John Stuart Mill famously wrote, about pushing principles to their logical limit, that “unless the reasons are good for an extreme case, they are not good for any case” (On Liberty). This is not obvious, for extremes often produce antinomic or non-generalizable results.
One may perhaps affirm that stealing $25 from Elon Musk without anybody knowing (I suspect that Musk rounds up his accounting figures to the nearest thousand) and giving it to a very poor family for a meal at McDonald’s would increase the latter’s utility more than it would decrease the former’s. But in less extreme cases, it becomes obvious (or so I argue with many if not most economists) that any concept of “aggregate utility” is meaningless because interpersonal comparisons of utility are scientifically impossible. As Anthony de Jasay kept repeating, it’s “my say-so against your say-so.” (See also my review of Lionel Robbins’s 1935 An Essay on the Nature and Significance of Economics.) Some principles or “laws”—such as the impossibility of interpersonal comparison of utility—may be considered absolutes, but they are only “relatively absolute absolutes” (to use James Buchanan’s expression) and may break down in extreme cases.
A Cretan’s statement that “all Cretans are liars” is antinomic: either it is true, which implies it is false; or else it is false, which implies it may be true. But, as Anthony de Jasay observes with his usual common-sense, down-to-earth approach, when we say in ordinary discourse that all Cretans lie, we are not literally meaning it; we metaphorically mean that most of them lie. This helps de Jasay build his argument for the possibility of private production of “public goods” in anarchy by ruling out extreme cases. If, concerning a given public good, all potential customers believe that none of them will free-ride (refuse to contribute or subscribe to the public good), then all of them will free-ride. In reality, some potential free riders will bet that some will free-ride and others not, and will cautiously decide to subscribe in case their own contributions could be decisive for the production of a public good they intensely want. (See de Jasay’s Social Contract, Free Ride, which I reviewed in Regulation.)
It may be a general phenomenon that, in our universe, extremes are puzzling or antinomic, at least for our limited minds. Mathematical infinity is an extreme that is difficult, if not impossible, to manipulate. But “tending toward infinity” is a useful concept. It is essential for calculating the present value of a perpetuity (or its special cases of a perpetual bond or a consol) as the recurrent coupon divided by the discount rate.
Thinking about a nearly omnipotent God may provide solutions to every problem, but an infinitely powerful God produces the “omnipotence paradox”: Can God create a rock so heavy that he cannot lift it? Aquinas answered that God is only omnipotent in “possible things,” in “whatever does not imply a contradiction.”—Summa Theologica, Part 1, Question 25, Article 3. So even God, it would seem, cannot go to the extreme of canceling logical contradictions. Needless to say that a human government cannot be omnipotent, but it can cause much damage by moving in that direction.
The problem remains to find where the extremes are and to identify the extreme cases that cannot be used to test a theory. In some instances at least, identification is possible—for example, when a variable goes from 0% to 100%, as for the proportion of the Cretans who lie or the voluntary subscribers to a “public good.”
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“Woman walking toward infinity”
READER COMMENTS
Craig
May 20 2025 at 12:05pm
Very interesting, indeed one might even say something along the lines of ‘take any point to its extreme and it becomes absurd.’ There is a legal adage that might be of interest to you “hard cases make bad law” which suggests that focusing on the extreme/unusual fact patterns when creating general rules can lead to bad laws when applied to many common situations.
Mactoul
May 21 2025 at 12:19am
JS Mill wrote many famous, not to say fatuous things and was soundly refuted by JF Stephen (Liberty Equality Fraternity). Unfortunately, England was then being carried along the tide of liberal utopian-ism, soon to throw off its liberal disguise and present as socialism of slow or fast variety. This evolution included Mill too who became notably socialist under influence of his wife.
Close attention given to the above-named book of JF Stephen can turn non-dogmatic assorted liberals and anarchists to liberals of less ideological and more prudential flavor.
Pierre Lemieux
May 21 2025 at 6:01am
Mactoul: Then, you should be happy for, if you read me correctly, I challenge the Mill quote. Moreover, as you must know, I generally oppose utilitarianism. My main point, though, is that even if one does not agree with certain beliefs of Mill, one should keep one’s mind open to other ideas of his. Who knows, one may be wrong.
Jose Pablo
May 21 2025 at 1:09pm
But “tending toward infinity” is a useful concept. It is essential for calculating the present value of a perpetuity
When calculating a perpetuity, we don’t merely approach infinity, we explicitly sum an infinite number of future cash flows. Every single one is taken into account.
The trick lies in the fact that when you add one more year to an already infinite number of years, you’re still getting the very same number of infinite years. Infinity plus one is still the same infinity.
This concept can be metaphorically extended to the current state of American governance. Add one more act of clownish nonsense to an already infinite stream of clownish nonsense… and, surprisingly enough, you still get the same infinite stream of clownish nonsense.
Or worse: consider the constant policy U-turns, the frequent contradictions among officials, and the ongoing institutional standoffs between Congress and the courts, and the sheer volume of political absurdity in today’s America seems to approach a larger infinity than Aleph-null (the cardinality of the perpetuity’s cash flows). In this arena we are now, very likely, in the realm of infinities of a higher order.
In fact, using the American government as a reference point, it is easy to visualize that some infinities (like the absurdity of Biden’s administration) are actually smaller than other infinities (like the absurdity of Trump’s).
nobody.really
May 21 2025 at 2:17pm
That shouldn’t be too hard to visualize in any event:
2/4 = 1/2
(2 * N) / (4 * N) as N -> infinity = 1/2
The numerator and the denominator both approach infinity, but the numerator remains half the size of the denominator.
Jose Pablo
May 21 2025 at 4:47pm
I’m not sure I follow your explanation, but in any case, half of infinity is still just infinity.
The set of even natural numbers and the set of all natural numbers have the same number of elements. They both have the same cardinality: aleph-null, which is the smallest form of infinity. Always an interesting concept when speaking about political nonsense.
Jose Pablo
May 21 2025 at 5:19pm
Corollary 1: My point is that the nonsense of Trump’s administration is not merely double that of Biden’s (which was already infinite), nor three times greater, nor even 2,454,376 times greater. All of which, by the way, would still represent the same infinity.
My point is that it is infinitely greater. So much so that the cardinality of its clownish nonsense is aleph-one, the smallest cardinality of an uncountable set.
Corollary 2: We are in the process of discovering that nothing prevents us from one day experiencing an administration (and it could very well be a Democratic one) whose clownish nonsense reaches a cardinality of aleph-two, or beyond.