Necessary but not sufficient: 596,000 google hits.
Sufficient but not necessary: 134,000 google hits.
Exercise for readers: explain the vast gap here. It seems like the two should be equally common, but when a friend of mine used "sufficient but not necessary" that sounded strange to me; that's what led me to Google, which shows that indeed this phrase is much less common than the reverse.
A random walk through mathematics -- mostly through the random part.
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16 comments:
Interesting! Here's one conjecture: Let P be some property. Let N be a property such that P => N, and let S be a property such that S => P. Then my guess is that, in general, there are many more choices for S than there are choices for N. So picking an S that is not necessary is somehow less surprising than picking an N that is not sufficient.
For example: under what conditions are all finite sets closed in a given topological space X? This is viewed as a very "light" restriction on topological spaces, so not many other properties are necessary for this to be true. But there are many sufficient properties that are much stricter.
It seems related, then, to the fact that mathematicians like to make things as general as possible. We like to start without many restrictions in a given discipline, and see where we can get; then we gradually impose stronger and stronger restrictions. There is usually a natural set of minimal restrictions, but not a natural set of maximal restrictions, so what we deal with tends to be close to the minimal - leaving lots of room for sufficiency without necessity.
This is all off the top of my head - I just thought this up now - but it is certainly an interesting question.
That's because " A is sufficient but not necessary for B" is synomymous with "A implies B," whereas "A is necessary but not sufficient for B" is synonymous with "not A implies not B," which is much more confusing.
Google's goal is to have the web indexed up to 5-grams. In this phrase, I would guess that it is the joint bigram frequency (sufficient-but, not-necessary) that is killing the results. Read Chapter 4.
Well, one non-math consideration is, "If it's not necessary, why are we doing it?" If you think about it in linguistic terms kinda, what you're saying is that it's kind of overkill. Something like, "To prove it's a square, you must measure all the sides AND angles, check that it's the rectangle with greatest area for a given perimeter, and then finally measure both diagonals." While that's sufficient to prove it's a square, that's not entirely necessary. =P
I'm kind of surprised the gap isn't bigger.
In math looking at "sufficient but not necessary" seems relatively rare; in the real world even rarer, doesn't it seem?
What are the qualifications to be president?
How far is it to the next town?
What time does the bar close?
Regular, ordinary questions, each directed to some minimum, some necessity...
Jonathan
The cases involving unnecessary conditions are usually less mathematically at issue (important/interesting in that there's a problem to solve) than the cases where there are necessary conditions.
That is, mathematicians talk about "necessary but not sufficient" a great deal more than the "sufficient but not necessary"; in particular there's the problem of finding what minimal set of further conditions might make it sufficient. Consequently, the first would be expected to turn up more in searches - because it's what mathematicians spend more time discussing.
Often the set of sufficient factors is a supersets of several necessary factors, at least for those factors of analytical interest to understanding important phenomena
In such cases, an example of something that is necessary but not sufficient can occur.
In such cases, an example of something that is sufficient but not necessary *cannot* occur.
An explanation: 5,850,000 hits for "necessary and sufficient" that rolls off your tongue very smoothly as a single word.
Weird google: "necessary and sufficient" -math -mathematics -geometry: 901,000 hits, "necessary and sufficient" -math -mathematics -geometry -algebra: 1,940,000 hits, but: +"necessary and sufficient" -math -mathematics -geometry: 1,950,000 hits. Now, how do you explain that?!
the "sufficient but not necessary" bit has a higher ratio of non-math to math pages (at least for the first few pages)
More from Google...
"necessary condition" = 1.87 million
"sufficient condition" = 1.54 million
I'd guess that it's just easier to reason or infer forward from a given premise A to B, than it is to reason backward and cook up an interesting B which implies the given condition A. There would be sound reasons for that from the standpoint of mathematical logic.
There are vast numbers of "sufficient but not necessary" conditions in mathematics, which we usually don't point out explicitly since no reasonably person could suspect otherwise. For example, when introducing Lebesgue's dominated convergence theorem, lots of books don't even bother to discuss examples showing that the hypotheses are only sufficient and not necessary (since it is easy and unlikely to confuse anyone).
The only time people really talk much about necessity and sufficiency is when they are trying to characterize exactly when some property (let's call it X) occurs. There are two natural strategies: you can either take the conjunction of a lot of necessary conditions for X, or the disjunction of a lot of sufficient conditions for X. Taking contrapositives shows that the first strategy works exactly as well for X as the second does for not-X, but somehow the second strategy seems much less common in practice. That strategy is exactly the case where it would be natural to say "sufficient but not necessary".
This has got to be psychological. If you're trying to understand mathematical objects with property X, it is somehow more satisfying to state universal properties holding for all of them, compared to constructions producing some of them. Logically, studying objects with X is the same as studying those that lack not-X, but psychologically it is different, so the negation symmetry doesn't tell us much.
The one big class of applications of the second strategy I see is classification theorems. For example, when people were classifying the finite simple groups, they made longer and longer lists, with the hope of attaining completeness. (Of course, the proof strategy was very different. I'm not talking about the proof.) Logically, each list had the form "to be a finite simple group, it is sufficient to be one of the following: ..." However, nobody talked about it in terms of sufficiency or necessity, since talking about completeness of the list is much more natural.
Can anyone think of any high-profile theorems that were arrived at by the second strategy (other than classification theorems)? The Nullstellensatz is a degenerate example: there is an obvious sufficient condition for an affine variety to have no points, and it turns out to be necessary as well. If there had been several different sufficient conditions, and it turned out that none was necessary individually but collectively they were, then that would be a non-degenerate example.
Stranger still, when I click your link for "necessary but not sufficient" I get 539,000 hits (57,000 fewer hits). And, less dramatically, 132,000 for "sufficient but not necessary" (2,000 fewer hits).
Is this the result of Google updating since your original post? Is it because I'm in Paris? Or something else?
Jesse,
I currently see 594k and 134k.
In general, these numbers tend to fluctuate, but within a fairly narrow band.
To me they are very different things, the first is a 'failure' of some type, while the second is an 'optional' pass.
In an example: the engine was necessary but not sufficient to keep up with traffic.
And: her gratitude was sufficient but not necessary.
In the first example, discussing failure is common place. The web, after all is a place to vent, is it not?
For the second example, if things are optional, we are less likely to judge them. If it wasn't necessary, caring if it's sufficient or not just isn't interesting.
Paul.
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