He starts by devising the thought experiment of a universe where time goes backwards. Naturally it follows the same sorts of rules as our own, modulo the sign change in time. If you imagine that it is an identical copy, even things like the forces will be the same (a little dimensional analysis: time appears squared in force, so the sign stays the same).
What doesn’t stay the same is entropy—the second law of thermodynamics. A famous illustration of this (not his) is of a movie of a man eating a steak. Run the movie backwards and the steak is un-swallowed, un-chewed, removed to the plate and assembled into a whole hunk of beef, un-cooked, un-butchered, and eventually un-slaughtered into a live cow. This is fairly old news, of course; people have been arguing about what makes time special for quite a while.
The author is a bit careless in his description of heat, but we understand what he means anyway. However, consider this section, which is the heart of his argument:
Tracing backwards, we find that, in the past, the farther back we go, the more we get a larger percentage of available energy in the universe, increasing at an ever greater rate. Therefore it follows that we must arrive at some definite time in the past—and that not at an infinite time back—when the available energy was 100% of the total energy of the universe. At a time probably not much farther back, all the motion in the universe must have consisted of molar motion of masses which, as we go back, must have increased in size till we arrive at a time when all the energy must have consisted of the energy of two halves of the universe moving together, each half of the universe being at a temperature of absolute zero and all its parts moving side by side at exactly the same velocity. This possibility, it is true, is somewhat corroborated by the fact that at present the stars are moving in two opposite directions, in two opposite currents, as it were, which may be supposed to be the remnants of the two original large groups of stars whose collision formed the present universe according to this hypothesis.
First, the stars don’t act like that today. Second, there is no reason to assume this sort of initial symmetry—there are other possibilities, with internal energy states. Kinetic energy isn't the only kind there is.
At the same time the two original halves of the universe cannot have been altogether mutually impenetrable, for in that case the result of the collision would but have made them rebound, though producing a great amount of internal heat-energy in each, and possibly breaking some small pieces off each. It would seem, then, as though the original halves of the universe must have consisted of separate dark stars, with a structure somewhat similar to the present universe. At the time of the collision, all the stars, even all the particles, in each semi-universe must all be moving together at the same speed and in the same direction.
He has to posit dark stars because emitting light is going to increase entropy. Once again he is assuming a symmetry that is one of many possibilities.
The second law of thermodynamics, then, must date from some sort of Great Collision out of which the present universe evolved. But what happened before this Great Collision? The answer would have to be, everything was at a temperature of absolute zero, there were two semi-universes which were moving towards each other, in each of which there was not even a trace of relative motion. Although each of the two semi-universes were in motion, yet within each there was no motion, no internal energy.
Once again, why this? Why not internal energy? It is reasonable to ask what happens before the "Big Bang," but not reasonable to demand an answer from physics.
But if such was the situation at the time of the Great Collision, it cannot have been so for an eternity past, unless we conceive of the law of gravitational attraction not to have been true in those times. Taking each semi-universe by itself, its reverse universe will also show the same conditions as we have already described, except that the semi-universes are moving away from each other, so that we can proceed in peace without danger from the impending Great Collision. Each semi-universe may, for the purpose of internal occurrences, be regarded as at rest. Gravitation will then draw all the stars of each semi-universe towards its center of gravity, till all of them fall in there. Reversing once more, so as to obtain the process as it must have been supposed to happen, we get the following result: Each semi-universe originally consisted of one great body; suddenly, somehow, that body exploded into pieces, which formed stars, each piece, though, remaining at a temperature of absolute zero. Finally, in each semi-universe. mutual gravitation of the stars slowed them down to relative rest. Just when this relative rest was reached, the two semi-universes collided, and out of this collision came our present universe. Thus we trace a little farther back to the Great Explosions; but these explosions cannot possibly be traced back any farther according to the known physical laws without violating the second law of thermodynamics. In consequence, if we wish to preserve the second law of thermodynamics, we must either dispense with some of the other physical laws, or as some physicists have done, intersperse a creation. In other words, the second law of thermodynamics cannot have been true for an eternity past, though it may be true for on eternity in the future. And even the assumption of a creation would be assuming a process different from the processes coming under the ordinary physical laws.
The model is goofy. The conclusion is fine. One way or another, the physical laws were different in the past. It is actually rather hard to avoid the conclusion that there was a creation—multiverses only shift the problem of where the rules of the game came from.
