Abstract
Husserl is well known for his critique of the “mathematizing tendencies” of modern science, and is particularly emphatic that mathematics and phenomenology are distinct and in some sense incompatible. But Husserl himself uses mathematical methods in phenomenology. In the first half of the paper I give a detailed analysis of this tension, showing how those Husserlian doctrines which seem to speak against application of mathematics to phenomenology do not in fact do so. In the second half of the paper I focus on a particular example of Husserl’s “mathematized phenomenology”: his use of concepts from what is today called dynamical systems theory.
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Notes
See Petitot et al. (1999); especially the contributions by Petitot, Van Gelder, and Varela.
See Tieszen (2005), ch. 6.
See Mancosu and Ryckman (2002).
See Mancosu and Ryckman (2002).
See Miller (1984).
See Tieszen (2005), ch. 3.
Though Husserl does not put it this way, P is a map on the set of possible intentional states which can be used to partition it into the infinite sequence of equivalence classes “perception, first-order imagination, second-order imagination, etc.” Note that the set of intentional acts is closed under P, no result of the presentation operator will ever fall outside the class of intentional states.
The equation can be interpreted in two ways, both of which are problematic. On one, more natural interpretation, i and s represent separately varying phenomena which constrain one another. This interpretation implies that as the intuitive weighting of a perceptual act increases its conceptual weighting diminishes (Compare any equation in which the sum of two quantities equals a positive constant; e.g. an equation which says that total number of predator and prey equals 100, so that as more predators enter a population there are less prey, and vice-versa.) But this is problematic, insofar as one very often learns more about an object as he or she sees more of it, which suggests that both the intuitive and conceptual components of an act can simultaneously increase. On another interpretation, the equation is simply describing a gradient from the case where i = 0 to the case where i = 1. Compare an equation which says that a square can be all white, all black, or some level of gray (white + black = 1). In that case we only have a single phenomenon which varies – let us say, amount of black coloring. The other variable, which represents absence of the first property, “tags along,” as it were, insofar as it does not represent an independently varying phenomenon. However, one would not normally introduce an “equation” to describe a simple gradient, and doing so in this case is misleading, for it implies the first interpretation.
The study of numerical representations and their properties is an area of research which began during Husserl’s lifetime (with Von Helmholtz) and continues today. See Narens (2002). On this type of analysis one would explicitly introduce a function from perceptual acts to numbers, which Husserl does not do (though, as we shall see, he was very much aware of the ontological distinction between real phenomena and their symbolic representation). As a result, he is sometimes not clear whether he is speaking of relations between actual mental states or between their symbolic representations.
Husserl is not clear whether the B i name the perceptual acts themselves, their degree of perfection, their “image” components, or some other “inner moment” of the acts.
Husserl bases the ordering relation on distances between perceptual acts. This assumes that the distances are more fundamental than the ordering relation. However, this need not be the case. In fact one often finds ordering relations without distances, as in the case of preference orderings. Distances are only meaningful in interval and ratio scales, which are stronger than ordinal scales (Narens 2002).
For formal treatment of Husserl’s theory of time-consciousness see Miller (1984).
For comprehensive discussion of Husserl’s theory and critique of science, including references to the earlier literature, see Tieszen (2005), especially chapter 1.
Husserl faced personal and professional crises in this era. He lost a son and a close student (Adolf Reinach) in the Great War and faced persecution by the Nazis in his last years. Husserl also faced a crisis in his own work – he was coming out of a decades-long period of public silence and personal insecurity, during which he had seen his philosophy eclipsed by such diverse trends as existentialism, psychoanalysis, behaviorism, and naturalism (see Welton (2000), introduction and ch. 5). Moreover, his former protégé Heidegger had turned against him, at first privately, but with increasing virulence in the late 20’s and 30’ – events which culminated, famously, in Heidegger’s denying Husserl the use of the university library at Freiburg and removing the dedication to Husserl from Being and Time (see Husserl 1997b).
For definitions and discussion of these terms, see (Husserl 1970, pp. 68, 292).
