, consequently, can be called “objective” in precisely this limited sense… this leads to the new conception of knowledge: knowledge is no longer understood in terms of an objectively true description of the intrinsic properties of the world; it is rather understood as the kind of knowledge that is needed to guide the future research of any agent, thus implying a weaker form of objectivity (251).
For Fuchs, the measuring device is analogous to a sensory organ, measurement is an experience. This leads de La Tremblaye to consider two notions of experience, one from Husserl, the other from William James, who influenced Chris Fuchs. de La Tremblaye argues that it is Husserl’s model of experience as involving a normative, intentional horizonal structure, that better coheres with the Qbist view. This shows a positive contribution phenomenology may offer to QBism: an explanation of the source of the Born Rule’s normativity. Another would be an adequate explanation of how it is that the rules of Bayesian probability can be objective via the intersubjective constitution of objectivity essential to Husserl’s model of the sciences.
In sum: this collection is promising though deficient in some respects. It will provide a number of starting points for a further development of a phenomenology of physics and provides the curious or sympathetic philosopher of physics something to chew on, but it is not a full meal. Many of the contributions would do well as additions to a graduate seminar or undergraduate course on phenomenology or the philosophy of science, with the materials on quantum mechanics showing the most potential for further development.
Heelan, P. A. 1987. “Husserl’s Later Philosophy of Natural Science.” Philosophy of Science 54 (3): 368-390.
Heidegger, Martin. 1977/1993. “The Question Concerning Technology.” In Basic Writings, David F. Krell (ed.). New York: HarperCollins.
———. 1962. Being and Time. John Macquarrie and Edward Robinson (trans.). New York: Harper and Row.
Ladyman, J. & Ross, D. et al. 2007. Every Thing Must Go: Metaphysics Naturalized. Oxford: Oxford University Press.
Nozick, R. 2001. Invariances: The Structure of the Objective World. Cambridge: Harvard University Press.
Ryckman, T. 2005. The Reign of Relativity: Philosophy in Physics 1915-1925. Oxford: Oxford University Press.
 Thanks to Porter Williams for reading the collection with me and sharing his thoughts with me, which allowed me to sharpen my own.
The Mathematical Imagination focuses on the role of mathematics and digital technologies in critical theory of culture. This book belongs to the history of ideas rather than to that of mathematics proper since it treats it on a metaphorical level to express phenomena of silence or discontinuity. In order to bring more readability and clarity to the non-specialist readers, I firstly present the essential concepts, background, and objectives of his book.
The methodology of this book is constructed on the discussion of concepts and theoretical perspectives such as Critical Theory, Negative Mathematics, Infinitesimal Calculus, expression and signification of silence and contradictions in language. Borrowed from the mathematics or from the thinkers of the Frankfurt School, each of these concepts becomes refined, revisited and transposed by Handelman in order to become operative outside of their usual context or philosophical domain. The term Critical Theory was developed by several generations of German philosophers and social theorists in the Marxist tradition known as the Frankfurt School. According to these theorists, a critical theory may be distinguished from a traditional theory as it seeks human emancipation from slavery, acts as a liberating tool, and works to create a world that satisfies the needs and powers of human beings (Horkheimer 1972). Handelman revisits what he calls a “negative mathematics”: a type of mathematical reasoning that deals productively with phenomena that cannot be fully represented by language and history, illuminating a path forward for critical theory in the field we know today as the digital humanities.
In The Mathematical Imagination, negative mathematics encapsulates infinitesimal calculation, logic and projective geometry as developed by Gershom Scholem (1897-1982), Franz Rosenzweig (1886-1929), and Siegfried Kracauer (1889-1966). These three German-Jewish intellectuals were connected to the thinkers of the Frankfurt School but distinct because they found ways to use math in their cultural theory. The negative mathematics found in the theories of Scholem, Kracauer or Rosenzweig (inspired by their famous predecessors Salomon Maimon (1753-1800), Moses Mendelsohn (1729-1786) and Hermann Cohen (1842-1918)), are not synonymous with the concept of negative numbers or the negative connotation of math that we see in the works of the other members of the Frankfurt School.
