ABSTRACT

A scrutiny of the contributions of key mathematicians and scientists shows that there has been much controversy (throughout the development of mathematics and science) concerning the use of mathematics and the nature of mathematics too. In this work, we try to show that arithmetical operations of approximation lead to the existence of a numerical uncertainty, which is quantic, path dependent and also dependent on the number system used, with mathematical and physical implications. When we explore the algebraic equations for the fine structure constant, the conditions exposed in this work generate paradoxical physical conditions, where the solution to the paradox may be in the fact that the fine-structure constant is calculated through different ways in order to obtain the same value, but there is no relationship between the fundamental physical processes which underlie the calculations, since we are merely dealing with algebraic relations, despite the expressions having the same physical dimensions.

Keywords: Geometry; Algebra; Numerical Uncertainty; Fine Structure Constant and Physical Uncertainty

1. Introduction

In this article, we study the implications of numerical uncertainty for the measurement of various physical magnitudes, such as the fine-structure constant and the speed, mass and charge of an electron. Numerical uncertainty occurs due to the need to engage in processes of algebraic approximation, and has profound implications for the measurement of physical magnitudes, which have been relatively neglected in the study of the relationship between mathematics and sciences such as physics.

The use of mathematics in science became so widespread that it is now difficult to imagine the formulation of scientific theories in many areas, such as physics, without the use of mathematics. Mathematics brought precision to the formulation of many theories within astronomy, and within the natural sciences, especially when there is the possibility of insulating causal mechanisms within an experimental context.

Given the widespread success of the use of mathematics as an instrument for formulating scientific theories, there has not been much scrutiny of the nature of the instrument which contributed so much to this success. Mathematics is taken to be a standard of precision, and an instrument which brings precision to the formulation of scientific theories.

However, a scrutiny of the contributions of key mathematicians and scientists shows that there has been much controversy throughout the development of mathematics and science, concerning the use of mathematics, and the nature of mathematics too. And a careful investigation of those controversies has important implications for the interpretation of scientific theories. For it shows that the instrument which is taken to be a standard of precision, namely, mathematics, has not always been used with full precision, a fact which introduces much uncertainty in the numerical estimations made within scientific measurement, and in the very formulation of scientific theories.

We start the article with a brief discussion of the use of geometrical and algebraic methods within mathematics, and afterwards address the implications of the use of those methods within scientific studies. We then show the implications of the use of algebraic or numerical approximations, and their path-dependent nature. Those implications are then scrutinized in more detail in the case of physics, more specifically when measuring the fine-structure constant and the relationship between the mass, speed and charge of an electron.

2. Geometry and Algebra

The mathematician Michael Atiyah [1] notes that behind the controversy between Newton and Leibniz over differential calculus, there were two mathematical traditions, one grounded on geometry, which Newton followed, and another one dealing with algebra, which Leibniz followed.

Newton believed that continuity was an essential property of Nature. But for Newton, only geometry can provide certain knowledge of a continuous reality. Algebra and arithmetic, when attempting to describe a continuous reality, can designate exactly the rational numbers, but provide only processes of approximation when attempting to describe real numbers such as the square root of a given prime number. However, Newton believed that processes of d to a greater extent than Einstein, deserving closer attention. This aspect is the philosophical implication that science has for our conception of reality. Whitehead [24] was later led to concluding that reality constitutes a process, as Heraclitus argued long ago. The fact that the electron’s charge and mass might change with its speed is another instance of this philosophical aspect. Free electrons, as a part of reality, are a process which is permanently changing, and probably its speed, mass and charge, cannot be taken to be always fixed.

The fact that reality is a continuous flow, raises the question of how to grasp knowledge of a reality which is permanently changing, where it is not only the speed of a particle, but also its mass and charge, which are changing. Plato’s solution, which was also followed by Whitehead and was already adopted before by the Pythagorean school, is to grasp the forms (geometrical figures and natural numbers) that reality assumes in this permanent flow. We can identify equations, like equation (1), which give us some knowledge of the process of change and try to reach conclusions based on mathematical operations.

Mathematical operations are, however, subject to uncertainty, which must be taken into account when using mathematics. Whitehead and Russell [25] tried to provide exact foundations for mathematics. Gödel’s incompleteness theorem undermined this prospect. Since Pythagoras and Plato, there has been a belief that reality possesses a mathematical structure to be discovered. Even Newton, who was aware of the fact that algebraic processes provide only approximations, still believed that a proper geometrical method (freed from algebra) could provide an exact description of reality, which Newton believed was continuous, rather than discontinuous.

For example, the Lorentz transformations can be described using the geometrical method that Newton used in his Principia, as the variation (the derivative) of the arc of a sinusoidal function of. This would eliminate numerical uncertainty by using a geometrical method instead of an algebraic method, following Newton’s philosophy, according to which Nature should be described in geometrical terms, rather than through algebraic processes that contain uncertainty because they only provide approximations.

These philosophical considerations are not an additional curiosity, but an essential ingredient for the development of mathematics and science. The results discussed above, concerning the relation between speed, mass and charge of particles, can only be properly addressed within a wider field, which encompasses those considerations. Amongst the discussions to take place, a central one is whether the reality we are studying is best seen in terms of processes or particles, as a continuous or a discontinuous entity.

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NOTES

^*Corresponding author.