These passages are late reflections of a long-standing concern on Husserl’s part that ideal constructions not be confused with directly perceived entities, and that reified methodological constructions not be confused with the conscious processes which give them sense. As Dallas Willard, citing Husserl’s 1891 review of Schröder, points out, the “theme of a lack of self-understanding – and even of self-deception – on the part of logic and logicians is no mere afterthought on the part of the later Husserl. Rather, it is one dealt with in detail in the early writings” (Willard 1979, pp. 144–145). Themes similar to those found in Crisis are also prefigured in Ideas 3 (Husserl 1980b).
For discussion see Willard (1980, p. 60 and following).
Roughly through the number ten (Husserl 2003, pp. 236–237).
In the Philosophy of Arithmetic number-concepts and (in part II) axiom systems were at issue. In the Logical Investigations, Husserl studies how the constructions of formal logic and language, in particular concepts, propositions, arguments, truths, proofs, and judgments, originate in processes of rational cognition. He later described the Investigations as involving a “turning of intuition back toward the logical lived experiences which take place in us whenever we think...to make intelligible how the forming of all those mentally produced formations takes place in the performance of this internal logical lived experiencing” (Husserl 1975, p. 14). Similar concerns are at work in Husserl’s late Analyses of Passive and Active Synthesis, Formal and Transcendental Logic and Experience and Judgment. In these works Husserl says, for example, that the logical categories of subject and predicate originate in perceptual experiences of things with properties. Formal and Transcendental Logic (part 1, chapter 1) contains a striking account of the phenomenology of logical inference. In his last work, the Crisis of European Philosophy, especially section 9, Husserl outlines a detailed phenomenology of scientific practice, as we saw above.
See Ideas sections 10 and 13, and Formal and Transcendental Logic sections 24–29.
However, Husserl does leave open whether one might pursue a kind of exact science of consciousness as a counterpart to descriptive phenomenology, which would describe perfect mental processes and their ideal structure “[There is at present no answer to] the pressing question of whether, besides the descriptive procedure, one might not follow – as a counterpart to descriptive phenomenology – an idealizing procedure which substitutes pure and strict ideals for intuited data and might even serve as the fundamental means for a mathesis of mental processes” (Husserl 1980a, p. 169).
Husserl makes these points vividly in connection with his critique of Descartes, who, on Husserl’s analysis, suffered from the “prejudice” of believing that “under the name ego cogito, one is dealing with an apodictic ‘axiom’, which, in conjunction with other axioms... is to serve as the foundation for a deductively ‘explanatory’ world science, a ‘nomological’ science... similar indeed to mathematical natural science” (Husserl 1960, p. 24).
See (Husserl 1991, pp. 45–46), where Husserl “deduces...in conformity with law” a formula describing the ordering of experiences in memory. Husserl gives a simpler example in Ideas, where he says “we can, for instance, predicate in a ‘blind’ way that 2 + 1 = 1 + 2; we can, however, carry out the same judgment with insight. The positive fact [that 2 + 1 = 1 + 2]... is then primordially given, grasped in a primordial way” (Husserl 1980a, pp. 350–351).
Husserlian manifolds are commonly taken to be models in the contemporary sense, though there are some differences; see (Smith 2002). Models are ordered pairs consisting of a domain of objects and an interpretation function which maps from the language to the domain. Husserl’s manifolds correspond to the domain; he does not, to my knowledge explicitly introduce an interpretation function. Moreover, Husserlian manifolds correspond to intended interpretations of a theory, whereas set theoretic manifolds can consist of any arbitrary set of objects which satisfy the corresponding theory.
As Husserl says, “The axiom system formally defining such a manifold is distinguished by the circumstance that any proposition... that can be constructed...out of the concepts...occurring in that system, is either... an analytic consequence of the axioms... or an analytic contradiction” (Husserl 1969, p. 96). We thus have the contemporary concept of a complete theory. It is worth emphasizing that not all mathematical theories are complete. Husserl himself realized this, taking definite manifolds to be an especially strong forms of manifold and the theories which describe them (“nomologies”) to be especially strong forms of theory.
Husserl thought of Euclidean geometry as the a priori science of “intuited world space,” Formal and Transcendental Logic, section 29; also see Analyses of Passive and Active Synthesis, section 31.
A differential equation is a dynamical system if it satisfies the existence and uniqueness theorem for differential equations.
See Beer (2000).
This map must meet two conditions in order to qualify as a dynamical system: (1) for all states s in S, ϕ(s, 0) = s, and (2) for all states s in S and times t 1, t 2 ϕ (s, t 1 + t 2) = ϕ (ϕ (s, t 1), t 2).
