Physical Theories: An Overview

Now that I have described how modern physics was founded by Galileo Galilei as a new science, I will follow the path of physics in the following and show how the legacy of the pre-Socratics and ancient mathematicians has borne fruit. After a decline of the ancient culture one could, after about 2000 years, revive it and reach new heights of knowledge. The fact that it took so long may has many reasons – if one can speak of reasons at all in history. However, I do not wish to take part in such considerations.

First, I want to give an overview of the theories that have been developed and established in physics over the course of time. The fact that they always had to prove themselves in competition with other theories than the “better ones” in each case was explained, for example, in (Honerkamp, 2017, p. 111ff). In the next chapters I will show what role the questions raised by the pre-Socratics have played in the development of these theories. In particular, I will discuss the development of the theories and their relationships to each other. Especially I will pursue the path along which there have always been mergers or unifications of theories. Thus, today we are only talking about two great theories and are out to recognize these two as parts of a single “theory for everything”. This would answer the most noble question of the pre-Socratics, the question of a “One”, albeit in a completely different way than one could imagine at that time.

The development and establishment of a physical theory always involved the explanation of phenomena of a certain type. We have already seen that the phenomenon of “motion” was the first theme that interested people in antiquity and again in the Renaissance. It is the most original and probably also the most general phenomenon that we know. 

We encounter the phenomenon of “light” in a similarly direct way.  So, it is no wonder that at the beginning of modern physics not only the phenomenon of motion was dealt with, but also experiments with light were carried out, as we know them from Isaac Newton, for example. A century later, people began to study other, seemingly quite different phenomena, electrical or magnetic. Finally, at the turn of the 20th century, they discovered sorts of radiation which obviously differed greatly from light rays, and among these new rays there were also different types.

The space of phenomena

In short, the history of physical theories is a history of discoveries in a “space of phenomena” where there have always been certain objects in play. In this space, one can identify large areas in which there are phenomena that seem to be so similar to each other that one might be tempted to invoke the same reason for their explanation. Thus, over time, the idea of fundamental forces acting between planets and the sun, electrons and other found or discovered objects emerged. Over time, four such interactions were distinguished: Gravitational, electromagnetic, strong and weak interaction. This distinction is still very helpful for an overview of the set of the theories developed in the 400 years since Galileo.

There are, however, two other categories in respect of which one should distinguish the theories. It is not only the interaction or the force that can be decisive for a phenomenon. This can also be the range on a length scale. Thus, phenomena can be distinguished according to the scale of length on which the phenomenon appears, whether in the world of the smallest dimensions, the largest dimensions or the middle dimensions.

Finally, an aspect will become important that has to do with our cognitive abilities, namely the question of whether we must describe the phenomenon as a complex one, namely as one in which it is not sufficient for an understanding to consider only a few objects with few important properties. Many or very many objects can represent a system that has new properties that are not inherent to the individual objects themselves, but only “emerge” through the interaction of the objects. Water, for example, has the property of being liquid. However, this does not apply to its components, the H2O molecules.

With these two characteristics, spatial size and complexity, we can already consider aspects that allow us to have an overview in the form of a landscape of phenomena. If one enters the characteristic length of some objects, which play a role in physical theories, into a coordinate system, in which this size is plotted against the complexity of the objects, then one obtains e.g. Fig.4.2.

Fig.4.2: Rough classification of certain objects according to size and complexity. Objects like planets can appear at different places, depending on how much of their properties you want to consider (R stands for order of magnitude, N for number of degrees of freedom or complexity resp.).

On this figure we can show how far we have explored the space of natural phenomena today with our physical theories. It also makes it clear that in addition to the physics of fundamental interactions, there is also a very large area of complex systems for which areas of physics such as thermodynamics or statistical mechanics, solid state physics, etc. are responsible. This should be kept in mind, even if we do not deal with it here and we mainly consider the wide range of spatial scales in the area of “simple” systems – from 10-15 to 1020 m.

The world of the middle dimensions in the range of about 10-4 to 1010 m is most accessible to us intellectually, because we ourselves, as participants of this world, can have direct experiences with it. Thus, the phenomena of this world are also the subject of the earliest physical theories; they are also called classical theories. The exploration of the space of phenomena thus started in the world of the middle dimension.

At the beginning of the 20th century, phenomena of the world’s smallest dimensions were discovered. One had to state that the concepts of the world of the middle dimensions are no longer suitable here. A completely different concept, a “quantum”, replaced the concept of a material object and gave physics on this scale the name “quantum physics”. At the same time a modern cosmology and astrophysics began to emerge. Today we hear of particularly spectacular discoveries in this field of the largest dimensions.