Handelman’s objective is to present his book on the path of Scholem, Kracauer and Rosenzweig using math and digital technology as a powerful line of intervention in culture and aesthetics. The Mathematical Imagination investigates mostly the position of these three German Jewish writers of the XX century concerning the relationship between mathematics, language, history, redemption, and culture in the XX century and extending his analysis to digital humanities. Mathematics is convened metaphorically in their theory of culture as pathways to realizing the enlightenment promises of inclusion and emancipation. The silence of mathematical reasoning is not represented by language but by the negative approach that is to say absence, lack, privation, discontinuity or division like in the conception of the infinite. One example of this productive negativity is to look at how mathematics develops concepts and symbols to address ideas that human cognition and language cannot properly grasp or represent, and surfs metaphorically with the concept of the infinite (Monnoyeur 2011, 2013). The infinite calculation is a generative spark for theorizing the influence of math in culture as differentials represent a medium between experience and thought. For Scholem, Rosenzweig, and Kracauer, these mathematical approaches provide new paths for theorizing culture and art anew, where traditional modes of philosophical and theological thought do not apply to modern life or situation of exile.
In The Mathematical Imagination, Matthew Handelman wants to give legitimacy to the undeveloped potential of mathematics and digital technology to negotiate social and cultural crises. Going back to the Jewish thinkers of the Weimar Republic, namely Scholem, Rosenzweig and Kraucauer, he shows how they found in mathematical approaches strategies to capture the marginalized experiences and perspectives of German Jews in Germany or exile at the beginning of the XX century. In doing so, he re-examines the critical theory of the Frankfurt School, specifically those philosophers who perceived in the mathematization of reason a progression into a dangerous positivism and an explanation for the barbarism of World War II. Handelman re-evaluates Adorno and Horkheimer‘s conception of mathematics, according to which math should not be treated as a universal science able to solve any problem because it is not able to rule the human world of culture, art and philosophy. For them, as for Adam Kirsch, who wrote in 2014 the article “Technology Is Taking Over English Departments” (published in New Republic), both mathematical and computational mechanization of thought exclude the synthetic moment of the intellect and cannot produce new or meaningful results.
The first chapter, titled “The Trouble with Logical Positivism: Max Horkheimer, Theodor W. Adorno, and the Origins of Critical Theory,” recounts the debate that took place between the members of the Frankfurt School — Max Horkheimer (1895-1973), Walter Benjamin (1892-1940), Theodor W. Adorno (1903-1969)—, and members of the Vienna Circle, such as Otto Neurath (1882-1945) and Rudolf Carnap (1891-1970). Mathematics, according to the Frankfurt School’s critical theory, is in apparent opposition to language, since there is a dialectical tension between two forms of thought, one expressed in mathematics that circumvents representation and the other mediated by language and representation. Adorno gave, through the tension between mathematics and other forms of knowledge, the political dimension that we find in his works and his confrontation with the Vienna Circle. For Adorno, the attempt in mathematics to abandon meaning, the ability to signify something else, constitutes the philosophical flaw of the logical positivists’ proposal to reduce thought to mathematics.
The second chapter, titled “The Philosophy of Mathematics: Privation and Representation in Gershom Scholem’s Negative Aesthetics,” revisits the relation between language and mathematics in the context of Kabbalist culture. In his writings on the language of lamentation, “On Lament and Lamentation,” Scholem explores the dilemma of saying the ineffable and the oscillations between spoken and unspoken language, in order to reconcile the paradoxes inherent in language (Scholem, 2014). At the heart of these paradoxes lies the deep dialectic between openness and secret, concealment and revelation. He underlines a common privative structure of communication in mathematics and laments that it negatively communicates language’s own limits, but it also reveals an aesthetic strategy. For Scholem, the philosophy of math deals with the problem of language by omitting its representation, and its inexpressibility represents the privation of life in exile with the possibility to recover a productive vision of mathematics. Math is done to speak purity, privation, a language without representation, and it deals with the shortcomings of language. According to Gershom Scholem, this fruitful approach lies beyond language within the sphere defined by the signs of mathematical logic. Scholem understands math, history, and tradition metaphorically, as characterized by silences and erasures that pave the way for the acknowledgment of historical experiences and cultural practices which rationalist discourses, majority cultures, and national, world-historical narratives may marginalize, forget, or deny.
