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# Etiqueta: Mathematics

## Dominique Pradelle: Intuition et idéalités: Phénoménologie des objets mathématiques, Puf, 2020

## Matthew Handelman: The Mathematical Imagination: On the Origins and Promise of Critical Theory

## Arkadi Nedel: Donner à voir, Tome 1: Les racines mathématiques de la phénoménologie husserlienne, L’Harmattan, 2019

## Jochen Sattler (Hg.): Oskar Becker im phänomenologischen Kontext, Wilhelm Fink, 2020

## Guillermo E. Rosado Haddock: Unorthodox Analytic Philosophy

## Mohammad Shafiei, Ahti-Veikko Pietarinen (Eds.): Peirce and Husserl: Mutual Insights on Logic, Mathematics and Cognition, Springer, 2019

## Matthew Handelman: The Mathematical Imagination: On the Origins and Promise of Critical Theory, Fordham University Press, 2019

## David Rowe: A Richer Picture of Mathematics: The Göttingen Tradition and Beyond, Springer, 2018

## Jeff Kochan: Science as Social Existence: Heidegger and the Sociology of Scientific Knowledge, Open Book Publishers, 2017

Intuition et idéalités: Phénoménologie des objets mathématiques

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The Mathematical Imagination: On the Origins and Promise of Critical Theory

Fordham University Press

2019

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Fordham University Press

2019

Hardback $95.00

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**Reviewed by**: Françoise Monnoyeur (Centre Jean Pepin, CNRS, Paris)

*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*, N*egative 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?

**Conclusion**

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.

**References**

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.

Donner à voir,Donner à voir, Tome 1: Les racines mathématiques de la phénoménologie husserlienne

Ouverture Philosophique

L'Harmattan

2019

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310

Ouverture Philosophique

L'Harmattan

2019

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Oskar Becker im phänomenologischen Kontext

Neuzeit und Gegenwart

Wilhelm Fink

2020

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216

Neuzeit und Gegenwart

Wilhelm Fink

2020

Paperback 149.00 €

216

Unorthodox Analytic Philosophy

Texts in Philosophy, Volume 27

College Publications

2018

Paperback £16.00

520

Texts in Philosophy, Volume 27

College Publications

2018

Paperback £16.00

520

**Reviewed by**: Jethro Bravo (UNAM/Husserl-Archiv der Universität zu Köln)

Guillermo E. Rosado de Haddock’s *Unorthodox Analytic Philosophy* (2018) is a collection of essays and book reviews representative of a Platonist understanding of analytic philosophy. In this sense, it is the counterpart of orthodox empiricist analytic philosophy, whose anti-universalism swings between negation and pragmatic forms of acceptance. In any case, this empiricism *cannot* be traced back to Gottlob Frege, as Rosado himself insists in this collection.[1]

In fact, the collection is strongly marked by the contentious approach to themes preferred by traditional analytic philosophy, like logic, mathematics and physics. I, as a philosopher formed in the phenomenological tradition founded by Edmund Husserl, was originally attracted to this book out of a hope for a possible critical exchange between both traditions. Alas, no such exchange is found here.

Nevertheless, the book speaks often of Husserl, but from the point of view of his objectivist efforts concerning logic and mathematics. Interesting topics include the simultaneous discovery of the (in Fregean terms) “sense-referent” distinction by both Frege and Husserl, Husserl’s distinction between “state of affairs” and “situations of affairs” (which I guess went unnoticed by many readers of Husserl), Husserl’s understanding of the relation between logic (syntactical) and mathematics (ontological-formal), which foreshadows that of the Boubarki School, or his acceptance of Bernhard Riemann’s views on geometry, who puts him at odds with the more antiquated Frege. He also touches upon Husserl’s notion of analyticity as a development of Bernard Bolzano, as well as Husserl’s very important understanding of mathematical knowledge as coming from a conjoined function between categorial intuition and formalization [as a side note, the treatment of categorial intuition is not so inexistent as Rosado thinks (152), one only has to look into Dieter Lohmar’s texts, who himself is a mathematician grown philosopher, just as Rosado likes to say about Husserl]. All these subjects, not excluding the case of Rudolf Carnap’s “intellectual dishonesty” in relation to Husserl’s ideas, which amounts to a sort of scandal in the philosophical realm, give a very interesting material for any philosopher -not just analytic philosophers.

