Jose Ferreiros

We discuss critically some recent theses about geometric cognition, namely claims of universality made by Dehaene et al., and the idea of a “natural geometry” employed by Spelke et al. We offer arguments for the need to distinguish visuo-spatial cognition from basic geometric knowledge, furthermore we claim that the latter cannot be identified with Euclidean geometry. The main aim of the paper is to advance toward a characterization of basic, practical geometry – which in our view requires a combination of experiments on visuo-spatial cognition with studies in cognitive archaeology and comparative history. Examples from these fields are given, with special emphasis on the comparison of ancient Chinese and ancient Greek geometric ideas and procedures.

This is a contribution to the philosophy of experimental work, engaging with questions posed by Hacking, Franklin, Pickering, Schaffer and Collins. It focuses on the dynamics of experimentation and offers a detailed argument that one finds no "regress" of the kind posited by Collins. In particular, we reanalyze the celebrated series of experimental investigations by Newton on optical phenomena, taking into account Schaffer's partial reconstruction, and we show how it must be supplemented to obtain a more complete picture.

The place of Richard Dedekind in the history of logicism is a controversial matter. The conception of logic incorporated in his work is certainly old-fashioned, in spite of innovative elements that would play an important role in late 19th and early 20th century discussions. Yet his understanding of logic and logicism remains of interest for the light it throws upon the development of modern logic in general, and logicist views of the foundations of mathematics in particular. The paper clarifies Dedekind's views and their relation to transcendental logic, makes explicit the role of the principle of comprehension (unrestricted) and offers a reconstrucion ot his dichotomy conception. The last section explores the differences with Frege (half epistemic, half metaphysical) making clear that there were different brands of logicism, and that logicism does not necessarily lead to singularism or “essentialistic objectualism”.

© The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] mathematics a reflection of some already-given realm? It would not matter whether we are talking about the empirical world in a Millian way, or the domain of a priori truths in Leibnizian or maybe Kantian style, or some world of analytical truths à la Carnap. Or perhaps — could mathematics be something more, or something less, than such a reflection? Might it be human, perhaps, dependent on our kind, on our forms of life? Could it just be a set of techniques and conceptions that we develop and employ to establish order and measure in the world around us, in our activities and our environment?Consider any basic mathematical notion that you may wish: e.g., number, function, or space. There is a long history to be told of changing conceptions of...

We offer an analysis of the disciplinary transformations underwent by mathematical or symbolic logic since its emergence in the late 19 th century. Examined are its origins as a hybrid of philosophy and mathematics, the maturity and institutionalisation attained under the label “logic and foundations,” a second wave of institutionalisation in the Postwar period, and the institutional developments since 1975 in connection with computer science and with the study of language and informatics. Although some “internal history” is discussed, the main focus is on the emergence, consolidation and convolutions of logic as a discipline, through various professional associations and journals, in centers such as Torino, Göttingen, Warsaw, Berkeley, Princeton, Carnegie Mellon, Stanford, and Amsterdam.

Dedekind used to refer to Riemann as his main model concerning mathematical methodology, particularly regarding the use of abstract notions as a basis for mathematical theories. So, in passages written in 1876 and 1895 he compared his approach to ideal theory with Riemann’s theory of complex functions. In this paper, I try to make sense of those declarations, showing the role of abstract notions in Riemann’s function theory, its influence on Dedekind, and the importance of the methodological principle of avoiding ‘forms of representation’ in shaping ideal theory. In order to emphasize the abstract viewpoint of Riemann and Dedekind, I compare their work with that of their great german contemporaries, Weierstrass and Kronecker; so, an influential ‘Göttingen group’ is confronted with the ‘Berlin school’ of mathematics. Some light is also thrown on the relation between Riemann and Dedekind, particularly with respect to Dedekind’s interest on Riemann’s ideas --in function theory, geometry and topology--, beginning in the 1850s and through later works. The abstract approach of the ‘Göttingen group’ was determinant for the turn to abstract mathematics in our century, and the direct influence of Riemann on Dedekind shows how this approach developed.

The present paper is a contribution to the history of logic and its philosophy toward the mid-20th century. It examines the interplay between logic, type theory and set theory during the 1930s and 40s, before the reign of first-order logic, and the closely connected issue of the fate of logicism. After a brief presentation of the emergence of logicism, set theory, and type theory (with particular attention to Carnap and Tarski), Quine’s work is our central concern, since he was seemingly the most outstanding logicist around 1940, though he would shortly abandon that viewpoint and promote first-order logic as all of logic. Quine’s class-theoretic systems NF and ML, and his farewell to logicism, are examined. The last section attempts to summarize the motives why set theory was preferred to other systems, and first orderlogic won its position as the paradigm logic system after the great War.

