Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...) expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning. We also investigate the emergence of category theory and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbiński and Tarski.

This paper explores what may be happening with information and also with happening itself: those versions of the event, of historicizing, deemed suitable to interrogating information. It pairs Claude Shannon’s foundational work on information theory with Martin Heidegger’s meditation on technology (and his evolving understanding of history), investigating each in light of the other. For Heidegger, the question concerning technology does not primarily concern an invention or tool, but a mode of revealing or truth. Similarly, Shannon’s invention is not a (...) device, but a new theory, a new way of viewing all media. Heidegger’s take on technology and Shannon’s theorization of information both ultimately rely, however, on events (of communicating and of appropriation, Ereignis, respectively). This permits, by way of conclusion, questioning how our own talk and thinking stand in information’s context, as well as our and Heidegger’s understanding of epochs and ages themselves. (shrink)

This thesis studies the enabling properties of ICT and their effects and potential for social change, and prepares the ground for a sustainability-oriented ethico-political assessment of this technology. It primarily builds on interdisciplinary scholarship to describe and explain the multifaceted co-evolution between the global deployment of ICTs and the emergence of the Information Society, understood as a socioeconomic restructuring of capitalism. Beyond the role of ICTs in this regime transition, the thesis delivers other philosophical insights about crucial aspects of ICT (...) development, applications and management. These include arguments about how we should conceptualize ICTs on the basis of their different roles in extending human communication and in performing or facilitating the remote control of humans and animals, machines and systems operations; about the entanglement between telecommunications, transport networks, urban development and work and organization; and about the relations between ICTs, culture and human values. Examples are offered to illustrate the potential that these empirical and philosophical lessons may hold for the construction of a framework for the ethico-political assessment of ICTs. (shrink)

The aim of this text is to reply to criticisms of the logics of evidence and truth and the epistemic approach to paraconsistency advanced by Barrio [2018], and Lo Guercio and Szmuc [2018]. We also clarify the notion of evidence that underlies the intended interpretation of these logics and is a central point of Barrio’s and Lo Guercio & Szmuc’s criticisms.

Scholars in both digital humanities and media studies have noted an apparent disconnect between computation and the interpretive methods of the humanities. Alan Liu has argued that literary scholars employing digital methods encounter a “meaning problem” due to the difficulty of reconciling algorithmic methods with interpretive ones. Conversely, the media scholar Friedrich Kittler has questioned the adequacy of hermeneutics as a means of studying computers. This paper argues that that this disconnect results from a set of contingent decisions made in (...) both humanistic and mathematical disciplines in the first half of the nineteenth century that delineated, with implications that continue to resonate in the present day, which aspects of human activity would come to be formalized in algorithms and which would not. I begin with a discussion of Nicolas de Condorcet, who attempted, at the height of the 1789 revolution, to turn algebra into a universal language; his work, I argue, exemplifies the form of algorithmic thinking that existed before the Romantic turn. Next, I discuss William Wordsworth’s arguments about the relationship of poetry and science. While Wordsworth is sometimes viewed as a critic of science, I argue that his polemic is specifically targeted at highly politicized projects like Condorcet’s that sought to supplant existing modes of thought with scientific rationality. Finally, I demonstrate the importance of Romantic thought for George Boole, creator of the logic system that would eventually form the basis of digital electronics. The reason Boole was able to succeed where Condorcet had failed, I argue, was that Romantic notions of culture enabled him to reconcile a mechanical view of mathematical reasoning with an organic view of the development of meaning—a dichotomy that remains a key assumption of computer interfaces in the twenty-first century. (shrink)

Justified by spectacular achievements facilitated through applied deep learning methodology, the “Everything is possible” view dominates this new hour in the “boom and bust” curve of AI performance. The optimistic view collides head on with the “It is not possible”—ascertainments often originating in a skewed understanding of both AI and medicine. The meaning of the conflicting views can be assessed only by addressing the nature of medicine. Specifically: Which part of medicine, if any, can and should be entrusted to AI—now (...) or at some moment in the future? AI or not, medicine should incorporate the anticipation perspective in providing care. (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)

Over the past ten years, we have worked in a collaboration between educators and computer scientists at the University of Illinois to imagine futures for education in the context of what is loosely called “artificial intelligence.” Unhappy with the first generation of digital learning environments, our agenda has been to design alternatives and research their implementation. Our starting point has been to ask, what is the nature of machine intelligence, and what are its limits and potentials in education? This paper (...) offers some tentative answers, first conceptually, and then practically in an overview of the results of a number of experimental implementations documented in greater detail elsewhere. Our key finding is that artificial intelligence—in the context of the practices of electronic computing developing over the past three quarters of a century—will never in any sense “take over” the role of teacher, because how it works and what it does are so profoundly different from human intelligence. However, within the limits that we describe in this paper, it offers the potential to transform education in ways that—counterintuitively perhaps—make education more human, not less. (shrink)

The complete arithmetization of the bivalued propositional logic is here presented and extended with original functions not hitherto included in other interpretations of propositional logic, as the algebraic logic or sets theory. An historical review of the former attempts of arithmetical representation of the propositional logic is presented.

The birth, growth, stabilization and subsequent understanding of a new field of practical and theoretical enquiry is always a conceptual process including several typologies of events, phenomena an...

In this chapter, an attempt is made to illustrate why the study of primary original sources is, as often stated, rewarding and worth the effort, despite being extremely demanding for both teachers and students. This is done by discussing various reasons for as well as different approaches to using primary original sources in the teaching and learning of mathematics. A selection of these reasons and approaches will be illustrated through a number of examples from the literature on using original sources (...) in mathematics education. In particular, focus is on empirical research findings when choosing illustrative examples. The chapter also includes a section on the background and academic forums of research on using primary original sources in mathematics education. The discussion on using original sources is briefly related to that of using history in mathematics education, and some theoretical constructs are drawn upon to assist this discussion. In the final section of the chapter, the past, present, and future of using primary original sources is discussed, in order to possibly identify new paths which research might follow. In particular, the section addresses the role of primary original sources in relation to the educational problems of recruitment, transition, retention, and interdisciplinarity. (shrink)