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)

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.

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)

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...

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)

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)

This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...) field, it lays down a much-needed marker of progress to date and provides a platform for informed and coherent future analysis and research of the subject. -/- The publication comes at a time of heightened worldwide concern over the standard of science and mathematics education, attended by fierce debate over how best to reform curricula and enliven student engagement in the subjects There is a growing recognition among educators and policy makers that the learning of science must dovetail with learning about science; this handbook is uniquely positioned as a locus for the discussion. -/- The handbook features sections on pedagogical, theoretical, national, and biographical research, setting the literature of each tradition in its historical context. Each chapter engages in an assessment of the strengths and weakness of the research addressed, and suggests potentially fruitful avenues of future research. A key element of the handbook’s broader analytical framework is its identification and examination of unnoticed philosophical assumptions in science and mathematics research. It reminds readers at a crucial juncture that there has been a long and rich tradition of historical and philosophical engagements with science and mathematics teaching, and that lessons can be learnt from these engagements for the resolution of current theoretical, curricular and pedagogical questions that face teachers and administrators. (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.

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.

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)