Few people in history have known such a drastic reversals of popularity as the 4th century BC Greek thinker Aristotle of Stagira. Aristotle’s influence was enormous in the ancient world, where the two aspects of his philosophy, naturalistic and speculative, alternated. In the Western Middle Ages, the two cornerstones of knowledge relied on Aristotle, the cultural authority, and the Bible, the religious authority. For the Bible there could be no contradiction between science and faith, since the object of science and the basis of faith both originate in God, and this led to the uncontested supremacy of aristotelic reasoning. On the opposite shore of the Mediterranean, the Aristotelian tradition was kept alive by the Arabs who, thanks to their interest in the natural sciences, produced numerous commentaries and translations of the Greek philosopher. Aristotle’s scientific theories did not collide, indeed they fit perfectly with the stories reported in the Koran; for this reason, Averroes, the greatest scholar of the Stagirite philosopher, affirmed that aristotelic thought should not be interpreted but accepted.

However, starting with the Renaissance, the principle of authority (expressed in the form ipse dixit, “he said it”, which admitted no further objection to the aristotelic dictate) started to be criticized, notably from natural scientists like Galileo, Harvey or Pascal, which realized that scientific investigation must pass to the sieve of logical reasoning and experience, and therefore “authority is useless”. Many of the aristotelic findings in science were found to be increasingly contradicted by the modern experiments, such as the notion that nerves start from the heart, or that embryo grows by adding parts, or the criticism of the atomic hypothesis on the basis of the logical impossibility of void. While the influence of Aristotle’s studies is still reflected in many great philosophers of the XX century, and elements of aristotelianism are still the object of active study today and continue to imprint different aspects of philosophy and religion, most scientists, and also a large part of philosophers, consider Aristotle and his thinkings a way of the past, that has not much to say about our current world. Aristotle’s approximations and deductive reasonings have no longer a place in modern science.

But we often talk approximations, e.g., to go from classical to relativistic, or from classical to quantum, and back. And I am not sure everybody is always perfectly aware of what that actually means. For example, it is commonplace to say that “quantum statistics merge into the classical distribution for high enough temperature”. Actually, when you look at the mathematical form of the quantum distributions, 1/(exp(E–μ/kBT)±1), and take the limit of a large kBT compared to the typical energy scale E, you will notice that it is actually the opposite. If the denominator of the exponential becomes large, the ±1 is never negligible, and you’ll never get the “classical” exp(μ–E/kBT) distribution. It actually looks rather that in order to get the classical limit, you should go instead to very low kBT… The subtlety is all in the temperature dependence of the chemical potential that, via the DeBroglie wavelength, is proportional to ln(T–3/2). Hence, “for a temperature large enough such that the particle spacing is much larger than the DeBroglie wavelength”, your chemical potential becomes (classically) large and negative. And this is what actually makes the ±1 in the quantum formulas to go away, and gives back the classical distribution.

Considering what actually are the approximations that make a certain theoretical model to work is crucial to develop a description of reality, I would say much more important than the ingredients of the model itself. We will never have a “monster” theory that can explain each and every phenomenon with the same degree of precision, just for the very reason that some terms in each theory necessarily become more or less important, according to the level of description we are looking for. A similar consideration can be done for classical gravity: Newtonian physics provides an effective model for understanding the physics of gravitation but, strictly speaking, it is wrong. Einstein’s theory of general relativity provides a better description of, e.g., Mercury’s trajectory. But if we restrict to a certain domain of small relative velocities, weak gravitational fields, etc., we get the Newtonian theory in the appropriate approximation, and we forget about Einstein and curvature. Understanding these relations is important, as it clarifies the meaning of relations between different stages of successful theories. It actually shows the very nature of our theories, that is, of how we understand and describe the world around us. In this respect, as much as also Einstein’s theory at some point – close to the Planck length scale – becomes wrong, we should say that Aristotle’s physics is not at all wrong per se, but it is valid in its domain of approximation, and gives back Newton when needed.

Aristotle’s physics is the correct approximation of Newtonian physics in the particular domain where we, humans, conduct our business. This domain is formed by objects in a spherically symmetric gravitational field (the Earth) immersed in a fluid (air or water). The fact that Aristotelian physics is to be properly understood as the physics of objects immersed in a fluid has been nicely emphasized in the works by Monica Ugaglia. For a modern student it may sound strange to start physics by studying objects moving in a fluid. But for the average lay person, it should actually sound strange to start from anywhere else: everything around us is immersed in a fluid, and touches some surfaces. So, why university physics starts by studying the movement of isolated bodies in void and without any friction? We can see that Aristotle’s physics is a quite correct description of these phenomena, and is consistent with Newtonian physics, in the same manner in which Newtonian physics is consistent with Einstein’s physics, when taken in its domain of validity.

