Some reflections on mathematics and mathematicians: Simple questions, complex answers

Abstract

SUMMARYIn this work we present some reflections on mathematics and mathematicians. Special emphasis is placed on the questions (1) what is mathematics? And (2) what is a mathematician? Some reflections and open questions are posed at the end of the work.(ProQuest: ... denotes formulae omitted.)... Mathematic does not have own existence. It is only an arbitrary code, designed to describe physical observations or philosophical concepts. Each can adapt it to its own needs. Dr. John Keyser, Ph.D. in Physics20. Introduction.Professions have played a key role in the development of disciplinarily, and vice versa. Within some disciplines the direct binding to a profession or a field have over time been loosened and (re)searching knowledge for its own sake has become a main driving force of a new, advanced kind of disciplinarily. For mathematics these historical shifts are symptomatic in the debates over the discipline's true nature. While the relationship between science, technology and mathematics historically the last 200 years has been rather symbiotic, mathematics today serve so many different professions and fields, that a unified, valid definition of its nature is hard to find.Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth from careful reasoning. These remain fruitful and important motivations for mathematical thinking, but in the last century mathematics has been successfully applied to many other aspects of the human world: voting trends in politics, the dating of ancient artifacts, the analysis of automobile traffic patterns, and long-term strategies for the sustainable harvest of deciduous forests, to mention a few. Today, mathematics as a mode of thought and expression is more valuable than ever before. Learning to think in mathematical terms is an essential part of becoming a liberally educated person.Much of mathematics is itself about mathematical objects. This is part of why mathematics can seem like an arcane and up-approachable field to an outsider. Fortunately in asking What is Mathematics? We are asking about the meaning and consequences of Mathematics as connected to the larger world, that is, we are asking what Mathematics means outside of its own world, in answering our question we can largely ignore many of the details of Mathematics3.Plato tries to clarify a position when indicating that the mathematical objects have their own existence, beyond the mind.Aristotle saw the mathematics like one of the divisions of the knowledge that was different from the physical knowledge and the theological one. He denied that the mathematics were a theory of an external knowledge, independent and unnoticeable. It associated to the mathematics with a reality where the knowledge obtains by experimentation, observation and abstraction. This position joint party that the construction of the mathematical ideas occurs through idealizations realized by the mathematicians like a result of its experience with objects in a specific context.The points of view of Plato and Aristotle have represented the great poles where the discussion has oscillated about the nature of the mathematics.But the absolutist vision entered crisis due to the discovery of some contradictions found in certain theorems that comprised of mathematical systems considered rigorous. For example, Russell demonstrated that the logical system of Frege was inconsistent. The paradox of the property of being an element of itself (a set is element of itself if and only if it is not element of itself) did collapsed its law number 15. But the mathematic one is certain and if all theorems are true, how can contradictions exist between their theorems? Something must be mistaken in the foundation of the mathematics. Paul Ernest proposes a socio-constructivist vision of the mathematical one4. In this vision it is considered that the mathematical truth is fallible and correctable, and that is the overhaul always open. …