These are non-issues. In the paradox that is the subject of this thread, we're concerned primarily with the quantities not the length of the tasks. — InPitzotl

This is where your mistaken. The paradox assumes a spatial form, created from mathematical principles, and names this form Gabriel's horn. Therefore the subject is this form which is created by mathematics, not the quantities which are used to create it. You need to divorce yourself from the numbers, from the mathematics, look at the form described, directly, and ask how is it to be measured.

This might be why you seem to be having so much difficulty understanding the problem. You want to use the same numbers which create the form, to measure the form. But when infinity is induced, this represents a failing in the capacity of the numbering system, so it is impossible to measure the form produced with the same numbers which produce it. That's what happens with the square root of two, and pi. There is a failure in the numbering system's capacity to measure the spatial form produced because of an inherent incommensurability. So we have a simple solution, we round off. Or, if necessary we can move toward employing more complex numbering systems, real numbers for example.

Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite. — InPitzotl

This is your false presumption, which is misleading you, that "the quantity of volume 'on the inside' is finite". If you would approach the problem with an open mind, rather than with what I see as a false presumption, we could probably make better progress in this discussion.

Here's what I propose. I'll justify my presumption, and you justify your presumption. I see that the spatial form which we are talking about, Gabriel's horn, is infinitely long. Therefore it is impossible, in theory, to precisely figure its volume. The volume therefore, in theory, is indefinite, being "infinite", unbounded, just like the extent of the natural numbers is "infinite", indefinite, or unbounded. We can however figure the volume of such forms, for practical purposes, by rounding off.

Will you justify your presumption that the volume is finite?

You are presuming that the paradox is solved by questioning the volume, but you haven't even shown a good reason to doubt the volume much less the flaw in the presumptions leading to the paradox. — InPitzotl

I told you already, Gabriel's horn is a spatial form with an infinite length. That's very good reason to doubt the volume. All you said was "You're reasoning by equivocation; 'the extent is infinite, therefore the volume is infinite'. Clearly there is no equivocation. A spatial form which has an unlimited (infinite) extension in one of its dimensions, will have an unlimited (infinite) volume accordingly

I thought token was an experience reserved for Saturday night.

The phrase "all the finite numbers" is itself such a bag. — InPitzotl

A phrase is a bag? Come on Pitzotl, you're reaching for straws. Get back to the subject.

No, you didn't address this, because if this were indeed the case... if you could paint infinitely thin, then you can paint an infinite area with a finite amount of paint. — InPitzotl

That's not true at all. Your interpretation of "infinite" is dreadful. You cannot paint an infinite area regardless of how much paint you have, because no matter how much painting you do there is always more to be painted. That's the issue with Gabriel's horn. It's infinitely long, so no matter how much paint you pour in the top, it never reaches the bottom.

It doesn't matter what you propose as the volume of the horn, you still cannot fill it with paint . Suppose you conclude it's 3.1 gallons,. You pour that in, but you haven't filled the horn because it hasn't reached the bottom.

But your proposed "real answer" doesn't address the extruded Koch snowflake, because that isn't infinitely long. The thing you're proposing isn't the real answer does address the Koch snowflake, because the infinite area on its perimeter is not a volume. — InPitzotl

We're discussing Gabriel's horn not snowflakes. How my answer relates to a snowflake is irrelevant. I don't see why you feel the need to bring up so many irrelevant issues.

You're reasoning by equivocation; "the extent is infinite, therefore the volume is infinite" simply doesn't follow. — InPitzotl

It seems you do not know the meaning of "equivocation".

Take it all with a grain of salt.

The method is to apply a limit. — InPitzotl

Yes, apply a limit to what is stipulated by the premise, as without limit. That is the mistake. Can't you see that it is stipulated that there is no limit to the length of the horn, therefore to apply a limit is to contradict the premise?

That bag has all of the finite numbers in it, but no infinite numbers. — InPitzotl

You cannot put all the finite numbers in a bag, because there is an infinite quantity of them. This example provides nothing of relevance.

