Abstract
This paper argues that the shared intersubjective accessibility of mathematical objects has its roots in a stratum of experience prior to language or any other form of concrete social interaction. On the basis of Husserl’s phenomenology, I demonstrate that intersubjectivity is an essential stratum of the objects of mathematical experience, i.e., an integral part of the peculiar sense of a mathematical object is its common accessibility to any consciousness whatsoever. For Husserl, any experience of an objective nature has as its correlate a “we,” which he terms the “community of monads”. Thus, even before mathematical objects gain expression, formalization, and axiomatization through natural and scientific language, from a phenomenological viewpoint their objectivity has its roots in raw pre-linguistic though intersubjective experience. Accordingly, I demonstrate the different senses in which the experience of mathematical objects is permeated by intersubjectivity, suggesting a picture of mathematical intersubjectivity as pre-linguistic common experience based on Husserl’s idea of a “community of monads”.
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Notes
Carr (1974, 86–87).
Citations from Husserl’s works in this paper are in English and accordingly references to page numbers are made to English translations. Abbreviations used for Husserl's works: CM for Cartesian Meditations, LI for Logical Investigations, FTL for Formal and Transcendental Logic, Crisis for The Crisis of the European Sciences.
Here I do not go into the philosophical considerations that led Husserl to change his terminology. In this study, however, I will opt to employ “eidetic intuition” when the topic touches upon eidetic variation. For a discussion of the problematics behind the idea of categorial intuition, see Lohmar (2002, 2006) and van Atten (2015).
Tieszen (2010) calls this unique status “Constituted Platonism,” implying that on the one hand, mathematical objects are in an important sense mind-independent and transcendent, but on the other, the objectivity which gives rise to this mind-independence and transcendence is constituted by consciousness (see also: Tieszen 1989, 2005). Rosado Haddock (2006, 2010) similarly claims that Husserl propounded an “ontological Platonist mathematics” that is opposed to constructivism. Mark van Atten, in opposition, determines that such a view “finds no support in transcendental idealism as Husserl developed and refined it after Ideas I” (van Atten, 2017, 277). He proposes the alternative of Brouwer’s intuitionism as an interpretation of the status of mathematical objects in Husserl’s phenomenology. Throughout this paper, however, the former view of a phenomenological form of Platonism will be further solidified by demonstrating that Husserl’s conception of a monadology supports a type of ‘realism’ or ‘Platonism’, if one accepts that the crux of the sense of objectivity is “being-there-for-anyone”.
The categorial intuition of the individual formal aspects, the original givenness of a multitude of individuals, is based on collecting and comparing. See: Husserl, Philosophy of Arithmetic; For a detailed analysis of Husserl’s account of the constitution of number see: Miller, 1982; Hopkins (2011).
Cf. “One occasionally reads in a treatise that the series of cardinal numbers is a series of concepts and then, a little further on, that concepts are products of thinking. At first cardinal numbers themselves, the essences, were thus designated as concepts. But are not cardinal numbers, we ask, what they are regardless of whether we ‘form’ or do not form them?” (Ideas I, §22, 42.). Husserl differentiates between the concept as a product of thought in a given time and place and the objective content of the concept itself, which is what it is prior to our act of thinking it: “the numerical objectivation is not the number itself, it… is an atemporal being” (Ibid.).
Smith (2003, 137).
It is important to note that geometry is a material eidetic science (Ideas I, §72), a “representative of material mathematics” (§73). Thus, mathematical essences can be either formal or material, but if they are material Husserl calls them “exact essences” and they are subordinated to formal mathematical essences, as well as to the ideal of a consistent and closed apophantic domain governed by axioms. Their exactness in this case derives from fixing them as an ideal limit (“in the Kantian sense” says Husserl) in relation to material inexact essences, and setting them down as axioms for a definite manifold of propositions, i.e., definite and complete axiomatic system (See Ideas I, §74). For discussions on the sense in which “formalization” is involved in the constitution of exact essences, see: Gurwitsch (1974), Drummond (1984), Seebohm (2015).
Though in this case, the application of geometry is, and can never be, direct, since the geometrical shape functions as an ideal Kantian “limit” in relation to the empirical “morphological” shape of the soccer field. The variation is guided by the formal rule of the “and-so-on” toward an ideal limit.
See Roubach (2021) for a discussion of the significance of Husserl’s conception of a mathesis universalis for his phenomenology in general.
Husserl advocates in this context, according to Hartimo (2019), a type of structuralism. Cf. Stewart Shapiro: “A structure is the abstract form of a system, highlighting the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system” (1997, 74).