Of course, the book contains other topics of interest, some of them original contributions from Rosado, like his definition of analyticity, which is strictly tied to his semantic treatment of the analytical-synthetical difference of judgements, or his many refutations of empiricism spread all over different essays. As I find the first one more attractive, I will sketch it out in what follows.

Rosado confronts the “traditional” identification of the following concepts: on the one hand, necessity, a priori and analytic; on the other hand, contingent, a posteriori and synthetic. To do this, he exposes three pairs of contrapositions, namely, necessity and contingency, which he characterizes as a metaphysic distinction, apriori and aposteriori, as an epistemological and analytic and synthetic, as a semantic (57-58). Rosado’s aim is to show in a comparison the inequivalence of the semantic notions with the other two (58), wherein the concept of “analyticity” comes to the fore. Rosado contrasts the definitions of analyticity given by Kant and Husserl. Although Husserl’s definition is regarded as more “solid” (59), it is not assumed. According to Husserl, a statement is analytic if its truth persists even when it is formalized. However, following this definition, some mathematical truths cannot be defined as analytic, e.g. “2 is both even and prime” (59-60). Therefore, Husserl’s notion, which seems to be more syntactical than semantical (60), cannot be followed. On the contrary, Rosado’s definition of analyticity is the following: “A statement is analytic if it is true in a model M and when true in a model M, it is true in any model M* isomorphic to M”, to which he adds the clause that the statement “does not imply or presuppose the existence either of a physical world or of a world of consciousness”. (61). In this sense, the Husserlian definition of “analytical necessity,” which is that of an instantiation of an analytical law, cannot be categorized as analytical. With this definition of analyticity, Rosado “attempts to delimit exactly what distinguishes mathematical statements from other statements” (72).

I think that in this context it is worth looking at the definition of necessity which is almost hidden in Husserl’s work. This definition is not metaphysical, but logical. In his *Ideas* I, necessity appears as a particularization of a general eidetic state of affairs and it is the correlate of what Husserl calls apodictic consciousness (Hua III/1: 19). On its turn, apodictic consciousness is the certitude that a given state of affairs cannot not-be or, to put it in Husserl’s words, “the intellection, that it is not, is by principle impossible” (96).

In the *Logical Investigations*, Husserl already exhibits this treatment with an interesting variation. In the third investigation, Husserl says that an objective necessity entails the subjective impossibility of thinking the contrary or, as he also puts it, the pure objective not-being-able-of-being-another-way, that is, necessity, appears according to its essence in the consciousness of apodictic evidence. Then he states that to the objective necessity corresponds a pure law, whereby necessity means to be on the ground of a law (Hua XIX/1: 242-243). We can then state that the comprehension of contingency is the exact opposite to that of necessity. That is, an objective contingency or a contingent object has the characteristic of being-able-to-be-in-another-way and the corresponding non-apodictic consciousness, both in the form of uncertainty and the possibility of thinking the contrary. However, this does not mean that objective contingency is unrelated to law or even that there are no necessary facts. As Husserl states in *Ideas* I, a contingent-object is limited by various degrees of essential laws and the necessity of existence of consciousness is grounded on an essential generality, through which we can recognize the mentioned subjective-objective characters (Hua III/1: 2; 98). Going beyond Husserl, not the object itself, but its being-contingent is an objective necessity based on the general eidetic law of contingency.

The treatment of the concept of analyticity by Rosado gains meaning in connection with the name he chose for “his philosophical endeavors since the 1970’s,” as an alternative to the term already taken by Karl R. Popper– “critical rationalism” (1). Rosado’s philosophy is analytic (I would not repeat why it is also unorthodox) because it has a strong tendency towards formalism in the sense of logical and mathematical analysis with the only exception being his lesser tendency to discuss physics. He believes that “you cannot do serious philosophy without taking into account the development at least of the three more exact sciences, namely, logic, mathematics and physics, but without committing to or presupposing in any sense the giant meta-dogma of empiricist ideology” (1), that is, that of the inexistence of “universals.”

