When did you experience Zeno's paradox?

The pre-Socratics: Zeno of Elea, the movement or how to search for the better

Views of a physicist

A pupil of Parmenides, Zeno von Elea (around -490 to around -430) still gains special attention today through the paradoxes with which he annoyed his philosophically interested contemporaries. We now understand paradoxes as arguments that lead to contradictions because an unclear or incorrect idea of ​​a term is involved. That was the case then for movement as well as for infinity. Today we have clear ideas about these terms and can resolve the paradoxes that made his discussion partners so insecure.

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

In reality, my writing is something of a support for Parmenides' thesis aimed at those who try to ridicule him.

Its four movement paradoxes are particularly famous. We want to deal with the third paradox here, because it is based on an error that was cleared up at the beginning of modern physics. It is about the apparent contradiction between the observation of a flying arrow and the assertion of Parmenides that this movement of the arrow only appears to exist, since beings remain in absolute rest.

A flying arrow, Zeno argues, is at a given time in a certain place that is always as big as the arrow itself. Since it is there in the “now”, it cannot be in motion either. So he is at rest at every moment, so the arrow actually stands still. The movement that we observe is only an appearance.

I do not find this argument convincing. Nevertheless, Zeno already shows a skepticism towards our everyday perceptions, albeit in a highly extreme form. In any case, Zeno seems to have the idea that in every “now” the state of an arrow is determined solely by a place, and he assumes that there is no movement. In contradiction to this, Aristotle says: “In the“ now ”neither rest nor movement can take place” (Mansfeld & Primavesi, 2011, pp. 383, no. 23). So he already sees that when describing the state of a moving body, Zeno only thinks of what is closest. Zenon's argument does not convince him either. But his counter-argument does not lead any further.

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

This was Galileo's discovery in the early 17th century. He studied the movement of a small ball as it rolls down a long, sloping wooden channel. In doing so, he not only found that the distance it traverses on the wooden channel increases with the square of time. He also extended the channel beyond the sloping part and observed that the ball continued to run on the horizontal channel, the less its run was impaired by unevenness in the ground. With ideal subsoil, it would have to keep running, he concluded. The movement remains as it is when there are no external influences on what is moving.

So movement is a state. In the medieval “impetus theory”, 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. Galileo, on the other hand, attributed a weakening of a movement to external influences, e.g. to friction. By ignoring external circumstances in this way, he was able to discover a principle of nature that would prove to be extraordinarily fruitful for the further development of physics. I will come back to this soon. But first we have to look at how a speed can actually be grasped in the “now”.

The current speed

Galileo did not yet have the possibility of calculating an instantaneous speed. In the mathematical field he was still at the level of the ancient Greeks, where geometry was in the foreground as a description of nature. However, his contemporary, the French philosopher and mathematician René Descartes, discovered how to convert geometric problems into arithmetic ones. An "analytical geometry" was created, which represented a great advance over ancient mathematics and with which one went beyond the level of the ancient Greeks for the first time.

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 extension. It was also possible to represent the position x (t) of the body as a function of the time t in a coordinate system.

An average speed in a period dt was easy to calculate by taking the ratio dx / dt, where dx is the distance covered in the time dt. It became difficult, however, if one wanted to extrapolate to the current speed, that is, if the speed had to be determined “now”. The time span dt should actually be zero, the distance dx as well, and the ratio 0 to 0 doesn't make any sense. Somehow you had to choose a very small time span dt, which should be arbitrarily small, but still unequal 0. These sizes had to have something to do with the “infinitely small”. They were called infinitesimal. It was not a clear idea, but with them it was possible to consistently calculate the ratio dx / dt in the “now”, the “differential quotient”. This "calculus" was developed independently by two great thinkers of the time for general functions f (x): Isaac Newton needed this knowledge for his considerations on motion. Gottfried Wilhelm Leibniz saw it as a purely mathematical problem that had to be solved if one wanted to determine the tangent at a point on the curve of a function in a diagram.

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

In mathematics it is sometimes the same as in physics and actually in 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 worry about the conceptual basics at some point. But it often takes some time to achieve satisfactory clarity.

The evolution of the theory of motion: One finds "looking for the better"

Knowing how to calculate the current speed from a time-dependent location coordinate was a prerequisite for a theory of motion in the language of mathematics. While Galileo had discovered a relationship between distance and time in the free fall, one theory was now about describing the location and speed of a body as a function of time.

The physicists and mathematicians of the time knew their ancient models very well. especially the elements Euclid of Alexandria, in which he brought the then known laws of geometry into a "logical order". Euclid thus set a standard 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 compulsory from the axioms, according to mathematical inference rules.

