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.
Sorensen, not Sorenson.
Misspelling fixed; thanks.
With some pain, I continued reading the material on this topic. Along the way, I found the term, “relatum”, (a term mentioned by Sorenson). This sent me to
https://www.oed.com/dictionary/relatum_n
which indicates that it reached its nadir in 1900 and then went on to a spectacular rise right through the 1980s where the time series graph unfortunately ended. The text said, “About 0.1occurrences per million words in modern written English.”
Sorensen, not Sorenson.
Sort of reminds me of the unexpected hanging puzzle (cf. Martin Gardner):
https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
Anon:
Yes, this has come up before. As Gardner explained, the unexpected hanging can be seen to be a special case of the liar paradox.
I don’t think that is true. The liar and the unexpected hanging have different resolutions.
David:
It seems there’s some debate on this issue. I was convinced by Gardner’s argument that the unexpected hanging is basically a liar’s paradox, because the hanging paradox relies ultimately on its one-day version (You are being sentenced to hang tomorrow. You will be hanged on a day when you don’t expect it to happen.), but, yeah, I realize that other people disagree on what is the crux of the paradox and how it should best be resolved.
Why not simply admit that every number is indeed interesting? If you don’t find a particular number interesting, that’s a statement about you, not the number. Dullness is not an attribute of a number, but of the lack of imagination in the imaginer. Hardy was a brilliant mathematician, but he didn’t see the interestingness of 1729.
I am reminded that in the play Proof, David Auburn transposes the famous Hardy-Ramanujan story about the cab number to his heroine. I got to meet Auburn on the opening night party after Proof opened and asked him how he dared to reuse such a famous story in his play. He sort of apologized, but pointed out that the number of theatregoers for whom that was a famous story was so small that he could probably respond to each of them individually.
Jonathan,
As discussed in Sorensen’s article and mentioned in my post, the terms “dull” and “interesting” depending on context. That said, there are certainly some numbers that are not interesting to anyone who has ever lived, a fact we can deduce by considering the finiteness of the human population and the finiteness of each person’s memory. I think the key insight in Sorensen’s article is that it can be interesting that a particular X is uninteresting, without implying that X itself is interesting.
Yeah, I’m not really disputing the philosophical point that interestingness can attach to the statement about interestingness without attaching to the number itself. Nor am I disputing that in practice the number 156.768694834723647596746586035883745 has never even been examined by anyone, much less found to be interesting by someone, it being the random (?) motion of my fingers just now on the top row of my keyboard. I’m only trying to expand the point about context to include *potential* context, and I am sufficiently impressed with human (and AI) perspicacity that I believe that under the gimlet stare of people with the right incentives, someone will somehow find something interesting about 156.768694834723647596746586035883745 to somebody (or something) qualified to judge interestingness.
Have you tried these calculators?
https://www.gematrix.org/?word=156768694834723647596746586035883745&view_rude=on
Sort of reminds me of “Ramanujan’s friends”.
https://x.com/PhysInHistory/status/1705587706057670779
Has that connection already been made, also. ;)
Anon:
Yes, Sorensen mentions the Ramanujan story in his article that is linked in the above post.
OK, professor. But, I just read a bunch of Nyad links and links of links. Probably more than you did. You probably have have me on the stats/math relevant link reading though. I did gaff those off. ;)
Head down, butt up, swimming freestyle. 10Ks in the pool. Your nose starts running and you just let it run. And it continues until the snot just runs out. And then…you just don’t have a runny nose any more. But watch out for calf cramps. Those are the worst. :(
It is worth reading John Baez on the Hardy/Ramanujan taxi story. He suggests Hardy may well have known that 1729 was interesting.
https://johncarlosbaez.wordpress.com/2022/01/30/hardy-ramanujan-and-taxi-no-1729/
Dmitri:
John Baez . . . the name rang a bell. I searched in my old emails and found this:
He never replied!
Sorry for not replying – I try to reply to all non-crackpot emails, but some emails slip through the cracks. I can’t reply on your blog anymore so I’ll do it here if that’s okay.
You wrote:
“But I think the motivation is more altruistic. I don’t know what’s in the Princeton Companion to Mathematics, but in general I don’t think that much, if any, prestige, comes from writing encyclopedia articles.”
We’re all eager to talk about what we’re interested in, and have people listen to us – there may even be some altruism in that. But the fact that this book is called the “Princeton” companion shows that it’s trying to wear its prestige on its sleeve: academics all know Princeton is an Ivy League school, that’s where the Institute for Advanced Studies is, that’s where Einstein and Oppenheimer and Godel worked, etc. When you write an article for the Princeton Companion of Mathematics you know you’ll be in an exclusive club of mathematicians who are recognized as “good”, and someone may notice you when they read your article. This is part of how you get “paid”…. while Princeton University Press makes the actual money.
