The smallest uninteresting number and fuzzy logic

I’ve tried to think of something interesting about the number 2013 and haven’t come up with anything. This reminds me of the interesting number paradox.

Theorem: All positive integers are interesting.

Proof: Let n be the smallest uninteresting positive integer. Then n is interesting by virtue of being the smallest such number.

The interesting number paradox is semi-serious, and so is the resolution I propose below. Both are jokes, but they touch on some serious ideas.

“Interestingness” is not an all-or-nothing property. Some numbers are more interesting than others, so perhaps we should use fuzzy logic to quantify how interesting a number is, say on a scale from 0 to 1.

For a given ε > 0, define as interesting the set of numbers whose interestingness is greater than ε. Suppose the interestingness of numbers trails off after some point. (Otherwise, if the interestingness dropped sharply, the first number after the drop would be interesting.) The largest interesting number then is barely interesting. The number one larger than a barely interesting number is even less interesting. So the proof of the interesting number paradox doesn’t apply in the continuous setting.

On a more serious note, many paradoxes in mathematics can be resolved by replacing a binary criterion with a continuous one.

For example, the sum of a trillion continuous functions is continuous, but the infinite sum of continuous functions may not be. How can that be? The problem is that we’re viewing continuity as an all-or-nothing property. If you have a series of continuous functions that converges to a discontinuous limit, the degree of continuity must be degrading. The partial sum after some large number of terms is continuous, but not very continuous. The modulus of continuity of each partial sum is finite, but is getting larger, and is infinite in the limit.

Classical statistics is filled with yes-no concepts that make more sense when replaced with continuous measures. For example, instead of asking whether an estimator is biased, it’s more practical to ask how biased it is.

Computer science is often concerned with whether something can be computed (i.e. exactly). But sometimes it’s more important to ask how well something can be computed. Many things that cannot be computed in theory can be computed well enough in practice.

Related post: How to solve supposedly intractable problems

26 thoughts on “The smallest uninteresting number and fuzzy logic

  1. @alfC, thanks for mentioning that Dawkins article. It’s well worth reading. Your link didn’t work for me, though. The problem might be the period at the end.

    Naive mathematical analogy: the Mandelbrot set is defined by whether a given recurrence diverges or remains bounded — yes or no. But the really pretty pictures are based on how fast the divergent parts diverge.

  2. Computer science is has “discrete” in its genes (!) ever since we found digital computers work better than analog ones.

    But even then, the more you go into “advanced” topics, the more you hear about polynomial time approximation schemes, and soft computing, and continuous numbers very much pop up.

  3. @Dave, thanks, I don’t know how to edit my post, maybe John can edit it.
    @Tomas, “discreteness” in Computer Science is ‘just’ a working hyphothesis (aka Church-Turin thesis) of nature, it can be violated at many different levels both in the principles and in the applications end of Computer Science (which in my opinion is a Natural Science like any other).

  4. “Suppose the interestingness of numbers trails off after some point. (Otherwise, if the interestingness dropped sharply, the first number after the drop would be interesting.)”

    This seems to be an unjustified supposition. E.g., the property of being prime might make a relatively uninteresting number more interesting than one less than that number.

    Furthermore, the interestingness could trail off to a non-zero number (so for some ε all numbers are sufficiently interesting).

    There may well be obvious problems with this counterargument (and I recognize that the semi-serious example is not the point).

  5. I wrote the proof in a hurry, and it could be tightened up. But it’s a joke, so I didn’t want to make it so serious that it’s no longer funny. Still, I do think there’s a serious principle in there.

  6. The smallest uninteresting number is the one about which Wikipedia has nothing to say. According to my research, it’s 275.

  7. ever since we found digital computers work better than analog ones.
    Digital computers do not work better than analog ones, it is just that with a digital computer you get repeatable, and thus reproducible, trials.

  8. Marking a number as interesting only because it’s the smallest (previously) uninteresting number only works once. So the next highest uninteresting number truly would be. On the 0-1 scale of interesting, being the lowest uninteresting number except for X has to be pretty close to 0.

  9. Nothing interesting at all about 2013?

    It’s the first year we’ve had with all four digits unique since 1987 — every year since then (until now!) has including at least one duplicated digit.

    It’s the first year where the digits can be rearranged to be sequential (0,1,2,3) since 1432, and we haven’t seen this particular sequence of digits since 1302.

    That’s all that comes to mind at the moment… but surely there are more. :)

  10. Digital computers do not work better than analog ones, it is just that with a digital computer you get repeatable, and thus reproducible, trials.

    `working better’ is a yes-no concept that make more sense when replaced with continuous measure ;-)

  11. The eminent numerical analyst Nick Trefethen, in his Numerical Linear Algebra book (w/ David Bau) talks about how the question “how solved” is more appropriate than “solved” in the fantastic introduction to iterative methods. He also has a book on pseudo-spectra (‘almost-eigenvalues’) that deals with very similar topics – normal operators (e.g., hermitian matrices) have pseudo-spectra that always look pretty much like the operator’s spectra; but not so for non-normal operators encountered in complex problems.

