On Jan 7, 2008 7:47 PM, Reid Nichol wrote: > You haven't pointed to an instance of an inconsistency in Mathematics. > Which, I'll point out, was what I explicitly asked for.
Let me speak in a more formal tone, so that perhaps I will be clearer. There are at least two useful mathematical systems that are mutually inconsistent. That is equivalent to my original claim about inconsistency in mathematics. I never said, "Mathematics is inconsistent." See below where, yet again, I provide an example of an instance of what I have been talking about. > Basically, you're referencing a choice in Mathematics that we have, > that we can go for either consistent OR complete. And you seem to be > saying that Mathematics is neither? You don't seem to understand the > issues involved and/or have incomplete knowledge/understanding of the > history of Mathematics. Is this addressed to me? Only very recently have I posted regarding completeness. My original claim that there are at least two useful mathematical systems that are mutually inconsistent has little to do with completeness. If this is addressed to me, I am beginning to better understand Stallman's claim that people on this list have been building straw men. You quote Ingo Schwarze at the bottom of your post, who compliments what I have said and also talks about completeness. From this you appear to have concluded that my claim that there exist useful but mutually inconsistent mathematical systems arises from an argument about completeness. Actually it has nothing to do with completeness. As far as I know, Ingo Schwarze brought up completeness for the first time in this discussion. You appear to be looking at what he said about me and assuming that it served as a summary of what I said. It does not. > "What is flabbergasting me" is that you haven't a clue and/or lack the > attention to detail to answer questions that were explicitly asked. > > Point of fact, Mathematics has been proven to have the option to be > either consistent OR complete. From what I've learned, we've chosen to > be consistent. Which, IMO, was a very very wise decision. If you > don't agree, point to a specific instance of an inconsistency in modern > Mathematics. A specific instance of an inconsistency in modern mathematics? Can you give me a specific instance of an inconsistency in your public library? Mathematics is not a system. It is a field of study. Different systems are studied. Some are consistent with one another. Some are not. Systems with mutually contradicting axioms are not mutually consistent systems, and yet may still be useful. I have already provided examples. I will do so again: Zermelo-Frankel set theory with the Axiom of Choice, versus Zermel-Frankel set theory with the negation of the Axiom of Choice. If you choose to continue to maintain that I am incorrect in my claim that there are multiple useful mathematical systems that are mutually inconsistent, please respond to that specific example. > Eliah Kagan wrote: > """ > Tony Abernethy's example of non-Euclidean geometries being > inconsistent with Euclidean geometry is a good one. > """ > > This is so very wrong it isn't even funny. You deserve to be ridiculed > publicly into oblivion for making such nonsensical statements. I'm sure that people aspiring to learn more mathematics will see that you have said that and conclude that you are the one who knows what you are talking about and that I am a nonsensical fool. > I mean seriously, Euclidean geometry assumes a perfectly flat plain > whereas non-Eucliden geometry does not. > Do you think they'll go in > different directions? Do you think that it is even remotely reasonable > to compare the conclusions after such a divergence without considering > limiting cases? I think systems with mutually inconsistent axioms are mutually inconsistent. This is not a *problem* and it does not make mathematics any less valid as a field. In fact, it is useful. But it is also true. > Though a couple of the statements you make after the above statement > are reasonable, you take it in a direction and make conclusions that > aren't (meaningless?!?!?). This mixture of reasonable with > unreasonable, including such logic makes such statements erroneously > compelling, which is very dangerous for those learning this stuff for > the first time. You insist on me giving examples even when I have already done so, repeatedly. I have acquiesced to your request. Now I would ask that you give specific examples of my unreasonable conclusions and specify why they are unreasonable. > Please stay away from making any statements on the > foundations of Mathematics in the future as you seem to be at least > partially ill equipped to speak on this topic. In other words, you > have enough knowledge and speak well enough to convince students/others > and perhaps yourself, but at the same time, lack the necessary > knowledge/logic to come to reasonable conclusions. That you disagree with me does not mean that I am wrong or dangerous. If you are bothered or feel threatened by my conclusions, that doesn't make them unreasonable. I am open to the possibility that I have made a substantial mistake in presenting my arguments. The existence of mutually inconsistent mathematical systems that are both useful is so basic in the field of mathematics that I (and all mathematicians) would be enormously surprised if it turned out not to be true, but I am open to arguments about that, too. (In the same way, I am open to arguments for the claim that there are no verbs in the English language--that doesn't mean my degree of belief in that is anything higher than zero.) I am not going to shut up just because you think my conclusions are unreasonable. I think this is off-topic for the OpenBSD-Misc list, but I am replying to the list to clarify misstatements about what I have said. Nothing that I have said here is even remotely controversial in modern mathematics. I think that you have misunderstood me, or you have conflated my claims with the claims of others, or you lack a basic understanding of formal systems, or I have failed to communicate effectively. If the latter is the case, I would be eager to correct my error, but you have not provided any specific examples regarding my alleged errors in reasoning. -Eliah