One important caveat to this discussion is that dynamical systems are essentially closed, in that in order for the map ϕ to be well defined we cannot allow external, unpredictable sources of input in to the system. But of course the mind and brain are open, in the sense that they are tightly coupled to the external environment. In ongoing work with a mathematician (Scott Hotton) I am analyzing such systems with the explicit goal of providing a framework for formalizing dynamical claims in phenomenology and neuroscience.
For recent discussion of the method of free variation, with reference to earlier works, see Tieszen (2005).
The calculus of variations is a technique for finding paths that minimize or maximize a function defined on a set of curves. In effect, the mathematician considers all possible curves connecting two points in an abstract space to find, for example, a path of least action across a surface. There are notable differences between free variation in phenomenology and the calculus of variations. Phenomenological free variation does not involve any explicit numerical calculation, and neither does it issue in a single solution to some equation (in the calculus of variations one solves a differential equation to identify a single path which maximizes or minimizes a function). Nonetheless, both techniques involve variation in a space of possibilities to secure mathematically rigorous results.
Both kinds of manifold are treated of explicitly in the manuscript “On Sets and Manifolds” of 1893. See (Husserl 1979), section II.
Husserl makes the point more famously in Ideas, where he describes the possible annihilation of the world as involving a disruption in the orderly stream of experience, whereby it would “lose its fixed regular organizations of adumbrations, apprehensions, and appearances” (Husserl 1980, p. 109).
In the most detail in Mapping Supervenience (unpublished manuscript), but also in (Yoshimi 2001).
I am grateful to Dagfin Føllesdal, Scott Hotton, Rolf Johansson, David Kasmier, Wayne Martin, Ronald McIntyre, David Woodruff Smith, Richard Tieszen, and Dallas Willard for helpful comments.
References
Abraham, Ralph, & Shaw, C. (1982). Dynamics – The geometry of behavior, (4 Volumes). Santa Cruz, CA: Aerial.
Barušs, I. (1989). Categorical modeling of Husserl’s intentionality. Husserl Studies, 6, 25–41.
Beer, R. (2000). Dynamical approaches to cognitive science. Trends in Cognitive Science, 4, 91–99.
Blecksmith, R., & Null, G. (1991). Matrix representation of Husserl’s part-whole-foundation theory. Notre Dame Journal of Formal Logic, 32, 87–111.
Chokr, N. (1992). Mind, consciousness, and cognition: Phenomenology vs. cognitive science. Husserl Studies, 9, 179–197.
Da Silva, J. J. (2000). Husserl’s two notions of completeness. Synthese, 125, 417–438.
Dreyfus, H. (Ed.) (1984). Husserl, intentionality, and cognitive science. Cambridge, MA: MIT.
Fine, K. (1995). Parts and wholes. In: B. Smith, & D. W. Smith (Eds.), The Cambridge companion to Husserl. Cambridge: Cambridge University Press.
Hill, C. O., & Haddock, G. E. R. (Eds.) (2000). Husserl or Frege? Meaning, objectivity, and mathematics. Chicago, IL: Open Court.
Husserl, E. (1960). Cartesian meditations. The Hague: Nijhoff.
Husserl, E. (1962). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy. First book: General introduction to a pure phenomenology. New York: Collier.
Husserl, E. (1969). Formal and transcendental logic. The Hague: Nijhoff.
Husserl, E. (1970). The crisis of European sciences and transcendental phenomenology: Introduction to phenomenological philosophy. Evanston, IL: Northwestern University Press.
Husserl, E. (1973a). Logical investigations. London: Routledge and Kegan Paul.
Husserl, E. (1973b). Experience and judgment. Evanston, IL: Northwestern University Press.
Husserl, E. (1975). Phenomenological Psychology. Lectures, summer semester 1925. The Hauge: Nijhoff.
Husserl, E. (1979). Aufsätze und rezensionen (1890–1910). Dordrecht: Springer.
Husserl, E. (1980a). Ideas pertaining to a pure phenomenology and to a phenomenological philosophy. First Book: General introduction to a pure phenomenology. Dordrecht: Springer.
