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.