Related papers: Hilbert's Program Then and Now
In the modern Bayesian view classical probability theory is simply an extension of conventional logic, i.e., a quantitative tool that allows for consistent reasoning in the presence of uncertainty. Classical theory presupposes, however,…
These five lectures on undecidability were given to students with a good level in mathematics but with no special knowledge on logic. The first conference presents the formalization of mathematics with a short historical survey, the…
In this paper a novel calculus system has been established based on the concept of 'werden'. The basis of logic self-contraction of the theories on current calculus was shown. Mistakes and defects in the structure and meaning of the…
The long-standing problem of time in canonical quantum gravity is the source of several conceptual and technical issues. Here, recent mathematical results are used to provide a consistent algebraic formulation of dynamical symplectic…
In this paper, a modified formulation of generalized probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, is presented and the guidelines to how to extend…
We initiate the study of parallel quantum programming by defining the operational and denotational semantics of parallel quantum programs. The technical contributions of this paper include: (1) find a series of useful proof rules for…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
We report on a pedagogical experiment to make mathematics easy by changing its philosophy. The Western philosophy of math originated in religious beliefs about mathesis, cursed by the church. Later, mathematics was "reinterpreted", in a…
The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms -- the…
This article is firstly a historic review of the theory of Riemann-Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert's 21st problem and Plemelj's work associated with it. The secondary…
This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the L\"owenheim-Skolem theorem, Craig interpolation, Beth's…
The study of computability has its origin in Hilbert's conference of 1900, where an adjacent question, to the ones he asked, is to give a precise description of the notion of algorithm. In the search for a good definition arose three…
Kolmogorov's Calculus of Problems is an interpretation of Heyting's intuitionistic propositional calculus published by A.N. Kolmogorov in 1932. Unlike Heyting's intended interpretation of this calculus, Kolmogorov's interpretation does not…
Quantum theory and functional analysis were created and put into essentially their final form during similar periods ending around 1930. Each was also a key outcome of the major revolutions that both physics and mathematics as a whole…
In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area…
Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…
This paper undertakes a foundational inquiry into logical inferentialism with particular emphasis on the normative standards it establishes and the implications these pose for classical logic. The central question addressed herein is: 'What…
The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…