For example, a lower limit of 1 and an upper limit of 2 refers to "all of the numbers in the bag that are greater than or equal to 1, and less than or equal to 2". By contrast, a lower limit of 1 and an upper limit of ∞ simply means: "all of the numbers in the bag that are greater than or equal to 1". — InPitzotl

This is not relevant either. It is stipulated in the Gabriel's horn example, that there is no lower limit. It is stipulated that the horn continues infinitely. That means no limit. It is therefore not a case of having an upper limit and a lower limit, and to represent it as such is a mistake. To impose a lower limit (such as zero) is to contradict the premise of the example. We are not talking about how many numbers there are between two numbers here, we are talking about an unlimited length. To impose a limit on that length, a point where the diameter of the horn reaches zero, is to contradiction the premise of the example.

Wrong. The real issue is very simple... areas aren't volumes. "Paint" tricks you into thinking they are. It is very interesting to note that you never actually tried to address this explanation, just as you never commented on the extruded Koch snowflake with a bottom having the same "issue". It's no wonder you're trying to play the if-you're-wrong-that-means-I'm-right card. — InPitzotl

I did addressed this. If the horn can go infinitely thin, then so can the paint. They must play by the same rules.

That sounds like a confused equivocation. A 1x1x1 cube by definition is 1 cubic units of volume, but it has an infinite number of points. — InPitzotl

Again, we're not talking about an infinite number of points within a confined space, so that is irrelevant. The horn is infinitely long therefore there is no confined space. If someone were to say to you that there is an infinite extension of the universe, would you think that this implies a confined space? If someone says to you, take this line which continues infinitely, would you think that they were talking about a line segment which extends between two points?

This seems to be where your misunderstanding lies. You want to make this into an issue of a confined, "limited" space, but it is clearly stipulated that the horn is infinitely long, therefore there is no such confined space.

For what? — NOS4A2

Whatever they can find him guilty of. Don't you think that the greatest witch hunt in history is bound to make a judgement of guilty at some point, and proceed with punishment, regardless of the person's actual guilt or innocence. That's what witch hunts do don't they?

When's he going to jail? Any bets?

Gotcha. Nice smack down. So you'd agree that society is a prerequisite for individuality, that the subjective narrative is influenced by society, if only because such narratives are frequently directed at others, but the ego (I) itself can't be reduced to social interaction because that's just retarded. Is that your view? — frank

I don't agree that society is a prerequisite for an individual. I think that this is a faulty form of holism which is demonstrably illogical. The fact that the fallacy of composition is a fallacy is the first indicator that the properties of one's individuality cannot be accounted by the society. And there is much evidence for this, especially in evolutionary theory whereby individualized properties are derived from genetic alteration, rather than from the social group. Biological evidence indicates that the individual with X properties is prior to the society with X properties. In general, I believe the idea that a human individual derives one's properties from one's position within a group, as a whole, is a misguided form of holism.

In reality, it is evident that the individual is a free willing soul, who creates one's own position in society through one's own intentional acts. This is not to say that the human being is absolutely free to choose whatever position one wants in society, because we know that the physical world imposes restrictions on our freedom. But we can consider that the restrictions imposed on us, by other human beings, are just an extension to the restrictions imposed on us by the physical world. This is because another human being's intent to impose restrictions on me is equally restricted by the physical world, as my will is restricted by the physical world. This means that the capacity of others to impose restrictions on me is mitigated by the restrictions imposed by the world on them, which are equal to the restrictions imposed by the world on me. This justifies a true and objective equality between us.

He says "well one over infinity that's zero, so you get nothing from that". — Metaphysician Undercover

t's right there under your nose and you can't see it. You read:
limx→∞1x=0limx→∞1x=0
...as "saying 1/infinity equals 0". But that's not what it says, and it's not what it means. I told you what it means, and showed you a link. — InPitzotl

I saw it, and heard it distinctly stated, on the Youtube video at the referenced point, he distinctly says "well one over infinity that's zero, so you get nothing from that". And, you said: your method is the same as the one on the video. You can insist, as fishfry stated, that this is the convention in such procedures, to take one divided by infinity as zero. But that is to round off, so we need to respect the fact that the solution to the question of the volume of the horn, is a rounded off solution.

