Mathematics of interest

Definition: A set is well-ordered (by some chosen ordering of the set) if any subset of it has a least element.

The integers are a well-ordered set. Real numbers are not a well-ordered set. The set of \{\textrm{cow},\textrm{pig},\textrm{ball peen hammer}\} with the ordering implied by \textrm{pig} \leq \textrm{ball peen hammer} \leq \textrm{cow} is well-ordered.

Theorem: All integers are interesting.

Proof: Suppose there are non-interesting integers. As the set of integers is well-ordered, there is then a smallest non-interesting integer. Such a number is obviously interesting; this is a contradiction; and the result follows.

Corollary: All rational numbers are interesting.

Note that as real numbers are not well-ordered, it is not intuitively obvious that all real numbers are interesting. However, there are interesting numbers arbitrarily close to any non-interesting real numbers, if they exist at all.

Corollary: The set of interesting real numbers is dense on the real line, and the set of interesting complex numbers is dense in the complex plane.

Open problems:

(1) Are there non-interesting real numbers? If so, is their set A dense in the real domain? What is the measure of A? Is it even a measurable set? Does anybody care?

(2) Durchfall and Pscheudonym have done work on the question of plain and sexy numbers. Is there anything to Pscheudonym’s Third Proposition, viz. “all sexy numbers are interesting”? What about the reverse assertion?

(The nameless reviewer has brought the author’s attention to a forthcoming paper by Durchfall and Ploetzlich that proves there are real numbers that are neither sexy nor plain; the interestingness of these “paraphiliac numbers” might be a good project for someone with the inclination.)

(3) The research of Bingo and Chainsaw into the question of the characterization of quasi-interesting functions of bounded mean oscillation (qiBMO) has found links between interestingness theory and the well-established field of the meanness of functions; most of the latter’s results are reviewed in Ostrog’s monograph “That’s one mean function, Bobby: On the emerging theory of mean and ungraceful functions” (Springer, 2009). Can similar characterizations be formulated in the context of real numbers, or do they inevitably lead to the trivial case of “Little Suzy’s Conjecture”, viz. that “All numbers are mean and non-interesting”? (The only result the author is aware of on the subject is that of Grinch and Sly, who showed by tedious counterexample that Mathemagician’s Lemma of Great Interest does not hold in the general case of Young groups, which resolves one side of LSC.)

(4) It is known that pi is interesting, but is it plain, sexy, paraphiliac, mean, ungraceful and/or lovey-dovey? What about the square root of two?

(The author has been supported by the Academy of Mathematics of Some Parts of Finland and by his undying burning hatred of the dean.)

* * *

(Old joke known wherever coffee is brewed, but now taken much farther.)

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