The affirmative answer to the title question is justified in two ways: logical and empirical. The logical justification is due to Gödel’s discovery that in any axiomatic formalized theory, having at least the expressive power of PA, at any stage of development there must appear unsolvable problems. However, some of them become solvable in a further development of the theory in question, owing to subsequent investigations. These lead to new concepts, expressed with additional axioms or rules. Owing to the so-amplified (...) axiomatic basis, new routine procedures like algorithms, can be reached. Those, in turn, help to gain new insights which lead to still more powerful axioms, and consequently again to ampler algorithmic resources. Thus scientific progress proceeds to an ever higher scope of solvability. The existence of such feedback cycles –in a formal way rendered with Turing’s systems of logic based on ordinal – gets empirically supported by the history of mathematics and other exact sciences. An instructive instance of such a process is found in the history of the number zero. Without that insight due to some ancient Hindu mathematicians there could not arise such an axiomatic theory as PA. It defines the algorithms of arithmetical operations, which in turn help intuitions; those, inturn, give rise to algorithmic routines, not available in any of the previous phases of the process in question. While the logical substantiation of the point of this essay is a well-established result of logico-semantic inquiries, its empiricalclaim, based on historical evidences, remains open for discussion. Hence the author’s intention to address philosophers and historians of science, and to encourage their critical responses. (shrink)

This paper examines an insoluble Cartesian problem for classical AI, namely, how linguistic understanding involves knowledge and awareness of u’s meaning, a cognitive process that is irreducible to algorithms. As analyzed, Descartes’ view about reason and intelligence has paradoxically encouraged certain classical AI researchers to suppose that linguistic understanding suffices for machine intelligence. Several advocates of the Turing Test, for example, assume that linguistic understanding only comprises computational processes which can be recursively decomposed into algorithmic mechanisms. Against this background, in (...) the first section, I explain Descartes’ view about language and mind. To show that Turing bites the bullet with his imitation game, in the second section I analyze this method to assess intelligence. Then, in the third section, I elaborate on Schank and Abelsons’ Script Applier Mechanism (SAM, hereby), which supposedly casts doubt on Descartes’ denial that machines can think. Finally, in the fourth section, I explore a challenge that any algorithmic decomposition of linguistic understanding faces. This challenge, I argue, is the core of the Cartesian problem: knowledge and awareness of meaning require a first-person viewpoint which is irreducible to the decomposition of algorithmic mechanisms. (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

This essay is an attempt to offer at least a partial answer to the question concerning Leibniz's motivation for the need to create a universal language. It is relevant, among other things, due to Leibniz's claim which contradicts the idea of a universal language, in which he clearly stresses that “every language, even the poorest one, can express everything”, as well as owing to Leibniz's historical contribution to the idea of formalization and to the mechanization of the reasoning processes. This (...) paper is also an attempt to reconstruct the paradigm of language research pursued in a specific period of time in 17th century language theory. (shrink)

This paper represents such an amateur approach; hence any comments backed up by professional erudition will be highly appreciated. Let me start from an attempt to sketch a relationship between professionals’ and amateurs’ contributions. The latter may be compared with the letters to the Editor of a journal, written by perceptive readers, while professionals contribute to the very content of the journal in question. Owing to such letters, the Editor and his professional staff can become more aware of the responses (...) of educated public to the journal’s output. (shrink)

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)