FYI:Concepts of objects. On the logical significance of Hegel’s criticism of infinitesimal
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Alex Shkotin
May 13, 2025, 12:26:03 PM5/13/25
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The concept in Hegel’s speculative-generic sense is the universe of
all concepts. Such a concept is a structured realm of possible things to
refer to in the very broad sense of “onta”, “entia” and “entities”. The
whole system of such domains is ‘the’ transcendental condition for
talking about something, hence for thinking about non-present objects
and possibilities. However, Aristotle already knew that there is no
‘comprehensive ontology’ of all such things:
We cannot define basic domains or concepts, like, for example, the pure numbers, by sortal predicates, as Frege’s logicism suggests. We rather need a phenomenological reflection on the local practices of coordinating (types of) representations (as self-produced expressions) and presentations (as passive sensations), by which we constitute the identities of entities fitting to their predicates in limited domains G of real and abstract entities. As definite
negations, the predicates must at least generically satisfy the Leibniz-Frege-principle of functional substitution. Since this formal, hence ideal, principle holds outside mathematics only for ‘good cases’, the logicist turn of modern Analytic Philosophy without the phenomenology of ‘continental philosophy’, so-called, is not yet a linguistic turn in philosophical analysis. It is no wonder that the greatest abstraction logicians after Aristotle and Descartes, namely Leibniz and Hegel, recognised this onto-logical problem of the concept of the object in the undefined nature of Newton's fluxions or the non-existence of all infinitesimal quantities before Abraham Robinson's invention of a new type of non-standard analysis. Even the most important phenomenological ontologists, namely Husserl and Heidegger, first studied differential geometry intensively, as G. Neumann proves for Heidegger in the book Phänomenologische Untersuchungen (Berlin: LIT Verlag 2025, p. 324). These facts and their significance are by no means sufficiently well known.
We cannot define basic domains or concepts, like, for example, the pure numbers, by sortal predicates, as Frege’s logicism suggests. We rather need a phenomenological reflection on the local practices of coordinating (types of) representations (as self-produced expressions) and presentations (as passive sensations), by which we constitute the identities of entities fitting to their predicates in limited domains G of real and abstract entities. As definite
negations, the predicates must at least generically satisfy the Leibniz-Frege-principle of functional substitution. Since this formal, hence ideal, principle holds outside mathematics only for ‘good cases’, the logicist turn of modern Analytic Philosophy without the phenomenology of ‘continental philosophy’, so-called, is not yet a linguistic turn in philosophical analysis. It is no wonder that the greatest abstraction logicians after Aristotle and Descartes, namely Leibniz and Hegel, recognised this onto-logical problem of the concept of the object in the undefined nature of Newton's fluxions or the non-existence of all infinitesimal quantities before Abraham Robinson's invention of a new type of non-standard analysis. Even the most important phenomenological ontologists, namely Husserl and Heidegger, first studied differential geometry intensively, as G. Neumann proves for Heidegger in the book Phänomenologische Untersuchungen (Berlin: LIT Verlag 2025, p. 324). These facts and their significance are by no means sufficiently well known.
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