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Precision, Ideal and Real

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How sharp is the tip of an imagined cone? How small is a Euclidean point?

Can we imagine infinitely precise measurement? It seems we could never physically achieve this.

Can a chess game be perfectly notated? Can it exist as symbols, minus board and pieces? Yes. It has been and is. What about copies of digital files? They too are perfect. The essence of digital media seem to be their perfect precision. And is this not because they in some way "transcend" the spatial?

@Reconstructo,

Reconstructo;172391 wrote:

How small is a Euclidean point?

A point is that which has no part. Euclid's Elements, Book I
Galileo used this to prove that all circles are the same size!

@ughaibu,

ughaibu;172393 wrote:

A point is that which has no part. Euclid's Elements, Book I
Galileo used this to prove that all circles are the same size!

That's what I thought, and that's why this spatial-continuous versus digital-discrete absorbs me so much. Our spatial intuition is of a different nature than our intuition/experience of integers, for instance. But numbers like pi or e seem to force against the boundary. What think ye?

@Reconstructo,

Reconstructo;172396 wrote:

What think ye?

It's all good fun.

@ughaibu,

ughaibu;172397 wrote:

It's all good fun.

Terse as usual. Of course I think terseness is great. Except I know you must have a flood of thoughts that would be valuable to me hidden away!

Do you have any sort of aesthetic reaction to math/geometry. You said it wasn't perfect sculpture for you, but is it something else? What draws you to it? Smile

@Reconstructo,

Reconstructo;172391 wrote:

How sharp is the tip of an imagined cone? How small is a Euclidean point?

Can we imagine infinitely precise measurement? It seems we could never physically achieve this.

Can a chess game be perfectly notated? Can it exist as symbols, minus board and pieces? Yes. It has been and is. What about copies of digital files? They too are perfect. The essence of digital media seem to be their perfect precision. And is this not because they in some way "transcend" the spatial?

About 20 years ago, we could write with atoms, moving them around and see them through a electron miscrosope. That's probaly the most accurate there is of messures.

..and please stop that "infinitive-mastrubational-abuse".

@Reconstructo,

Another view is that the ideal does not and cannot exist, and to pursue it is one of the origins of totalitarianism. Pursuit of the ideal means neglect of the actual. The reality is the actual situation we have to deal with, and is often neglected or misinterpreted in our pursuit of the ideal.

Krishnamurti:

Quote:

The idealist is the man with an idea, and it is he who is not revolutionary. Ideas divide, and separation is disintegration, it is not revolution at all. The man with an ideology is concerned with ideas, words and not with direct action; he avoids direct action. An ideology is a hindrance to direct action'.

Commentaries on Living, Second Series, Page 19.

@Reconstructo,

Reconstructo;172391 wrote:

Can we imagine infinitely precise measurement? It seems we could never physically achieve this.

Not physically, no (although conceivably some physical constants are infinitely precise, even if we can never know them). But in terms of Eudoxus's theory of measurement (which is how the ancient Greeks resolved the problem of the irrational), we can easily give an example of an infinitely precise 'measurement' of an irrational (indeed transcendental) number. For this purpose, consider multiplication of positive integers as if it were addition in a physical dimension of measurement. Now apply Eudoxus's definition to the ratio of 3 to 2 in this dimension of measurement (i.e., in our modern notation, the number log_2(3), which is easily shown to be transcendental, by an appeal to the powerful Gelfond-Schneider theorem). The 'sum' of n copies of 2 is the power 2^n, and the 'sum' of m copies of 3 is the power 3^m. The 'ratio' of 3 to 2, according to Eudoxus's theory of equal ratios, is irrational, because we never have 2^n = 3^m (by unique factorisation, "The Fundamental Theorem of Arithmetic"). In modern terms, the number log_2(3) is greater or less than the rational number m/n according as 2^n < 3^m or 2^n > 3^m. In a sense, we 'know' log_2(3) absolutely precisely, because we have this criterion. (These are just my own thoughts, and by no means a standard piece of mathematics. I like to think that Eudoxus himself would have liked it, even though nobody else seems to!)

