I confess Pure Mathematics seems an Applied Science to me. Childhood activities familiar to many of us who have mathematical talent include counting, assembling similars together, and early disambiguation. We have joy in these activities. Preparing for revision of mathematics due to updating the definition of if, I was faced with the same question I faced in the winter semester of 1996 when taking on the responsibility of bringing mathematics to market. At what level to start, imposing thereby which implicit standards for humanity? For bringing mathematics to market the standards I stand for have basic binary logic, as I learned from reading Quine’s Elementary Logic, be the floor of human conversation, plus fallacy recognition as cognitive self-defense, brought down to grade eight reading comprehension levels, thereby casting the largest supportive net for humanity I might. Thus in 2011, reconsidering the same question, this time for handling our update of mathematics with derivative disciplines, I began by updating the definition of money since money has familiarity.
Where does mathematics begin? We in the sciences shared our pursuit of the atom, the indivisible concept, the axiom, the root source from whence light and all matter. A Set Theoretic approach may propose starting with the Propositional Calculus. Arithmetic Calculi including single variate, multi-variate, vector, integral, and differential, frequently start with Boolean Algebra and maps, then functions. Introductory Linear Algebra may leap into groups and fields, so intent on methods of calculation in matrix form as to omit mention of the optional absence of continuity in the representation of information. Brainstorming a random tour of Mathematics in lieu of beginning with the beginnings of mathematical logic could produce anything from the Riemann Hypothesis, whether P= NP, to Statistics, Finance, Accounting, Algorithm Design, Computer Architecture and applications in the Social Sciences, Rather than randomly tour derivative disciplines of Mathematical Logic, instead we could begin with the Propositional and Predicate Calculi, the Foundations of Mathematics from Sets to Formal Number Theory, Computability, Decidability and some advanced preliminaries. However, sixteen years of bringing Mathematics to Market prompts me to opt for world driven agendas. Abstractions of these approaches meriting attention include analysis of what constitutes an acceptable Logic system, what in Logic is definable, whether Arithmetic inspires Logic, ior whether to lead with Relations.
The approach I select in live application, from the moment of redefining money, includes settlement of the classical paradoxes then subsequent challenge of the axioms, thereby establishing the beginnings of a new way of seeing Set Theory. Before developing this system, I prefer review select introductions to Mathematical Logic including William Van Orman Quine and Bertrand Russell. Three important dissertations are gathered together in Provability, Compexity Grammars, American Mathematical Society Translations, ISSN 0065 9290, series 2, volume 192, translated to English from Russian with preface written by Sergei Artemov. Lev Beklemishev, author of Classification of Propositional Provability Logics, establishes definability constraints for positive, constructive provability logics. Mati Pentus, author of Lambek Calculus and Formal Grammars, proves formal languages arising from Lambek grammars have liberty from context, thereby establishing dual systems the Chomsky hierarchies V the Lambek categorical grammars. My work aims to show Lambek grammars have plausibly constructive symbiotic relations with nature. IE given dual system proposals, the biosphere has its existence preferences. Nikolai Vereshchagin, author of Relativizability in Complexity Theory, shows dual solutions show up due to relativization to different oracles, therefore responds with a general framework for formulating relativizability criteria and for limit analysis.
Proposed as its own authoritative discipline undomesticated from Philosophy ior Science, Mathematical Logic has a familiar sequence in its opening preamble which aims to systematically build from establishment of the concept of truth to establishment of the concept of numbers. Rereading these introductions recently, with the world-supplied driving question Who Is Who?, I see these painstaking introductions to such intuitively clear concepts as evidence of half-acknowledged battles in an undeclared war. From the perspective of the supply chain, what in the world pre-existed mathematical minds seemingly inspired to devote lifetimes to splicing (axiomatic) truth from near-truths, conditional truths, meta-truths, inductive and deductive truths, organizationally quantified truths, relational truths, and similar systems for discerning isness, prior to having sufficient defined reality to define the concepts zero, one and successor. IE what disruptive force in conversations pre-exists such patient, detailed, thoughtful, positive construction of what there is to disambiguate, for whom, prior to saying what zero is, what one is, and what is next. Succeeding...