Limits still aren't rounding off. — InPitzotl

"Infinite" means unlimited. When you apply limits to the unlimited, you are either contradicting or rounding off. Which charge do you prefer? If you will not accept the fact that you are rounding off, then the paradox arises due to the contradiction.

Rounding off implies that there's a stated answer a, and a real answer b, and that a is not b but is "close enough" to it. — InPitzotl

Yes, and that describes exactly what is the case with the volume of Gabriel's horn. The "real answer" is that the horn is, stipulated by the stated premise, as infinitely long. And, the volume of an infinitely long container cannot be determined. The "close enough" answer is produced by applying the limits of integration, which works with approximations. The appearance of a paradox arises, because "close enough" fails the task intended by the proposition "infinite". In other words, there is no such thing as "close enough" when we're talking about the infinite.

So the limit here is met by the value 0 exactly. — InPitzotl

To impose a limit on the infinite is to contradict. So it is not the determined value, 0, which is wrong. That value is correct according to the terms of application, but the judgement that the situation is suited to the application, is wrong. The example stipulates that the horn is of infinite length. Therefore to impose a limit on its length is a mistaken procedure.

In common practice, this is not a problem. We don't often encounter infinitely long things which we want to figure the volume of. But when we encounter an infinitely long thing proposed in a thought experiment, it is of paramount importance that we stay true to the premises, and have one consistent way of interpreting "infinite", or else someone will claim that the results are paradoxical.

LOL! Look who's in denial!
Saying that 1/infinity equals zero is obviously an instance of rounding off. There's nothing wrong with rounding off. We do it all the time with pi, square roots, etc.. That's how we get the job done by rounding off. If we couldn't round off, we couldn't get the job done in many instances. So you shouldn't be embarrassed by it. You should be embarrassed by insisting that it's not an instance of rounding off, when it clearly is, though.

No, the PLA wasn't supposed to cover that, nor does it cover that.

It's very common to be misled. I do see what you mean though. But in the context of a discussion of colour and colour experiences, Richard does not see what you mean by 'red', though I do. And this is using 'see' in its visual sense. I am obstinate about that. — unenlightened

Well, I'll have to beg your pardon, and I hope you'll accept my apology for interrupting a discussion you were having with a number of other people. I can easily understand how you took me in the wrong direction. You didn't seem to have ever grasped what I was objecting to in the first place.

What I disputed was what you appeared to be asserting, that the difference between Richard's colour experience, and Unenlighten's colour experience, is non-existent, just because you apprehend it as insignificant.

I find this an entirely agreeable explanation, except that I take it one step further, and say that things that make absolutely no difference should be treated as non-existent. So I never speak of X or Y at all, but only of red apples and blood and green grass and colourblindness and such. Subjectivity disappears from the conversation, because there are no words for X or Y and can be none. There are apples and grass and colours, and blindness, and we agree abut that. — unenlightened

Notice your use of "absolutely" to qualify "no difference". That there is "no difference". is simply a subjective opinion, and you use "absolutely" in an attempt to make this judgement sound more authoritative. Furthermore, your mention of colourblindness indicates that you actually recognize such differences as being very real. Therefore you clearly acknowledge the reality of these differences, and your attempt to make them disappear by saying that they make no difference is a foolish error which will inevitably lead to misunderstanding when you assume that another person apprehends a specified thing in the same way that you do.

After that, our discourse went in two different directions. I wanted you to see that since these differences are real, then a person's experience of colour itself can be considered as a thing with properties, which we can talk about. It is necessary to assume it as a distinct thing with properties, in order to support such differences. So we can talk about that thing and its properties, and what make's one person's experience of colour different from another's, being distinct things. However, we need to allow that your experience and my experience are distinct things. If we premise that they are one and the same thing, then the fundamental laws of logic, identity, non-contradiction and excluded middle, get in the way, preventing us from talking about the reality of these differences. You seemed to insist that the only type of thing we ought to talk about is a thing which we can see with the eyes.

