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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.

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.

The mathematician Michael Atiyah [

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 approximation cannot provide the certainty required by science. Thus, he thought that geometry was the more appropriate tool for describing Nature - see Guicciardini [

Atiyah [

The Cartesian axes presuppose the idea of a continuous geometrical line, where a real number corresponds to each of the infinite points of the continuous line. The geometrical idea of a real line presupposes a continuum of points, but algebraic and arithmetical operations can only provide an approximation (as close as we like) to some of those points. While an exact formulation of rational numbers can be easily obtained, we cannot reach the real numbers that are presupposed by the Cartesian axes (such as the square root of a given prime number) in any other way, other than through a process of arithmetical approximation. Thus, Newton thought that geometry must be studied without Cartesian axes, which introduce numerical discontinuities, and thus uncertainty, into science, and science ought to reach certain knowledge.

However, the development of mathematics followed the Cartesian perspective, rather than Newton’s perspective. Until the beginning of the nineteenth century, the Cartesian method was dominant in the European continent (where Leibniz’s notation was adopted) while Newton’s geometrical approach was dominant in Cambridge and England. And throughout the nineteenth century, the Cartesian approach became dominant even in Cambridge and in England.

The discontinuities in the real line were addressed afterwards through the contributions of Dedekind, Cantor and Zermelo, which led to the completion of the algebraic project started with Descartes, and the numbering of the real line that was implicit in the Cartesian axes. This perspective, often termed as mathematical “Platonism”, assumes that numbers are existing entities.

Criticisms of this perspective continued, not least through mathematicians like Kronecker, for whom only the natural numbers were exact, and real numbers were constructed through arithmetical operations, rather than “Platonic” entities that already exist. But criticisms such as Kronecker’s were marginalized, and the “Platonic” approach of Cantor and Zermelo, developed by Hilbert too, became the standard approach within mathematics. The Cartesian project led thus to the perspective which is now dominant within mathematics.

Scientists have often used Cartesian mathematics when formulating their results, without further discussion of the problems of the use of Cartesian axes within geometry that were perceived early on by Newton. For example, Einstein writes, in his book The meaning of Relativity:

“I shall not go into detail concerning those properties of the space of reference which lead to our conceiving points as elements of space, and space as a continuum. Nor shall I attempt to analyse further the properties of space which justify the conception of continuous series of points, or lines. If these concepts are assumed, together with their relation to the solid bodies of experience, then it is easy to say what we mean by the three dimensionality of space; to each point, three numbers, x_{1}, x_{2}, x_{3} (co-ordinates), may be associated, in such a way that this association is uniquely reciprocal, and that x_{1}, x_{2} and x_{3} vary continuously when the point describes a continuous series of points (a line)” [

Einstein is clearly aware that further discussion of the assumption of a continuity of points of the Cartesian axes is necessary. But he does not go into detail on this issue, and simply uses the Cartesian coordinates, unlike Newton, who felt the need to abandon the Cartesian perspective, and ground his approach within pure geometry.

However, if we scrutinize the implications of Newton’s perspective on mathematics for Einstein’s theory, we reach interesting conclusions, which are connected to the idea of uncertainty, which was advanced by Heisenberg quickly after Einstein made the remark above. Newton’s point was that arithmetical and algebraic operations provide only approximations to real numbers. Indeed, if we want to compute a square root of a non-squared number, we can reach a degree of approximation as close as we want. But since we can only make a finite number of operations, there will be a given degree of uncertainty concerning the final result, which depends on how far we decided to go in our process of approximation. And this uncertainty has important implications for Einstein’s relativity theory too, as we shall see.

Arithmetical operations of approximations lead to the existence of numerical uncertainty, which depends upon the operations made, and the number system used. The uncertainty of the outcome is closely linked to the numeric base used in operations. The fraction 1/10, for example, can be represented in the decimal number system as 0.1, but in binary format becomes the regular binary decimal 0.000110011001100110011 ... which is not exact. What happens is that 0.1, despite being accurate in the decimal system, ceases to be accurate on the binary base and cannot be represented in a finite way. Thus, we can only reach approximations to this quantity in a binary based calculation system.

Therefore, a calculation that leads us to the real number 1 does not correspond necessarily to the natural number 1. That depends on the set of rules used throughout successive approximations. In other words, we could get the number 1 in various ways. The number 1 can be obtained as the product of n times^{2} times^{3} times

and so forth, but such a result is nothing more than an approximation. There is no uncertainty in the value of nwhich are just natural numbers, but there is uncertainty in the calculation of

Each of those operations, whose exact outcome would be the natural unit, causes errors, that is, numerical uncertainty, when we generate this number. Both the addition and subtraction and the product are defined in N, while Q, the set of rational numbers, is generated only by the introduction of the operation of the division of natural numbers. In this context, any rational number q can be algebraically generated by an infinite set of data from an operation, whose statistical distribution has uncertainty k, and where q is the mean value of the range

This also means that in any calculation process in R there remains numerical uncertainty when determining the value of a natural number, which depends on the sequence of operations through which we proceed. The question that follows concerns how to measure this numerical uncertainty. We use here, as an estimate of the numerical uncertainty in the generation of a real number, the standard deviation of the statistical distribution of all products of natural numbers by their inverse, which lead to the number we wish to calculate.

