Eric Schwitzgebel recently took up the question of whether an infinitely extended life must be boring. The discussion ended (when I looked at it) with Eric’s fruitfully suggesting that we look at various cognitive architectures and their capacities for boredom over the long run.

No doubt there are many kinds of minds. Let’s radically simplify the problem, in hopes of arriving at a precise answer for at least one case. (After all, if a mind without much to think about can escape boredom, then presumably a more amply stocked mind can too.) The mind I want consider thinks only of natural numbers and number theory (algebraic and analytic). Its “perceptions” consist in presentations of random natural numbers. Will it be bored?

First a digression. The mathematician G. H. Hardy went to visit his friend and fellow mathematician Ramanujan in the hospital:

I remember going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi–cab No. 1729, and remarked that the number (7 × 13 × 19) seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”

Hardy supposes that some numbers are dull (and presumably also that some are not). 1729 he thinks is dull, but it isn’t. It satisfies what for a number theorist is a very interesting proposition. 1729 = 13 + 123 = 103 + 93,

Ramanujan, in his “lost” notebooks, had in fact recorded a way to generate infinitely many numbers which (i) differ from a cube by 1; (2) are the sum of two cubes. For details, see M. Hirschhorn, “Ramanujan and Fermat’s Last Theorem”.

which is already interesting; even more interesting is the fact that it is the *least*number to be a sum of cubes in more than one way.So it is possible that our number-obsessed

*Gedanken-experimentische*mind will have an occasional interesting experience. But sadly most numbers, it would seem, are dull, by which I mean that other than writing it out there is nothing to be said about it. Such a number is said to be*random*in the Kolmogorov-Chaitin sense. More precisely, a number*n*is K-C-random with respect to a programming language*L*if the “complexity” of the number —See Gregory J. Chaitin, “Randomness and mathematical proof”,

the length of the shortest program whose output is that number — is close to *Scientific American*232.5 (1975): 47–52; available here). As stated the complexity of a number depends on*L*, but there are ways of defining complexity so as to make it independent of the precise features of the programming language, provided that the language is “natural” see Delahaye and Zenil, “Towards a stable definition of Kolmogorov-Chaitin complexity”, arXiv:0804.3459.*n*.Things are not looking good for our mathematical mind. It will be bored most of the time. But remember that this mind has a full command of number theory. Even though its “data” consists only in natural numbers, it can still ask itself whether the Goldbach Conjecture is true, whether there is an odd perfect number, whether all the zeros of the Riemann zeta function in the complex plane lie on the line

*Re*(*z*) = 1/2. Those are interesting questions, and difficult too.But we are considering a mind of infinite duration. For such a mind

*l’avenir dure longtemps*indeed. There are infinitely many provable theorems in number theory, even if you take into account Gödel phenomena. But most of them may well be dull. The question would then be whether there are infinitely many*interesting*provable (or provably unprovable) theorems.It seems to me that the continuing interest of the natural numbers to mathematicians is owed to the ongoing production of new

*concepts*with which to formulate conjectures and theories about them. The Riemann hypothesis can be stated only in a mathematics that includes complex numbers and infinite series — the mathematics of Euler, if not of Riemann himself. The Fermat conjecture, though it can be stated in the language of Peano arithmetic, turned out to require very advanced concepts and methods for its proof. Were we stuck with the concepts and methods of Diophantus, I think we would have grown tired of the conjecture long ago. Instead even very simply stated problems in number theory have retained their interest for centuries, as does number theory itself. But can the production of interesting concepts and methods go on forever?I think that what I'm proposing is a variant of what Williams considers to be an illusion, that an eternal existence of intense intellectual inquiry would not end up in boredom or in the ceasing-to-be of the individual inquirer (see Eric’s entry for the reference). I don’t find Williams’ argument on the point persuasive; it hinges on whether for an eternal mathematician the satisfactions of inquiry will “relate to

*him*, and not just to the enquiry”. If a certain inquiry is the task of greatest interest to me, and the source of my deepest satisfactions, what more is needed to make it relate to*me*? It would seem that for Williams I as an individual can be sustained only by what has my fingerprints on it, my smell. Numbers, of course, don’t relate to you or me in that way; but nothing that does, it seems to me, has any chance of being eternal.
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