Sure chief, one over infinity is zero, and that's not a matter of rounding off. Tell me another.

Then looking is a misleading way to express, which is what I told you a while back. — unenlightened

Actually it's a very common use of the word "look", you're just obstinate, refusing to look at anything unless it's in front of your eyes.

But what happens if there are no individual perspectives going into the production of the map? — frank

I don't see how that's possible. Have you ever seen a group of people working together before. It's all individual perspectives going into the production.

IOW, if what we take to be individual perspectives are actually all cultural constructs? Is our map then also a purely cultural construct? What would the implications of that be? — frank

There is a small number of people who work together to produce a map. There is a very large number of people who make up a culture. It would be a fallacy of composition to say that the culture constructs the map. We also have the same type of fallacy if we try to say that the individual's perspective is the perspective of the culture. It's plainly and simply illogical.

Ok. It's just that real maps really are objective accounts. — frank

You might think so, but it is always subjects who decide what to put on maps, therefore maps are inherently subjective.

Suppose there's an array of one hundred items in front of you, and you draw a map locating the relative position of twenty items. Regardless of how true your representation of the locations of those items is, your map is subjective, because you have chosen which items to map.

But saying we don't see the world as it is isn't right, either. We do see things as they are - the sugar in the bowl, the tree in the garden. Sure, we don't see it all, but we do see enough to get by. — Banno

The issue is the process by which, what we do see, is selected from everything else, just like in the example of drawing a map, above. There is a process which makes us see crystals of sugar rather than molecules of sugar. But we smell and taste molecules. And I don't think we sense atoms in any way. Since this process is properly "of the subject", then the world which we see is subjective. Agreement between us, because the process is similar between similar beings, makes our descriptions intersubjective. But this does not change the fact that the aspects sensed are selected for, by the living systems of the subjects, so it does not make it any less subjective.

And I will. Do the test linked above, and find out how well you can hold a simple image in your mind for a few seconds. — unenlightened

Looking at something with the mind does not mean to hold an image of it, it means to think about it. There are many different ways of thinking about things, such as establishing associations and relationships. This is how language facilitates thinking, by replacing the image with a symbol. Then the mind can make associations between symbols (words or numbers) without the difficult task of holding visual images. Now we're not only talking about visual images, but aural images, as well. I'm sure you have the capacity to bring these aural images of words into your mind, from memory. And while you're looking at these aural images, consider how easy it is to hold the visual images of numerals and other mathematical symbols in your mind.

Gabriel's horn has infinite surface area, but holds a finite volume. — InPitzotl

The volume of the horn is only determined as finite when the infinite radius is rounded off to zero at some determinable length, as is demonstrated by the YouTube video. Otherwise the infinite length of the horn ensures that the volume is infinite.

I've got a few but too most of them are banned or inactive.

I already discussed that here. — InPitzotl

At that point you're basically just mapping points to points, and there are plenty enough points in a tiny droplet of paint to map to the infinite surface area of the horn. — InPitzotl

In other words you want the paint to follow different rules for the inside of the horn, than for the outside of the horn, allowing a finite volume of paint to cover an infinite surface area on the inside, but not the outside. What's the point, if you can just make up whatever rules you want?

Why can't we just do the same thing with the infinite surface area on the outside of the horn as well? We can just say that any tiny amount of paint can be spread out over the infinite outside surface area, just like you say for the inside.

Do banned followers count?

The problem is, there's only a finite portion of the horn that can fit "whatever the layer is", and there's an infinite portion of the horn too thin to have "whatever the layer is" be the thickness of "whatever the layer is". Because of this, you cannot multiply "whatever the layer is" by infinity and get anything meaningful. — InPitzotl

Obviously, if we can assume that the horn is infinitely long, with an infinitely small diameter, we can also assume that the paint can go infinitely thin. That this is not an issue of the real physical properties of a real horn, or the real physical properties of real paint was determined in the first couple of replies in the thread. Do you agree that if the horn is allowed to go infinitely thin, then the paint must play by the same rules, and be allowed to go infinitely thin as well?

