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how long is the coast of britain?
    #9967847 -
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this is a pretty old paper (1967) but i found it interesting. if nothing else, it shows the vast gulf between the obvious and the true.

cheers!


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Re: how long is the coast of britain? [Re: Krishna]
    #9967865 -

oh, and if you're too lazy/busy to read it, the answer is basically ∞


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Re: how long is the coast of britain? [Re: Krishna]
    #9967946 -

Krishna said:
oh, and if you're too lazy/busy to read it, the answer is basically ∞


Incorrect, of course!  :awesome:

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Re: how long is the coast of britain? [Re: Krishna]
    #9968818 -

Not if I use a really big meter stick... :wink:


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Once, men turned their thinking over to machines in the hope that this would set them free.
But that only permitted other men with machines to enslave them.

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Re: how long is the coast of britain? [Re: trendal]
    #9968826 -

You know it's funny, I was just thinking about this (that exact paper, too!) last night.

I think a lot of people like to say that the coastline is infinite...but the first line of that paper completely invalidates such reasoning.

Quote:
Geographical curves are so involved in their detail that their lengths are often infinite or more accurately, undefinable.



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Once, men turned their thinking over to machines in the hope that this would set them free.
But that only permitted other men with machines to enslave them.

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Re: how long is the coast of britain? [Re: trendal]
    #9969573 -

Well what is the line that is distinguished between finite and infinite. To imply that the two models somehow meet, that one or the other could be chosen seems somewhat ridiculous.

If we concede the point that it is infinite, and also encompassed by a finite perspective, it seems to have more to do with scale than size. If we really consider this, it is really the infinite that undermines the finite; that the definable finite is only just cutting corners. Ultimately, the undefinable precedes definable. You could say this about a coastline, or the flat edge of a ruler.

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Re: how long is the coast of britain? [Re: daytripper23]
    #9970056 -

Is that supposed to mean something?

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Re: how long is the coast of britain? [Re: trendal]
    #9970131 -

hey trendal! long time no see :wave:

and yes, the truth is the answer isn't "really" infinity - but more accurately something like an asymptote towards infinity. i'm not that well versed on fractal maths, but found this paper interesting as it demonstrates Mandelbrot's process of coming to an understanding of fractals (or, as he calls them here, "self-similar" patterns).

why i find this interesting is why i find xeno's paradox interesting - it is "obviously" wrong (we do every day go from point a to point b within a finite amount of time) and yet philosophically/mathematically/logically sound.

@daytripper - i'm not sure if i get what you're saying. when you write,
Quote:
, it is really the infinite that undermines the finite; that the definable finite is only just cutting corners. Ultimately, the undefinable precedes definable

i read this as the finite being a filtered instantiation of the infinite? which makes sense - we could even think of something like matter being a (4-dimensional) filtered instance of infinite energy (so that given infinity, we take a 4-dimensional "snapshot" of it at a particular point/vector in spacetime, and get, e.g., the laptop i am typing this on). so as finite, cognizing beings stuck within the internal perspective of spacetime, we experience the finite as "real" and the infinite as a metaphor, whereas in "reality" the finite is a metaphorical chunk of the infinite. this syncs with, e.g., Kant's claim that space & time are a part of the architecture of consciousness itself - when we try to imagine infinity, we can only do it in finite terms (eg, take this line, double it, double it, double it, double it, etc), whereas from the "external" perspective (that we cannot access as finite beings) it might better be understood as being the other way around.


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Re: how long is the coast of britain? [Re: ChuangTzu]
    #9970135 -

:lol: I thought it, but choose not to say it.


Daytripper, you seem like a cool guy but honestly - I can never understand wtf you are talking about :tongue:

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Re: how long is the coast of britain? [Re: Krishna]
    #9971207 -

There is a very interesting episode of NOVA that discusses fractals and mentions how they are used to calculate lengths of coast line.  Check it out:

http://www.pbs.org/wgbh/nova/fractals/program.html


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Re: how long is the coast of britain? [Re: cortex]
    #9977617 -

Yeah, this has been pointed out to me quite a bit lately, I guess I have to work on that. I am actually an English major, so go figure.

Anyways, what I was trying to say is that an finite vs infinite argument is really just a matter of perspective, particularly in scale. I think this is what Krishna describes well as the "filtered instantiation of the infinite". Krishna, correct me if it doesn't fit.

