Abstract
Why is nature amenable to mathematical description? This question has received attention in the philosophy of science but rarely from a phenomenological perspective. Nevertheless Husserl’s late essay “The Origin of Geometry,” which has received some critical scholarly attention in recent years, contains the beginning of a striking answer. This answer proceeds from Husserl’s main claim in that essay, which he also makes in the Crisis of the European Sciences, that the original meaning of science has been covered over or “sedimented” by concepts that obscure the true intentional core of scientific meaning. In the first three sections of this paper I develop Husserl’s central insights about mathematics in light of two contemporary critiques of his project of “reactivation” of the original, sedimented meaning of science. In the latter two sections, I argue that accepting Husserl’s account of the original meaning and development of science offers a promising explanation of why nature is amenable to mathematics. This explanation hinges on a conception of the objects and methods of mathematics and the mathematized physical sciences as accomplishments, that is, as constituted contents of consciousness.
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Notes
The manuscript, composed in 1936, is published as Beilage III in Biemel’s edition of the Crisis (Hua VI). The title comes from Eugen Fink’s publication of the manuscript (omitting the two opening paragraphs) under the title “Die Frage nach dem Ursprung der Geometrie als intentionalhistoriches Problem” in Revue Internationale de Philosophie, Vol. 1, No. 2 (1939), a title that is clearly adopted from the beginning of the third paragraph of the essay (HuaVI, p. 365). In his translation, Carr (1970, p. 353n) notes that the opening paragraphs of the essay imply that the text was intended for inclusion in the Crisis, a claim that gains plausibility through the consideration that although it comes to us as a freestanding text, the essay clearly elaborates and serves as the basis for much of Husserl’s transcendental-historical treatment of modern science in the book, especially in §§ 8–9.
Jacob Klein characterizes Husserlian phenomenology as radical rather than archaeological, since Husserl was concerned with seeking the roots (rizomata) from which things grow into perfect shape (arche) rather than the perfect shape itself (Klein 1940, p. 147). Nevertheless, I retain the term “archaeology” to emphasize the goal of uncovering through phenomenological analysis that is characteristic of Husserl’s philosophy.
In addition to the dubious comparisons to Freud and Hegel, Hacking’s statement that Husserl was defending the objectivity of the sciences is confused. For the “objective science” that is assumed in the natural attitude is precisely the conception of science that Husserl seeks to critique and it is the dualism of subject and object that is supposed to be overcome through the reactivation of the original sense of geometry.
This orientation—away from factual history and towards historically constituted a priori conditions in the phenomenological reconstruction of the history of exact science—motivates Husserl’s conception of the historical task of the philosopher. The task is to critique science “from the inside” of the tradition of science’s teleological becoming rather than to discern facts from “outside” the becoming in which the philosopher’s own thought evolved. For this task, Husserl rejects the outside explanations of factual history, i.e., the determination of the external causal series that gives rise to science. For only when understood from the inside does this teleology present itself as a philosophical task that is truly our own, that is, as a way of understanding, from the perspective of the infinite task of philosophy, the tradition as it has been handed down to us (Hua VI, p. 72/70-71).
Although it is beyond the scope of this paper, Husserl conceives his archaeological project not just as an explanation of the sciences but as an investigation into the possibility of philosophy. This presupposes a conception of philosophy as an infinite task, i.e., as episteme or “rigorous science” rather than as doxa or mere cultural criticism. Indeed Gurwitsch has gone so far as to claim that Husserl’s late work is meant to help contemporary philosophers understand their own specific roles within the infinite task of thinking (Gurwitsch 1966, p. 403).
This point about the commensurability of geometrical objects was made forcefully to me by Duane Lacey during a colloquium at the United Arab Emirates University in February 2011. Professor Lacey impressed upon me that in Euclid the concepts of relation and of rationality are closely associated. The word logos was translated into Latin as ratio, and indicates that two objects being compared are commensurable. Similarly, “irrational” translates alogon, which indicates the incommensurability of some object with a measure. A magnitude can be irrational only in relation to an object composed of incommensurable unit lines.