In this work, we represent numerical uncertainty by

In the calculation of a real number by the aforementioned operations, the resulting error, evaluated through the standard deviation, is associated with the approximation produced in the operations involved. Regardless of the type of approximation performed or numeric base used, any number on which we operate always has a finite number of digits, given our finite ability for computing numbers. This error is not, in most cases, equivalent to the calculated error by the theory of errors where

The numerical uncertainty given by ^{a}k between groups of numerical uncertainty. It should be noted that these uncertainties are associated with the concrete realization of operations (division and product) of numbers that generate the unit and as such, are not effectively the natural unit, but an approximation to the unity in real numbers. In order to visualize the behavior to which we are referring, in the following graph (

As we can see in figure 1, numerical uncertainty^{p}^{ }points, with

Potentiation introduces certain rules for small natural numbers, since until n = 100 we have that

If n is a natural number and k the finite uncertainty of a real n generated by algebraic operations, we can find that

acts on the uncertainty of the previous number n is the same or vanishes, that is:

Obviously radicals can also produce numerical uncertainty, which means that

In general,

If we focus on the expression

The fine-structure constant α is a physical constant which characterizes the magnitude of electromagnetic force and was defined by the first time by Arnold Sommerfeld [_{0}

the vacuum permittivity, h the Planck’s constant and c the speed of light in vacuum.

For some time now, physicists of the international scientific community have questioned themselves whether the so-called universal constants are actually “universal variants”, that is, capable of assuming new values as time goes by. If the fine-structure constant, even being an empirical constant, had a lower value, the density of atomic matter in the Universe would also be lower, with weaker connections under lower temperatures. If, on the contrary, the fine-structure constant were larger, the smaller atomic nuclei would not exist due to electric repulsion between protons. This interpretation of alpha can predict a physical outcome; even we do not assign it a real physical meaning. Others interpretations where proposed: α is the ratio of two energies or the ratio of the velocity of the electron in the Bohr model of the hydrogen atom to the speed of light, among others.

Richard Feynman, referred to the fine-structure constant in these terms:

“There is a most profound and beautiful question associated with the observed coupling constant, α - the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to π or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don't know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly!” [

In fact, the numerical uncertainty Δ(0.08542455) is much lower than the numerical uncertainty of Δ(137.03597).The numerical error of 0.08542455 measured as the standard deviation of 1000 calculations of the type ^{−16} and the standard deviation associated to a thousand calculations of ^{−12}. Let us assume then that 137.03597 is the average of the numbers generated in the previous operation and that if we take into account numerical uncertainty it will lead to an interval from137.03597 − 2.50239E^{−12} to 137.03597 + 2.50239E^{−12}. In order to calculate the value of α mentioned by Feynman, for the number 137.03597 we have then two possibilities:

^{−16}and 1.40267E^{−15}, respectively.

If the difference of 2 in the last decimal case related to the calculation of α, is connected to the calculation path, then it is possible that the two values are coincident with the numerical intervals obtained through their numerical uncertainty. In fact: (0.08542455 − 7.08393E^{−16}) − (0.0854245500000008 − 1.40267E^{−15}) = 0, which explains that the difference between the two values of the fine structure constant pointed out by Feynman may be due to the numerical uncertainty generated by the operations.

The fine-structure constant can also be calculated as

Since

From equation 1, is possible to compute the product _{e} will be the mass of the traveler electron with speed greater than the speed in the hydrogen atom. If this is possible what happens with the electron charge in movement? Must this product be in all circumstances constant?

Recently Nair et al. [

Apparently, the reason why dividing the percentage of visible light shining through π gives the fine structure constant is that the electrons in graphene behave as if they have lost their mass. That could mean that the fine structure constant could physically represent more than a simple ratio between the speed of the hydrogen electron and the speed of light in the vacuum.

Webb et al. [

Nasseri [^{−17})α, where α is known the fine-structure constant. This reduction is indeed very small.

When we calculate the numerical uncertainty for the fine-structure constant^{−16} which is slightly larger than the calculated change of α from Nasseri [

Let us assume here an hypothetical physical relationship between the product

If this is possible, equation (1) shows that if the relativistic mass of the electron tends to infinity, due to a speed increase of the electron to light speed, its electrical charge would go down to zero. The idea that mass may reach infinity is also suggested by Einstein’s expression for the equivalence of mass-energy, which shows that the inertial mass of a particle varies with the relation_{o}_{ }is the mass of a particle at rest and m its mass when it moves at constant speed v.

