## Monday, September 04, 2017

### Rules and Exceptions

As a mathematician nothing grates me more than the expression "The exception that proves the rule". Either we bake the exception into the rule (all primes are odd except two) or the exception in fact disproves the rule.

According to Wikipedia, "the exception that proves the rule" has a legitimate meaning, a sign that says "No parking 3-6 PM" suggests that parking is allowed at other times. Usually though I'm seeing the expression used when one tries to make a point and wants to dismiss evidence to the contrary. The argument says that if exceptions are rare that gives even more evidence that the rule is true. As in yesterday's New York Times
The illegal annexation of Crimea by Russian in 2014 might seem to prove us wrong. But the seizure of Crimea is the exception that proves the rule, precisely because of how rare conquests are today.
Another example might be the cold wave of 2014 which some say support the hypothesis of global warming because such cold waves are so rare these days.

How about the death of Joshua Brown, when his Tesla on autopilot crashed into a truck. Does this give evidence that self-driving cars are unsafe, or in fact they are quite safe because such deaths are quite rare? That's the main issue I have with "the exception that proves the rule", it allows two people to take the same fact to draw distinctly opposite conclusions.

1. I think the expression is usually used to say "The fact that this exception is interesting/surprising shows that the rule is typically true."

I agree that it's ripe for a abuse, but I also think that it's useful. The Joshua Brown case is actually a very good example. Self-driving cars drive a lot; if they kill someone, it's major news; and we all know of exactly one example.

1. But that is just sloppy wording, then. Wouldn't you want to say something like "the exception highlights the rule"?

2. Sayings can be sometimes be interpreted either way. My other favorite example is the latin proverb (which you probably don't care about, as you don't seem to mind if other languages disappear http://blog.computationalcomplexity.org/2014/12/the-ethics-of-saving-languages.html) is https://en.wikipedia.org/wiki/De_mortuis_nil_nisi_bonum, whose other version is de mortuis nil nisi bene, where one means something like "of the dead speak only good," while the other means "of the dead speak only well," i.e., the truth, i.e., don't ignore mentioning the bad stuff.

3. First, this is not the first clue that the editors at the New York Times are not always up to scratch. This use is silly. You cannot prove a "rule" by contradicting it. But I am not sure that the misuse of an idiom is news.

There seems to be two ways to justify this saying in some circumstances:

First, the sign example as a legal use: a stated exception verifies that a rule typically exists. If I cannot modify a building without a permit, it implies there is some permitting process.

Second, the phrase could be used to mean "the exception tests the rule, and finds it valid." For example, if we say that math is useful. But then we find number theory. It's the exception that proves the rule. Even number theory turns out to be useful. So I guess all math must be.

4. Here is a quote from Bertrand Russel's book The Philosophy of Logical Atomism (1918):
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"Vagueness is very much more important in the theory of knowledge than you would judge it to be from the writings of most people. Everything is vague to a degree you do not realize till you have tried to make it precise, and everything precise is so remote from everything that we normally think, that you cannot for a moment suppose that is what we really mean when we say what we think."
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In a similar vein, in Bertrand Russell's essay "Mathematics and the Metaphysicians" (1919) we read:
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"A book should have either intelligibility or correctness; to combine the two is impossible, but to lack both is to be unworthy."
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Continuing this theme, page vii of mathematician Henry George Forder's monograph The Foundations of Euclidean Geometry (1927) affirms
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"The virtue of a logical proof is not that it compels belief but that it suggests doubts."
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Forder's maxim is quoted and extended Morris Kline's book Mathematics: the Loss of Certainty (1988)
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"[...] The proof tells us where to concentrate our doubts."
—
As a practical application of these maxims, recent research in respect to Quantum Supremacy — see for example Aaronson and Chen "Complexity-Theoretic Foundations of Quantum Supremacy Experiments (arXiv:1612.05903) and a Google-supported survey "Characterizing Quantum Supremacy in Near-Term Devices" (arXiv:1608.00263v2) — provides a concrete guidance for affirming that the Extended Church-Turing Thesis (ECT) is true, both fundamentally and practically, in QED-governed universes (like ours).

5. I prefer `the exception that TESTS the rule'

Example: ALL hall of fame baseball players were in the major leagues by the age of 24 and began as very good players.

There are exceptions but they really do help to verify the rule since the exceptions are ... exceptional

One obvious one: Jackie Robinson was in his late 20s
when he got to play in the majors. There were racial barrier
to his playing before then.

Other exceptions are playes who served in the military and,
at the very beginning of baseall in the 1880's--- some players could not have played in their early 20's since there was not baseball to play.

There may be a few few other exceptions that can't be explained but they are... really rare.

My point is not about baseball its about a good example of how the exceptions TEST the rule.

6. I'd heard (but don't know where) that this expression actually comes from an older use of proves which was understood more as tests. For instance, shooting a suit of armor with a old fashioned handgun was a proof of the armor (i.e. it put the armor to the test). It both makes sense of how proof came to have its modern meaning (the evidence that a test was successfully passed) as well as the saying.

On this understanding an exception puts the rule to the test because it threatens the rule not because it justifies the rule. If wikipedia didn't have this explanation maybe its all BS but I always found it made a lot more sense than the idea the exception justified the rule.

7. Peter G. and Anon. 11:18am are correct. The original expression in Latin is "exceptio probat regulam" where "probat" means tests (closer to English probes) not proves.

An adjacent issue is that "reproves" means "admonishes" not "proves again" and the word we have from that is reprobate. When a poor rule is exposed by a desirable event it classes as an exception, one can say the exception reproves the rule.