The scholarship provides diverse views on this subject but can be roughly divided into two positions. The first one does not see the transition from direct intuition of essences to deduction from axioms, and back, as inherently problematic. See, e.g., Sokolowski (1974), Tieszen (1989, 2005), Rosado Haddock, (2006), Seebohm (2015). The second position views the mathematical manifold as a region uniquely defined by a formal deductive system, and as such detached from the lifeworld. Given this situation, the problem is then to explain how Husserl relates the objects of the mathematical manifold to the mathematical essences of formal ontology. Miller (1982) emphasizes the role of Husserl’s later concept of “sedimentation”. Hopkins (2011) thinks that Husserl fails to solve this problem, since he does not succeed to “desediment” the meaning of symbolic mathematics on the basis of the lifeworld. Hopkins claims that he was bound to fail due to his lack of awareness to unclarified ontological assumptions concerning the origin of number. According to Hopkins, these ontological assumptions, however, were uncovered by Jacob Klein. Lastly, Hartimo (2012, 2019) can be viewed as a middle position. She claims that Husserl’s method of Besinnung focuses on describing, clarifying, and revising different normative activities. Accordingly, the gap between formal mathematics and formal ontology should be understood on the grounds of two different scientific activities with different norms. She provides what she terms a “pluralistic” account, claiming that different parts of what we call mathematics are characterized by different criteria of evidence without any unifying norm apart from the aspirations of rationality and science as a whole. It is worth mentioning that in any case most studies, however, use the concept “formal” without clarifying the sense proper to the given context and as a result lead to equivocations of the meaning of “formal”. Miller (1982) is a notable exception.
Here, to the best of my knowledge, I depart from Husserl’s own explicit position on the matter; Regarding the perfect definiteness of axioms, it is unclear to what extent Husserl was aware of Gödel’s incompleteness theorems. In effect, Gödel’s theorems amount to a refutation of Husserl’s vision of a closed definite axiomatization of the mathesis universalis. We will subsequently see, however, that the crux of Husserl’s phenomenology allows it to remain unaffected by the difficulties raised by Gödel.
Tieszen (2005, 32).
Seebohm (2015, 187).
Ibid. 190.
Derrida (1989, 77).
Though my aim here is not to focus on a critique of the interpretation of Husserl’s conception of mathematics as intuitionism, this point also besets the attempt to associate Brouwer’s intuitionism with Husserl’s phenomenology, since for Brouwer, “an account should be rendered of the phases consciousness has to pass through in its transition from its deepest home to the exterior world in which we cooperate and seek mutual understanding. This account does not imply mutual understanding and in some way may remain a soliloquy” (Brouwer, 1949, 1235; Quoted by van Atten 2017, 273). Brouwer claims that intuitionism focuses on essential structural properties of the mind, and so language serves to communicate mathematical constructions that are privately subjective but communicable. The focus on non-particular, essential, properties of the mind is supposed to account for the intersubjective validity of these mathematical constructions. As long as language is the only “bridge” between subjects, however, I cannot see how Brouwer can overcome the problem of relativism with respect to mathematical constructions, not to mention the problem of solipsism (this is intensified even further when one recalls that intuitionism does not presuppose any epochē).
“Preface to the General Science,” 1677, in Leibniz (1951), 15.
“We need not be surprised then that most disputes arise from the lack of clarity in things, that is, from the failure to reduce them to numbers” (“Towards a Universal Characteristic,” 1677, in Leibniz (1951), 24).
“Preface to the General Science,” 1677, in Leibniz (1951), 16.
“Towards a Universal Characteristic,” 1677, in Leibniz (1951), 20.
“And if someone would doubt my results, I should say to him: ‘Let us calculate, Sir,” and thus by taking to pen and ink, we should soon settle the question” (“Preface to the General Science,” 1677, in Leibniz (1951), 15).
“…in thinking of ourselves, we think of being, of substance, simple or composite, of the immaterial and God himself, conceiving that what is limited in us is in him without limits” (“The Monadology”, §30, 1714, in Leibniz (1951), 539).
Ibid. §78, 549.
In this respect, I think that Andrea Altobrando’s brief description of the constitution of the objective world by the monads is imprecise. He writes: “A full-fledged account of objectivity should work as a kind of algorithm of a function in which the factors are all the different subjects, with all their “really” (reell) immanent as well as all their intentional elements, including the ones which pertain to the relationships between the different subjects, and, finally, all the relationships between all, both immanent and intentional, elements of all subjects” (Altobrando, 2021, 294). But as we will see, it is misleading to understand Husserl’s monadology as some kind of algorithm.
As Gödel states in a draft from around 1961: “In what manner, however, is it possible to extend our knowledge of these abstract concepts, i.e., to make these concepts themselves precise and to gain comprehensive and secure insight into the fundamental relations that subsist among them, i.e., into the axioms that hold for them?… The procedure must thus consist, at least to a large extent, in a clarification of meaning that does not consist in giving definitions” (Gödel, 1995, 383). Gödel also thought that a proper development of Husserl’s idea of a phenomenological monadology holds promise to provide solid foundations for mathematics. For a discussion of what a Gödelian monadology (in the Husserlian sense) might have looked like, in the context of the incompleteness theorems and Gödel’s criticism of both Hilbert and Carnap, see Tieszen (2012).
On intuition of mathematical concepts in the context of Gödel, see Tieszen (2002).
Cf. Gödel (1995, 385): “Namely, it turns out that in the systematic establishment of the axioms of mathematics, new axioms, which do not follow by formal logic from those previously established, again and again become evident… it is just this becoming evident of more and more new axioms on the basis of the meaning of the primitive notions that a machine cannot imitate.”.
Sokolowski (1974, 76–77).
Ibid. 77.
Lohmar (2010, 89).
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Cohen, N. Mathematical Objectivity and Husserl’s “Community of Monads”. Axiomathes 32 (Suppl 3), 971–991 (2022). https://doi.org/10.1007/s10516-022-09648-w
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DOI: https://doi.org/10.1007/s10516-022-09648-w