Newton formulated his theory on this model. Galileo's idea that motion can be a state was included as a first axiom in his theory of motion, which is also known today as “Newtonian mechanics”: “A body remains at rest or in linear, uniform motion when no forces act on him. "

Of course, something about space and time must have been said beforehand in the definitions so that one knows what straight-lined-uniform is supposed to mean. So you have to know what a straight line is, and you have to say something about the passage of time, before you can speak of a uniform speed, that is, one that is constant in direction and magnitude. Only then can one speak of this special movement in the axiom and postulate that this movement continues when there is no external influence on the moving body.

In a second axiom, Newton then logically describes a procedure for formulating a mathematical equation in the event that an external force is now acting on the body. With a suitable mathematical expression for the force, taking into account the circumstances at hand, one can then calculate all movements in heaven and on earth from such an equation of motion.

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

Here is the opportunity to speak of two other theories of motion, on the one hand a theory that Aristotle had formulated about 2,000 years earlier, and on the other hand a theory that Albert Einstein developed a good 300 years later and soon became the "special theory of relativity" was called. These three theories can be used to demonstrate very nicely how, in the course of time, people did in fact find “searching for the better”. This story is not a special case. Many such examples can be found. But let's first characterize the other two theories:

Aristotle was a great systematic, and so he first distinguished the movements in the sky from the movements on the earth. He divided the earthly movements into movements of living beings, into natural and finally into forced movements. For each type of movement he gave a different reason. The movements in the sky showed the eternal order. With natural movements, the "disturbed order" was restored, e.g. smoke rises and a stone falls to the ground, because light has its place above and heavy below. With a forced movement, a force has to act constantly, otherwise it would come to a standstill.

This Aristotelian theory of motion lasted for more than 2,000 years. It was still being taught in the academies in Galileo's day, and Galileo had dealt intensively with it himself until he finally overcame it.

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

The theory of motion developed by Albert Einstein, the special theory of relativity, is again better than Newton's theory. The motivation for the development were problems with the idea, which has prevailed since antiquity, that the whole universe is filled with a subtle substance, an "ether". This should also be the carrier of the electromagnetic waves, which at that time had only been known for about two decades. The ether should also mark the absolute rest and one wanted to measure the movement of the earth against it. Whenever and however you did it, you couldn't see any such movement.

In a sense, Einstein took the bull by the horns. He made this negative result the principle of his new theory: "The speed of light is independent of the speed of the light source in every inertial system." That means: however I move relative to the light source, I always measure the same speed for it Light.

This theory is also built up as an axiomatic-deductive system. As such, it is even particularly "elegant", because it is based only on this principle, as is still the case on a "principle of relativity", which is already known from Maxwell's theory for electromagnetic phenomena. A plethora of phenomena could then be predicted; some of these differ strikingly from our everyday experiences, but have now 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 calm, but instead an absolute speed: the speed of light as measured in a vacuum. It is an upper limit for the transfer of effects.

When comparing these two theories one finds that the special theory of relativity is an extension of Newtonian mechanics, in the sense that the statements of both theories agree better, the lower the speeds to be considered are compared to the speed of light. For speeds that come close to the speed of light, the amazing phenomena mentioned above are predicted, all of which have since been confirmed.

A useful measure of the “goodness” of a theory is its scope. Newton's theory already had a very wide range of validity, because it can be used to explain all movements that are "non-relativistic", that is, sufficiently small compared to the speed of light of approx. 300,000 km / sec. The theory of relativity could also be used here. But that would be unnecessary, just more difficult. If you now consider increasingly higher speeds, the statements of the two theories will differ more and more. One leaves the area of ​​validity of Newton's theory, but remains in the area of ​​validity of the theory of relativity. In this sense it is an extension of Newtonian mechanics and thus the better theory.

If we use the picture of an evolution of theory, then one can say that the Aristotelian theory survived 2,000 years because there was no other theory which could be dangerous to it. With Newton's theory, however, such a dominant competitor emerged that it soon “died out”. The special theory of relativity is then a further development of Newton's theory, so that there are now two theories, both of which have their own habitats. Wherever these overlap, every theory can come into its own.

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

Josef Honerkamp was professor for theoretical physics for more than 30 years, first at the University of Bonn, then for many years at the University of Freiburg. He has worked in the fields of quantum field theory, statistical mechanics and stochastic dynamic systems and is the author of several text and non-fiction books. After his retirement in 2006, he would like to devote himself even more to interdisciplinary discussions. He is particularly interested in the respective self-image of a science, its methods as well as its basic starting points and questions and can report on the views a physicist comes to in view of the development of his subject. Overall, he sees himself today as a physicist and "really free writer".