Following the original point of view of Carlo Rovelli, let us look at Newton’s equation for the motion for a body of mass m, the famous F=ma (force and acceleration to be intended as vectors). This equation is an extreme simplification of our reality, where it should be rather written as: F= –mgz + Vρz – Cρvn + Fext, with z a vector pointing to the center of the Earth, V the volume of the body, and ρ the density of the medium in which the body moves with a (vector) velocity v; C is a friction (drag) coefficient that depends on the shape of the body, with the exponent of the drag force being about n=1 for slow motion (low Reynolds) and rather n~2 for fast motion; and Fext are any other external forces that may be acting on the body. If Fext is zero, Aristotle would call this the “natural” motion of the body; otherwise, it would be a “violent” or unnatural motion. Let us firstly look “aristotelically” at a simple situation, of a body with zero velocity in natural motion. The equation is ma= –(mg–Vρ)z = V(ρ–ρb)z, with ρb=mg/V the density of the body. It is evident that the body will start moving up or down in the medium, with a positive or negative velocity according to its density difference. Therefore, ipse dixit, water moves down in air, while air moves up in water. A body immersed in a substance of the same kind, like water in water, stays at rest: it lies at its “natural” place. Wood, which according to Aristotle is a mixture of air and water (his chemistry was indeed limited), can move up in water and down in air: its “natural” place is lower than air, but higher than water. This was indeed taken in the antiquity as the explanation of why boats float, hence it followed that a boat could not be built from metal. Aristotle’s theory is clearly incomplete here. According to Archimedes (one century later), the quantity V in the above equation is not just the volume of the body, but the overall volume of water it displaces. When the Greek sailors understood Archimede’s principle, they started building much larger ships in the 3rd century BC, with the hull protected by sheets of metal.

Now, consider the full equation of motion in one dimension for a moving body: ma= –mg + Vρ – Cρvn, that is more clearly, dv/dt= –(g–Vρ/m) –(Cρ/m)vn. This is a differential equation of the form v’=–A–Bvn, whose solution simply gives a constant speed (that is, v’=0) when v=(A/B)1/n. Hence, we see immediately that the velocity of the body ultimately attains a steady value of [(mg – Vρ)/Cρ]1/n. To be more precise, there is indeed an initial time in which the velocity is not constant, but for any practical purposes such a time is of the order of (mg/Cρ)-1/n, and it is always relatively short to be observable (at least with the instruments of ancient sciences). Then, since the buoyancy term Vρ is usually much smaller than the weight mg, the steady state velocity turns out to be just proportional to (mg/ρ)1/n. As Aristotle properly observed, a body subject to gravity attains a constant velocity while moving in a fluid, provided of non-zero density (the impossibility of void, or horror vacui); this speed is faster for a less dense fluid (air vs. water); and a heavier body should always get to a faster speed than a lighter one. O Galileo, Galileo, did you ever actually drop your two balls from the top of the Pisa tower?

In the case of “violent” motion, for which Aristotle states that the body will come to rest in a finite time once the external force stops acting on it, the equation of motion reads: ma = -Cρv2 + Fext, or, d2x/dt2= –(Cρ/m)(dx/dt)2 + Fext. When the external forces cease to act upon the body, Fext=0, this equation is easily solved as y’= –Ay2 with y=x’, obtaining a solution with x(t) proportional to the logarithm of t. Hence the derivative of x(t), that is the velocity v(t), goes to zero as 1/t, and the body comes at rest. So, again, Aristotle physics remains correct until the regime of observation is the appropriate one, so as to say, a steady state in which all the initial and final transients are not easily accessible to experimental determination. Galileo, with his ingenious experiments of balls rolling on the inclined plane, was able to study just these early stages of motion, in which indeed the buoyancy and drag forces have not yet settled in, and the only force acting is gravity. By lowering the inclination angle, Galileo properly isolated the effect of the force (acceleration), attaining such slowfalling speeds that the initial phase was actually measurable, and he thus obtained the celebrated “progression of odd numbers” for the distances traveled in equal times (that is, the Newtonian equation of motion x”=at2/2), independently on the mass of the rolling ball. Galileo’s findings resisted the time and ushered in the revolution of experimental method in science. But Aristotle was not entirely wrong. As well as he was not entirely right.

Aristotle’s theories in biology are of quite little value today, despite some brilliant intuitions; he often reported second-hand notions, or let himself to wild speculations. But the relentless, systematic work of observation, dissection and classification that he conducted for two years on the island of Lesbos has represented the basis of biology and taxonomy for two millennia, until Linnaeus and Darwin came on board. Reading his books about animals is always a source of immense pleasure, for both their poetic and scientific value. Although many of his observations have obviously been superseded by the advent of modern microscopic techniques, he introduced comparative anatomy, pattern discovery, and a general scientific method of inference, and his vision of the “soul” makes him strangely close to modern systems’ biology. In his monumental book Perì Zòon Genéseos  (The Generation of Animals, 760b), he ends a long discussion about the mystery of the reproduction of bees, with the most modern statement about the scientific method: “Such appears to be the truth, judging from theory and from what are believed to be the facts; the facts, however, have not yet been sufficiently grasped; if ever they are, then credit must be given rather to observation than to theories, and to theories only if what they affirm agrees with the observed facts.” Ipse dixit.

Aristoteles vindicatus

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