Comment by adrian_b | Hacker News Reader

Comment by adrian_b | Hacker News Reader

> Of course, we don’t teach about computable numbers in school. Instead, the most common “upgrade” from ℚ are reals:

While "computable" numbers are a recent concept, already for a few centuries, since the early 18th century, mathematics has taught about another set of numbers intermediate between rational numbers and "real" numbers: the algebraic numbers, which are a subset of the computable numbers.

Like the "real" numbers, the "complex" numbers have also been partitioned since that time into "complex" integer numbers, "complex" rational numbers, "complex" algebraic numbers, "complex" transcendental numbers.

Everything that is now discussed in terms of "computable" and "non-computable" numbers, was previously discussed in terms of algebraic numbers and transcendental numbers.

While "computable" numbers is a more general concept that more precisely defines the limit between what is countable and what is not, the practical importance of this concept is reduced, because few of the computable numbers that are not algebraic are interesting, the main exceptions being the numbers that are algebraic expressions containing "2*Pi" and/or "ln 2".

> few of the computable numbers that are not algebraic are interesting, the main exceptions being the numbers that are algebraic expressions containing "2*Pi" and/or "ln 2".

I don’t think this is true at all. For example: the solution to a generic PDE that has no closed form solution at some point of import is likely transcendental, not algebraic, but definitely computable. (Think, say, Navier-Stokes being used for weather predictions in some specific place.)

True, but with such numbers you will normally not do anything else except computing an approximate value of them.

They are not comparable with numbers like 2*Pi or various irrational nth roots that can appear in a lot of relationships and formulae in symbolic computations.

That is what I meant by "interesting", i.e. the necessity of using symbols of such numbers, obviously for use in symbolic computations, since in numeric computations you would never use the actual numbers, but only some approximations of them.

What I have said is equivalent to saying that there are only a few transcendental numbers for which you need symbols.

The number of symbols that are really needed is much less than the number of symbols that happened to be used during the history. For instance a single symbol related to Pi is needed, and it would have been much better if it was a symbol for 2*Pi, not for Pi. When using decimal numbers, one may want to use the value of the decimal logarithm of "e", or of its inverse, but there exists no need whatsoever to use decimal numbers anywhere, this is just a historical accident. Etc., there are various other examples of superfluous constants, which are not needed in any practical application, unlike "2*Pi" and "ln 2", which are ubiquitous (because they appear in the derivation formulae for the trigonometric and exponential functions).

> True, but with such numbers you will normally not do anything else except computing an approximate value of them.

That's what I think people do with other numbers like "pi" at the end of the day, no? :)

> That is what I meant by "interesting", i.e. the necessity of using symbols of such numbers, obviously for use in symbolic computations, since in numeric computations you would never use the actual numbers, but only some approximations of them.

It's very much an encoding problem, I think. Though we probably, on aggregate, use "unnamed computable numbers" implicitly on the order of as much as we use "named computable numbers" the former just has way more of a "tail" of uses where the "encoding of the symbol" is, e.g., "here's the PDE you use to compute this number"!

(It gets a little weird since we're kind of not distinguishing between the approximation that can be used to construct said numbers to arbitrary precision vs the specific program instance that constructs one specific approximation, but the idea is mostly there.)

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