Rossella Lupacchini

One of Wittgenstein’s most quoted passages from his Remarks on the Philosophy of Psychology concerns Turing’s “machines” and says verbatim: “These machines are humans who calculate. And one might express what he [Turing] says also in the form of games.” This passage not only captures the kernel of Turing’s conceptual argument for the adequacy of his definition of “computability”, as presented in his article On Computable Numbers (1936), but also helps clarify Turing’s idea of “mechanical intelligence.” Indeed, the notion of game provides an ideal means to focus on similarities and differences between Turing and Wittgenstein’s views of mechanical procedures, mathematical understanding, and thinking activity. The live encounter between Ludwig Wittgenstein and Alan Turing took place in Cambridge in 1939, when Wittgenstein’s Lectures on the Foundations of Mathematics were regularly attended by Turing. Interestingly, during the conversations between the two, Turing seems to play the role of the Wittgenstein of the Tractatus, to allow the present Wittgenstein to reassess what he deplores as mistaken or misleading in his early work. As for Turing himself, his reflection on thinking machines from the late 1940s demonstrates the significance of his dialogue with Wittgenstein.

This volume draws attention to the encounter between physics and Japanese philosophy during the last century. While a remarkable global network of Japanese philosophy has been growing and enhancing connections with the arts, religion, hermeneutics, aesthetics, the prevailing opinion is that there is no common ground for a meaningful dialogue between theoretical physics and Japanese philosophy. With a special focus on Nishida Kitarō's engagement with scientific thought, this book invites readers to appreciate both the significance of foundational questions arising from relativity and quantum theory for Japanese philosophy, and the insightful influence of certain East Asian philosophical notions on theoretical issues in modern physics. The idea for this volume came up during the conference ‘Philosophy and Physics between Europe and Japan, 1922-1953’, held at the University of Naples Federico II in October 2023. Bringing together contributions from scholars of multiple areas of expertise, this book highlights, on the one hand, the scientific sensitivity of certain philosophers of the Kyoto School, and on the other, the role played by their analyses in the conceptual development of some Japanese physicists. This text identifies new research crucial to refining the understanding of theoretical issues at the heart of modern physics as well as to Japanese philosophy from an inter-cultural perspective. It appeals to researchers and students working at the crossroads of theoretical physics and Japanese philosophy.

Despite its extraordinary predictive power, quantum mechanics has been hailed as a paradoxical, self-contradictory theory of nature. How does it question the intelligibility of physical worldview? The wave-particle dualism, the incompatibility of physical quantities, the complementarity between the space-time description and the causal description of phenomena question key-notions of the traditional metaphysics, such as substance and cause, but they also call attention to the vital dialectical contrast between the continuous and the discrete, the infinite and the finite, consciousness and matter, and to the essential relational character of measuring, seeing, and knowing. Does quantum physics also question the Western way of thinking? The aim of this article is to show how quantum monadology not only breathes new life into Leibniz’s and Nicholas of Cusa’s monads, but also echoes Nishida’s ‘dialectical monadology’ and orients our gaze towards a metaphysics of universal harmony, i.e., a metaphysics of the dialectical universal or a metaphysics of indeterminacy.

‘Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means.’ Along this line, in The Open World, Hermann Weyl contrasted the desire to make the infinite accessible through finite processes, which underlies any theoretical investigation of reality, with the intuitive feeling for the infinite ‘peculiar to the Orient,’ which remains ‘indifferent to the concrete manifold of reality.’ But a critical analysis may acknowledge a valuable dialectical opposition. Struggling to spell out the infinity of real numbers mathematicians come to see the active role of emptiness. Pondering over the essence of self-awareness, the Japanese philosopher Nishida Kitarō comes to see the ‘place’ where it abides as absolute nothingness. Thus, the two ways of seeing coalesce into a perspective in which infinity and nothingness mirror each other.

The invention of “artificial perspective” revealed the ideal character of Euclidean geometry already in the Renaissance Europe of the fifteenth century. To the extent to which it made painting a “science” relying on mathematical rules, it made mathematics an “art” independent of the “geometry of nature.” It was the artistic vision emerging from perspective drawing that paved the way for scientific abstraction. However, it was only in the nineteenth century that the discovery of non-Euclidean geometry compelled mathematics to ponder the visual evidence of its principles and the reliability of its abstract concepts. At that time, it was the mathematical vision that first championed the rights of ideal forms to a higher level of abstraction and, therefore, oriented science and art towards new representational spaces.

§1. Mathematics and the physical world. Genuine scientific knowledge cannot be certain, nor can it be justified a priori. Instead, it must be conjectured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics.This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, “the universe is written in the language of mathematics”. Galileo's introduction of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on supposedly necessary a priori principles, to its modern status as a theoretical, conjectural and empirical science. Instead of seeking an infallible universal mathematical design, Galilean science usesmathematics to express quantitative descriptions of an objective physical reality. Thus mathematics became the language in which we express our knowledge of the physical world — a language that is not only extraordinarily powerful and precise, but also effective in practice. Eugene Wigner referred to “the unreasonable effectiveness of mathematics in the physical sciences”. But is this effectiveness really unreasonable or miraculous?Numbers, sets, groups and algebras have an autonomous reality quite independent of what the laws of physics decree, and the properties of these mathematical structures can be just as objective as Plato believed they were.

Both Hilbert's axiomatics and Cassirer's philosophy of symbolic forms have their roots in Leibniz's idea of a 'universal characteristic,' and grow on Hertz's 'principles of mechanics,' and Dedekind's 'foundations of arithmetic'. As Cassirer recalls in the introduction to his Philosophy of Symbolic Forms, it was the discovery of the analysis of infinity that led Leibniz to focus on "the universal problem inherent in the function of symbolism, and to raise his 'universal characteristic' to a truly philosophical plane." In Leibniz's view, the logic of 'things' cannot be separated from the logic of 'signs,' as "the sign is no mere accidental cloak of the idea, but its necessary and essential organ."Every 'law' of ..