Husserl, E. (1980b). Ideas pertaining to a pure phenomenlogy and to a phenomenological philosophy. Third book: Phenomenology and the foundations of science. Dordrecht: Springer.
Husserl, E. (1981). Shorter works. Notre Dame: University of Notre Dame Press.
Husserl, E. (1983). Studien zur arithmetik und geometrie. Dordrecht: Springer.
Husserl, E. (1991). On the phenomenology of the consciousness of internal time. Dordrecht: Springer.
Husserl, E. (1997a). Thing and space: Lectures of 1907. Dordrecht: Springer.
Husserl, E. (1997b). Psychological and transcendental phenomenology and the confrontation with Heidegger 1927–1931. Dordrecht: Springer.
Husserl, E. (2001a). Die ‘bernauer manuskript’ über das zeitbewußtsein (1917/18). Dordrecht: Springer.
Husserl, E. (2001b). Analyses of passive and active synthesis. Dordrecht: Springer.
Husserl, E. (2003). Philosophy of arithmetic. Dordrecht: Springer.
Krysztofiak, W. (1995). Noemata and their formalization. Synthese, 105, 53–86.
Majer, U. (1997). Husserl and Hilbert on completeness. Synthese, 110, 37–56.
Mancosu, P., & Ryckman, T. A. (2002). Mathematics and phenomenology: The correspondence between O. Becker and H. Weyl. Philosophia Mathematica, 10, 102–129.
McIntyhre, R. (1986). Husserl and the representational theory of mind. Topoi, 5, 101–113.
Mensch, J. R. (1991). Phenomenology and artificial intelligence: Husserl learns Chinese. Husserl Studies, 8, 107–127.
Miller, I. (1984). Husserl, perception, and temporal awareness. Cambridge, MA: MIT.
Narens, L. (2002). Theories of meaningfulness. Mahwah, NJ: Erlbaum.
Perko, L. (1996). Differential equations and dynamical systems (2nd ed.). New York: Springer.
Petitot, J. (1999). Morphological eidetics for a phenomenology of perception. In F. Varela, et al. (Eds.), Naturalizing phenomenology. Stanford, CA: Stanford University Press.
Petitot, J., Varela, F., Pachoud, B., & Roy J-M. (Eds.) (1999). Naturalizing phenomenology. Stanford, CA: Stanford University Press.
Sharoff, S. (1995). Phenomenology and cognitive science. Stanford Humanities Review, 4, 189–204.
Smith B. (Ed.). (1982). Parts and moments: Studies in logic and formal ontology. Munich: Philosophia Verlag.
Smith, D. W. (2002). Mathematical form in the world. Philosophia mathematica, 10, 102–129.
Smith, D. W., & McIntyre, R. (1982). Husserl and intentionality. Dordrecht: Reidel.
Tieszen, R. (1989). Mathematical intuition: Phenomenology and mathematical knowledge. Dordrecht: Kluwer.
Tieszen, R. (2005). Phenomenology, logic, and the philosophy of mathematics. Cambridge, MI: Cambridge University Press.
Tragesser, R. (1977). Phenomenology and logic. Ithaca, NY: Cornell University Press.
Tragesser, R. (1984). Husserl and realism in logic and mathematics. Cambridge: Cambridge University Press.
Van Gelder T. (1996). Wooden iron? Husserlian phenomenology meets cognitive science. Electronic Journal of Analytic Philosophy 4.
Welton, D. (2000). The other Husserl: The horizons of transcendental phenomenology. Bloomington, IA: Indiana University Press.
Willard, D. (1979). Husserl’s critique of ‘extensional’ logic: A logic that does not understand itself. Idealistic Studies, IX, 142–164.
Willard, D. (1980). Husserl on a logic that failed. The Philosophical Review, 89, 46–64.
Willard, D. (1984). Logic and the objectivity of knowledge: A study in Husserl’s early philosophy Athens, Ohio: Ohio University Press.
Yoshimi, J. (2001). Dynamics of consciousness: Neuroscience, phenomenology, and dynamical systems theory. Doctoral dissertation: UC Irvine.
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Yoshimi, J. Mathematizing phenomenology. Phenom Cogn Sci 6, 271–291 (2007). https://doi.org/10.1007/s11097-007-9052-4
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DOI: https://doi.org/10.1007/s11097-007-9052-4