On the dialectical foundations of mathematics

This paper tracks the systematic dialectical determination of mathematical concepts in Hegel's Encyclopädie der philosophischen Wissenschaften (1830, 1817) and investigates the insights that can be gained from such a perspective on the mathematical. To begin with, the determination of Numbers and arithmetical operations from Being shows that the One and the successor function have a qualitative base and need not be presupposed. It is also shown that even for infinite Intensive Magnitudes (cardinals) there exists an Extensive Magnitude through which they gain meaning. This makes the 'bad' in Hegel's 'bad infinity' a trifle problematic. Finally, if 'Dasein' is interpreted as the whole of perception in the present, Place can be viewed as the spatial Now, Motion as the passage from Place to Now and Matter as the actual (as opposed to observed) Presence of the natural realm

Proof Pearl: Faithful Computation and Extraction of ?-Recursive Algorithms in Coq

Basing on an original Coq implementation of unbounded linear search for partially decidable predicates, we study the computational contents of ?-recursive functions via their syntactic representation, and a correct by construction Coq interpreter for this abstract syntax. When this interpreter is extracted, we claim the resulting OCaml code to be the natural combination of the implementation of the ?-recursive schemes of composition, primitive recursion and unbounded minimization of partial (i.e., possibly non-terminating) functions. At the level of the fully specified Coq terms, this implies the representation of higher-order functions of which some of the arguments are themselves partial functions. We handle this issue using some techniques coming from the Braga method. Hence we get a faithful embedding of ?-recursive algorithms into Coq preserving not only their extensional meaning but also their intended computational behavior. We put a strong focus on the quality of the Coq artifact which is both self contained and with a line of code count of less than 1k in total

A proof of Bertrand's postulate

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development of number theory. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base. 1

Apperceptive patterning: Artefaction, extensional beliefs and cognitive scaffolding

In “Psychopower and Ordinary Madness” my ambition, as it relates to Bernard Stiegler’s recent literature, was twofold: 1) critiquing Stiegler’s work on exosomatization and artefactual posthumanism—or, more specifically, nonhumanism—to problematize approaches to media archaeology that rely upon technical exteriorization; 2) challenging how Stiegler engages with Giuseppe Longo and Francis Bailly’s conception of negative entropy. These efforts were directed by a prevalent techno-cultural qualifier: the rise of Synthetic Intelligence (including neural nets, deep learning, predictive processing and Bayesian models of cognition). This paper continues this project but first directs a critical analytic lens at the Derridean practice of the ontologization of grammatization from which Stiegler emerges while also distinguishing how metalanguages operate in relation to object-oriented environmental interaction by way of inferentialism. Stalking continental (Kapp, Simondon, Leroi-Gourhan, etc.) and analytic traditions (e.g., Carnap, Chalmers, Clark, Sutton, Novaes, etc.), we move from artefacts to AI and Predictive Processing so as to link theories related to technicity with philosophy of mind. Simultaneously drawing forth Robert Brandom’s conceptualization of the roles that commitments play in retrospectively reconstructing the social experiences that lead to our endorsement(s) of norms, we compliment this account with Reza Negarestani’s deprivatized account of intelligence while analyzing the equipollent role between language and media (both digital and analog)

Speaking in circles: completeness in Kant's metaphysics and mathematics

This dissertation presents and responds to the following problem. For Kant a field of enquiry can be a science only if it is systematic. Most sciences achieve systematicity through having a unified content and method. Physics, for example, has a unified content, as it is the science of matter in motion, and a unified method because all claims in physics must be verified through empirical testing. In order for metaphysics to be a science it also must be systematic. However, metaphysics cannot have a unified content or method because metaphysicians lack a positive conception of what its content and method are. On Kant's account, metaphysicians can say with certainty what metaphysics does not study and what methods it cannot use, but never how it should proceed. Without unified content and method systematicity can only be guaranteed by some either means, namely, completeness. Without completeness metaphysics cannot have systematicity and every science must be systematic. Completeness can only be achieved if we severely limit the scope of metaphysics so that it contains only the conditions for the possibility of experience. This dissertation defends the claims made about the centrality of completeness in understanding Kant's conception of metaphysics as a science in two ways. First, the first two chapters point to a substantial body of textual evidence that supports the idea that Kant was directly concerned about the notion of completeness and links it to his conception of metaphysics as a science. Chapters 3 and 4 consider some possible objections to thinking that metaphysics as a science can be complete, giving special consideration to Gödel's incompleteness theorem. Chapter 5 explains why, if this position is as clear as this dissertation has argued, previous scholars have failed to acknowledge it. Giving a full answer to this question requires considering the general neglect of the "Doctrine of Method" section of Kant's primary theoretical text, The Critique of Pure Reason. The Doctrine of Method contains many of the passages which most directly support my thesis. Chapter 6 explains why scholars have ignored this important passage and argues that they should not continue to do so