In other words, we come to the inevitable conclusion that the subsistence of the irreversible second law of thermodynamics in the same universe as the reversible laws concerning the motion of particles is a paradox, both from that point of view and from the fact that this second law, pushed to its logical conclusion, leads back to a mysterious creation which denies all physical laws whatever.
No. He did not show entropy to be a paradox "from that point of view," and pushed to its logical conclusion it shows not paradox but incompleteness.
You can think of entropy in terms of the number of possible states for a system of particles in a closed system. (The log of the number, if you want to get technical.) If you have many particle states with equivalent energy, a little interaction will share out the particles in lots of those different equivalent states. Why we experience time the way we do is not known, of course, but it agrees nicely with the "arrow of time" defined by increasing entropy, which has led some to speculate that this is the reason we see time happening as we do.
In Chapter V he asserts that
To help us towards a solution of this paradox, we must first find out what the probabilities actually lead us to conclude. We have already seen that, in a given case, the chances are even as to whether energy will run down or build up. There are also small chances of a neutral condition, in which energy remains, on the whole, at the same difference of concentration as before. But the probability of this neutrality is negligible, and we may say that the probabilities are, that in 50% of the cases the second law of thermodynamics will be obeyed, and in 50% of the cases it will be reversed. If such is the case, the universe as a whole will be neutral; that is, taking all the occurrences over all of time and space, there will be no tendency in one direction or the other.
We have not "seen that … the chances are even as to whether energy will run down or build up." On the contrary, we simply have a risible model and some bald assertions. He may claim this as a possible model, but he may not claim that he has proved that this is possible, or even likely.
In chapter VI he asks how we would observe pockets of 2nd Law reversal, and concludes there are two: regions where effects seem to precede causes in a teleological way (does he really mean that entropy is reversed if there seems to be purpose in events?!!) or if "small causes" produce big events (because energy would be gathered rather than dissipated).
In short, we may say that, in general, events in the reverse universe appear as though they were living phenomena; and the general events of the reverse universe may be taken as the type of negative phenomena, of the reversal of the second law of thermodynamics. … We may therefore conclude: first, that inanimate phenomena, when reversed, become animate: second, that animate phenomena, when reversed, lose the appearance of animation; and third, that animate phenomena, when reversed, lose this appearance because, when reversed, they tend to follow the second law of thermodynamics. The logical conclusion from these would be: that inanimate phenomena are positive tendencies, and follow the second law of thermodynamics, while animate phenomena, on the contrary, are negative tendencies and tend to reverse that law. Thus we have found where our part of the universe contains reversals, and come to a solution of our paradox.
OK, full stop. Sorry, but this is weirdo land.
UPDATE: I should be more specific. First, a "reversal of entropy" is perfectly possible if the system is not closed; and a living organism typically acquires energy from outside in the form of food. This is elementary, and has been understood for a long time. Second, he uses the completely undefined weasel word "tendency" which leaves you with no way to quantify what he is talking about, and therefore no way to physically test to see if what he is saying is true. If he were talking about love or philosophy I'd use different criteria for evaluating what he says, but he claims to be talking physics; and we're all about measuring physical things in this part of the campus.
Chapter 8 shows no particular understanding of nuclear fusion or nuclear stability. The date of the book being 1920, perhaps this is not so surprising.
Chapter 11 discusses theories of the origin of life on Earth, without a hint of a suspicion that both the theories he favors and those he doesn’t alike fail to explain the origin of the life ours is supposed to have come from. The panspermia="cosmozoa" is particularly silly in this respect: how is "life on Earth comes from meteors from Mars" an explanation?.
The online copy is defective: Chapter 12 points to Chapter 11.
Chapter 13 is about astronomy, and includes howlers such as that a nova comes from nowhere with no pre-existing star. He apparently knows better, as his description of a 1902 nova shows, but he concludes that the star "had all the necessary heat, but that, until that day, the second law of thermodynamics was, for some reason, not operative on it." To be fair, Eddington's work on stellar fusion wasn't until later that decade.
Later chapters are predicated on the assumption that life is a reversal of the 2nd law. How exactly this is to be arranged on a local scale he never explains.
In the Conclusion he admits that "I may also state that I cannot supply any satisfactory answer to most of the objections stated in Chapter XVIII." That is an understatement.
My conclusion is that he knows how to use big words and namedrop, but his arguments are lacking, there's no effort to translate generic concepts into testable math, and he gets into crank-land when he starts in on the nature of life. I'll forgive his not knowing future details about stars and quantum mechanics, since I suspect his memory, like mine, only works one way in time.