Philosophy of Mathematics: Making a Fresh Start

Mathematical Thinking and Learning, 2019

Abstract At a time in the history of mankind when man became aware of his environment the changes he observed in the phenomena prompted him to wonder. Then he tried to speculate on what could be at the base of reality. This quest for knowledge came in two distinct forms: First some consider it only in the material aspect of being where as others based it on both material and immaterial aspects. Though it is relatively old for people of the developed nations, the discipline is also relatively new in the sense that the study comes with its discoveries in consonance with the age that studies it. The task of this paper is to examine the philosophy of mathematics and its relevance to national development.

The present doldrums position and state of decadence, internal differences, external aggression (geographical and ideological), lack of self-confidence and dependence, illiteracy, political instability, economic disaster, lack of knowledge and wisdom, back benchers in science and technology, education, medicine, trade and business, banking system and defensive incapability of Muslim World prompted me to look at our principal sources of inspiration, which are, the Qur’an, Sunnah of the Prophet (SAW), and examples of the “enlightened Caliphs” and see what is Islam’s view about seeking knowledge, technology and inventions in general and mathematics’ education in particular. We will discuss the nature of mathematics and its scientific status. We will highlight the position of mathematics in Islamic classification of knowledge. We will also discuss the current state of mathematics and future suggestions. We have gathered together some of these impressions; these are all tentative, nothin...

I. Prologue: Scientific and Cultural Bases of Mathematics ORIGINS AND HISTORICAL PERSPECTIVE The present account is largely based on the principle that mathematics is a scientific activity with social-historical roots, in opposition to a static view that takes it as existent per se, quite independent from human culture. The first stand, on whose side one counts famous forerunners, sees mathematics in a position similar to that of other sciences, to wit, both a function and agent of society and of the scientifically explainable universe. The precise language of mathematics and the beauty of its universal formulas, apparently conceived to perfection, would seemingly be responsible for the platonic-like vision of the discipline as a pure manifestation of the human or divine spirit, which exists independently of a given cultural state of mankind or of its social development. The nature of the present document moves away from the above vision and tries to convey the idea that the mathema...

2015

can be traced back to the (Kantian) conviction that mathematics does not depend on human experience, but is accessible to the reine Vernunft. The truth of mathematical statements and the status of mathematical knowledge according to this conviction should not depend on contingent facts about humans or human society. This view has not only left traces in philosophy of mathematics, but can also be seen in the practice of 2 · Benedikt Löwe, Thomas Müller, and Eva Müller-Hill sociology: Compared to sociological research dealing with other sciences, sociology of mathematics is severely underrepresented.1 On the other hand, some of the central questions of philosophy of mathematics have an empirical core, and some of the statements that one finds in philosophical texts about mathematics are empirical claims. For instance, consider the question of the connection between philo-sophical views and mathematical achievements. According to main-stream philosophical belief, many working mathemati...

Paradigmi 33(2), pp. 23-42, 2015

In order to deal with the question of the applicability of mathematics this article distinguishes between natural mathematics, that is, innate mathemat-ics, and artificial mathematics, that is, mathematics as a discipline. It argues that natural mathematics is applicable to the world because the systems of core knowledge of which it consists, being a result of biological evolution, fit in certain mathematical properties of the world. On the other hand, the basis for the applicability of artificial mathematics to the world is Galileo’s philo-sophical revolution, the decision to confine physics to the study of some properties of the world mathematical in character, that is, of a kind currently dealt with in mathematics. But, like the applicability of natural mathematics, also the applicability of artificial mathematics depends on our makeup, and hence ultimately on biological evolution. This puts constraints on the ap-plicability of mathematics to the world.

LOGIC AND POSSIBILITIES TO ANSWER THE QUESTION: WHAT IS MATHEMATICS? (Atena Editora), 2022

This article derives from an experience carried out in the classroom, in order to answer some questions relevant to the usefulness of mathematical knowledge. It brings a suggestion for the Introduction to Reflection on “What is Mathematics”, its applications, its logical structure, its functionality as a language. It starts with this questioning and ends with the exploration of a problem, evolving from a case of trial and error to a logical-combinational way of solving certain problem situations. In the analysis and elaboration of the solution, I approach both basic numerical/operational, as well as exploring concepts/characteristics of Simple Series, as is the case of Arithmetic Progressions, all without losing sight of the investigative characteristics that mathematics provides us, depending on how you approach it. As a contribution to this part, I work with some ideas on Mathematical Research in the Classroom, from Mathematics teachers: João Pedro da Ponte, Joana Brocardo and Hélia Oliveira. I also talk about: the Language Obstacle, the logical structure that a language must have, the symbolic completeness required to efficiently communicate an idea, the interpretation of facts in Problem Solving, the analysis of data and its properties and the generalization of the possible results. To do so, I will dialogue with the teachers: Júlio César de Melo e Sousa (Malba Tahan) and Luiz Carlos Pais, in addition to a quick conversation with the psychologist, anthropologist and sociologist Carlos Rodrigues Brandão about Education. I conclude, justifying algebra and the value of generalizations, with a very simple example: Mathemagic, to paraphrase Walt Disney.

Synthese, 1991

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematicai theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.

Resonance, 2018

In this article, I discuss the relationship of mathematics to the physical world, and to other spheres of human knowledge. In particular, I argue that Mathematics is created by human beings, and the number π can not be said to have existed 100, 000 years ago, using the conventional meaning of the word 'exist'.