Related papers: To be or not to be constructive, that is not the q…
Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…
When teaching an elementary logic course to students who have a general scientific background but have never been exposed to logic, we have to face the problem that the notions of deduction rule and of derivation are completely new to them,…
In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that…
Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent…
It is well known that most constructive and predicative foundations aiming to develop Bishop's constructive analysis are incompatible with a classical predicative development of analysis as put forward by Weyl in his $\textit{Das…
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 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…
In this paper, we present a paradox arising from the acceptance of the Law of Excluded Middle (LEM) within classical mathematics. Specifically, we construct a nonzero analytic function on a connected open subset of the complex plane whose…
We provide a systematic, thorough treatment of the foundations of probability theory and stochastic processes along the lines of E. Bishop's constructive analysis. Every existence result presented shall be a construction; and the input…
There is a received wisdom about where to draw the boundary between classical and nonclassical for various types of quantum processes. For multipartite states, it is the divide between separable and entangled; for channels, the divide…
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…
A cyclic proof system gives us another way of representing inductive definitions and efficient proof search. In 2011 Brotherston and Simpson conjectured the equivalence between the provability of the classical cyclic proof system and that…
Eugene Wigner famously argued for the "unreasonable effectiveness of mathematics" for describing physics and other natural sciences in his 1960 essay. That essay has now led to some 55 years of (sometimes anguished) soul searching ---…
This paper describes an axiomatic theory BT for constructive mathematics. BT has a predicative comprehension axiom for a countable number of set types and usual combinatorial operations. BT has intuitionistic logic, is consistent with…
One of the most challenging and fascinating issues in mathematical and theoretical physics concerns identifying the common logic, if any, which underlies the physical world. More precisely, this involves the search of the possibly-unique…
Recently, a novel intuitionistic reconstruction of the foundations of physics has been primarily developed by Nicolas Gisin and Flavio Del Santo drawing on naturalism. Our goal in this paper is to examine and develop the philosophical…
Large Language Models (LLMs) have demonstrated impressive capabilities in structured reasoning and symbolic tasks, with coding emerging as a particularly successful application. This progress has naturally motivated efforts to extend these…
We define constructive truth for arithmetic and for intuitionistic analysis, and investigate its properties. We also prove that the set of constructively true (first order) arithmetical statements is Pi-1-2 and Sigma-1-2 hard, and we…
It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by double-negation, while the…
Machine learning models increasingly function as representational systems, yet the philosoph- ical assumptions underlying their internal structures remain largely unexamined. This paper develops a structuralist decision framework for…