However, if we allow either the mass or the electrical charge to reach infinity (where the other factor of the product

Any of the conditions exposed here generates 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 relation between the fundamental physical processes which underlie the calculations, since we are merely dealing with algebraic relations, despite the two expressions having the same physical dimensions.

This issue raises problems in mathematics and philosophy which were present in the mind of scientists from Newton to Einstein, as noted above. Newton believed that Nature was continuous, and thus used geometry, rather than algebra, since algebra provides only approximations which contain discontinuities. For Newton, only through geometry one can obtain a representation of nature where uncertainty is not introduced through algebraic processes of approximation. Einstein uses Cartesian axes in his analysis without discussing its underlying presuppositions.

The introduction of algebra introduces numerical uncertainty which, if Newton is right, may introduce uncertainty and discontinuity where it does not exist. Or it may be the case that Newton was wrong and Nature is discontinuous. In that case, algebra, rather than geometry (numbers, rather than figures), capture an aspect of reality, rather than introducing uncertainty into a certain reality. Whatever is the case, we can certainly benefit from a study of the nature of the instrument we are using when studying Nature and the Universe, in order to infer the extent to which we are capturing an aspect of reality, or imposing our framework into reality.

When we admit speeds of the electron greater than the velocity of the electron in the hydrogen atom, a loss of mass of the electron (has observed by Nair et al. [

Davies [

With these issues in mind, let us explore the uncertainty associated to the algebraic expression of equation (1). The physical and experimental uncertainty when calculating the c, ε_{0, }h and e constants is frequently changing due to improvement of the accuracy of laboratory processes. These physical constants can take many dimensional forms, such as the case of the speed of light, or be dimensionless, as the fine structure constant α.

We have, at least so far, no reason to believe that the constants

Now, in Millikan’s experiences, the measurement of the electrical charge of the electron is determined through variables which depend upon the electron’s mass [

If we admit a relativist change in the electron’s mass, through equation (1), we are led to the conclusion that there is also a relativistic variation in the electron’s charge, which should lead, through the product

where m_{e}_{ }is again relativistic.

This shows that positive numerical uncertainty associated with ^{−120} C^{4} kg. Numerical uncertainty is here the absolute value of the difference between

It is assumed that the two members of equation (1) are equal, term by term, with the same nil result, except if we use approximations. A study of numerical uncertainty, measured as the difference between the two members of equation 1, shows that its statistical distribution is Gaussian, with a mean of −7.6687E^{−122} C^{4} kg and a standard deviation of 4.9359E^{−120} C^{4} kg. That means, for example, that a percent error of measurement of the electron’s charge of 8.71E^{−33} C, increases the numerical uncertainty by a factor of 10.3, which is still very small when compared to experimental uncertainty.

The most likely value of the difference between

where n is part of N, but n assumes only specific values of N. The rule of the values assumed by n is connected to the numerical uncertainty produced.

If equation (1) is physically valid, then there is no reason to believe that equations (2) and (3) are not valid too. Thus, the electron may well be above the speed of light in the vacuum, even if the uncertainty surrounding

does not increase greatly, or if

Once again we have, essentially by equation (3), quantic numerical uncertainties, as was observed early in the expressions of the numerical approximation and calculus path dependence.

A numerical uncertainty of this type, for values above the speed of light, only results in a basic randomness in the electron’s movement, as a particle electrically charged, and a limited statistical predictability of the behavior of this apparently quantic system.

We can see the variation between n’s (from equation (3)) which lead to almost zero uncertainty as n increase, and independent of the speed of the electron. It is easy to see that there is not a clear tendency, but as speed increases, the behavior appears to be similar to that of uncertainty of the type

By the equation (3), apparently, there is no convergence (or a tendency for nil uncertainty) to zero as the speed of the electron approaches the speed of light. If there is any such convergence, it exists only for values above the speed of light.

If convergence is of the type

Einstein’s equation restricts the electron’s movement to a speed below the speed of light in the vacuum, which would mean that the behavior observed above has no physical meaning, but Bertozzi’s [

Thus, if equation (1) would have physical meaning the mass of the electron, as its charge, would be relativistic variants. This does not challenge Einstein’s relativity theory. It only means that the speed of light in the vacuum may not be the highest existent speed in the Universe because the electron may travel at a faster speed, due to its physical mass singularity.

We can measure the speed of an electrically charged particle through Cherenkov’s radiation, or Cherenkov’s effect [

These experiments suggest an interaction between electrical charge (which may exist in excess) and the speed of the particle. If the mass and charge of the electron change with its speed, when the electron passes from vacuum to other environments at a high speed, its speed will diminish, and the observed physical effect will be an apparent loss of electrical charge.

We have no algebraically or physical reasons for denying the possibility that equation (1) reflects a physical relationship, which seems to translate a clear dependence between the electron’s mass and charge, leading to the appearance of another level of uncertainty, connected to the simultaneous determination of their mass and charge.

Whitehead [

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 [

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

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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