On 2011-05-10 18:12, Mathieu Bouchard wrote:
> On Mon, 9 May 2011, Bryan Jurish wrote:
>
>> sqrt(2) ? exp(1) ? pi ? ... certainly each of the "usual suspects" has
>> a discrete specification, but I've always been a bit suspicious of the
>> hardcore constructionist approach to irrational numbers
>
> Of course, infinitely long patternless sequences of digits make a lot
> more sense (???).
Of course :-D
Since they're patternless, they're incompressable (in the
Kolmogorov/Chaitin sense), so they can only be realized by a
non-terminating process (i.e. in an infinite number of discrete
computation steps). I can dig the idea of a non-terminating process,
and I feel about the reals like some people of my acquaintance feel
about deity: it's comforting to know that they're around, but I don't
want to deal with them directly (at least not anytime soon) ;-)
>> (while at the same time finding it extremely attractive to my
>> engineering/hacker instincts). ok, so these are probably not
>> "measurable" in the sense you mean either, but they are *thinkable*,
>> and that (I think) is the whole point (or as it were, the whole
>> hypotenuse, curve, circle, etc) ;-)
>
> or rather, it's the whole tangent that gets you away from the topic ;)
Indeed.
> There are lots of facts about the universe that are not knowable.
>
> Analogue audio theory is made with «Real» numbers because that's what
> fitted best to explain the experiments that had been made. Irrational
> numbers are an artifact of our manners of thinking, and uncountable sets
> of «Real» numbers are even more so artifacts.
>
> It doesn't mean that those artifacts don't exist in the physical world,
> it means that we had to invent those concepts by ourselves because we
> can't perceive them from the physical world.
Very Kantian of you, if I may say so. Historically, you're certainly
right; but I'm more of a Platonist bent on this one: our (to be more
precise Frege's) having come up with a logically consistent framework
for talking about uncountably infinite sets -- whatever its motivations
-- means that such sets are, always were, and always will be; at least
to the extent that our theory really is internally consistent. External
(physical) reality doesn't enter into it all.
marmosets,
Bryan
Extra credit bonus question: does the empty set exist?
--
Bryan Jurish "There is *always* one more bug."
jurish(a)uni-potsdam.de -Lubarsky's Law of Cybernetic Entomology