What’s So Great About Consistency?
May 10, 2005 § 7 Comments
Human beings are rational animals. We go to great lengths to make our arguments appear consistent. We know instinctively the truth from a lie as we speak it. We know instinctively that if we contradict ourselves we have undermined the very foundations of whatever it is we are trying to say. But why is that?
When we lie we are taking a falsehood and giving it the appearance of truth; we are placing a skin over falsehood (typically the skin of our own credibility) to make it appear true. When we are inconsistent we are doing the same kind of thing. We may not be culpable for doing it if we do it unawares, but nevertheless we are taking a falsehood and dressing it up as truth.
Mathematicians/logicians don’t tend to say much about inconsistent systems, because from a logical point of view they are not very interesting. Inconsistent premeses are an “everything all at once” generator: when we start from inconsistent premeses we can in principle reason our way to any conclusion whatsoever (we can also reason our way to its opposite!). Therefore an inconsistent system doesn’t tell us anything about what is true or false. In the eternal Platonic realm of mathematics, where every true answer already exists “out there” to be discovered, an inconsistent system is meaningless and therefore nonexistent.
In human terms inconsistency is quite a different matter, though.
When we discuss, argue, and make decisions it is a process that we do in time, not in the eternal Platonic realm of mathematical truth. Most of our intellectual lives are lived on what I will call orbits, along particular lines of reasoning rather than standing above it all with a comprehensive grasp of the entire system as a whole. The effect of inconsistent premeses on how we live our intellectual and spiritual lives is that it can appear that we are reasoning consistently, because in a local sense along our particular chosen orbit we are reasoning consistently. But in reality we are subject to the a-rational whims of our passions. We may sincerely think that we are standing on principle, but in fact it isn’t the truth that we are following but something else entirely that, ironically, has nothing to do with how we have reasoned our way to a conclusion. And someone else can reach the opposite conclusions that we do with equal sincerity.
The most common way to make an inconsistent line of reasoning appear consistent is to equivocate: to use the same word in multiple places as if we mean the same thing, but without really meaning the same thing in those different places.
Consistency is important, it seems to me, to anyone who doesn’t want to lie to himself.
<>the eternal Platonic realm of mathematics, where every true answer already exists “out there” to be discovered<>>>I am not at all sure that this realm exists.>>According to a certain school of thought, the “excluded middle” can be denied, i.e., <>not<> that for every closed sentence S, either S is <>true<> or S is <>false<>.>>According to another (somewhat more controversially) mathematical truths are about the interaction between the intuition and the passage of time, so that many statements about infinite processes can be stated positively but are incoherent when stated negatively. (For example, “this program halts on input 0” seems to be much more “meaningful” when true (wait and see!) than when false (er, still waiting…).)>>So Platonism might be true, but only if you agree that not every apparently-coherent mathematical question is really meaningful. In particular, S might be meaningful but not-S might not. >>The answers are “out there” sure enough, but you gotta know which are the questions.
<>I am not at all sure that this realm exists.<>>>Well, I am using “Platonic” and “exists” in the very loosest sense to mean “objective, independent of being willed or perceived by any particular human mind”. I don’t categorically deny everything in the intuitionist and formalist programs. I don’t for example have a fixed opinion about actual infinities. (Hell, I’m not even a mathematician or a philosopher of math). But I am an ontological realist.
Good, that’s settled.>>I think “I am an ontological realist” in this setting means roughly “I assert the law of non-contradiction”, a denial of which seems to me to entail refusing to think altogether.>>(As in:> “I deny the law of non-contradiction.”>> “I agree, let’s go have a beer.”>)>>I picked up on your “Platonic” word instead of your “out there” word: your emphasis is on the truth of true statements independent of the knower, right?>>But the problem is that knowledge requires a knower, statements a stater, in order to be knowledge, statements, at all. So you can’t abstract completely.
<>…your emphasis is on the truth of true statements independent of the knower, right?<>>>Quite right.>><>But the problem is that knowledge requires a knower, statements a stater, in order to be knowledge, statements, at all. So you can’t abstract completely.<>>>Absolutely. It seems to me that there cannot be any knowledge without God, because knowledge requires a knower and yet its truth-status is independent of any finite human knower. It is inconceivable that there is no Logos.
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