Now, I think that philosophy does not need to unconditionally consider the latest developments of logic, physics and mathematics. This is clear, insofar as philosophy should not be identified with these specialized and highly technical enterprises. Philosophy’s endeavors can and must have another sense, namely, that of the examination of the fundamental concepts of scientific (in a broad sense that not only includes formal and natural sciences, but also the material eidetic sciences and the rigorous humanities) and everyday knowledge.[2] But this approach must also embrace our practical and emotional understanding in general too.

In fact, this concept of philosophy was present in Husserl since his *Habilitationsschrift*, which Rosado, in accordance with his Platonic point of view, sees as a “dead born child” (87). However, the most significant aspect of this very early text of Husserl does not lie in his unclear position regarding psychologism [through which, however, we can learn a lot in regard to philosophical thinking and which I would not call “mild psychologism” as Rosado does (87, 147, 162)], but in his use of the psychological analysis to clarify the phenomenal character and the origin of a fundamental concept in mathematics, namely, that of the number. For Husserl, philosophy was from the very beginning of his career a psychological analysis, which searches for the “concrete phenomena” related to a concept and the psychical process through which this concept is obtained, namely, abstraction (Hua XII: 292; 298-299). As Husserl’s analysis shows, this search is also carried out in intuition and by testing conflicting theses. In fact, Husserl’s famous argumentative style of the *Prolegomena* makes its first appearance in his *Habilitationsschrift*.

Moreover, the concept of a psychological analysis in Husserl’s *Habilitationschrift* is clearly distinguished from that of a mathematical, logical or even metaphysical analysis (291-292). In this line of thought, I agree with Rosado’s constant affirmation that Husserl’s logic and mathematical ideas do not lose their validity after the so-called “transcendental turn”. However, if we have to talk about a “turn” instead of a penetration of former intentions, or, on the other side, of an unchanged validity of logic and mathematics instead of a modification of this same validity by clarification of its phenomenal character and origin such that it cannot stem from logic or mathematics themselves, then this is not so easily dismissed.[3]

Also, the more developed concepts of categorial intuition and formalization as epistemological groundings of mathematics can only be examined through a phenomenological analysis, for they are processes of consciousness. We have here a more advanced case of the clarifying function of Husserlian phenomenology. Nevertheless, this contribution of phenomenology to the understanding of mathematics is not highlighted by Rosado as something that comes from outside mathematics itself, and in fact, outside any “objectifying science”.

In this same sphere of themes, it appears to me that the famous discussion of the *Prolegomena* presupposes a peculiar attitude of analysis that cannot be understood as pre-phenomenological, as Rosado understands it (150). If we agree with Husserl when he states that the dogmatic scientist does not question the givenness of his objects but just deals with them without further trouble (cf. Hua III/I: 54-55), then the problem of the recognition of universals and the confrontation with logical-psychologism is a problem that originates in the critical or epistemological attitude and its solution demands the clear exercise of reflection and the distinction of the different “data” given to consciousness. I believe that this is not only the true understanding of the discussion in the *Prolegomena*, but also that this is clearly seen in the study of the origins of this discussion in Husserl’s prior philosophical endeavors. Husserl’s philosophy started as a psychological analysis in the sense of his master, Franz Brentano, and only through the imperfection of this psychology in which there was no clear demarcation between psychological objects and logical objects the critique of psychologism became possible. To put it another way: without the prior reflective attitude towards consciousness and the confusion caused by conflating logical objects with psychological objects, i.e., without psychologism, there is no possibility of distinguishing both spheres of objects or to exercise any critique in relation to the psyche and the logical, which will be in fact missing. And the only way to solve this theoretical conflict is by means of a clear reflective analysis, in which the objects of each side are distinguished as they are given in their different sorts of acts of consciousness. The common idea that Husserl’s phenomenology is a consequence of the critique of psychologism seems to me to be false. In truth, it is the other way around.[4]