The third chapter analyses the relation between infinitesimal calculus and subjectivity/motion in Franz Rosenzweig’s Messianism. Rosenzweig’s (1886-1929) major work, The Star of Redemption (1921), is a description of the relationships between God, humanity, and the world, as they are connected by creation, revelation, and redemption. He is critical of any attempt to replace actual human existence with an ideal and, for him, revelation arises not in metaphysics but in the here and now. He understands knowledge not as what is absolutely proven, but rather what individuals and groups have verified through their experience. For Rosenzweig, verification did not mean that ideas substantiated in experience automatically counted as knowledge; neither does it imply that theoretical statements become meaningful when verified by experience, as Carnap later argued. He analyzes thus how concepts such as subjectivity, time, and redemption are central to critical theory and avoided by the official languages of philosophy and theology. Rosenzweig’s thought is an example of how cultural criticism can borrow from mathematics to illuminate its concepts without mathematizing culture. For instance, the way infinitesimal calculus linked nothingness with finitude represented a tool that could be used to reorient epistemology around the individual subject. For him, mathematics possesses the ability to resolve a fundamental problem for both theology and philosophy, which is the creation of something from nothing. Calculus is motion over rest, reveals multiplicities of subjectivity and representation, and shows how the theoretical work done by mathematics offers epistemological tools useful for cultural criticism. These tools could help theorists to think through concepts that remain obscure in aesthetics and cultural theory, as fractal geometry illuminates the theory of the novelty. Mathematics helps us to construct more capacious versions of these concepts as well, and conceptual tools exist that allow us to intervene more immediately in a project of emancipation, in the service of theories of culture and art, and where they are at work.
Chapter fourth presents geometrical projection and space in Siegfried Kracauer’s Aesthetics. In The Mass Ornament, written in 1921 but published in 1960, Siegfried Kracauer reads the ephemeral unnoticed and culturally marginalized phenomena of everyday city life as an ornament. His attention to the quotidian leads him to decipher in urban life a hidden subtext referring to biblical figures that comfort his experience of intellectual exile. Improvisation constitutes a key category in Kracauer’s critical engagement with metropolitan experience and modern culture; improvisation, with its invocation and representation, lies at the confluence of Kracauer’s preoccupation, the contemporary cityscape. In this book, he decodes the surface meanings of the new city phenomena in their shallowness, personal and political significance. These collected essays dream wild about the ultimate meaning of the banal and the beautiful in cities and gather a diverse range of observations such as boredom and bullfights, dance crazes and detective novels, to reviews of sociology, theology and Biblical translation. The Mass Ornament offers an opportunity to reflect historically on culture and connects the theoretical or philosophical discourse to the passing flux of fashion and the inexorable demands of quotidian life in the city. As a report from the past, this book invites us to renewed reflection on the relation between theory and history, fashion and tradition. Kracauer, in relation to the entire range of cultural phenomena, includes fascinating portions of history and situates man’s relation to society and time. By rearranging the language and textual space as a projection of rationalization, Kracauer explores the point of transference where geometric projection and the metaphors of space become a natural geometry in cultural critique. For Kracauer, geometry is a bridge across void because the mathematical study of space bridges the void between material reality and pure reason. The logic of mathematics informed his readings of mass culture, which sought to advance, rather than oppose, the project of the Enlightenment. For him, geometry enabled a literary approach to cultural critique in which the work of the critique helped to confront the contradictions of modernity and, through such confrontation, potentially resolve them. In The Mass Ornament, geometric projection turned into a political mode of cultural critique, projection, and the metaphors of space became aesthetically operative in the exploration of the rationalized spaces of the modern city.
In his final historical book, titled The Last Things Before the Last (1969), Kracauer presents mathematics as a web of relationships between elements abstracted from nature (Kracauer, 1969). The surfaces Kracauer describes are not an objective reality in the sense of the natural sciences describe them; surfaces exhibit innate breaking points built into by the phenomenology of his approach of a reality stripped of meaning. For Kracauer, the study of history had to mediate between the contingency of its subject matter and the logic of the natural sciences. Nonetheless, this type of cultural critique, enabled by negative mathematics, must resonate with those of us who live in a world of new media, one ever more mediated and controlled by computers and other digital technologies. Kracauer assessed popular culture on its own terms, with a mind open to new technology and communications, and articulated a still valid critique of popular culture.
In his last chapter, titled: “Who’s Afraid of Mathematics? Critical Theory in the Digital Age,” Handelman concludes that digital technology with textual analysis is engaged in social emancipation and can give an answer to the crisis in the humanities. In his analysis of Gershom Scholem, Franz Rosenzweig, and Siegfried Kracauer’s project, he develops the concept of Negative Mathematics in the tradition of Maimon, Mendelson, and Cohen to show how certain mathematical features and concepts can express the unexpressed part of language. In this endeavor, he focuses on infinitesimal calculation and reveals how culture, emancipation and social life can benefit from mathematics. That is to say, the seemingly tautological repetition of mathematics or digital technologies can act as a cultural aesthetics and interpretative medium. Handelman considers that mathematics and digital technology are by nature able to be a tool of liberation and emancipation if a good use is made of them. According to Handelman, if critical theory accepts the way Horkheimer and Adorno associate mathematics with instrumental reason and politics of domination, it risks giving up the critical potential of mathematics and any other interpretive tool such as technology or computer science.
