1

A little goofy, but this question occurred to me in the context of the following example. While answering a strictly-programming question at https://stackoverflow.com/questions/44482735/#44491340 I generated the following list of six numbers that are the only integers between 1 and 10million whose representations as numerals in both base 3 and base 4 are "palindromic"...

        10(base10) = 101(3) = 22(4)
       130(base10) = 11211(3) = 2002(4)
     11950(base10) = 121101121(3) = 2322232(4)
    175850(base10) = 22221012222(3) = 222323222(4)
    749470(base10) = 1102002002011(3) = 2312332132(4)
   1181729(base10) = 2020001000202(3) = 10200200201(4)

Okay, so that maybe seems a slight curiosity, but also seems pretty meaningless, i.e., having (as far as I can tell) no number-theoretic significance whatsoever. But, what, it occurred to me, does "meaningless" rigorously mean? We can introduce the perfectly well-defined characteristic function (aka relation) $f:\mathbb{N,N^2}\to\{0,1\}$ such that $f(i;j,k)=1$ iff $i\in\mathbb{N}$ is "palindromic" in both bases $j,k$ (and $=0$ if not). So that's a perfectly legitimate $f$, but also perfectly (or approaching perfectly) "meaningless".

But how can you characterize that more rigorously? For example, I can rigorously characterize a sequence of numbers as "random" by their Kolmogorov complexity (or by various other rigorous measures). On the other hand, we can pretty safely characterize this "palindromic function" as meaningless by "inspection". And that intuitively seems pretty much true, i.e., it's meaningless. But there doesn't seem to exist any rigorous approach to such a characterization. Can you conjure one up?

John Forkosh
  • 712
  • I disagree that $f$ is "meaningless", you even described its meaning yourself! – mrp Jun 15 '17 at 08:29
  • @mrp In that case, I'd think, any function $f:\mathbb{N\to N}$ would be "meaningful" if not random by, say, Kolmogorov complexity. That is, if I can write a computer program that calculates $f(n)$ given $n$ (that's shorter than an enumeration of $(n,f(n))\forall n$), then $f$'s "meaning" is just a description of that program's behavior. In this case, a "palindrome program". But there are countably infinitely many such computable functions, presumably not all "meaningful" (I'd presume only finitely many meaningful). So that further suggests (at least to me) "description"$\not\equiv$"meaningful" – John Forkosh Jun 16 '17 at 21:59
  • It seems to me that your proposal runs into a variant of the interesting number paradox. If there were meaningless functions $f : \mathbb N \rightarrow \mathbb N$, then we could find the least (under some order) such function, and its meaning would then be that it is the least meaningless function. Unless you have a much clearer explanation of what "meaningless" should mean, I think that this question is meaningless. – mrp Jun 19 '17 at 14:18
  • @mrp Yeah, that seems relevant, and funny. But I can suggest a clearer definition, but not a formal one. Just consider http://researchguides.uic.edu/c.php?g=252299&p=1683205 whereby a function/relation is "meaningful" based on the number of times it's used/referenced/cited in other publications. So that's a meaningful definition of "meaningful", and even intuitively reasonable, but completely informal. – John Forkosh Jun 21 '17 at 05:25

0 Answers0