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).

This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine history and philosophy of mathematics, especially in order to evaluate properly the history of mathematics in India, in particular the history of the calculus.The pivotal goals of the book are to oppose the Eurocentric account of the history of science in general and mathematics in particular; to avoid the usual philosophical idea of the centrality of proof for mathematical knowledge, in favour of the traditional Indian notion of pramāṇa [validation] encompassing empirical elements and emphasizing calculation; to analyze the thousand-year-long development of infinite series in India, starting in the fifth century; and to show ‘how and why the calculus was imported into Europe’ from about 1500. The result is a picture in which inputs from the Indian subcontinent and epistemology are the driving forces of the history of mathematics, as people in the European subcontinent struggle to adopt new calculation techniques from the East in spite of Western philosophico-religious biases . Thus, a ‘first math war’ involved the algorismus de numero indorum, adopted for practical reasons, which forced Europeans to modify their epistemology of number and quantities. A ‘second math war’ revolved around infinite series and the calculus, since the background of Western epistemology created ‘spurious difficulties’ about infinities and infinitesimals, partially resolved with theories of real numbers. And a ‘third math war’ is …

El artículo intenta promover una recepción más amplia de los trabajos recientes sobre filosofía de la actividad científica experimental. Primero se comentarán los orígenes y las características de la tradición teoreticista predominante, criticando sus presupuestos y sus "miserias". Se analizará luego la función de los instrumentos, proponiendo una tipología de la actividad experimental, aunque elemental --esperamos-- útil. Tras analizar la estructura del experimento, empleando contribuciones de Pickering y otros, discutiremos la dinámica de la experimentación: los procesos de formación de datos. Ésta es, obviamente, la cuestión más crucial y debatida, de la que depende la especificidad y fiabilidad de los métodos científicos. /// This paper attempts to promote a more widespread reception of recent work on the philosophy of experimental scientific activity. First, we comment on the origins and character of the predominant theoreticist tradition, offering critical remarks on its assumptions and "poverty". Then we analyze the function of instruments, proposing a coarse but hopefully useful typology of experimental activity. After analyzing the structure of experiment, drawing on work by Pickering and others, we discuss the dynamics of experimentation --the processes of data formation. This is obviously the most crucial and disputed issue, on which the specificity and reliability of scientific methods depends.

Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo—Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect

I offer a critical review of several different conceptions of the activity of foundational research, from the time of Gauss to the present. These are (1) the traditional image, guiding Gauss, Dedekind, Frege and others, that sees in the search for more adequate basic systems a logical excavation of a priori structures, (2) the program to find sound formal systems for so-called classical mathematics that can be proved consistent, usually associated with the name of Hilbert, and (3) the historicist alternative, guiding Riemann, Poincaré, Weyl and others, that seeks to perfect available conceptual systems with the aim to avoid conceptual limitations and expand the range of theoretical options. I shall contend that, at times, assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images. To round the discussion, I mention some traits of an alternative program that investigates the epistemology of mathematical knowledge.

The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor's very novel research, giving particular attention to a key contribution, the Grundlagen (Foundations of a general theory of sets) of 1883, where those motives are articulated in some detail. Evidence from other publications and correspondence is pulled out to provide clarification and a detailed interpretation of those ideas and their impact upon Cantor's research. Throughout the paper, a special effort is made to place Cantor's scientific undertakings within the context of developments in German science and philosophy at the time (philosophers such as Trendelenburg and Lotze, scientists like Weber, Riemann, Vogt, Haeckel), and to reflect on the German intellectual atmosphere during the nineteenth century.

A general introduction to the celebrated foundational crisis, discussing how the characteristic traits of modern mathematics (acceptance of the notion of an “arbitrary” function proposed by Dirichlet; wholehearted acceptance of infinite sets and the higher infinite; a preference “to put thoughts in the place of calculations” and to concentrate on “structures” characterized axiomatically; a reliance on “purely existential” methods of proof) provoked extensive polemics and alternative approaches. Going beyond exclusive concentration on the paradoxes, it also discusses the role of the axiom of Choice, issues of predicativity, and goes on to present the ideas of intuitionism and Hilbert's program. The essay closes with Gödel's contributions and the aftermath.

In 1887–1894, Richard Dedekind explored a number of ideas within the project of placing mappings at the very center of pure mathematics. We review two such initiatives: the introduction in 1894 of groups into Galois theory intrinsically via field automorphisms, and a new attempt to define the continuum via maps from ℕ to ℕ in 1891. These represented the culmination of Dedekind’s efforts to reconceive pure mathematics within a theory of sets and maps and throw new light onto the nature of his structuralism and its specificity in relation to the work of other mathematicians.