The Significance of Evidence-based Reasoning in Mathematics, Mathematics Education, Philosophy, and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines

Specker Sequences Revisited

Specker sequences are constructive, increasing, bounded sequences of rationals that do not converge to any constructive real. A sequence is said to be a strong Specker sequence if it is Specker and eventually bounded away from every constructive real. Within Bishop’s constructive mathematics we investigate non-decreasing, bounded sequences of rationals that eventually avoid sets that are unions of (countable) sequences of intervals with rational endpoints. This yields surprisingly straightforward proofs of certain basic results from constructive mathematics. Within Russian constructivism, we show how to use this general method to generate Specker sequences. Furthermore, we show that any nonvoid subset of the constructive reals that has no isolated points contains a strictly increasing sequence that is eventually bounded away from every constructive real. If every neighborhood of every point in the subset contains a rational number different from that point, the subset contains a strong Specker sequence