I prefer to explain math and physics in venues that people can easily access for free. I don’t think this is because I’m more altruistic. It’s because I’m more annoyed at how big companies exploit academics to work for free – whether it’s their altruism, their urge for prestige, their desire for self-promotion, or whatever.
Of course, thanks to LibGen we can now read the Princeton Companion for free. That may change the calculus – I’ll have to think about it. I don’t think it existed back when I was making my decision.
Note that I follow John on Mastodon, and have had a few convos with him where he’s explained some weirdness in more esoteric physics etc. I also have linked to a blog post of his about dimensionless numbers and logarithms a few times to help people understand dimensionless ratios. He’s certainly got a lot more prestige than I do, but I do feel like within our areas of expertise we have similar attitudes about disseminating information. Thanks John for being a guy on the internet who’s doing good stuff and disseminating info in venues we all can access.
Not sure where to reply, but will do it here.
1. I never thought of the story as showing rapid Ramaj insight. More showing the opposite, that he had thought of these things before. Everything about him was lots of detail and working of formulas, even going back to the drill book he used to learn from (much panned by analysts, but love by Ramaj).
2. If anything there’s a general cognition point that it’s a lot easier to solve riddles by knowing a lot of them than pure riddle solving ability. Feynman tells a great story about this. “It was the bubonic plague.” This actually generalizes to a bigger pedagocical and even cognitive psychology point where I find myself in opposition to a lot of young Ph.Ds and Internet experts who deprecate drill and advocated long, unproductive struggle. I just don’t think they are generally right about how meat-computers (even high IQ ones like them) solve problems. A lot more of creativity is mixing and matching and knowing ahead of time than pure Greek god spark of inspiration. Maybe some of each, sure. But a lot less than they think.
3. I think your article was excellent in showing us why 1729 is important and some perspective on why Ramaj knew what he did. Not just a random thing.
4. But I’m not sure (say less than 50% betting) that Hardy recognized the two cube sums prior to telling Ramaj the number. It might not have jumped out to him and/or was not something he had worked on previously. Sure, maybe he would have got it with some time and a challenge to find something interesting. But not necessarily even the first integer sum of two cubes. I just don’t think it’s that blazingly obvious. Plus he had not thought about all the other earlier numbers in that direction. This is not to put Hardy down in any sense. He was fully capable of Euler-like jumps in algebra or Cleo-like definite integrals (although I think Cleo cheated)
P.s. I’m a civilian, not a mathematician.
John:
Thanks for the reply. Better late than never!
Regarding the general point of that discussion, I guess I’d say that you are being altruistic when expressing your annoyance by posting–sharing your thoughts for free to the world. You want to help people understand the world better.
I similarly think it’s altruistic of me to write books and publish research articles rather than just sitting on my ass all day . . . ummm, ok, I don’t have a standing desk so I actually am sitting on my ass when writing this, but you get the point . . . I want to help people understand, I want to clarify people’s thinking, just like that person in the xkcd cartoon who’s bothered by something being wrong on the internet.
I agree that my motivations aren’t entirely altruistic: on net, I probably make money from all this writing (not always, I suspect, as sometimes I piss off powerful people), and there’s that warm feeling of knowing that people think well of me. But that’s fine. I looked up “altruistic” in the dictionary and found this definition: “showing a wish to help or bring advantages to others, even if it results in disadvantage for yourself.” In this case, the disadvantage is that writing books and articles requires a lot of concentration and effort. The fact that it can feel good to do something altruistic does not remove the altruism; the feeling-good-about-helping-others thing is a motivation to do good things, and I don’t think it should be disparaged.
Somewhat relevant here is the classic article by Dale Miller, The Norm of Self-Interest. See discussion here.
To be fair, Andrew, I hate emails of the form “check out this thing I did.” My very large “to do” pile is always in danger of tipping over, and these emails are like somebody wandering by and just dropping a few more manila folders onto it. Like … thanks?
I am more sympathetic when people make some effort to say something about my own work, or what the connection is, or something like that.
Anyway, Gelman and Baez are two of the best science bloggers out there so I am looking forward to the team-up where Doc Ock traps you in the garden of forking paths, and the only way out is to combine statistics and higher category theory.
Totally OT but:
https://www.cnbc.com/2024/12/30/the-4-word-phrase-you-should-never-say-to-your-kid.html
A sociologist from Columbia has found the key to not harming your children! Given that he is a “scientist” in the discipline of irreproducible results – a fake scientist – *and* from the fake US 2nd-Best university, I say we take him at his word!
