According to the second fundamental theorem... change has to take place in such a way that the total entropy of the particles increases. This means a… - Ludwig Boltzmann

It can never be proved from the alone, that the minimum function H must always decrease. It can only be deduced from the laws of probability, that if the initial state is not specially arranged for a certain purpose, but haphazard governs freely, the probability that H decreases is always greater than that it increases. It is well known that the theory of probability is as exact as any other mathematical theory, if properly understood. If we make 6000 throws with dice, we cannot prove that we shall throw any particular number exactly 1000 times; but we can prove that the ratio of the number of throws in which that number turns up to the whole number of throws, approaches the more to 1/6 the oftener we throw.

[W]e will consider the , rather than the velocity of the molecules. Each molecule can have only a finite number of values for its kinetic energy. As a further simplification, we assume that the kinetic energies of each molecule form an ...<math>0,\epsilon,2\epsilon,2\epsilon,...p\epsilon</math>We call <math>P</math> the largest possible value of the kinetic energy, <math>p\epsilon</math>. ...after the collision, each molecule still has one of the above values of kinetic energy.

Let us now tum to the second matter in dispute between us [about the rejection of philosophy in virtually all senses]. That the majority of students don't understand philosophy doesn't bother me. But can any two people understand philosophical [i.e. metaphysical] questions [and agree upon what it is that they understand]? Is there any sense at all in breaking one's head over such questions? Shouldn't the irresistible pressure [Drang] to philosophize be compared with the nausea caused by migraine headaches? As if something could still struggle its strangled way out, even though nothing is actually there at all? My opinion about the high, majestic task of philosophy is to make things clear, in order to finally heal mankind from these terrible migraine headaches. Now, I am one who hopes not to make you angry by my forthrightness, but the first duty of philosophy as love of wisdom is complete frankness. Through my study of Schopenhauer, I am learning Greek ways of thinking again, but piecemeal.