Why I Am Not a Mathematician: Heidegger, Mathemata, and The New York Statesman’s Straw Man
Dr Jonathan Kenigson, FRSA
Note: I have no ill-will toward the editorial staff of the Statesman, but the preparation of my manuscript was obviously deficient and was scarcely readable as presented. Please refer to the work here. It is my hope that UK readers forgive the American dialect of my narrative. This was present in the Statesman and I reproduce it here, with some modifications for clarity. In Brief: I am not a mathematician because I am first a poet, albeit an inarticulate one. Euclidean Geometry is a muse sufficiently lovely that the reproach of mathematics should not be ascribed to her.
Because res extensa is ontologically basic for Descartes, corporal entities are categorized by extension and dimension. Spatiality is the essential characteristic by which objects may be assimilated into the subject’s experience; discovery consists in objectifying experience and forcing it to disclose the a-priori about it (Heidegger 1996). The objectifying mode of understanding becomes the sole and total mechanism by which knowledge about which objects can be understood (Adamczewski 1968; Heidegger 1996). To apply this thinking to paradigmatic physical systems yields a more nuanced perspective on the critique. Newtonian mechanics is a paradigmatic abstractive of the interactions of gravitation among massive bodies. It is not the facticity of the reliance of such mechanics on abstract relations that is objectionable for Heidegger, but the presupposition of the mathemata that “universal” gravitation is ontologically prior to the Being in represents (Karsten 2010). The mathematical odyssey is not flawed in its reliance upon abstraction, but rather on the postulation that Being can be dissected into the beings that characterize subjectiticy and objectivity – objects on a distended space that are separate from their mutual relationality in Being. Theoria in modern mathematics after Galileo is precisely the dissecting tendency – the position of subjects on an inert mental space which is the violence of mathemata as a tendency.
Subjecticity is absent in Heidegger because selfhood is not grounded in a coincidentia oppositorum. Selfhood is a purely factical construction determined by the internalization of a dualistic metaphysics and a construction of the facticity of thrownness in Being (McDaniel 2013). By ‘thrownness,’ it is intended as the inscrutable factical orientation of Dasein in time – between past and present (Adamczewski 1968; Heidegger 2009). The ready-to-hand nature of knowing cannot be reduced to the facticality of phenomena, because, while such reduction forces a disclosure of some a-priori mathemata about a being, it must be an apophantic (derivative and assertive) characterization (Heidegger 1996). The apophantic character of subjecticity is in the identification of dismembered otherness that is characterized in the axiomatization of phenomenal reality. Primordial understanding cannot be obtained from such a mode of inquiry: because the object and subject separate in the present-at-hand mode of relation, primordial ontology is of necessity self-reflective and focused on the concernful realization of Dasein’s being (Adamczewski 1968). Mathemata as a mode of understanding the world is essentially epistemological reduction. For Heidegger, numbers, as well as geometric figures, are imbued with relations (relata) describing the relations of the manifold (Karsten 2010; Roubach 2008). The postulation of axiomatization as a totalizing characterization of figure or number creates a system of objectification. The figure is presumed complete in-itself in its taxonomic heritability of descriptions given by the axiom scheme. But the disclosive nature of the figure is overlooked in the mathematical process – the predefined relations native to the mathematical program informing simplifications in calculation at the expense of the ready-to-hand mode of relation (Elden 2001). The platonic construction founded in modern mathematics that defines the figure as a continuum misses the possibility that there may exist a potential infinitude of self-disclosive entities among the points that are not defined by the relations formally defining the figure (Elden 2001). The relational characterization of the figure is consequently grossly incomplete, although not inaccurate.