The infinity of Britain's coastline is understood as a line "approaching" infinity, by evidence of the apparently ever increasing nooks and crannies of the beach. Basically this is a good example because it is conceivably quantitative. But the more crannies we see is not actually the line expanding, it is just an increasing depth of perspective. If the line were really ever expanding, then would it come full circle? This is what I meant by the "finite encompasses the infinite". But then, what is described as finite is really just a perceptual notion of scale, only arbitrarily connected to reality.

So I don't think you can ever draw a line between the finite and the infinite, because that is to indicate that both of these concepts are somehow conceived from a single reality. I like the idea that Krishna describes, sort of a covalent metaphor between these realities. Pretty cool.

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Re: how long is the coast of britain? [Re: daytripper23]
    #9977726 -

I don't think its anymore complicated than the fact that the real coastline is made of atoms and is not a fractal.  Its a confusion of the mathematical model with reality.


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Re: how long is the coast of britain? [Re: daytripper23]
    #9977775 -

> Anyways, what I was trying to say is that an finite vs infinite argument is really just a matter of perspective

Less perspective and more semantics.  Mathematically, infinity is not a quantity, but instead is a concept.  Many people try to erroneously use infinity as a quantity when debating philosophical concepts.  It all boils down to infinite series can have finite limits (infinite series can converge), thus even though the distances between two points on Britain's coast can forever be broken down into smaller parts, the sum of all these infinite parts is a finite number.


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Just another spore in the wind.

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Re: how long is the coast of britain? [Re: Seuss]
    #9978443 -

Daytripper, what I understood of your second post seems to reveal you fall victim to confusing uncertainty in measurements with reality.


Just cuz we don't know things percisely doesn't mean anything regarding the reality.  When people first encounter a problem such as this one and think about it they often confuse the problem they have in measuring the length with the existance or nature of the lenght itself- two very different things.  If you think analytically about every day objects though, the problem persists- and we wouldn't say such things are infinite just cuz we are  uncertain about their length.  A stick that is 1.2 meters long isn't exactly 1.2 meters and we are incapable of knowing how long it is exactly.  This is true even though we know it is 1.2 meters long (as such is defined).
 

Just cuz the stick's exact length is impossible to know doesn't mean it is infinite or the stick doesn't have a length.  In the end we will define the length of teh coast by some terms which will themselves define the measurement.  It will be the summation of the atoms' electron clouds which comprise the perimiter.


So we have a bunch of different arcs- a finite number.  We don't know the number, but their will be a finite amount of these arcs which cannot give rise to an infinite coastline.

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Re: how long is the coast of britain? [Re: johnm214]
    #9979847 -

And no matter what definition you use, the coastline of Britain is longer than that of Africa.

True story! :awesome:


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I know... that just the smallest
                                                part of the world belongs to me
You know... I'm not a blind man
                                                    but truth is the hardest thing to see

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Re: how long is the coast of britain? [Re: johnm214]
    #9980414 -

So you would claim that particular reality (1 island) is immanent? My question would be how do you make this claim outside of what is arbitrary?

For instance, you could say it is this many atoms large, but I could just as well say that the "unit" is one island. You could say "one" can be referred a certain amount of molecules, and that to a certain amount of atoms, but the problem is you will never find a true basis however deep you go.

If reality were really atomistic, it definition of being would stop at a point. You can understand the paradox, because the atom is no longer the smallest unit we have seen anymore, yet we still call this approach atomistic. Why? Because we need to pretend there is a true basis, a final unit that is not undermined.

Just like when you imagine the scaling of the coast, you cannot imagine that the chaotic definition would ever stop. Thus it really is getting "longer and longer" in a sense. What is changing you would say, is the scale, but if the basis for definition is consistently undermined (as it does in our perception becomes longer) by an increasingly minuscule part, how can you claim any? If there really is no end to scale, there is no basis for any particular scope, as it would be ever increasing.

I would say that I am a step back, making one less claim than you John.

Yes we recognize "oneness", in that it is perceptually obvious that the coastline comes back to meet itself, but this is just another atomistic quality that has no real basis. One "island" seems to encompass and contradict any notion of contained infinity, and so we assume that finiteness precedes infinity. Thats how we go about conceptualizing infinity isn't it - without end. But I would say that the infinite, which we can recognize as the increasing nooks and crannies of an island, must precede the finite, just as we assume that being and perception are arbitrarily imposed upon nothingness.

