A very interesting discussion by Roy Sorensen of the interesting-number paradox

There’s this mathematical joke that all numbers–more precisely, all positive integers–are interesting. As Roy Sorensen puts it:

Mathematicians are fond of Edwin Beckenbach’s (1945) argument:

A. If some [positive] integer is not interesting, then there is a least such integer.

B. If some integer is the first uninteresting integer, then that fact makes the integer interesting.

C. Therefore, all integers are interesting.

The sophism has attracted no philosophical commentary because of a trivializing resemblance to Berry’s paradox and the sorites.

I’d not heard of either Berry’s paradox or the sorites–you can look them up yourself on wikipedia. Indeed, you can find a whole list of paradoxes of self-reference.

Sorensen’s article is fun and thought-provoking. The full reference is Roy Sorensen (2011). Interestingly dull numbers. Philosophy and Phenomenological Research 82, 655-673.

Here’s how he resolves the paradox:

My main objection is to premise B (`If some integer is the first uninteresting integer, then that fact makes the integer interesting’) of Beckenbach’s sophism. . . .

I [Sorensen] agree with Beckenbach that some numbers are interesting. . . . But I also think that there are infinitely many uninteresting integers. Since the dull integers must start somewhere, there must be a first one — even if the vagueness of `dull’ makes it impossible to specify which it is. . . .

Beckenbach bases premise B on the fact that any instance of D will imply E:

D. It is interesting that n is the first uninteresting integer.

E. Therefore, n is an interesting integer.

My counter-explanation of the unsoundness is that the inference is invalid. . . .

Sorensen’s key point is that a number can be embedded in an interesting statement without itself being interesting. Suppose, for example, we say that each of the integers from 0 through 20 are interesting, as are 23 and 24, but that 21 and 22 are dull (with the terms “dull” and “interesting” depending on context; the set of numbers that are interesting to a sports fan could differ from the set of numbers that are interesting to a mathematician; and also conditional on the level of focus, as the harder you look the more likely it is that you can find something interesting; but that doesn’t affect the paradox, it’s just a matter of definition). It’s arguably an interesting statement that 21 is the smallest dull positive integer, but that doesn’t make 21 itself interesting.

Sorensen gives several examples of that sort of thing:

An uninteresting fact can embed an interesting fact. (For instance, it is interesting that the coastline of Norway is longer than the coastline of the United States but it is not interesting that this fact is interesting.) The case for D centers on the dual of this embedding principle: An interesting fact can incorporate a dull fact.

Indeed, it can be interesting that a fact is dull . . . `873 is the difference between the squares of two consecutive integers’ looks interesting. But actually this fact is not interesting; any odd number greater than 1 is the difference between the squares of two consecutive integers. The dullness of `873 is the difference between the squares of two consecutive integers’ is interesting because this dullness is explained by an interesting generality.

Undistinctiveness is just one genre of instructive dullness. The monotony of `The decimal expansion of 1/9 is .111….’ is a sign that it is a non-terminating fraction. The enervating patternlessness of the decimal expansion of π is a sign that it is a transcendental number.

In an early example of computer program, Alan Turing analyzed chess into a sequence subtasks. The more menial he made the sub procedures, the more interest he added to the overall effect. Turing’s chess programs breathed new life into homuncular models of psychological processes. . . .

In defense of conceptual analysis, I say the dullness of an identity statement often promotes the interest of the analysis. Consider contested identities. Students at first deny 1 = .999…. The teacher then points out that 1/3 = .333…. and 1/3 x 3 = 1. This conjunction of trivial truths makes most students regard 1 and .999…. as alternate numeric representations of the same number. They stop viewing 1 = .999…. as a near miss and start regarding it as trivially true. In the final analysis, the interesting fact is not that 1 = .999…. Just the opposite! The interesting fact is that `1 = .999….’ is dull.

Or, for an even simpler example, you can write an article with about the color green, without that article itself being green.

The questions become more subtle when we move between mathematics and the social world (including the very use of base 10 as a social convention):

A dull number can be denoted by an interesting numeral. In hexadecimal (base 16), 570005 is denoted by DEAD. . . .

Once we become sensitized to the distinction between using an expression and merely mentioning it, we become more discriminating about the means by which a number can inherit interest from a fact. . . .

In mathematics, inheritance is restricted to internal relations. The interest of `92 is the number of different arrangements of 8 non-attacking queens on an 8 x 8 chessboard’ is assigned to chess rather than 92 because chess is an alien relatum.

People relate interestingly to numbers but the interest of these relationships attaches to people. The interest of `The grandmaster Bobby Fisher died at 64, the number of squares on a chessboard’ attaches to Fisher, not 64.