Memory is not looking at experiences, because one can remember in the dark. I remember the last time I was in the chip shop, the smell of hot fat and vinegar, the soft shine of the stainless steel counter and the bubbly battered fish hot under the lights. But I am looking at the words appearing on the computer screen and smelling the clean washing just out from the dryer. — unenlightened

I said "look at them with our minds". Obviously I wasn't talking about looking at them with your eyes. And the point was that when we recall memories we are looking at things with our minds.

Of course you can continue to deny that it's possible to look at something with one's mind, and therefore deny that there's anything being looked at with the mind, but I don't see the point in such a denial. Philosophy is an attempt to understand these things which we look at with our minds, so denial that they even exist is a step in the wrong direction, a mistake.

Gotta watch out for those people with dual personalities, they could gang up on you. Or, the two could be completely opposed and go into an endless argument with each other, trying to make a spectacle of themselves.

It is not a mistake. because it cannot be recognised as a thing. "I see a red apple" means I see a thing in the world that is red. There is nothing in my interior world that is red. But "an experience of red" suggests that the red is in my head in my interior world (whatever that is). But I don't see colour in my experiences, because I never look at them - my eyes point outwards not inwards. It is a linguistic construction that is mistaken for a thing The experience of seeing cannot be seen and thus cannot be coloured. Only what is seen is coloured and never the experience of seeing. — unenlightened

Again, I think that's obviously a mistaken proposition. We all "look" at our experiences, we look at them with our minds. If we didn't we'd have no memory, as that's what memory is, looking at our experiences. So when you look at a red apple, and turn away and later talk about it, you are looking at your experience of having seen a red apple and this is what allows you to talk about it. When you understand that this is the reality of the situation, that your mind (your "interior world") is necessarily the medium between the red apple, and talking about the red apple, then the interior world is necessarily a "thing" which we can talk about. Otherwise we cannot account for our capacity to talk about remembered things.

I think you'll find that it's very clear that you don't see colour with your eyes. "Colour" refers to a generic concept, grasped only by your mind, while your eyes see particular instances of different colours. It really doesn't matter that people say "I see colour with my eyes" because our common expressions do not represent what is really the case, they are just formed around facilitating communication. We also say that the sun comes up, and the sun goes down. We need to get beyond what the common expression appears to mean, to understand what is really the case in that situation referred to by "I see colour with my eyes".

It doesn't matter how many times less than a layer's worth of paint you've got inside, the horn can only have an inner layer that thick if it's thick enough on the inside. And only a finite portion of the horn so qualifies. — InPitzotl

The physical properties of the paint being incompatible with an infinite horn, was already rejected as not the subject of this discussion. If we were discussing whether the molecules of paint could fit down inside an infinitely small tube, we might just as well reject the infinitely small tube as a nonsensical proposition in the first place.

I find this an entirely agreeable explanation, except that I take it one step further, and say that things that make absolutely no difference should be treated as non-existent. — unenlightened

This is a mistake. If it can be recognized by you as a thing, then by that fact, it has made a difference, and cannot be excluded as non-existent. In other words this is willful ignorance, denying the existence of something you recognize as existing. It's nothing more than saying I have recognized the existence of this thing but since it makes absolutely no difference to me (I care nothing about it), I can claim it as non-existent. The problem being that whether or not something makes a difference is a subjective judgement, and the fact that you don't care about it doesn't mean that no one cares about it..

But that inside Gabriel's horn you refer to is less than 1/3000 units across beyond 3000; so if you fill it, you're filling it with less than a layer's worth. It's less than a thousandths of a layer's worth beyond 3,000,000; less than a billionth beyond 3,000,000,000,000. — InPitzotl

I don't see how this is relevant. Are you forgetting that it's infinitely long? It doesn't really matter how many time less than a layers worth of paint you're putting in there, it's infinitely long, so whatever the layer is, it will be multiplied by infinity, and therefore enough paint to make an infinite number of those "less than layers worth" layers on the inside of that horn..

