I said: > (There are also multiple useful, > mutually-inconsistent formal systems in both fields.)
Duncan Patton a Campbell said: > Provably so? Reid Nichol said: > I'd love an example of Math being inconsistent. Quite frankly, I'd be > surprised if this is true. Tony Abernethy's example of non-Euclidean geometries being inconsistent with Euclidean geometry is a good one. The statement "Mathematics is consistent," is not false. It is meaningless. At least if you try to consider it mathematically. It is sort of like saying, "the public library is consistent." In mathematics, there are mathematical systems. Mathematical systems have axioms. Axioms are statements that, within a particular system, are accepted without proof. Using a mathematical system doesn't mean you believe the axioms--it just means that you are willing to see what happens when you suppose that they are true. A set of statements is consistent if the conjunction of all the statements in the set is not a contradiction. (Also, the empty set is consistent.) Otherwise the set is inconsistent. A mathematical system is itself consistent if the set containing all and only axioms of that system is consistent. Otherwise the system is inconsistent. Two or more mathematical systems are mutually consistent if the union of their sets of axioms is consistent, and mutually inconsistent otherwise. Statements A and B are dependent if and only if either provably follows from the other. Otherwise they are independent. The axioms of Euclidean Geometry are provably consistent. The Parallel Postulate, which states that parallel lines intersect nowhere, is provably independent of the other axioms of Euclidean Geometry. Adding in the Parallel Postulate gives you a geometry describing a flat space. Adding in its negation or statements stronger than its negation (i.e. statements from which its negation follows, but which do not follow from its negation) give you geometries describing other spaces. Both Euclidean and non-Euclidean geometries (such as those in which the parallel postulate does not hold) are used by mathematicians. A similar situation exists where ZFC (accepting the Axiom of Choice) and ZF-C (accepting the negation of the Axiom of Choice) systems are mutually inconsistent extensions of ZF (Zermelo-Frankel) set theory. Both ZFC and ZF-C are used by mathematicians. Separate from the matter of inconsistent systems, there are also fundamental questions in mathematics about how precise or absolute our math really is. What I have just done is to sketch a proof. It is a proof about mathematical systems. To do this proof formally, I need a formal metasystem that handles mathematical systems as mathematical objects. How do I then justify my metasystem? To justify a claim formally, I prove it. How do I justify that I have proved it? Ultimately all formal reasoning rests on informal reasoning. In physics, the obvious example is that General Relativity is inconsistent with quantum mechanics (or if you don't think QM is a system, then with any system based on QM, e.g. QED, QCD). The hope is that a unified field theory can be formulated that makes accurate predictions about gravitation at high energies at the quantum level. To speak fast and loose, this would represent a rewriting of General Relativity to make it consistent with what we know about quantum mechanics, in the same sense that Newtonian physics has to be rewritten to turn it into quantum mechanics. And yet, General Relativity is still hugely useful. Not only does it predict cosmic observations with great accuracy, but your GPS wouldn't work without it (the Earth's gravitational field has an effect on the spacing of signal pulses, and that effect has to be accounted for). In informal language on this list, Richard Stallman has certain ideas about what "contains" and "recommends" mean. Theo de Raadt and most other list contributors have a different idea. Defining these terms in different ways, these people come to different results. The results are inconsistent because the definitions are inconsistent. In the way I'd use the words, I don't think OpenBSD contains or recommends any non-free software. I say this because, for Stallman's notion of recommending by reference to make sense, a compilation must at least recommend whatever it contains (e.g. OpenBSD recommends its kernel). But I don't think that presenting non-free software as an option to users constitutes recommendation. Since this is the only way that anyone (e.g. Stallman) has suggested that OpenBSD recommends non-free software, I don't think there is any real recommendation. If this is true then by the contrapositive law OpenBSD doesn't contain non-free software either. That is *not* a proof--just an outline of my thinking. See, it makes sense to me that one might think that presenting non-free software as an option constitutes recommendation. As a somewhat parallel case, I don't think that presenting contraception as an option in sex education classes constitutes recommendation, but I can see why some people do. Since I don't think there's anything wrong with recommending contraception, I don't really care whether or not it does, and so if someone who thinks that contraception is morally wrong tells me that I am doing something wrong by advocating presenting it as an option in sex education classes, then I will disagree with their belief that I am recommending contraception and also with their belief that doing so would be wrong. My understanding is that Stallman believes that any use of non-free software outside specific circumstances where one is developing free software replacements is morally wrong. So he thinks that to indicate to people who might be tempted to engage in this wrong that it is possible and tell them how to do it is also wrong. Consequently he decides not to endorse a project that he believes does that. Stripped of the original dispute about the use of language, this makes sense...though I'd point out that the connection between his potential endorsement and someone actually using non-free software is pretty indirect. I do not believe that the last paragraph is a straw man argument. If I got Richard Stallman's position wrong, it is because, not being him, anything I say about his position is subject to possible mistakes. I am CC'ing this to RMS because I have actually said something relevant to the discussion he started, and so that he may, if he wishes, correct anything that I have incorrectly attributed to him. -Eliah
