[PD-ot] Real numbers (WAS: [PD] CVs)

Bryan Jurish jurish at uni-potsdam.de
Wed May 25 22:44:28 CEST 2011


moin Mathieu,

apologies for moving to pd-ot without a direct reply... my bad.

On 2011-05-21 21:53, Mathieu Bouchard wrote:
> On Tue, 10 May 2011, Bryan Jurish wrote:
> 
>> 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 don't understand that. Let's say a infinitely long programme just
> starts spitting its own digits one after the other. Can't you say that
> each digit is being spitted out in O(1) steps ? What are the
> Kolmogorov-Chaitin assumptions of what a computer is ?

Complexity of a string (e.g. number) = length of minimal (Turing
machine, input) pair required to output that string, encoded in the base
of your choice (usually binary): idea is that you can feed the encoded
(program,input) pair to a universal TM and watch what happens.  So yes,
I suppose each digit is being spit out in O(1) steps for an infinitely
long description, which indeed a truly patternless (incompressable) real
would require.  The pitfall of course being the `infinitely long
program' part, since if the program as such were infinitely long it
wouldn't be a TM anymore, at least not in Turing's sense... but if you
need an infinite description to generate the desired output (number)
anyways, then you might as well shove all the nasty patternless data to
the input part of the description pair, and then you can work with a
trivial program (cat) which just copies its input to its output: then
you've got a finite TM and incompressability, at the price of "only" an
infinite input sequence.

> As far as I'm concerned, an infinite computer is impossible, so it
> doesn't make much sense to me to postulate O(42) or O(log n) read-time
> for a digit in nth position in the memory.
> 
>> I can dig the idea of a non-terminating process,
> 
> I can't. It makes me think about the bloody Crown of England.

:-D

>> 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) ;-)
> 
> But among themselves... would the Bible's God invite an unnamable,
> unspecifiable number for dinner ?

Unlikely, I think.  Descartes' God would, without a doubt.  Leibniz'
too, if it could spare some attention from the umpteen gazillion monads
it has to keep track of.  Spinoza's probably not -- just way too laid
back: probably wouldn't even lift an eyebrow (metaphorically speaking of
course) if such a number dropped by to hang out a bit and shoot the
poop, though.

>>> 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:
> 
> I don't have enough of a philosophy background to associate myself with
> one or the other. I never did read Kant and forgot much about Platon.
> I'm pretty sure, though, that my main influence has been a lot of books
> about Physics. They didn't talk about that topic, but imho a true
> scientist must read between the lines about things like this.

I agree.  Sadly, most physicists I know tend not to do so to any great
degree; the exception being a 70-year-old experimental molecular guy
who's probably got a more solid background in philosophy than I do.

>> our (to be more precise Frege's) having come up with a logically
>> consistent framework for talking about uncountably infinite sets --
>> whatever its motivations --
> 
> Motivations for a lot of «pure math» topics tend to be « wow, it's
> amazing that those sentences make sense at all and are truer than nearly
> all things in life, even though we have no clue what they refer to ! ».

That sounds about right :-)

>> External (physical) reality doesn't enter into it all.
> 
> Amen.
> 
>> Extra credit bonus question: does the empty set exist?
> 
> There exist ontologies for whichever conclusion you want to reach.

Best answer I've heard for that one yet!

marmosets,
	Bryan

-- 
Bryan Jurish                       "There is *always* one more bug."
jurish at uni-potsdam.de       -Lubarsky's Law of Cybernetic Entomology



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