@Reconstructo,

I would have thought that the indeterminacy of objects on the quantum scale would be fatal to infinite precision. Atoms were conceived as perfect points, but they have been found not to be.

@jeeprs,

jeeprs;172428 wrote:

I would have thought that the indeterminacy of objects on the quantum scale would be fatal to infinite precision. Atoms were conceived as perfect points, but they have been found not to be.

Only if you care to measure the position and the momentum simultaneously. If you care not about momentum, then, in theory, you could have approaching infinite indeterminacy in momentum to find a pretty darn precise position.

@Huxley,

jeeprs;172417 wrote:

Another view is that the ideal does not and cannot exist, and to pursue it is one of the origins of totalitarianism. Pursuit of the ideal means neglect of the actual. The reality is the actual situation we have to deal with, and is often neglected or misinterpreted in our pursuit of the ideal.

Krishnamurti: Commentaries on Living, Second Series, Page 19.

I think that's true about revolutions. The beginnings of human revolutions are often alcohol.... not much thought. But then everybody stands around thinking: what did we just do? Ideology.

But about perfection, I've thought of that in regard to motherhood. The only perfect mother is imaginary. On the other hand, each of us had the perfect mother to create us. The desert is the perfect mother for a cactus.

Huxley;172439 wrote:

Only if you care to measure the position and the momentum simultaneously. If you care not about momentum, then, in theory, you could have approaching infinite indeterminacy in momentum to find a pretty darn precise position.

So you can't have precision without a lack of focus on the background.

Digital transmission is more accurate, but not because the signal doesn't degrade as much as an analog one, but because it's so easy to reform the signal at the destination. It's the magic of encoding. So when you hear your friend's voice on the phone, you're hearing a computer's translation of a digital code. A computer has generated a voice that sounds to you very familiar. Is there any difference between that and actually hearing your friend? Is reality digital down inside like a movie?

@Reconstructo,

When a man buys anything, feeling ripped off or making a steal, he likes what he has gained. But an artist... when is an artist ever satisfied with his gains?

@Reconstructo,

Reconstructo;172391 wrote:

Can a chess game be perfectly notated? Can it exist as symbols, minus board and pieces? Yes. It has been and is. What about copies of digital files? They too are perfect. The essence of digital media seem to be their perfect precision. And is this not because they in some way "transcend" the spatial?

Perhaps this is because they exist in different realities, realities we have created for then. Ever law a computer and a chess game follows were created by humans, and therefore they dont follow any laws that humans cannot comprehend. For example, a computer can never draw a perfect circle, but he can draw one as perfect as possible.

ughaibu;172393 wrote:

Galileo used this to prove that all circles are the same size!

This is not really possible =)

Reconstructo;172396 wrote:

That's what I thought, and that's why this spatial-continuous versus digital-discrete absorbs me so much. Our spatial intuition is of a different nature than our intuition/experience of integers, for instance. But numbers like pi or e seem to force against the boundary. What think ye?

Well, computers arent very happy about such numbers either. A computer cant draw a perfect circle exactly because it cannot calculate Pi fully, since its an infinite number =)

jeeprs;172428 wrote:

I would have thought that the indeterminacy of objects on the quantum scale would be fatal to infinite precision. Atoms were conceived as perfect points, but they have been found not to be.

Quantum particles may ultimately have a previsible behavior that we are yet to learn to predict. But I believe "there is always a smaller particle", and that is fatal to infinite precision.

jeeprs;172417 wrote:

Another view is that the ideal does not and cannot exist, and to pursue it is one of the origins of totalitarianism. Pursuit of the ideal means neglect of the actual. The reality is the actual situation we have to deal with, and is often neglected or misinterpreted in our pursuit of the ideal.

I agree. So much foolishness is done in the name of surrealistic dreams.

Arjuna;172448 wrote:

Is reality digital down inside like a movie?

I think so. I dont see any reasons to not believe our reality is something analogous to a computer program before a superior lifeform.

@jeeprs,

jeeprs;172417 wrote:

Another view is that the ideal does not and cannot exist, and to pursue it is one of the origins of totalitarianism. Pursuit of the ideal means neglect of the actual. The reality is the actual situation we have to deal with, and is often neglected or misinterpreted in our pursuit of the ideal.

