[surement]

> "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety.

The point of the article is that most reals are useless in practice; pi and e might be transcendental and the transcendental numbers are uncountable, but they are both computable, and the set of computable numbers is countable.

[throwaway5752]

Is pi computable? I would have thought it would be a good halting problem example. I am admittedly not strong in modern developments in computability. Was aware of Chaitin and some of his work prior to this, but that's about the limit.

If his point is that the universe is finite and finite methods are a more correct basis for physical sciences, then I'm open to that even if I'm not particulary interested (theoretical math objects are perfectly interesting in their own right to me). But if there's more to it than that, I'd appreciate the help.

[hawkice]

A computable number one where there is a finite length representation of the number, namely, there is the program that, given a number of digits, can output the number it describes accurate to that many digits. There are a tremendous number of ways to calculate pi and e.

[throwaway5752]

I've taken a minor personal interest in some of Ramanujan's more quickly converging sequences for pi. Giving myself a crash course in computable analysis/computable reals. Wish there was more out there about them (particularly limitations vis a vis traditionally defined reals).

[marplebot]

Agreed, I think that one of the consequences of the article is that computable numbers are generally a substitute for how most people think about the reals (e.g. the fundamental theorem of algebra is true for computable numbers extended with the square root of -1, not just complex numbers).