Classical physics, quantum physics and cosmology: this is a classification that can also be described as physics of medium, smallest and largest dimensions. Cosmology today has not yet required its own conceptual apparatus, as quantum physics does, which is why it is also added to classical physics, if one wants to emphasize the methodological aspect.

With a distinction regarding the spatial size alone one has of course not yet exhausted the space of phenomena. There are other quantities which, measured by the conditions of our world of daily experiences, can be small or large. Particularly prominent in this context is speed; but the strength of fundamental forces, in particular of gravity, will also be significant for the nature of physical theories. The landscape sketched in Fig.4.2 must therefore only be imagined as a slice from the whole space of phenomena.

The exploration of this space of phenomena resembles the exploration of our earth in the time of the great discoveries in the 16th century. One spoke thereby of the discovery of the “world”, although it were only new ranges of the planet earth, which one discovered at that time gradually. Today one knows almost every corner of the earth and “reaches for the stars”. 

Thus, one also knows all laws of nature in the world of the middle dimensions, but only on the fundamental level. The more complex the systems are in these dimensions, the less familiar they are to us today. But the more we limit ourselves to the fundamental side, the further we have advanced into the world of the smallest and also the largest dimensions.

Even if it were possible in several years or decades to establish a theory for all fundamental interactions, physics would not be at its end. In the direction of complex systems there are still many questions waiting for an answer. The transition to chemistry, biology and cognitive science will be fluent. Also, in the exploration of life and consciousness one will not be able to ignore physical conditions.

The theories of classical physics

Classical physics is dominated by three large phenomenon areas: the phenomenon of motion and the two areas in which we encounter the fundamental forces of gravity and electromagnetism, respectively.

Phenomena from these areas have been known since ancient times. “Nothing is older than motion,” we may quote Galileo once again. For the pre-Socratics, motion or non-motion always played a role, Aristotle distinguished different types of motion and formulated a first kind of theory of motion. Even in the Middle Ages there were always natural philosophers who wanted to trace the nature of motion.

Gravity was also an everyday phenomenon. With Aristotle it was a quality that made all bodies of the sublunar world strive for the centre of the world. Also, the sphericity of the earth was later explained by such a “natural striving”.

We read less about magnetic and electrical forces in early sources, but magnetic and electrical phenomena were already known in ancient times. If you rubbed an amber, it would attract dust or shreds of wool. Iron was attracted by a magnetis stone and it was discovered that splinters of such stones always rotate in a north-south direction.

The physical theories, which today explain all basic phenomena from these three phenomenon areas, are

– for motion: Newton’s and Einstein’s theory of motion,

– for gravitation: Newton’s and Einstein’s theory of gravitation,

– for electrical and magnetic phenomena: Maxwell’s theory of electromagnetism.

Normally, Newton’s theory of motion and gravitation is subsumed under the name classical mechanics. Newton’s theory of gravity essentially consists of a law for the forces between two material bodies. Newton was able to use this law to explain the motion of the planets within the framework of his theory of gravity.

Einstein’s theory of motion is the special theory of relativity, Einstein’s theory of gravity is the general theory of relativity. Both represent extensions of the corresponding Newtonian theories to a larger range of phenomena: for motions to “higher” velocities, for gravitation to “higher” velocities and to “stronger” gravitational forces. It will still be necessary to make precise what the terms “higher” or “larger” mean in each case.

Maxwell’s theory of electromagnetism serves to explain all electrical and all magnetic phenomena as well as those phenomena in which electrical and magnetic effects are mutually dependent. It is the result of a unification of two earlier theories, one for electricity and one for magnetism.

Gravity and electromagnetic forces act over long distances. Otherwise, we wouldn’t have felt it all the time. They are therefore called long-range, in contrast to the short-range forces that were only discovered in the world of the smallest dimensions. They hold the world “together at its innermost” and their reach does not go beyond that. Of course, the long-range forces can affect even at short distances, electromagnetic forces are even quite important for understanding structure of atoms. However, gravitational forces at the level of atoms have not yet been registered. The masses of the building blocks of the atoms are obviously much too small.

The theories of quantum physics

The establishment of Maxwell’s theory, in particular by the discovery of electromagnetic waves in 1886, increasingly drew physicists to the question of how an electric current and how electromagnetic radiation can be generated in matter. In the end, the question of the structure of matter stood in the center of attention.

From a pre-Socratic Leukipp and his follower Democritus one knew the concept of an atom, a smallest indivisible particle (άτομος gr. indivisible). The chemists used this idea in the 19th century with great profit for the explanation of the laws in the reactions of different chemical elements. But there were also vehement opponents, because one had not yet “really seen” an atom and the idea of indivisibility only raised new questions.