I am also not convinced that there is a Platonism of ideas in Husserl, as Rosado thinks (4). It is true that Husserl acknowledged the distinct givenness of ideal objects and that he defended his independence from empirical objects. However, this acknowledgement and defense do not make Husserl a Platonist (not even a structural one). So long as logic and mathematics, to mention two “ideal” sciences, deal with their respective subject matter, the sense and limits of their ideal objects are not in question. But when the epistemological problems start to confuse the mind of the scientist, that is, when he reflects on the relation of his objects with knowledge, then his acquiescence fades away. Now, even when the critical reflection on the mode of givenness of mathematical and logical objects shows that these objects are not to be confused with empirical data, this recognition does not amount to Platonism. On the contrary, their mere givenness, that is, the possibility of having something as “ideal objects” persists as a theoretical problem to be decided within the epistemological-phenomenological attitude. The sheer acceptance of the independent existence of these objects, that is, Platonism, cannot be conceded. On the contrary, just as realism of nature succumbs to the phenomenological analysis, so do Platonic ideas. It should be noted that Husserl was neither a psychologist in his early development, nor a Platonist at any moment of his career.

To conclude, I still would like to point out that although Rosado is well aware that for Husserl, first philosophy meant epistemology in the sense of transcendental phenomenology (145), he tries to downplay this determination by contraposing Husserl’s own definition of logic as first philosophy in his 1908 lectures on old and new logic (143-144). There, Husserl states, in effect, that the new logic is “first philosophy” (Hua M6: 7). Nonetheless, this same logic is understood as a dogmatic-positivist discipline in *Formal and Transcendental Logic*: logic can only be a truly philosophical logic says Husserl, as if remembering his lectures of 1908, or first philosophy, when it stays true to its original sense, already present in Plato, i.e., to the broader idea that ends in transcendental phenomenology as transcendental logic (Hua XVII: 17 ff.) Here again, the use of such beloved philosophical tags proves itself deceitful, for this enterprise resembles the empiricist’s traditional aim of exposing the origin of concepts in intuition.

**Bibliography**

Rosado Haddock, Guillermo E. 2018. *Unorthodox Analytic Philosophy*. Texts in Philosophy 27. College Publications. Lightning Source: United Kingdom.

Husserliana

II: *Die Idee der Phänomenologie. **Fünf Vorlesungen*. 1950. Hrsg. Walter Biemel. Martinus Nijhoff: Den Haag.

III/1: *Ideen zu einer reinen Phänomenologie und phänomenologische Philosophie. Erstes Buch: allgemeine Einführung in die reine Phänomenologie*. 1976. Hrsg. Karl Schuhmann. Martinus Nijhoff: Den Haag.

XII: *Philosophie der Arithmetik. Mit ergänzenden Texten (1890-1901)*. 1982. Hrsg. Lothar Eley. Martinus Nijhoff: Den Haag.

XVII: *Formale und transzendentale Logik. Versuch einer Kritik der logischen Vernunft*. 1974. Hrsg. Paul Janssen. Martinuns Nijhoff: Den Haag.

XIX/1: *Logische Untersuchungen. Zweiter Band. Erster Teil. Untersuchungen zur Phänomenologie und Theorie der Erkenntnis*. 1984. Hrsg. Ursula Panzer. Martinus Nijhoff: Den Haag.

Materialen

6: *Alte und neue Logik. Vorlesung 1908/1909*. 2003. Hrsg. Elisabeth Schuhmann. Springer: Dordrecht.

[1] I want to thank R. Andrew Krema for the review of the English of a penultimate version of this text.

[2] I took the idea of everyday knowledge hearing Dieter Lohmar’s lectures about modern epistemology.

[3] In fact, the problem digs deeper, because with the phenomenological clarification we attain the true understanding of the basic objects of science (cf. Hua XVII: 18 or Hua II: 22) or even of non-scientific attitudes, for example, of the world as being a horizon.

[4] I own this line of thought to an idea shared to me by my teacher and friend Antonio Zirión Quijano, who once conjectured that phenomenology does not comes from the critique of psychologism, but that this very critique indeed presupposes phenomenological analysis. If I have been true to Zirión’s intentions in my present development of his seminal idea, any possible error is of course my responsibility, not his.

Peirce and Husserl: Mutual Insights on Logic, Mathematics and Cognition

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Springer

2019

Hardback 103,99 €

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Logic, Epistemology, and the Unity of Science, Vol. 46

Springer

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The Mathematical Imagination: On the Origins and Promise of Critical Theory

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A Richer Picture of Mathematics: The Göttingen Tradition and Beyond

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Science as Social Existence: Heidegger and the Sociology of Scientific Knowledge

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Open Book Publishers

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