The original proposal of H. H. Pattee (1971) of basing quantum theoretical measurement theory on the theory of the origin of life, and its far reaching consequences, is discussed in the light of a recently emerging biological paradigm of internal measurement. It is established that the "measurement problem" of quantum physics can, in principle, be traced back to the internal material constraints of the biological organisms, where choice is a fundamental attribute of the self-measurement of matter. In this light, which (...) is shown to be a consequence of Pattee's original suggestion, it is proposed that biological evolution is a gradual liberation from the inert unity of "subject" and "object" of inanimate matter (as "natural law" and "initial conditions"), to a split biological existence of them and, as a consequence, the "message of evolution" is freedom, rather than complexity in itself. Some classical philosophical systems are brought into context to show that the epistemologies of several strictly philosophical systems of the social sciences are well acquainted with the problem and their solutions support our conclusions. (shrink)

The Turing Machine presents itself as the very landmark and initial design of digital automata present in all modern general-purpose digital computers and whose design on computable numbers implies deeply ontological as well as epistemological foundations for today’s computers. These lines of work attempt to briefly analyze the fundamental epistemological problem that rose in the late 19th and early 20th century whereby “machine cognition” emerges. The epistemological roots addressed in the TM and notably in its “Halting Problem” uncovers the tension (...) between determinism and uncertainty, regarded here as the primal and inherent features of machine cognition. Resumen La Máquina de Turing se presenta como el hito y el diseño inicial de un autómata digital presente en todas las computadoras digitales modernas de propósito general y cuyo diseño en números computables establece bases profundamente ontológicas y epistemológicas para las computadoras de hoy. Estas líneas de trabajo intentan analizar brevemente el problema epistemológico fundamental que surgió a finales del siglo XIX y principios del XX mediante el cual emerge la “cognición de la máquina”. Las raíces epistemológicas que se abordan en la TM y, en particular, en su “Problema de detención” ponen al descubierto la tensión entre el determinismo y la incertidumbre, consideradas aquí, como las características primordiales e inherentes de la cognición de la máquina. (shrink)

The identification of an informal concept of ‘effective calculability’ with a rigorous mathematical notion like ‘recursiveness’ or ‘Turing computability’ is still viewed as problematic, and I think rightly so. I analyze three different and conflicting perspectives Gödel articulated in the three decades from 1934 to 1964. The significant shifts in Gödel's position underline the difficulties of the methodological issues surrounding the Church-Turing Thesis.

We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill’s calculi, and focus on their fragments including multiplicative (Lambek) connectives, additive conjunction and disjunction, brackets and bracket modalities, and the! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut (cut elimination theorem). We also prove (...) algorithmic undecidability for both calculi, and show that categorial grammars based on them can generate arbitrary recursively enumerable languages. (shrink)

The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that (...) non-mechanical steps of intuition are needed to transcend particular formal theories. Thus, there is a substantive point in comparing Turing’s views with Gödel’s that is expressed by the assertion, “The human mind infinitely surpasses any finite machine”. The parallelisms and tensions between their views are taken as an inspiration for beginning to explore, computationally, the capacities of the human mathematical mind. (shrink)

Chomsky’s highly influential Syntactic Structures ( SS ) has been much praised its originality, explicitness, and relevance for subsequent cognitive science. Such claims are greatly overstated. SS contains no proof that English is beyond the power of finite state description (it is not clear that Chomsky ever gave a sound mathematical argument for that claim). The approach advocated by SS springs directly out of the work of the mathematical logician Emil Post on formalizing proof, but few linguists are aware of (...) this, because Post’s papers are not cited. Chomsky’s extensions to Post’s systems are not clearly defined, and the arguments for their necessity are weak. Linguists have also overlooked Post’s proofs of the first two theorems about effects of rule format restrictions on generative capacity, published more than ten years before SS was published. (shrink)

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...) developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic. (shrink)

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a (...) Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers. (shrink)