The mathematician as Dasein is inherently involved in mathematics in a manner much more profound than as a mere systematizer, observer, or discoverer. The mathematician’s presumed subjecticity is apophantic of the participation-in-Being necessitated by the thrownness of Dasein in the primordial world (Adamczewski 1968). Dasein is not a distinction between subject and objects of knowledge, but a continual process of participation – “Being In the World” (Carman 2003). Mathematical relationality is a-spatiotemporal, abstractive, and non-illustrative of the contexts in which mathematics is daily operative. The subjectum is founded in sorge (concernful dealing) for Being – a process not inimical to formalization, but not bound or ontically transcended by it. The presupposition of sorge within Being is in its lostness in the proximal relations of Being that disclose themselves through process (Heidegger 1996). Plane geometry is not merely and totally in-itself, but also in its relations to the everyday contexts which led to its eventual abstraction from daily experience through the present-at-hand mode of being (Elden 2001). The “line” formed by the crease of a paper; the “circle” formed by the disk of the moon; and the figures representing these forms in mathematical practice are hardly equivalent (Elden 2001). Circumscriptive forms of reality-reduction are ultimately decouplings of primordial relationality (Heidegger 1996). The existential essence of the mathematician – as Dasein – is, much in the same manner as the Heisenberg Principle, an interactive reality. By observing “reality” (in the Cartesian sense), one invariably changes it, because no matter how inescapably dramatic the epoche, one cannot remove oneself from the ‘inert’ matter of the object (Carman 2003). No such distinction is ontologically warranted. Inasmuch as there is no projectivity in the distinction between the mathematician and mathematics, there is also no defensible ontological distinction between the practitioner and the practice; one is always among the other, and not over and against the other (Adamczewski 1968; Roubach 2008). The existential essentiality of mathematics is thus intimately grounded in the existential nature of the mathematician. There is no dialectic between observer and observed. Being-In-The-World is not prescriptive of the predicament of the subject-object mode of relation, which is, in Heideggerian terms, a derivative mode (Heidegger 1996).
How, if at all, can mathematics be rescued from dualism? How can subjecticity cease to be the sole basis for mathematical practice? Can art abnegate the hegemony of subjecticity for mathematics as it can be held to do for Dasein (Lopez & Rosario 2009)? As with many other aspects of Heidegger’s thought, Being and Time is an outstanding exponent of his mature philosophy before ‘the turn’ (Abbott 2010; Gier 1981). It is in Being and Time that Heidegger first pronounces the isometries between the thought of Galileo in physics and Descartes in epistemology and explored some alternatives to mathemata and its attendant addiction to subjecticity (1996). Because a phenomenological ontology of Being is the only way Dasein can be delivered from platonic metaphysics (see McDaniel 2013), it follows that mathematics, a domain in which Dasein is really and presently active, can only be reconstituted using the same means (Heidegger 1996). Phenomenology must transcend the traditional opposition of eidos and phenomenon, and the Heideggerian agenda is to replace the cogito with Being-In-The-World (Ihde 2010). The cogito’s destruction should spell the timely death of platonic ontology and its fulfillment in the age of techne, and the beginning of a program of unconcealment (Heidegger 1996). The obviation of time in the platonic tradition (see Heidegger1977) culminates in the ultimate irony of the Calculus – a mathematics concerned with change in-time but presumed to be totally separate and external to it. It is temporality which unifies existential experience and makes Being the possible object of sorge. Being-In-The-World is also the mathematician in practice, just as it is the artist in the process of creation. Isness is ontologically prior to the mathemata, and cannot yield a-priori knowledge of it. The artistic, with its temporal orientation, can be presumed to rescue mathematics if it only discloses the temporality of the mathematical (Heidegger 1996). Mathematical physics is the product of an age, not a set of timeless truths; time itself is the discloser of truths through Being, and not the arbiter of the truth of Being (Carman 2003; Heidegger 1977).
It is fitting to regard the Heideggerian perspectives on mathemata as methodologically self-contradictory; his own abstraction in dealing with the isomorphism between Cartesian and mathematical ontology leave great latitude for interpreting how projectivity in mathematics could be redefined to permit a greater practical authenticity (Carman 2003). Characteristically, one should regard an ideological illness within a discipline as not native to the discipline but the ontological agenda of Dasein – the practitioner (Adamczewski 1968; Heidegger 1996). I contend that a Heideggerian mathematics would not in any way abolish the abstraction and proof process undertaken within the discipline but would rather abolish (in Heideggerian terms) the hubris native to believing that mathematics – or theoria, for that matter – can form a totalizing system of explanation for first causes or definitions in the universe (Heidegger 1996). The axiomatization of systems is not to be abolished, but rather the presumed disposition that avers that Being obeys axioms. For instance, the very search for a Grand Unified Theory (GUT), or “Theory of Everything” in cosmology would, in the Heideggerian ontology, be fraught from the start: even if one could presume to describe all physical interactions in the cosmos, and to categorize all particles that could ever exist in it, one would still not have succeeded in subjecting Being to Idea. One would have succeeded in subjecting one scheme of axioms to another – a task that seems disturbingly exiguous in comparison to the presumed aims of the project. The study of Being is, at its essence, always a phenomenological investigation of the being that posits it (Ihde 2010).
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