Well, the "does a tree make a sound when it falls in the forest with nobody there to hear it?" question is really about a different issue than this matter of what is the truth-value of dictionary definitions of words. When Bishop Berkeley posed that original question and said "no," he was asserting an idealistic philosophical perspective that I doubt few of the readers of this blog are particularly sympathetic with, although a lot of mathematician readers are probably bigger Platonists than they might be willing to admit (Did you "discover" that proof?).

Regarding dictionaries and common usage and Caterpillar assertions of "I can make a word mean whatever I want it to mean," well, certainly dictionaries do ultimately simply report reasonably common usages, with the possibility of these simply expanding for any given word. Most of the time these new usages simply evolve in spontaneous and oddly linked ways through similarities between the newer and the older usages.

Things get a bit odd when we have consciously made changes of meaning a la the Caterpillar. Sometimes these are ironic or hip or whatever, often playing off or against an established meaning ironically. This can lead to confusion if the new meaning gets added on with the older ones, especially if the new meaning is somehow logically or factually at odds with older meanings. This means that users of the word will have to be careful about contexty and audience, if they wish to avoid confusion. Sometimes this is conscious, such as "bad" meaning "really coolly good," although I doubt that usage has made it to the OED, if it ever will. Others with deeper historical roots are mysteries. Thus, why do both "inflammable" and "flammable" mean the same thing? And then we have words that have evolved to mean just the opposite of their original meaning, such as "pretty," whose Old English root, "praetig," apparently meant more like "ugly," although I may be slightly off on that one. But, such cases are definitely out there. Should the original person to use some form of "pretty" to mean what it does today have been punished, and why did his or her listeners go along with such an extreme change of meaning (which probably happened sort of gradually anyway)?

As a sideshow, I would note the death of Rasputin, whom some were not so certain was really a man either, although rather than a demi-god possibly like Socrates, some of those doubters thought that he was more like a demon, and I am also unaware of anybody getting involved in such a debate when he refused to die according to the usual causes.

In any case, he was killed by a group of tsarist nobles who were upset about his apparent control over Tsar Nikolai II and his family. So, they invited him to dinner. He was poisoned, he was shut, he was beaten and knifed. None of this did the trick. It required taking him outside and forcing him into an icy river where he presumably both drowned and froze to finally do him in.

Oh, and regarding infinitesimals again, some have argued that the old medieval dispute about how many angels can dance on the head of a pin was really a debate about the existence or nonexistence of infinitesimals.

Regarding calculus, it is possible to accept infinitesimals and thus view dx/dy as meaningful in an absolute sense, and not just as the outcome of a limit process. This is what is done in nonstandard analysis, in which infinitesimals are the reciprocals of superreal numbers that are infinite (although not equal to or equivalent to the infinite cardinals). This is in fact how both Newton and Leibniz thought of the matter. For Leibniz, a monad was a point surrounded by infinitesimals.

However, while infinitesimals are smaller than any positive real number, they are not equal to zero, strictly speaking. Therefore, they are irrelevant to the discussion of the steps in the Heinlein exercise, which is dealing with the actual zero. Indeed, this exercise is a reminder as to why dividing by zero is ruled out. Allowing it allows absurdities such as this exercise. 0/0 can be anything.

My late father was once asked by a young woman in the audience at one of his public lectures, "Is zero a real number?" He replied, "one of the finests, my dear, one of the finest."

Well, the deeper issue is "Must we rely on the Peano axioms?" I shall not get into all the Godelian issues that can arise, but I will note that by suitable reinterpretations, one can indeed pose real world cases where an "apparent two plus another apparent two" do not equal "apparent four," without being utterly ridiculous. The problem is that such cases are not readily amenable to being easily put together into useful axiomatic systems. There may be something better out there than Peano, but Peano seems to work pretty well an awful lot.

As for "what is really true?" Well... . . . .

Well, the real reason why it is useful in arithmetic to accept that 2+2=4 is that this is part of a deeper relation in the arithmetic field regarding relations between the three basic arithmetic operations: addition, multiplication, and exponentiation. Thus, 2 is the solution to the following question: what is x such that x plus x equals x times x equals x to the x power? And, of course, all of these operations equal 4.

J. Thomas is right. Religion can evolve in primitive man for purely selfish reasons. Group selection may be relevant only for competition between religions. Hence, all the prisoners' dilemma arguments and tit for tat and all that is irrelevant. The vocal heretic is the loner who loses group support and thus has a very difficult time surviving.

So, back in the primitive tribe, surely there were plenty of people who had doubts about their tribes Ugu Bugu god. But as long as one did not wish to get tossed out of the tribe by the chieftain who at least appeared to worship and pray to Ugu Bugu, it paid the individual to go along to get along. All hail Ugu Bugu!

I think James Bach was on the right track here, but did not take this far enough. Eliezer's interlocuter was not able to really articulate his argument. Properly argued, probability is completely irrelevant.

So, let us contemplate the position of a serious, hard science creationist, and I hate to say it, but such people exist. So, this individual can fully agree and admit that how a given body grows and develops depends on its DNA structure, so that indeed it is not surprising that different species that appear morphologically and behaviorally to be somewhat similar, such as various canine species or feline species, or for that matter chimps and humans, even if the older creationists got all in fits about having a monkey for an uncle, and so forth, will have very similar DNA structures.

The issue then is how did this come to be. The evolutionist says that it is due to evolution from common ancestors and so forth. The scientific creationist says, "no," this simply reflects that the intelligent designer set them up this way because DNA controls the growth of individual entities, so similar appearing and behaving species will have more similar DNA, and God (or The Intelligent Designer) made it this way fully consciously in accord with the laws of science, which presumably the same Entity is also fully aware of, whether or not this Entity in fact set up those laws his or herself.

It is well known that capable leaders consciously surround themselves with advisers who hold competing views, with at least some able to tell the leader when things are not going well. We have just been seeing a counterexample of this in an important real world position for the last seven years...

Ah, but A is still A, no matter what any of you may say... :-).