The sad thing is so many generations of kids have grown up not knowing that they were supposed to give up on life and turn into a puddle of ship when parents expressed disappointment in them. It’s sad to think of the incalculable harm that I myself have surely been subjected to as a result. When my mum told us she was disappointed in us, usually the next thing out of her mouth was “Don’t you back-talk me young man! Your father will be home in an hour!!” I thought I was supposed to stick up for myself but now thanks to an amazing fake scientists at a fake university, I understand that you only get the lollipop if you turn into a puddle of ship when you’re criticized.
Anon:
I clicked on the link, and . . . yeah, that guy is notorious! In all fairness to Columbia, he’s a professor in the business school, not in the sociology department. Not that our sociology department is perfect (recall this story), but the b-school professor mentioned in your linked article has perpetrated junk science before. See this story from 2016:
Should this paper in Psychological Science be retracted? The data do not conclusively demonstrate the claim, nor do they provide strong evidence in favor. The data are, however, consistent with the claim (as well as being consistent with no effect)
and this from 2023:
It’s worse than you might think: Passive corruption in the social sciences.
So, yeah, I wouldn’t take this study seriously, even beyond how ridiculous it is on its face, as you note. It’s very much in the Ariely-Gino tradition.
And, yeah, it’s horrible that it got this kind of uncritical press. That’s what these researchers are best at: fooling credulous reporters. I guess they often do this first by fooling themselves. And, yes, I checked: he has a Ted talk and has been featured many times on NPR.
As expected, the study is nurture uber nature. And in the “don’t discipline too harshly” direction. Dr. Spock anyone?
That’s what I bet I would find, before following the links! Good Bayesian hunches. Dog bites man story.
—
I’ve just been reading a bunch on twin studies. Which in addition to Pavlov’s dog are probably some of the best work in psychology.
Net, net: It’s probably not even 50-50 on nature versus nurture. More like 80-20 nature wins. (Of course you’d have to define your terms and math and blabla.) But big picture identical twins raised in radically different environments had more similar personalities, jobs, IQ, political and religious views, etc. effing etc. than fraternal same sex twins raised together.
re “DEAD” in hexadecimal
Many hexadecimal numbers containing DEAD are interesting from their computer uses.
DEADBEEF was especially famous.
https://en.wikipedia.org/wiki/Magic_number_(programming)#DEADBEEF
The idea that describing something as “interesting” implies a context embedding is wonderfully clarifying.
This post reminded me of one of my favorite sociology papers, “That’s Interesting!: Towards a Phenomenology of Sociology and a Sociology of Phenomenology” by Murray Davis (citation at bottom).
A central claim of the paper is that interesting propositions are “easily translatable into the form: ‘What seems to be X is in reality non-X’, or ‘What is accepted as X is actually non-X’.”
The catch is that “what is accepted as X” depends on the audience receiving the idea.
A nice illustration of this (~p329) is how often ideas that are interesting to lay audiences are uninteresting to expert ones and topics interesting to expert audiences are often boring to lay ones. Davis thinks this occurs so frequently because expert communities form by developing consensus around claims that negate conventional wisdom. But in as much as experts take expert claims for granted, they will no longer interest the experts. Experts will then only be interested in claims that negate expert assumptions, i.e., claims that negate the negation of the conventional wisdom. This can end up seeming like the conventional wisdom lay audiences were used to in the first place (e.g., economists finding that self-interest doesn’t always maximize the collective good).
Davis, M. S. (1971). That’s Interesting!: Towards a Phenomenology of Sociology and a Sociology of Phenomenology. Philosophy of the Social Sciences, 1(2), 309-344. https://doi.org/10.1177/004839317100100211
Nick:
Thanks for the reference. Your comment reminds me of the point that Basbøll and I make in our paper, When do stories work? Evidence and illustration in the social sciences, where we argue that a story can be useful by being anomalous (representing aspects of life that are not well explained by existing models), in which case you can flip the equation: if a story is interesting, what are the assumptions or understanding of the world to which it is a surprising exception?
The same thing goes for statistical graphics, as I discuss in my paper, A Bayesian formulation of exploratory data analysis and goodness-of-fit testing: the purpose of exploratory data analysis is to learn from the data beyond what was expected, hence in considering a predictive check, one can reflect upon “what was expected,” which indeed will depend on the observers and their context.
Great papers.
I wholeheartedly subscribe to the idea that a statistical model is a particular style of story and a story is a particular style of statistical model (usually severely underspecified).
Strangely, I still find it useful when working to regularly alternate my mindset between storyteller and analyst. I think posing as the storyteller helps me imagine my audience while posing as a the analyst helps me attend to detail. I wonder if social scientists trained to think less dichotomously feel less need to toggle back and forth between mindsets.