We have now read about a decision by a federal appeals court in Texas that has ruled that 13 abortion clinics in the state must close because their facilities do not have all of the accoutrements required of full-fledged hospitals.
http://www.nytimes.com/...
The main issue appears to have been whether a “large fraction” of Texans who are females of reproductive age would have a substantial obstacle to getting abortion services.
What does that mean?
The facts are that about 17/100 or roughly 1/6 of women living in Texas who are of reproductive age will have to drive more than 150 miles if the clinics are closed. Apparently, everyone agreed that driving that far was a “substantial obstacle”. But the court did not think that 1/6 was a large fraction, and it is that fact that inspired this diary.
Now, of course, if you think about positive real numbers, any one of which can be represented as a (proper or improper) fraction, they range from 0 to infinity. Furthermore, if you consider two such numbers, if one is further from zero than the other, it is larger; if it is closer to zero than the other, it is smaller. So we are on firm ground when the question is whether one fraction is larger than another.
Also, the universe of discourse in this case is such that infinity is off the table: we are speaking only of fractions ranging from zero to one. However, this doesn't help us when it comes to figuring out what a “large fraction” is, although the observation regarding “larger fractions” mentioned above continues to apply.
I believe that in the case of these fractions we can say that 0/anything (which is equal to zero, of course) will never be a large fraction, because every fraction in the universe of discourse is either of the same magnitude or of a larger magnitude. Similarly, we can conclude of fractions where the numerator and denominator are equal (i.e., equal to one), that they will always be large, because every other fraction in the same universe is smaller.
So we now at least have one small fraction and one large fraction. Yay us.
But what about the infinite number of other fractions F such that 0 < F < 1 ? How many of them are “large fractions” if any?
I'm completely flabbergasted here, because of the infinity aspect of the situation. I believe that the subjective label “large” that we want to understand depends on the context. If, for example, the State of Texas has to pay $1 billion for each woman denied abortion services, then 17/100 would probably appear to be a much larger fraction than if they didn't have to pay anything. Similarly, if a plague infected and wiped out one Texan out of six, that would also appear to be a pretty damn large fraction. But you can turn these things around: if a vaccine saved 5/6 of all Texans from the plague, that number might appear to be a horribly small fraction.
A comment in the article I cited referred to this as the “large fraction test”. What a stupid test. It really comes down to values. Regardless of the magnitude of a fraction, if that proportion of something (call it “W”) is harmed, then if W is of high value, it will be a large fraction, but if W is of low value, it will not be a large fraction.
In this case, that 1/6 of W(omen) of reproductive age having to confront a substantial obstacle to receive abortion services has been adjudged not to be a large fraction says nothing about fractions, and everything about the perceived value of women.