> Sorry, the question is whether Hegel's Logic is consistent. See
> Bertrand Russell and the Brit rejection of Hegelianism on that one.
> Completeness is out for any system of logic that incorporates simple
> arithmetic.
How will seeing "Russell and the Brit rejection of Hegelianism" help?
The fundamental point at issue was Hegel's concept of "organic unity" i.e. of "internal relations" (a concept which, by the way, Hegel is elaborating in the passages from the Science of Logic you've quoted). G.E. Moore claimed (e.g. Principia Ethica, , 2nd ed., pp. 84-5) the concept was self-contradictory. In this he was mistaken. The conclusion depended on implicitly adopting an atomic ("ordinary language") meaning for the concept of a "part" and then proceeding to demonstrate that this was inconsistent with treating the part as internally related to the whole.
The mistake in this argument was pointed out by Frank Ramsey in the context of defending Whitehead's version of the idea of internal relations from similar criticism by Russell and Johnson. He says of the criticism that it mistakes "for a fundamental characteristic of reality what is merely a characteristic of language." (Ramsey, Foundations of Mathematics, p. 117)
That the claim of inconsistency was mistaken was eventually acknowledged by Russell himself. He says he was persuaded of this by Whitehead who also showed him how to disentangle "mathematical logic" from ontological atomism. He was unable, however, to free himself from the atomism: "I am persuaded that the world is made up of an immense number of bits, and that, so far as logic can show, each bit might be exactly as it is even if other bits did not exist."
"I began to develop a philosophy of my own during the year 1898, when, with encouragement from my friend G.E. Moore, I threw over the doctrines of Hegel. If you watch a bus approaching you during a bad London fog, you see first a vague blur of extra darkness, and you only gradually become aware of it as a vehicle with parts and passengers. According to Hegel, your first view as a vague blur is more correct than your later impression, which is inspired by the misleading impulses of the analytic intellect. This point of view was temperamentally unpleasing to me. Like the philosophers of ancient Greece, I prefer sharp outlines and definite separations such as the landscapes of Greece afford. ...
"It was Whitehead who was the serpent in this paradise of Mediterranean clarity. He said to me once: 'You think the world is what it looks like in fine weather at noon day; I think it is what it seems like in the early morning when one first wakes from deep sleep.' I thought his remark horrid, but could not see how to prove that my bias was any better than his. At last he showed me how to apply the technique of mathematical logic to his vague and higgledy-piggledy world, and dress it up in Sunday clothes that the mathematician could view without being shocked. This technique which I learned from him delighted me, and I no longer demanded that the naked truth should be as good as the truth in its mathematical Sunday best.
"Although I still think that this is scientifically the right way to deal with the world, I have come to think that the mathematical and logical wrappings in which the naked truth is dressed go to deeper layers than I had supposed, and that things which I had thought to be skin are only well-made garments. Take, for instance, numbers: when you count, you count "things," but "things" have been invented by human beings for their own convenience. This is not obvious on the earth's surface because, owing to the low temperature, there is a certain degree of apparent stability. But it would be obvious if one could live on the sun where there is nothing but perpetually changing whirlwinds of gas. If you lived on the sun, you would never have thought of counting because there would be nothing to count. In such an environment, Hegel's philosophy would seem to be common sense, and what we consider common sense would appear as fantastic metaphysical speculation.
"Such reflections have led me to think of mathematical exactness as a human dream, and not as an attribute of an approximately knowable reality. I used to think that of course there is exact truth about anything, though it may be difficult and perhaps impossible to ascertain it. Suppose, for example, that you have a rod which you know to be about a yard long. In the happy days when I retained my mathematical faith, I should have said that your rod certainly is longer than a yard or exactly a yard long. Now I should admit that some rods can be known to be longer than a yard and some can be known to be shorter than a yard, but none can be known to be exactly a yard, and, indeed, the phrase 'exactly a yard' has no definite meaning. Exactness, in fact, was a Hellenic myth which Plato located in heaven. He was right in thinking that it can find no home on earth. To my mathematical soul, which is attuned by nature to the visions of Pythagoras and Plato, this is a sorrow. I try to console myself with the knowledge that mathematics is still the necessary implement for the manipulation of nature. If you want to make a battleship or a bomb, if you want to develop a kind of wheat which will ripen farther north than nay previous variety, it is to mathematics that you must turn. You will kill a man with a battle-ax or with a surgeon's knife; either is equally effective. Mathematics, which had seemed like a surgeon's knife, is really more like the battle-ax. But it is only in applications to the real world that mathematics has the crudity of the battle-ax. Within its own sphere, it retains the neat exactness of the surgeon's knife. The world of mathematics and logic remains, in its own domain delightful; but it is the domain of imagination. Mathematics must live, with music and poetry, in the region of man-made beauty, not amid the dust and grime of the world.
"I said a moment ago that, in revolt against Hegel, I came to think of the world as more like a heap of shot than a pot of treacle. I still think that, on the whole, this view is right; but I gradually discovered that some things which I had taken to be solid shots in the heap did not deserve this dignity. In the first flush of my belief in separate atoms, I thought that every word that can be used significantly must signify something, and I took this to mean that it must signify some thing. But the words that most interest logicians are difficult from this point of view. They are such words as 'if' and 'or' and 'not.' I tried to believe that in some logicians' limbo there are things that these words mean, and that perhaps virtuous logicians may meet them hereafter in a more logical cosmos. I felt fairly satisfied about 'or' and 'if' and 'not,' but I [41] hesitated about such words as 'nevertheless.' My queer zoo contained some very odd monsters, such as the golden mountain and the present King of France - monsters which, although they roamed by zoo at will, had the odd property of nonexistence. There are still a number of philosophers who believe in this sort of thing, and it is their beliefs which have become the philosophical basis of Existentialism. But, for my part, I came to think that many words and phrases have no significance in isolation, but only contribute to the significance of whole sentences. I have therefore ceased to hope to meet 'if' and 'or' and 'not' in heaven. I was able, in fact, by the roundabout road of a complicated technique, to return to views much nearer to those of common sense than my previous speculations.
"In spite of such changes, I have retained a very large part of the logical beliefs that I had fifty-five years ago. I am persuaded that the world is made up of an immense number of bits, and that, so far as logic can show, each bit might be exactly as it is even if other bits did not exist." (Russell, "Beliefs: Discarded and Retained" in Portraits from Memory pp. 38-42)
Ted Winslow