But other questions also came to mind. In the “golden years of physics” from 1895 to 1898 further rays were discovered, such as X-rays, cathode rays, α-, β- or γ-rays. Finally, there were the heat rays, a phenomenon that had been known for some time: All bodies become red, light red and finally white-yellow with constant heating; and one feels that heat emanates from them.

The origin and nature of these rays had to be understood. It was a very fruitful time for physics, and during this time the idea of atoms should establish itself as building blocks of matter, but only as a milestone on the way to ever smaller building blocks. One could then obtain an explanation for all these rays and thereby gain a consistent picture of the structure of matter and the atom.

In an attempt to develop this picture into a consistent theory success was only achieved after daring to describe the discovered relationships between the experimental results with a completely different mathematical conceptual apparatus.

A test case for each approach of a theory for the structure of an atom was the calculation of the possible energy states of a hydrogen atom. The success or failure of a mathematical calculation thus now decided on the success of a theory in this world of the smallest dimensions. The building blocks of an atom such as electrons, protons or neutrons could then no longer be regarded as particles in the sense of classical physics and the living world. They were soon called “quanta”, like the energy packages Max Planck had talked about in a lecture on 14 December 1900 when he gave an explanation of thermal radiation. By the way, the date of this lecture is regarded today as the birthday of quantum physics.

The quantum theories responsible for all the fundamental phenomena of the world’s smallest dimensions are first of all

– quantum mechanics, to a certain extent the replacement for classical mechanics

– quantum electrodynamics, the continuation of electrodynamics on the atomic level.

An explanation of the origin and nature of α- and β-rays could only be achieved by introducing two completely new types of forces, the “strong” force, which is responsible for the binding of the building blocks of the atomic nucleus, and the “weak” force, which can cause the transformation of a neutron into a proton, but also serves to describe the decays of other later discovered “particles”. Thus

– theories of weak and of strong interaction

were created. These two theories were constructed according to the model of quantum electrodynamics. This soon led to the desire to describe these three interactions within a unified theory. As an intermediate step the

– theory of electroweak interaction,

a unification of the electromagnetic and weak interaction was found, and finally

– the unified theory of electrical, weak and strong interaction, the so-called standard model.

Today, this model is regarded as the basic quantum theory. Quantum mechanics now plays the role of a theory for a limited range of phenomena in which relativistic effects do not have to be taken into account and in which there is no decay and no generation of particles.

Figure 4.3 shows the development of the individual theories over the course of time.

Fig.4.3: Timetable for the emergence of physical theories of fundamental interactions

Zenon of Elea, the Motion or how to find seeking the better

A disciple of Parmenides, Zenon of Elea (around -490 to around -430) still attracts special attention today because of the paradoxes with which he annoyed his philosophically interested contemporaries. We now understand paradoxes to mean argumentations that lead to contradictions because an unclear or incorrect idea of a concept is at play. That was the case for both movement and infinity at the time. Today we have clear ideas about these terms and can resolve the paradoxes that have so unsettled his discussion partners. 

Zenon wanted to support the theses of his teacher Parmenides with his considerations. At least this is how we read it in Plato’s dialogue PARMENIDES, in which he lets Zenon say: (after (Mansfeld & Primavesi, 2011, pp. 365, No. 5):

In reality, my writing is something of a support for the Parmenides’ thesis, which is aimed at those who try to make him ridiculous.

His four movement paradoxes are particularly famous. We want to deal here with the third paradox, because it is based on an error, the enlightenment of which stood at the beginning of modern physics. This is about the apparent contradiction between the observation of a flying arrow and the Parmenides assertion that this movement of the arrow is only apparently present, since the being remains in absolute silence.

A flying arrow, Zenon argues, is at a given time in a certain place, which is always as big as the arrow itself. Since he was there in the “now”, he could not be in motion then. So, he is at rest in every moment, the arrow actually stands still. The motion we’re watching is a fake.

I don’t find that argument convincing. After all, Zenon already shows a scepticism towards our everyday perceptions, albeit in the most extreme form. In any case, Zenon seems to me to have the idea that in every “now” the state of an arrow is determined by a location alone, and he assumes that there is no motion. Aristotle says in contradiction: “In the “now” neither rest nor movement can take place” (Mansfeld & Primavesi, 2011, pp. 383, No. 23). So he already sees that Zenon only thinks of the most obvious when describing the state of a moving body. Zeno’s argument does not convince him either. But his counterargument does not go any further as well.