  12. Your “proof” plays with the definition of interesting. What if, being the smallest thing of some set would not be interesting at all?
    You are just saying that between all integers, there exists a difference ;-) which is kind of a trivial truth.
    I like your post, just in case there are any doubts after reading my post ^^

  13. Xeno’s paradox is another example of just this danger, I think. If you divide things into enough discrete components that are continuous, you can get confusing results – like the paradox which says no movement is possible because you cannot pass from point A to point B without passing through every bisecting point of which there are an infinite number that never quite gets you to B.

  14. 2013 is pretty uninteresting, but thanks to wolframalpha, I’ve learned that it can be represented as bbb in base 13. bbb is a pretty cool number. Biotic Baking Brigade FTW!

  15. Interesting-ness is not a property of numbers. It is a relation between a person and a number, who may or not have an interest in it. With that, the paradox disappears.

  16. @John: Base 13 seems to be the key. In Adams’s Hitchhiker’s Guide to the Galaxy, the Ultimate Answer “42” is correct for the Ultimate Question “What do you get when you multiply 6 by 9?” — in base 13.
    @Harlan: Zeno’s paradoxes get a bad rap these days, IMHO. It’s mostly forgotten that he actually proposed a set of 6 paradoxes, which had to be considered all at once. Individually, they seemed to show that space must be discrete, space must be continuous, time must be discrete, time must be continuous, motion must be discrete, and motion must be continuous. Collectively, they showed that argumentation up to that time regarding the nature of space, time, and motion had been inconsistent and sloppy. People who dismiss Zeno by saying “Now we know that an infinite number of terms can add to a finite sum” have missed the point entirely.

  17. 2013 is the first year since 1987 to have 4 different digits.

    I do like the base 13 representation, though.

  18. Your twisting of the original “smallest uninteresting positive integer” into the continuous setting also reverses this to “largest interesting number”. This reversal is not handled clearly in your discussion, and is partly why “the proof of the interesting number paradox doesn’t apply in the continuous setting.” is strange. I think that the continuous setting you present does show “All positive integers are interesting.” in the limit.

    The positive integers (like the natural numbers) have a well-order. This means “that every non-empty subset of S [positive integers] has a least element in this ordering”. Going bigger this is not true, for example the subset of even numbers are non-empty but do not have a largest element.

    The original paradox shows that the set of non-interesting numbers is empty, since if it were non-empty then it has a least element Q, and that this Q would therefore be interesting. Proof by contradiction.

    Let me chase your reasoning to see where I go:

    Generalising to the interestingness of a positive integer “x” being a non-negative real number “F(x)” and consider a threshold ε > 0. This lets you define a sequence of interesting/non-interesting subsets as ε gets smaller, I(ε) = {x | ε 0 for all positive integers x.

    “Suppose the interestingness of numbers trails off after some point” forces all such interesting subsets I(ε) to be finite (and for small enough ε the I(ε) is non-empty), thus there must exist a largest interesting number L(ε). But this largest interesting number depended on ε and as ε decreases you get an non-decreasing sequence of L(ε). Because the real numbers are not a well-order and F(L(ε)) > 0 you can have infinitely long decreasing ε sequences and infinitely long strictly-increasing sequences of L(ε).

    You then consider ε > 0 and the non-interesting number 1+L(ε). These also form an ever-increasing sequence as ε decreases. I note that 1+L(ε) was never discussed as being the smallest element not in the finite subset I(ε). Let me grant you that I(ε) is {0,1,2,…L(ε)} so that 1+L(ε) is this smallest element. Note that the interesting set I(ε) is finite and the non-interesting set is non-empty by definition, not by virtue of any proof. It is not that the original proof does not apply, it is that your continuous case has defined things to be this way.

    Now one has the proof that the limit the infinite sequence of finite sets {1…L(ε)} as L(ε) increases is the infinite set of natural numbers, and that there are no non-interesting numbers in the limit ε goes to 0. This concept and proof is quite similar to the rest of the examples you discuss. A property that is true of each item in a sequence is not true of the new item which is the “infinite limit” of the sequence of the original items.

    How does this proof in the continuous setting work? We are trying to prove that there are only interesting number in the limit ε goes to zero. For contradiction assume there remains a non-interesting number Q with F(Q) greater than zero. In the infinite decreasing sequence of ε to zero it must become less that F(Q) at some point: εQ < F(Q). So I(εQ) is in the sequence of finite sets and Q is in I(εQ). Since ε1 0 eventually holds. Varying ε does not change F(x), so there each I(ε) ends in some finite drop from F(L(ε)) to F(1+L(ε)). Perhaps you mean that this drop decreases as ε decreases?

  19. The factors of 2013, 3/11/’61 are the date that Operation Trinidad was presented to President Kennedy and called for the Invasion of Cuba via the Trinidad beaches. This was rejected and 4 days later the plans for the Bay of Pigs were presented. The rest is history.

    My interest in text analytics shows me no number or concept is an island. With enough data everything has some sort of connection. I like the idea of the degree something is interesting is scaled between 0 and 1 with nothing ever being exactly 0 or exactly 1.

  20. Little-known fact: The golden ratio is the most irrational number.

    Why? Because its optimal rational approximations (i.e. continued fraction approximations) are worse than those of any other number, in the sense of having a larger relative error.

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