Handelman poses the question: what happens if we allow mathematics to speak with analogy and image, to work with the integral of tradition, the continuity and derivative of truth? What if we applied mathematics more directly to cultural criticism? What possibilities, if not also, dangers, arise in using mathematics as an instrument of cultural thought?
Handelman’s choice to focus on Scholem, Rosenzweig, and Kracauer’s approach to mathematics in order to reveal pathways through the apparent philosophical impasse and an opportunity to realize the Enlightenment promise of inclusion and emancipation is exhilarating. His endeavor to build on the thought of these three lesser-known German-Jewish intellectuals of the interwar period can help move today’s debates that pit the humanities against the sciences. By locating in mathematics a style of reasoning that deals productively with something that cannot be wholly represented by language and history, The Mathematical Imagination illuminates a path forward for critical theory in the field we know today as the digital humanities. Furthermore, this volume explores mathematics as more than just a tool of calculation but one that is a metaphorically powerful mode for aesthetics and cultural analysis. Handelman reintroduces critical theory in the benefice of mathematics as access to culture and expression of the inexpressible. In other words, Handelman revitalizes a forgotten field of research at the intersection of language, math, history, and redemption, so as to capture the irrepresentable presence and interpretation of the complementarity of silence, and the language to express what was forgotten by the official language and culture. He also questions Adorno and other members of the Frankfurt School as unremitting opponents to mathematics. Instead, negative mathematics offers a complement to the type of productive negativity that Adorno, in particular, had located originally in the Hegelian dialectic. Negative mathematics reveals prospects for aesthetics and cultural theory neither as a result of being opposed to language, as Adorno and Horkheimer suggested, nor because it uses the trajectory of history or the limit of tradition. Instead, negative mathematics constitutes its own epistemological realm alongside history and mysticism, illuminating, based on its problematic relationship to language, in the dark corners and hidden pathways of representation. In this sense, it is positive because it deals successfully with what cannot appear in normal use of language or disappears behind official discourse. To this point, Handelman maybe meets the critical and social purpose of the Frankfurt School and fulfills his ambition to produce a theory both critical and mathematical, and even digital. If we take the Frankfurt School main critique regarding mathematics, according to which mathematical and computational mechanization of thought excludes the synthetic moment of the intellect and thus cannot produce new or meaningful results, we have to question then if Handelman’s negative mathematics can actually produce new and meaningful results? Handelman’s negative mathematics does not propose a general way to social critique as a block but rather opens space for the expression of what is suppressed, forgotten, hidden or impossible to realize because of official culture. Silences, disruption, movement, fashion, improvisation, news and materiality occupy the world of culture and are brought to existence by adapted mathematical processes. In this sense, the special treatment of mathematics does not repress the synthetic moment of the intellect but gives a voice to what could not exist before. Common, traditional, usual and politically dominant ideologies cannot resist or foresee this new critical mathematical cultural theory. Of course, this perspective is limited and is not enough to prepare a general critique of society as the thinkers of the Frankfurt School pursued it but improves significantly cultural and critical analysis.
Matthew Handelman noticed that many humanists nowadays have turned to mathematics and digital technologies and tries to forge new paths for modernizing and reinvigorating humanistic inquiry. The Mathematical Imagination presents mathematics and digital technologies as providing a key to unlock the critical possibilities hidden in language to give a voice to silenced communities. Handelman’s book improves cultural and critical analysis, and results into a new and thought-provoking Critical Theory bridging humanities and digital/mathematical technologies. His methodology and ideology are deliberately provocative, and he intends to develop a post-academic approach to fix the weaknesses of traditional and official discourse. His endeavor is also fruitful from the perspective of the history of the science as it shows the relation between various mathematical processes, such as the infinitesimal calculation and everyday phenomena that remain unexplored.
Horkheimer, M. 1972. Critical Theory. New York: Seabury Press.
Kirsch, A. 2014. “Technology Is Taking Over English: The False Promise of the Digital Humanities.” New Republic, May 2, Article 117428.
Kracauer, S. 1969. History: The Last Things Before the Last. New York: Oxford Univ Press.
Monnoyeur, F. 2011. Infini des philosophes, infini des astronomes. Paris: Belin.
Monnoyeur, F. 2013. “Nicholas of Cusa’s methodology of the Infinite.” Proc. Conference on History & Philosophy of Infinity, Cambridge: University of Cambridge. DOI: 10.13140/RG.2.1.1595.0881
Scholem, G. 2014. “On Lament and Lamentation.” In Ferber I. & Schwebel P. (Eds.), Lament in Jewish Thought: Philosophical, Theological, and Literary Perspectives, 313-320. Berlin/Boston: De Gruyter.