No mental stuff here. — Banno

You don't say?

No, the horn is not closed. There are many online calculus tutorials and classes available that explain the theory of limits. — fishfry

It's quite simple. The radius is y, which is represented at the one limit as y=1/x=0. From someone who adamantly argues that = means "the same as", I don't see that you have anything to argue. The diameter is taken to be 0 at that point in the solution. This means that the horn is closed at that point, represented within that method of figuring out the volume. There's nothing to discuss, it's clear and obvious.

The nature of "a limit" is irrelevant and just a ruse. What matters is the values that are plugged into the equation, the values which bring about the apparent paradoxical conclusion. If y=0 then the radius is zero at that point, which is what is employed in that method. If the premise is that the radius never reaches zero, then the solution does not truthfully adhere to the premise.

No, the reason for the appearance of a paradox is that the shape has finite volume and infinite area... — InPitzotl

The shape is only said to have finite volume because of the method employed to determine the volume. As explained above, this method assumes a point where the radius of the shape is zero. Therefore this method contradicts the premise of the problem, which states that the horn continues infinitely without reaching a zero radius.

If we were to properly consider that infinite extension of the horn, then despite the fact that it is an infinitely small circumference, being an infinitely long extension of that circumference gives us an infinitely large volume. That volume could not ever be filled.

But the reason that is, as already mentioned by andrewk early on in this thread (and apparently severely underappreciated), is that painting areas with paint requires some thickness of paint. — InPitzotl

Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside.

If the argument is that the thickness of the paint prevents it from going into that tiny channel, then we're just arguing physical properties, which has already been dismissed, as not what is to be discussed. Therefore if we are assuming that the paint can go around the tiny surface on the outside, it can also go around the tiny surface on the inside, because we are not discussing the physical properties which would prevent the thought experiment. If that were the case, we'd just reject the whole idea of an infinitely long horn as ridiculous in the first place.

1∞=01∞=0 in the extended real number system, which is always the implicit domain of integration problems. — fishfry

So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox. To figure the volume of the horn requires that zero is taken as the limit, rather than the unlimited "infinity".

And to be fair, and to hold you to your own words, you ARE objecting, because you have denied that the volume of the horn is pi, when in fact it is exactly pi. — fishfry

The volume of the horn is figured to be pi only when the infinitely small radius, as stipulated by the premise, is taken to be zero, as required for calculating the volume.

If I'm not mistaken, the part you object to is when he's proving that the cross-sectional area is infinite; that is, the area under 1/x from 1 to infinity. — fishfry

The part I am bringing your attention to (and I'm not objecting to it, I'm bringing your attention to it, as relevant to the so-called paradox), is where he determines the limits to the radius of the horn. The radius is given as represented by y, which is equal to 1/x. Then when he plugs the values into the equation, 1/x becomes 1/infinity, which he says is zero.

I agree but Attention from where? from your conscious mind or from the subconscious ie, over-mind? I would say the latter. — Ken Edwards

Think about when you let your mind wander, daydream, it's similar to dreaming while you're asleep, the thoughts can proceed in all sorts of different ways. You can think bout anything. I would say that this is as close to subconscious as I can get and still be conscious. So when my attention gets directed, and my thoughts get focused, my mind gets pulled away from the subconscious, and directed toward my conscious activities.

But, a caveat. 2 or more events must occur before you can direct your conscious attention to it. If you are touched that would activate your sense of touch without the participation of the conscious mind. If it were something that was very hot that would instantly turn on a series of alarm bells and you would take violent action without the participation of the conscious mind. — Ken Edwards

But there are five senses, and they can all be sensing at the same time. If the conscious mind focuses on the products of one sense, wouldn't this be the conscious mind which directs the thoughts in this way? If something very hot touches me and I take violent action, I think this is at the direction of my conscious mind.

Who or what is "I". Your conscious mind obviously. It would be normal for me to use a slightly different vocabulary. That's why when something touches me I would say: "It would attrect my attention" which would be a general statement refering to all aspects of my response. Rather than: "Direct attention to it" — Ken Edwards

If, your conscious mind is what you refer to with "I", then I think there is inconsistency to say that you might "take violent action without the participation of the conscious mind". This would not be you who is taking that action?

