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

[Mathieu Bouchard]

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 ?

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.

> 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 ?

>> 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.

> 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 ! ».

> 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.

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