Math and Modernity: Critical Reflections

Besides the birth of new revolutionary concepts and methods, and of new areas of research, mathematicians, logicians, and philosophers have put into question the foundations of the discipline itself and the whole meaning of “mathematical truth.” Before then, at the end of the eighteenth century, mathematics was mainly concerned with explaining the “real world” and its laws. At the beginning of the “modern era” things started to change, sometimes slowly, other times abruptly. Abstract mathematics was no longer intimately related to the real world and its description. This abstract approach, both on research and on mathematical education, generated critical reactions in the mathematical community, and some “modern” ideas were rejected or neglected after several decades of experimentation.

Carnap and Beth on the Limits of Tolerance

Rudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance does not suffice to remove all obstacles to the employment of infinitary rules.

Mathematical Determinacy

This chapter addresses the second major challenge for the extension of conventionalism from logic to mathematics: the richness of mathematical truth. The chapter begins by distinguishing indeterminacy from pluralism and clarifying the crucial notion of open-endedness. It then critically discusses the two major strategies for securing arithmetical categoricity using open-endedness; one based on a collapse theorem, the other on a kind of anti-overspill idea. With this done, a new argument for the categoricity of arithmetic is then presented. In subsequent discussion, the philosophical importance of this categoricity result is called into question to some degree. The categoricity argument is then supplemented by an appeal to the infinitary omega rule, and an argument is given that beings like us can actually follow the omega rule without any violation of Church’s thesis. Finally, the chapter discusses the extension of this type of approach beyond arithmetic, to set theory and the rest of mathematics.

Shadows of Syntax

What is the source of logical and mathematical truth? This volume revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. In Shadows of Syntax, Jared Warren offers the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. He argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims. In Part I, Warren explains exactly what conventionalism amounts to and what linguistic conventions are. Part II develops an unrestricted inferentialist theory of the meanings of logical constants that leads to logical conventionalism. This conventionalist theory is elaborated in discussions of logical pluralism, the epistemology of logic, and of the influential objections that led to the historical demise of conventionalism. Part III aims to extend conventionalism from logic to mathematics. Unlike logic, mathematics involves both ontological commitments and a rich notion of truth that cannot be generated by any algorithmic process. To address these issues Warren develops conventionalist-friendly but independently plausible theories of both metaontology and mathematical truth. Finally, Part IV steps back to address big picture worries and meta-worries about conventionalism. This book develops and defends a unified theory of logic and mathematics according to which logical and mathematical truths are reflections of our linguistic rules, mere shadows of syntax.

From Logic to Mathematics

Part II (chapters 3-7) of the book developed and defended an inferentialist/conventionalist theory of logic. In this, the opening chapter of part III, it is explained why the extension of part II’s approach from logic to mathematics faces significant philosophical challenges. The first major challenge concerns the ontological commitments of mathematics. It is received wisdom in philosophy that existence claims cannot be analytic or trivially true, making it difficult to see how a conventionalist account of mathematics could possibly be viable. The second major challenge concerns mathematical truth. Unlike (first-order) logical truth, mathematical truth, even in basic arithmetic, is computationally rich. There are serious challenges for conventionalists in trying to capture our intuition that mathematical truth is fully determinate, in light of this feature.

Why anything rather than nothing? The answer of quantum mechanics

Many researchers determine the question “Why anything rather than nothing?” as the most ancient and fundamental philosophical problem. Furthermore, it is very close to the idea of Creation shared by religion, science, and philosophy, e.g. as the “Big Bang”, the doctrine of “first cause” or “causa sui”, the Creation in six days in the Bible, etc.Thus, the solution of quantum mechanics, being scientific in fact, can be interpreted also philosophically, and even religiously. However, only the philosophical interpretation is the topic of the text.The essence of the answer of quantum mechanics is:1. The creation is necessary in a rigorous mathematical sense. Thus, it does not need any choice, free will, subject, God, etc. to appear. The world exists in virtue of mathematical necessity, e.g. as any mathematical truth such as 2+2=4.2. The being is less than nothing rather than more than nothing. So, the creation is not an increase of nothing, but the decrease of nothing: it is a deficiency in relation of nothing. Time and its “arrow” are the way of that diminishing or incompleteness to nothing.

“Mathematical Paradise”: The Parallel Social Infrastructure of Post-war Soviet Mathematics

This article examines the response of the Soviet mathematical community to the geographical restrictions, physical barriers, political and administrative pressures, and conceptual constraints that they faced from the 1950s through the 1980s. Many talented mathematicians with “undesirable” ethnic or political backgrounds encountered discrimination in admission to universities, employment, travel to conferences abroad, etc. The mathematical community in response created a parallel social infrastructure, which attracted young talent and provided support and motivation for researchers excluded from official institutions. That infrastructure included a network of study groups (“math circles”), correspondence courses, math competitions, specialized mathematical schools, free evening courses for students barred from top universities, pure math departments within applied mathematics institutions, and a network of open research seminars. A community emerged in which mathematics became a way of life, work and leisure converged, and research activity migrated from restrictive official institutions to the private spaces of family apartments or dachas. In the informal community of Soviet mathematicians, a specific “moral economy” operated, which relied on a network of friendly connections and on an exchange of favors. The various external constraints further strengthened personal ties, encouraged mutual help, and fostered close friendships in the community. Although excluded from elite privileges, the “parallel world” of Soviet mathematics cultivated an ethos of noble rejection of career ambitions, material rewards and official recognition in order to pursue the highest ideals of mathematical truth. This way of life, which opposed the bureaucratic spirit of official institutions, was often perceived by its participants as a “mathematical paradise.”

One Green Quantum Computing Tablet Per Child

A Paper on the design for the One Quantum Tablet Per Child, the new E-Paper in Graphene and organic polymer on PLA substrate, with a Quantum Ising Glass architecture with spin wave based STT, Quantum Internet for last mile connectivity and a distributed QPU-GPU-MCU architecture for robust scalable fault tolerant computing.Keywords: Graphene, OLED, MCU-QPU-GPU, integration, E-Paper, Quantum Cloud, Quantum Operating systems, E-Learning, Quantum Tablet, Mathematical Truth, Light as New Age Religion.