The arrow paradox dissolves when one knows that the state of a body in space is determined by a location and (!) by a velocity (or impulse) at any time, in any “now”.

This was Galilei’s discovery in the early 17th century. He studied the motion of a small ball as it rolls down an inclined long wooden channel. Not only did he find that the distance she travels on the wooden channel increases with the square of time. He also extended the channel beyond the inclined part and observed that the ball on the horizontal channel continues to run the longer the less its movement is affected by unevenness of the ground. He concluded that, with ideal ground, it would then have to continue to run forever. The motion remains as it is if there are no external influences on what is moving.

So movement is a state. In medieval “impetus theory” the movement was still a process: an “impetus” had to work constantly. This would be given to the body at the beginning, kept the movement upright, but was also slowly used up, so that it gradually came to a standstill. Galilei, on the other hand, attributed a slowing of a motion to external influences, e.g. friction. By refraining from external circumstances, he was able to discover a principle of nature that would prove extraordinarily fruitful for the further development of physics. I’ll come back to that soon. But first we have to look at how a velocity in the “now” can be grasped in concrete terms.

The instantaneous velocity

Galilei did not yet have the possibility to calculate an instantaneous velocity. In mathematics he was still at the level of the ancient Greeks, where geometry as a description of nature stood in the foreground. But his contemporary, the French philosopher and mathematician René Descartes, discovered how geometric problems could be converted into arithmetic problems. An “Analytical Geometry” was created, which represented a great advance compared to ancient mathematics and which for the first time enabled one to go beyond the status of the ancient Greeks.

One now learned to describe the location of a point in a coordinate system and to see such points as locations of material bodies, if one abstracted from their expansion. One could also represent the place x(t) of the body in dependence on the time t in a coordinate system.

An average velocity in a time period dt was easy to calculate by forming the ratio dx/dt, where dx may be the distance covered in the time dt. But it became difficult if you wanted to extrapolate to the instantaneous velocity, i.e. if you had to determine the velocity in the “Now”. The time span dt should actually be zero, the distance dx thus also, and the ratio 0 to 0 makes no sense. One had to choose somehow a very small time span dt, which should be however arbitrarily small, but still unequal 0. Somehow these quantities had to have something to do with the “infinitely small”. They called them infinitesimal. It was not a clear idea, but they succeeded in consistently calculating the ratio dx/dt in the “now”, the “differential quotient”. This “infinitesimal calculus” was developed independently by two great thinkers of the time for general functions f(x): Isaac Newton needed this knowledge for his reflections on motion. Gottfried Wilhelm Leibniz regarded it as a purely mathematical problem which had to be solved if one wanted to determine the tangent in a point of the curve of a function in a diagram.

For some time such calculations were very popular; they inspired many new ideas and questions. At the end of the 18th century, mathematicians were no longer satisfied with the justification of such calculations with infinitesimals. The Italian mathematician Lagrange found a method for calculating the differential quotient without having to use the term infinitesimal. In the 1960s, a new type of number could finally be defined using hyperreal numbers in a so-called non-standard analysis. A clear definition of the infinitesimal was now possible: they were certain hyperreal numbers.

Mathematics is sometimes like physics and actually every science: New concepts are not always clearly defined at first. But you can already use them and when you notice that they are “good for something”, you start to take care of the conceptual basics at some point. But it often takes some time for satisfactory clarity to be achieved.

The Evolution of Motion Theory: One Finds “Seeking the Better”

Knowledge of how to calculate the instantaneous velocity from a time-dependent spatial coordinate was a prerequisite for a theory of motion in the language of mathematics. While Galilei had discovered a relationship between distance and time in free fall, one theory was now concerned with describing the position and velocity of a body as a function of time.

The physicists and mathematicians of that time knew their ancient models very well. Especially the ELEMENTS of Euclid of Alexandria, in which he brought the then known laws of geometry into a “logical order”. Euclid has thus set a benchmark for what a mathematical or physical theory should look like. At the beginning there are definitions, conventions and axioms. According to this, all statements of the theory must be logically deducible from the axioms according to mathematical rules.

Newton formulated his theory according to this model. Galileo’s idea that motion can be a state was the first axiom he incorporated into his theory of motion, which is now known as “Newtonian mechanics”: “A body remains at rest or in straight-line, uniform motion if no forces act on it.

Here, of course, something must have been said about space and time in the definitions beforehand, so that one knows what uniform motion means. So you have to know what a straight line is, and you have to say something about the course of time before you can speak of a uniform velocity, one that is constant in direction and size.  Only then can one speak of this particular motion in the axiom and postulate that this motion remains if there is no external influence on the moving body.

