Interesting number paradox
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The interesting number paradox arises from attempts to classify numbers as "interesting" or "dull". 1729, for example, is often considered an interesting number because it is the smallest number expressible as the sum of two positive perfect cubes in two different ways.
However, attempting to classify all numbers this way leads to a paradox (strictly speaking, an antinomy). In a classification of numbers as to whether they had interesting properties or not, there would be a smallest number with no interesting properties. For instance, all the numbers up to 37 may have interesting properties, but 38 might not. This in itself would be an interesting property of the number 38, so it would no longer be dull. This is a classic paradox of self-reference or proof by contradiction.
Consideration shows that the smallest or largest dull number is required to be a surreal number which cannot be mathematically defined. These dull numbers are, in essence, so dull that nobody knows what they are, nor cares what they might be.
Other interesting paradoxes involving dull numbers are:
- each number in the set of dull numbers is interesting for not being in the set of interesting numbers
- each number in the set of interesting numbers is dull, because there are many other interesting numbers
One of the problems with this proof is that we have not properly defined the predicate of "interesting". But assuming this predicate is defined, is defined with a finite, definite list of "interesting properties of positive integers", and is defined self-referentially to include the smallest number not in such a list, a paradox (strictly speaking, an antinomy of definition) arises. The Berry paradox is closely related, arising from a similar self-referential definition.
Liouville's number is an example of a candidate for an uninteresting number. It arises from Liouville's construction of transcendental numbers, but it was neither the first nor the simplest such number constructed; the construction evidently generates many other similar numbers in the same way. Its constructed property of transcendentality is shared with almost all other numbers.
It may be noted that some dull numbers necessarily have fewer uninteresting properties than others, as human knowledge is countable and finite. While there may be infinitely many unknown properties, these properties are interesting due to being unknown. Consequently, there exists a set of the most dull numbers (having only one uninteresting property) as well as a set of the least dull numbers (having the largest subset of the uninteresting properties). These sets are therefore interesting. Note that no dull number may have zero properties, as having zero properties is itself a property (though if such a number was found, it would undoubtably be extremely interesting).
See also
Further reading
- Martin Gardner, Mathematical Puzzles and Diversions, 1959 (ISBN 0226282538)