Plutarch remarks that “The Pythagoreans also have a horror for the number 17, for 17 lies exactly halfway between 16, which is a square, and the number 18, which is the double of a square, these two, 16 and 18, being the only two numbers representing areas for which the perimeter equals the area”. This is an interesting fact about Pythagoreans. Their horror does not add to the interest of 17 (though 17 may accrue interest from the mathematical relationship that troubled the Pythagoreans).

In 1866, sixteen year old B. Nicolò I. Paganini found the small amicable pair (1184, 1210). . . . Paganini’s amicable pair is interesting in that it partly answers `Which are the amicable pairs?’. But it is more interesting as evidence for the psychological question `How reliable were the great mathematicians?’. Mathematicians are reluctant to credit a pair of numbers with the interest that attaches to contingent facts about it.

Another obstacle to crediting interest to Paganini’s numbers 1184 and 1210 is that they are interesting as a pair. Interest in a pair need not pass down to the individuals comprising the pair (just as a husband and wife can each be bores and yet be interesting as a couple).

Also this:

The robustness of interesting dullness is manifested by the sheer volume of commentary on boredom. Philosophers marvel at the power of this motivational vacuum. . . . Social scientists agree that just as there can be sober studies of inebriation, there can be interesting studies of tedium, repetition, and apathy. . . .

When astronomers explain away coincidences with an identity hypothesis, loss of wonder is experienced as insight. Demystification is a sign of explanatory progress.

The balance of boredom

Sorensen’s article is full of striking insights, for example:

Satiation differs from boredom in that you can exit. The gorged gourmand just leaves the restaurant. But the dishwasher is obliged to stay. World weary, the dishwasher can only escape into daydreams and diversions. Boredom correlates with understanding. So there is some temptation to compress Heidegger into two lines: To understand everything is to be bored by everything. So everything is boring.

But boredom can also be produced by incomprehension or a slight distraction (too small to be recognized as the true cause of one’s inability to focus). The laggard is too far behind to make sense of the lesson. The prodigy is too far ahead to find the lesson stimulating. The interested student lies in between, challenged but not overwhelmed.

Prudent students monitor their boredom to check whether they have deviated from this balance. The vain misconstrue the boredom of incompetence as the boredom of mastery.

Relation to the philosophy or sociology of science

The discussion in the second half of Sorensen’s article reminds me of some things we’ve talked about regarding bad science:

1. My false theorem. I proved a false theorem once! Here’s the original article (Andrew Gelman and T. P. Speed (1993), Characterizing a joint probability distribution by conditionals, Journal of the Royal Statistical Society B 55, 185-188), and here’s the correction notice, from 1999. Embarrassingly for us, the falseness of our claim was shown by a one-line counterexample. Here’s the relevance to the present discussion: I always referred to this as our “false theorem,” but then someone pointed out that it’s not a theorem if it’s false! So what word to use? “Conjecture” doesn’t seem quite right, because we were offering it as a theorem, not a hypothesis. “Claim” is better, but that doesn’t convey that we were not just saying that a particular statement was true; we were claiming we’d proved it. I think this difficulty in explaining is “real,” not just a matter of the lack of a good word in English for “claimed proof.”

2. Evidence vs. truth. We’ve talked about this many times, for example here, here, and here. I think this is a big issue with problematic science, that researchers will make a claim that might well be true, but they don’t offer good evidence. When a scientific paper P is published claiming to demonstrate statement X, what the paper is really claiming is not “X is true” but rather “P contains strong evidence in favor of the truth of X.” When an outsider (such as me!) criticizes the paper’s “methods,” we’re typically arguing against that second claim, i.e. we’re saying that P does not contain strong evidence in favor of the truth of X. The original authors of the paper will typically respond with some version of, “We believe that X is true,” which might be fine, but I think it impedes the discussion for them to not first accept that P does not contain strong evidence in favor of the truth of X–or at least to address the outsider’s criticism on that level.

3. Big if true. This comes up a lot, for example studies of extra-sensory perception, or the claim that women are three times more likely to wear red or pink clothing during certain times of the month, or claims that subliminal messages can cause huge opinion swings, or claims of a stolen election in 2020, or various other topics we’ve covered in this space over the years: these claims are implausible on their face, and a careful look at the published evidence offered in their support do not change this assessment. But, if they were true, they’d be interesting! Thus, as Sorensen discusses, these are statements whose interestingness depends on their truth value.

P.S. I just wrote this post this morning. The next slot on the schedule is in May, but I bumped today’s scheduled post and stuck in this one instead, because the topic is so interesting (to me). Really.