In a second axiom, Newton then logically describes a procedure for formulating a mathematical equation for the case that an external force now acts on the body. With a suitable mathematical expression for the force, one can then calculate all motions in the sky and on earth from such an equation of motion, taking into account existing circumstances.

This Newtonian theory of motion, briefly outlined here, was regarded as the only ideal of a scientific theory for over 200 years from the end of the 17th century, and its structure as an axiomatic-deductive system represented a model for future sciences.

Here is an opportunity to speak of two other theories of movement: on the one hand a theory that Aristotle had formulated about 2,000 years earlier, and on the other a theory that Albert Einstein developed some 300 years later and was soon called “special relativity theory”. One can very beautifully demonstrate with these three theories how indeed in the course of time people found “seeking the better”. This story doesn’t describe a special case. One can find many such examples. But let us first characterize the other two theories:

Aristotle was a great systematist, and so he first of all distinguished the motions in the sky from the motions on earth. He divided the earthly motions again into movements of living beings, into natural and finally into forced motions. He gave a different reason for each type of movement. The motions in the sky showed the eternal order. In natural motions the “disturbed order” was restored, e.g. smoke rises to heaven and a stone falls to earth, because light has its place above and heavy has its place below. In a forced motion, a force must always act, otherwise the motion would come to a standstill.

This Aristotelian motion theory had more than 2,000 years of existence. It was still taught in the academies during Galilei’s lifetime, and Galilei had studied it intensively himself until he finally overcame it.

Newton’s theory finally completely replaced it, for it is obviously “better”. From the equations of motion of Newton’s theory, it was possible, with a certain expression for the force which a body exerts on another body due to its mass, to derive the three Kepler laws for the motion of the planets around the sun, even to accurately predict the return of a comet. So with fewer assumptions one can explain more phenomena. The theory also makes predictions that can be tested and indeed have been confirmed. To this day, it is indispensable for calculations in everyday motions.

The theory of motion developed by Albert Einstein, the special theory of relativity, is in turn better than Newton’s theory. The motivation for the development were problems with the idea that the whole universe is filled with a subtle substance, an “ether”, which has prevailed since ancient times. It should also be the carrier of the electromagnetic waves, which at that time had only been known for about two decades. Ether should also mark absolute rest and one wanted to measure motion of earth versus the ether. Whenever and however you did it, you couldn’t see such motion.

Einstein took the bull by the horns, so to speak. He made this negative result the principle of his new theory: “The speed of light in any inertial system is independent of the speed of the light source”. So, this means that no matter how I move relative to the light source, I always measure the same speed for the light.

This theory is also structured as an axiomatic-deductive system. As such, it is even particularly “elegant”, because it is based only on this principle, as on a “principle of relativity” already known from Maxwell’s theory of electromagnetic phenomena. A wealth of phenomena could then be predicted; some of them differ strikingly from our everyday experiences, but in the mean time they have all been proven experimentally.

The hypothesis that there should be something like an ether was dropped. He was no longer needed. There is no absolute rest, but instead an absolute speed: the speed of light as measured in a vacuum. It is an upper limit for the transmission of effects.

When comparing these two theories, one finds that the special theory of relativity is an extension of Newton’s mechanics, in the sense that the smaller the speeds to be considered are in relation to the speed of light, the better the agreement between the statements of both theories. For velocities close to the speed of light, however, the astonishing phenomena already mentioned above are predicted, which have all been confirmed in the meantime.

A useful yardstick for the “goodness” of a theory is its scope of validity. Newton’s theory already had a very large validity range, because with it one can explain all motions which are “non-relativistic”, i.e. sufficiently small compared to the speed of light of approx. 300,000 km/sec. Here one could also try the theory of relativity. That wouldn’t be necessary, it’d just be harder. If one now considers increasingly higher speeds, the statements of the two theories will differ more and more. One leaves the scope of Newton’s theory but remains within the scope of the special theory of relativity. In this sense it is therefore an extension of Newton’s mechanics and thus the better theory.

If we look at the image of an evolution of theory, then we can say that the Aristotelian theory survived 2,000 years because there was no other theory that could become dangerous to it. But Newton’s theory was such a dominant competitor that Aristotle’s theory became extinct. The special theory of relativity is then a further development of Newton’s theory, so that there are now two theories that both have their own habitats. Where these overlap any theory can come into its own.  

Aristotle – Newton – Einstein: Aristotle has submitted, Newton and Einstein “found seeking the better”. Who knows when someone will come and find something better, and what further insights we will gain about motion and thus about space and time. Only one thing seems clear to me after 2,500 years: The way of Xenophanes to find “seeking the better” is also the better way to knowledge.