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?
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.
_______________________________________________________________________ | Mathieu Bouchard ---- tél: +1.514.383.3801 ---- Villeray, Montréal, QC
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
On Wed, 25 May 2011, Bryan Jurish wrote:
apologies for moving to pd-ot without a direct reply... my bad.
You did send me a direct reply as well, but I couldn't reply to it on pd-ot until I was subscribed there.
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...
At UdeM, there is/was a course called Informatique Théorique, for which I'm pretty sure I was told that it contained many infinite TM tapes, one for each infinite cardinal or something like that... several levels of uncountability. But I might be misremembering.
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.
oh ok. I didn't study TM theory enough to know those distinctions. To me, the equivalence principle is good to know, but actual TM mechanics are somewhat like learning the Brainfuck language or anything similar : much more time-consuming than assembly language, for none of the benefits of assembly language !
I can dig the idea of a non-terminating process,
I can't. It makes me think about the bloody Crown of England.
:-D
It's still not terminated. Can I « force-quit » now ?
Leibniz' too, if it could spare some attention from the umpteen gazillion monads it has to keep track of.
Is that less than the number of virtual quarks ?
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.
I'm not a physicist either. My last physics course was in grade thirteen... which was in 1996. But overall, I have more of a science background than a philosophy background, and I often read in physics manuals belonging to other family members... for fun.
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!
Ever looked at ZFC and stuff ? (Zermelo Fried Chicken or whatever)
The saga of the questions of the family « what's the smallest number that is greater than the smallest infinity ? » and « is that the same as the number of different real numbers ? » led to all sorts of answers like « you need an extra axiom for that » which basically means that you can make up the answer that you want.
In some way, your question makes me think of that.
In the end, the conclusion you will want to reach will depend on what awaits beyond that conclusion. You will have to take into account what is going to be the purpose of the concept of existence, to see what are the requirements for the concept, the constraints on stuff you make up.
_______________________________________________________________________ | Mathieu Bouchard ---- tél: +1.514.383.3801 ---- Villeray, Montréal, QC