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This book invites readers to see mathematics not just as formulas and rules, but as the deepest expression of human thought. It begins by exploring the timeless idea of mathematics as a universal language, contrasting its precision with the…
There are growing uncertainties surrounding the classical model of computation established by G\"odel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows…
The present article introduces ptarithmetic (short for "polynomial time arithmetic") -- a formal number theory similar to the well known Peano arithmetic, but based on the recently born computability logic (see…
This work proposes a complete algebraic model for classical information theory. As a precursor the essential probabilistic concepts have been defined and analyzed in the algebraic setting. Examples from probability and information theory…
Constructor theory seeks to express all fundamental scientific theories in terms of a dichotomy between possible and impossible physical transformations - those that can be caused to happen and those that cannot. This is a departure from…
This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we…
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of…
This paper has several purposes. We present through a critical review the results from already published papers on the constructive semigroup theory, and contribute to its further development by giving solutions to open problems. We also…
The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which…
Despite ample evidence that our concepts, our cognitive architecture, and mathematics itself are all deeply compositional, few models take advantage of this structure. We therefore propose a radically compositional approach to computational…
We observe some puzzling linguistic data concerning ordinary knowledge ascriptions that embed an epistemic (im)possibility claim. We conclude that it is untenable to jointly endorse both classical logic and a pair of intuitively attractive…
We introduce a notion of Kripke model for classical logic for which we constructively prove soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications.
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
We develop a constructive theory of continuous domains from the perspective of program extraction. Our goal that programs represent (provably correct) computation without witnesses of correctness is achieved by formulating correctness…
Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof…
The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become…
In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with the…
The classical simulation of physical processes using standard models of computation is fraught with problems. On the other hand, attempts at modelling real-world computation with the aim of isolating its hypercomputational content have…