Edited by daytripper23 (03/16/09 07:02 AM)

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Re: how long is the coast of britain? [Re: daytripper23]
    #9980474 -

I said before I thought you were confusing reality with our ability to discern it.

Now I think you are  also conflating different words or concepts.


We spoke of length.  Saying "one island" long is meaningless as it doesn't equate with length.  length is a distance between two points.  If by "one island" you mean the length of that island than sure we could use it.

Quote:
Just like when you imagine the scaling of the coast, you cannot imagine that the chaotic definition would ever stop. Thus it really is getting "longer and longer" in a sense. What is changing you would say, is the scale, but if the basis for definition is consistently undermined by an increasingly minuscule part, how can you claim any ostensible definition?


Again, you seem to conflate reality with our uncertainty or ability to observe it.

Just cuz we cannot conceive of a way to be infinitely percise doesn't matter- this is true with everything including how tall you are- but it doens't matter.

The fact is while I can't say how tall I am exactly I do know I am a finite distance tall (unless we are using crazy definitions of height here) and what's more I can say I am taller than you or vice versa.

Also, refer to Seuss' discussion.  Something can get larger and larger forever and not grow to infitity.  Here we have finite components of finite dimensions (again unless weird definitions of length or whatever). 

In this case all we need to know is that the length will be the sum of some number of parts that are finite in number, whatever we define them to be.  As such they are finite in all proportions and the perimeter is also finite. 

Unless we are using weird definitions we will end up measuring the length across various atoms, which will be a bunch of arcs.  A finite number of arcs is a finite number.

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Re: how long is the coast of britain? [Re: johnm214]
    #9980497 -

Hmm Ive got to goto class, Ill be back.

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Re: how long is the coast of britain? [Re: daytripper23]
    #9980889 -

Quote:
I said before I thought you were confusing reality with our ability to discern it.



And I am surprised that you think that you can be unconfused:

I do not think you can really draw this line. It is one thing to recognize the ignorance of our circumstance, but another to assume this recognition actually defines the margins of existence. So I am asking you what you base this real or finite reality upon, if not these arbitrary definitions.

Quote:
We spoke of length.  Saying "one island" long is meaningless as it doesn't equate with length.  length is a distance between two points.  If by "one island" you mean the length of that island than sure we could use it.



That was precisely my point, one "island" is just as arbitrary as one atom, or one inch, or whatever it is in scope you arbitrarily choose. The word "atomistic" might imply that there is a certain basis, but this is just a history of our ignorance that we are past by now. Smaller and smaller units than the atom have been found, yet we still refer to this reduction as atomistic. So why don't we find a better term? I would assume that we finally recognize that it whatever unit we choose will indefinitely be undermined. The atom-istic approach is really just "istic" or "ism"; that is, "Essentialism" our relative and arbitrary imposition upon reality.

I am saying your idea of "preciseness", i.e your perception, is your only substantiation of finiteness (the atom), or what is "real". You are assuming that you have a precise height, even if (though) we can't measure this. But how do you assume this other than an utter leap of faith?

I understand the argument, "just because we cannot precisely define or observe reality, doesn't mean it does not immanently exists." I agree. But I would not take the additional step to assume such an existence, as I take it you are. You say I am conflating our ability to observe reality with reality itself, but I am merely recognizing that this is our only basis of knowledge. So my question for you, is what basis for finite reality do you have outside of perception? What evidence are you basing this claim on, if not our arbitrary definitions? How can you assume that reality is definite if this is not through observation? The burden of proof is on you.

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Re: how long is the coast of britain? [Re: daytripper23]
    #9981051 -

okay, well if you don't assume reality exists, then I guess that is where we part ways.  I think the very act of scietific inquiry presumes such, and the discussion kinda breaks down outside of these presumptions.  I agree we cannot be sure we are measuring reality, and I further say it doesn't matter, but to say reality doesn't exist is too far.  We exist, ergo the universe exists.

I generally presume we are measuring reality with experiments, and really don't care whether we are or not, as I think it a scientifically useless question.  Untill we have evidence our observations are crap we might as well presume they aren't, and so long as they are repeatable they seem good enough.


And the units by which we measure length are immaterial.

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