Krishnamurti: Commentaries on Living, Second Series, Page 19.

Exactly. And this is why it is important to distinguish between the ideal and the real. To show that man is the collision of the discrete and the continuous. And to show this is easy, if one cares to look.

---------- Post added 06-03-2010 at 01:59 PM ----------

Arjuna;172448 wrote:

Digital transmission is more accurate, but not because the signal doesn't degrade as much as an analog one, but because it's so easy to reform the signal at the destination. It's the magic of encoding.

Exactly. Digital is not sensuous. It's made of pure concept. It's just that we have to convert it into electricity, etc. in order to send it. The digital is of a different nature altogether than the analog. For me, the beauty and clarity of digital concept is ideal sculpture, the most perfect sculpture. For whatever reason, not many notice or care. Platonic Form exist, I say. And it's not boring. Smile

---------- Post added 06-03-2010 at 02:02 PM ----------

Huxley;172439 wrote:

Only if you care to measure the position and the momentum simultaneously. If you care not about momentum, then, in theory, you could have approaching infinite indeterminacy in momentum to find a pretty darn precise position.

Ah, but approaching infinite is not good enough, in my opinion. Because much of my point is that we can imagine impossible precision. But only spatially. When it comes to adding number to this, we see the contradiction. It would take an infinitely long number to measure infinite precision. Smile

---------- Post added 06-03-2010 at 02:03 PM ----------

jack;172474 wrote:

When a man buys anything, feeling ripped off or making a steal, he likes what he has gained. But an artist... when is an artist ever satisfied with his gains?

I don't know. I sometimes think he is. But I guess he goes back for more, because he likes the first gain so much. Maybe you're right, then.

---------- Post added 06-03-2010 at 02:05 PM ----------

manored;172501 wrote:

Perhaps this is because they exist in different realities, realities we have created for then. Ever law a computer and a chess game follows were created by humans, and therefore they dont follow any laws that humans cannot comprehend. For example, a computer can never draw a perfect circle, but he can draw one as perfect as possible.

Yes, that's just it. Spatial precision is one thing, and numerical precision is another. Our intuition of space is of a different nature than our intuition of pure number. This hit me a few months ago, and that's when I got obsessed with math. Smile

---------- Post added 06-03-2010 at 02:07 PM ----------

manored;172501 wrote:

Well, computers arent very happy about such numbers either. A computer cant draw a perfect circle exactly because it cannot calculate Pi fully, since its an infinite number =)

And this issue of pi was specifically what made me realize the collision of the continuous and the discrete. Do you know about the number e? It's as fascinating as pi, maybe even more so. Pi is a sort of eternal number, because the circle is a good symbol for eternity. But e is the symbol of growth and decay....or time. My avatar equals -1. If we want to get all poetic we could say that Time to the Power of Imaginary Eternity equals Negative One.

@Arjuna,

Arjuna;172448 wrote:

Is reality digital down inside like a movie?

There are those who think so. Digital physics. :flowers:

@manored,

manored;172501 wrote:

ughaibu;172393 wrote:
Galileo used this to prove that all circles are the same size!
This is not really possible =)

Why not? The proof demonstrates that the circumference of any Euclidean circle is equal to a Euclidean point.

@Reconstructo,

if it is not precise - you have not gone far enough. Look deeper and take more into consideration. Never accept a blunt edge - why that just gives people room for movement. Close all gaps. Sharpen up!!

@ughaibu,

ughaibu;172715 wrote:

Why not? The proof demonstrates that the circumference of any Euclidean circle is equal to a Euclidean point.

And doesn't that say something about proof? Smile

@Reconstructo,

Reconstructo;172807 wrote:

And doesn't that say something about proof?

What do you think it says, if anything?

@ughaibu,

ughaibu;172822 wrote:

What do you think it says, if anything?

Our spatial imagination is in conflict with our sense of digital quantity. A point has no extension. So we are adding zeroes, right? But then how do we imagine a circle that's made of nothing? A point without extension is a fascinating paradox. It's like the infinitesimal in spatial terms.

That something so counter-intuitional can be proven shows that not all is well when it comes to our relating the continuous with the discreet.

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