Indeed, because of its enduringness Aristotle considers intellectual activity and contemplative pleasure to be divine. See Nicomachean Ethics X.7, pp. 1860–1862 in Barnes (1984).
Husserl here leaves open the question of whether a counterpart science to this singular geometry is possible. It seems to me that there is reason to believe that Husserl holds that phenomenology is just such a counterpart, a scientific method that treats the plena directly, rather than treating them indirectly as correlates of the “real” shapes of nature. The analogy between geometry and phenomenology is underwritten by Husserl’s description of mathematical natural science as apodictic thinking that proceeds in a stepwise manner towards an infinite world (Hua VI, p. 19/22). In Ideas I, Husserl describes phenomenology as a science, the singular method of which is reduction, a progressive application of the epoché that suspends content not immanent in the directly experienced phenomenon (Hua III, pp. 59–60). Both mathematical physics and phenomenology are then stepwise procedures that proceed on the infinite horizon of science. But while mathematical physics looks for the formal “reality” behind the appearances, phenomenology suspends such transcendent interpretations of phenomena in an effort to penetrate “to the things themselves.”
The arithmetization of geometry in particular was the crucial mathematical development that served as a condition of the possibility of physical measurement. Not only did arithmetization allow for the treatment of measured values to be related to the geometrical models of physics, but it allowed for the increasingly sophisticated symbolic procedures to be developed and then deployed in those models. In a philosophical analysis that in many respects echoes Husserl’s transcendental historical treatment, Hans Jonas argues that “[t]he ostensible return to the Pythagorean-Platonic ‘geometrization’ of the world at the beginnings of modern sciences somehow masked a novel approach, which only the ‘algebraization’ of physical description increasingly revealed. Algebra applied to geometry, rather than classical geometry itself, became the mathematics of the new physics. This alone would suggest that it no longer dealt with the intuitive objects of Greek ontological speculation” (Jonas 1966, p. 67). As argued in the second section, Husserl similarly holds that the shift from intuition to calculation leads to an “emptying of meaning” of mathematics, in the sense that it becomes possible to treat mathematical thinking itself as an algebraic, mechanical process (Hua VI, p. 44/44-45).
Since \(c{\Delta}t^{'} = 2\sqrt {l^{2} + \frac{{\left( {v{\Delta}t^{'} } \right)^{2} }}{2}} \quad {\rm and}\;\; {\Delta}t^{'} = \frac{2l/c}{{\sqrt {1 - {\raise0.7ex\hbox{${v^{2} }$} \!\mathord{\left/ {\vphantom {{v^{2} } {c^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${c^{2} }$}}} }} = \frac{{\Delta}t}{{\sqrt {1 - {\raise0.7ex\hbox{${v^{2} }$} \!\mathord{\left/ {\vphantom {{v^{2} } {c^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${c^{2} }$}}} }}\).
See Kogut (2001, pp. 7–10) for full development of the model.
The mirrors of the model are neither good descriptions of a counterfactual world nor bad descriptions of the actual world. Rather they abstract away from the bodily nature of mirrors in order to determine laws about what the physicist is interested in, viz., the relation of physical systems under the postulates of the model. Hence the motion under examination is also idealized: it is assumed that there is no gravitational interference, that light moves through a vacuum rather than through any particular medium, and that it moves continuously along a straight line. Clearly such assumptions are a matter of convenience: e.g., by assuming a vacuum, the model essentially treats a limit case of all cases of the movement of light, i.e., one in which the light travels at the maximum speed specified in the second postulate. Similarly, the straight line traveled by light abstracts away from its wave nature. For the model to work, one must simply realize that at the scale described by the model such behavior is “negligible.” This kind of idealization is so transparent that there is no need to analyze it philosophically.
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Humphreys, J. Husserl’s Archaeology of Exact Science. Husserl Stud 30, 101–127 (2014). https://doi.org/10.1007/s10743-014-9148-y
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DOI: https://doi.org/10.1007/s10743-014-9148-y