[throwaway5752]

Doesn't this basically rehash stuff covered 100 years ago by Hilbert, Whitehead & Russell, and Godel? If it wasn't such an eminent author, I would give it a pretty solid eye-roll. As some other poster noted in a link they provided, "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety.

And again, even the point about describing the world in 1s and 0s at the end seems to me to be repetitive of Whitehead & Russell's Principia Mathematica which (and please correct me) used logic to construct the integers?

[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).

[chriswarbo]

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

pi and e are computable; we even know a bunch of programs which will compute them (to arbitrary precision; Google for "spigot algorithms"). Computable numbers are countable, so we can identify them with the natural numbers, and there's no need for "reals". For example, we might think about the tapes of a binary universal turing machine: we can enumerate them (empty; 0; 1; 00; 01; etc.), and hence the tapes/programs are isomorphic to the natural numbers; the computable numbers are the results of running those programs (this is easier to imagine with a separate "monotone" tape for writing "output").

What about the rest of the "uncountably infinite transcendentals"? I would claim that such an argument is too hand-wavey in the form you've presented it. Can you write down a uniquely determined transcendental number which is uncomputable?

Such numbers can be defined; Chaitin's "omega" is an example. However, it's difficult to decide whether such numbers "exist" or not: we can assume that the physical world "exists", by definition; things which can't exist physically, but can be explored mathematically, like pi and e, can be said to exist, although it's a leap of faith; uncomputable numbers can't really be explored mathematically, or in any way that we know. Can we still say they "exist"?

Consider that the digits of omega: whatever formal system you use to study it, the digits you "discover" are nothing more than a scrambled representation of your formal system's axioms; since the axioms are assumed rather than proven, the "discovery" is actually a circular argument.

> Whitehead & Russell's Principia Mathematica which (and please correct me) used logic to construct the integers

It's easy to construct the integers "using logic". For example, Peano constructed the naturals like this (more or less):

    The symbol '0' is a natural
    The symbol 'S', followed by a natural, is a natural

We can count through Peano's naturals as '0' (0), 'S0' (1), 'SS0' (2), 'SSS0' (3), etc.

We can likewise construct the integers like this:

    A natural, followed by the symbol ':', followed by a natural, is an integer

We can interpret the first natural as positive, the second as negative, and the resulting integer as their sum. Hence '0:0' is 0 (0 + 0), and so is 'SSS0:SSS0' (+3 + -3); 'SSS0:0' is 3, '0:SSS0' is -3, and so on.

It's not too hard to construct something logically. The problem is what assumptions you're willing to make. Whitehead and Russell wanted to prove everything using only the assumptions of set theory; that's why it took them hundreds of pages to prove seemingly trivial theorems about integers.

Goedel's incompleteness theorem(s) showed that no set of assumptions is complete and consistent; hence proving some result using set theory is no better than proving it with some other system. In particular, if it's easier to construct a theory of integers (like the one above) than it is to derive one from set theory (like Whitehead and Russell), we might as well take the easy path :)