Why I Am Not a Mathematician: Heidegger’s Charge
Dr Jonathan Kenigson, FRSA
The study of subjectity is one of the most well-trodden domains of ontology; it is indeed hackneyed to raise the myriad objections to Cartesian dualism found, it seems, in every quarter of the Continental tradition for the past 30 years (Polt 1999). Heideggerian perspectives on Cartesian ontology have been particularly well-studied, especially in the contexts of language, mind, and substance (Adrin 1999). However, comparatively few studies have sought to apply a Heideggerian perspective to the study of mathematical ontology (Gall 2014; Karsten 2010). In the current brief work, I seek to relate Cartesian and Heideggerian notions of subjectity to the mathematical sciences, arguing that Heideggerian ontology of mathematics is grounded in the Being-In-The-World construct, and that Heidegger advances a notion of projectivity for the mathematical sciences long before this notion became fashionable in general epistemology. I will explore Heidegger’s critique of mathematical ontology in terms of his corresponding critique of ontotheology – the platonic perspective of subject that finds its most mature expression in Penses (Adrin 1999; Heidegger 1996; Karsten 2010).
It was Descartes whose epistemological foundation for all sciences consisted of the cogito – or, in Sartrean existential terms, a positing of essentiality preceding existence (Heidegger 1996; Richardson 2003). The cartesian subject consists of res cogitans and eschews any direct grounding in res extensa; it is precisely cogitation and its attendant reflexive perceptions that constitute the single, irreducible ground for epistemological reality (Guignon 2006; Heidegger 2008; Rorty 1991). Cogitation, then – in the sense of systematic, directed mental action – is the foundation – or hypokymenon – for subjectity (Malpas 2006). For Heidegger, the extant subject in the Cartesian tradition is the fulfillment of a platonic ontological agenda – a triumph of mental over actual substance, and a triumph of platonic over Aristotelian understandings of Being (sein) (Guignon 2006; Heidegger 1996). In Aristotelian and Scholastic ontology, sein is grounded in both physical and mental substance – a perspective articulated in the Nicomachean Ethics and carried forward in the Augustinian holism of Being. A consequence of Cartesian ontology is not only the division of the cogitating self from the object of selfhood, but also a general division between subjects and objects that finds its form in the atemporal objectivism of the modern mathematical sciences. Heidegger’s challenge to the mathematical sciences’ inherited notion of the subjectivizing tendency is bipartite: First, modern mathematics, Heidegger contends, is the inheritor of an entire tradition of mathemata, and is not capable of penetrating sein to its most profound levels (Harman 2002; Heidegger 2001). Additionally, the emphasis upon the use of technology to subjugate the natural world and other Dasein is an inheritor of the mode of thought grounded in the mathemata (Heidegger 1977; Karsten 2010).
For Heidegger, subjecticity in mathematical practice is manifest in the presupposition of the ontological character of sein found first and foremost through experiential reality – and that the knowledge one can expect to attain from mathematical discourse, while not flawed, is not sufficiently fundamental to constitute a foundation for understanding the subjecticity of sein or Dasein in the world (Gall 2014; Heidegger 2001; Shapiro 2007). Galilean science marked a departure from the ancient to the modern by demanding that phenomenological analysis is precisely mathematical analysis, and that mechanics is the sole arbiter of ontological claims (Heidegger 2008; Seel 1994). For Heidegger, only entities that can be quantized are accorded ontological status – an approach radically different from Aristotelian physics, in which the universe of knowledge is comparably relational but irreducible to mathematical formalism (Harman 2002; Karsten 2010). The modern approach is charged by Heidegger to make nature nothing more than interchangeable forces. Axiomatization of mathematical systems in self-disclosive deductive forms is equivalent to reducing truth to propositional truth – a reduction which, much like Husserl’s epoche, is given unfounded primacy (Heidegger 2001; 2008).
The indwelling of Dasein in sein is the most fundamental, relational, ontological property in Heidegerrian anthropology; much as the Copernican Revolution in Kantian thought turned the cogito to be the center of sensory experience, it is the embedding of man in the present-to-hand relationships of reality that is the expression of Dasein’s self-awareness (Adluri 2013; Heidegger 1977). Dasein is the only being capable of positing a res cogitans, the only being with the ability to pose the question of his own being. Epistemology grounded in subjecticity is inherently incapable of apprehending Being as it is in its relationships and lived reality (Heidegger 1996). Speaking phenomenologically with respect to mathematics, subjecticity is the foundation of the mathematical formalization of relata – the determination of definitive relations among aspects of scientific theories of reality. Cartesian ontology is the ontology of modern mathematics. The relata themselves are permeable to mathematical analysis only inasmuch as ideality trumps actuality in modern science – the predictions of natural laws are held to apply categorically and in all situations, and to be fundamentally explanatory of phenomena as well as predictive of them. Instead of a phenomenological ontology of Being, platonic metaphysics bequeaths mathematics with a totalizing, penetrating window into an eidos which its axioms create and sustain (Heidegger 2008). The structures of modern mathematics found especially in Hilbert’s Formalist program obtain an ontological priority which for Heidegger is not philosophically founded (Karsten 2010). For instance, the development of the theory of functions has imbued the research paradigm with a reliance upon function as the foundational ontological component of mathematical reality and the chosen structure by which formal systems can be constructed. Heidegger is not against structuralism per se; rather, he stands against the totalizing agenda of reductionism found in Principia Mathematica, which, he believes, is the culmination of the Cartesian project (Heidegger 2008).
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