Scientific Perspectivism and the Methodology of Modern Mathematical Physics

Perspectival realists often appeal to the methodology of science to secure a realist account of the retention and continued success of scientific claims through the progress of science (e.g. Massimi, 2016). However, in the context of modern physics, the retention and continued success of scientific claims is typically only definable within a mathematical framework. In this paper, I argue that this concern leaves the perspectivist open to Cassirer’s (1910) neo-Kantian critique of the applicability of mathematics in the natural sciences. To support this criticism, I present a case study on the conservation of energy in modern physics.

Mathematical Indispensability and Arguments from Design

The recognition of striking regularities in the physical world plays a major role in the justification of hypotheses and the development of new theories both in the natural sciences and in philosophy. However, while scientists consider only strictly natural hypotheses as explanations for such regularities, philosophers also explore meta-natural hypotheses. One example is mathematical realism, which proposes the existence of abstract mathematical entities as an explanation for the applicability of mathematics in the sciences. Another example is theism, which offers the existence of a supernatural being as an explanation for the design-like appearance of the physical cosmos. Although all meta-natural hypotheses defy empirical testing, there is a strong intuition that some of them are more warranted than others. The goal of this paper is to sharpen this intuition into a clear criterion for the (in)admissibility of meta-natural explanations for empirical facts. Drawing on recent debates about the indispensability of mathematics and teleological arguments for the existence of God, I argue that a meta-natural explanation is admissible just in case the explanation refers to an entity that, though not itself causally efficacious, guarantees the instantiation of a causally efficacious entity that is an actual cause of the regularity.

Contextualized education in the teaching of Mathematics: a case study at the Kindergarten and Elementary School Manoel Rodrigues do Nascimento

Teaching mathematics as a concept formation process requires rethinking the teacher's role, the conditions for organizing pedagogical work, the way of thinking, feeling and acting in education, the characteristics and interests of students. It needs the entire school community's involvement, presenting the content in a contextualized way, through a problematic situation, compatible with a real situation, which has elements that give meaning and construction to the mathematical content. This work sought to broaden the understanding of contextualization in the teaching of mathematics since working with content that is not related to the context in which the student is inserted is not attractive. To this end, a survey was conducted based on a semi-structured interview with the teachers and students of that school, in addition to observations and pedagogical practice. According to the results, the teachers' conceptions are identified and perceived, showing that the methodologies used by them in a contextualized way provide meaningful learning, since teaching is an active, evolving process about contextualization and its contribution to the learning of the students of this school, the research also shows how the contextualized teaching of mathematics is evaluated and the importance of bringing innovative, motivating and meaningful situations to classrooms. Therefore, it is understood that Contextualized Education in the teaching of mathematics enables the construction of the student's knowledge according to his reality, mainly due to the applicability of Mathematics, stimulates creativity, investigation, critical analysis of results and contributes significantly to the teaching-learning process.

On the Logic of Being and Wigner's Astonishment Regarding the Applicability of Mathematics

The Nobel Prize winning Physicist, Eugene Wigner, famously posed a powerful challenge (1960) by asking why is mathematics so effective, especially in the physical sciences. It is possible that the reason for the effectiveness of mathematics is not because mathematics is in any way causative, but instead because mathematics studies the structure of logical possibility and constraint. When plugged into a possible world, mathematics gives us the tools to analyze the logically possible outcomes. Therefore, when a possible world that is expressed mathematically sufficiently aligns with reality, mathematics becomes effective at expressing relationships and outcomes.

The System of Physics

Chapter 10 considers the structure of Proclus’ rarely discussed Elements of Physics and its original contribution to the understanding of physics in antiquity. It is argued that the purpose of the treatise is not only a systematic arrangement of the arguments scattered throughout Aristotle’s works on natural philosophy, using the structure of Euclid’s Elements as a model. Proclus also aims to develop a universal theory of motion or physical change that establishes the first principles as definitions, formulates and demonstrates a number of mutually related propositions about natural objects, and culminates in establishing the existence and properties of the prime mover. Unlike modern physics, which presupposes the applicability of mathematics to physics, Proclus shows that the study of natural phenomena in the more geometrico way can be a systematic rational science arranged by means of logic rather than mathematics.

The Re-Recognition of the Relationship between Mathematical Knowledge and Ability Based on Key Competencies and Values

In the current situation that ability is more and more valued, teaching process should take the key competencies of mathematics as the frame, realize the vividness, rigour and applicability of mathematics knowledge, identify these relationships that mathematical knowledge is the carrier of ability,mathematical ability is the sublimation of knowledge,mathematical knowledge and ability should be integrated as well as mathematical thought is the bridge of mathematical knowledge and ability so as to better cultivate key competencies of mathematics.