I agree that there are all sorts activities carried out without the participation of the conscious mind, pumping of blood for example, and breathing, but we're talking about thinking activities.

We can't really go by what you would say, or how you would say it, because we're looking for the truthful description, not the habitual one. That you'd be more likely to say "it would attract my attention" rather than "I would direct my attention to it", is irrelevant. I would be more likely to say that the sun rises and it sets, than to say that the earth spins so that I'm facing the sun and then it continues to have me facing away from the sun at night. How we describe something in our common way of speaking is often not a good representation of what is really going on.

So I would rather include the entirety of myself, conscious and subconscious within "I", and recognize that some actions are direct by the subconscious part of me, and some by the conscious part of me. There could be some overlap, or a grey area. But what about thinking? Isn't all thinking directed by the conscious part? We could go back and take another look at the mind wandering, daydreaming, and dreaming itself. Is it really appropriate to call this 'thinking'? I don't think so. So what is it? Can we say that it is mental activity which is being directed by the subconscious part? If not, what is it? It's clearly not completely random, so it's somewhat directed, but not directed by the conscious mind.

But, I think any thought can be interrupted or cancelled and a new thought provoked or intruded or substituded. — Ken Edwards

Wouldn't you agree that these actions imply direction?

You sure you want to throw in your lot with a bloke with an eccentric notion of equality, Mww? — Banno

Do you really think that upholding the difference between being equal, as two human beings are equal, and being one and the same thing, like I am the same as myself, and no one else, is an eccentric notion?

Check my reference above, 6:25 in the video, where one over infinity is taken to be zero. Otherwise you do not get pi.

Plus....here we go again with the goalposts; I never said words used by only one person. I said words subjectively invented, which implies one person, but does not imply use, that being merely a possible consequent. While it may not make sense to invent a word then not use it, that doesn’t mean the use is necessary because of invention. The use is necessary for something else, which, again, presupposes the invention. — Mww

This relates directly to the faulty dichotomy which people like Banno seek to create. That dichotomy stipulates that either a language must be public (comprehensible to all), or private (comprehensible to only one person. The dichotomy is faulty because those who employ it do not distinguish between what is potentially comprehensible to others, and what is actually comprehensible to others.

So the invented, private language, which is not actually comprehensible to others, but potentially comprehensible to others, requiring an act of teaching, or some other form of learning, is excluded as not truly private, because it is potentially comprehensible. Then the role of the private language within the overall public language is dismissed because the notion of "private language" is rejected as unreal by that premise that it is potentially comprehensible. However, since the invented, private language, is actually incomprehensible to all others, prior to being learned by others, and it is a true language employed by the individual, the rejection of private language is unjustified.

Is that what you're objecting to? That the area under 1/x from 1 to infinity is infinite? Or what mathematical fact are you objecting to? — fishfry

I'm objecting to the method employed by the person in the YouTube clip, which replaces the stated infinite limit, (approaches zero) with zero, as I referenced above. By that method, the equation for the volume of the horn resolves to pi, as stated in the op. If you have a method to figure out the volume of the horn without that substitution, then you might present it. If not, then we probably don't disagree, and we're just wasting our time talking past each other.

I suggest that the proper representation is that the volume is necessarily indefinite, rather than finite, and there is no paradox. This means that the amount of paint required to fill the horn cannot be determined. Therefore no act of pouring a determined amount of paint into the horn will fill it

The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity. — fishfry

I think the real issue is that it's cumbersome to talk about limits when the subject is infinite, because it's contradictory. Is the horn closed (limited), or is it infinite (unlimited). Clearly the premise is that it is unlimited, infinite, and any mathematical axioms which deal with it by imposing a limit, are not truthfully adhering to the premise. And that's why the appearance of a paradox arises.

Let me change that to: "a result of a past act of thinking" — Ken Edwards

OK, so we have identified two features, the act called thinking, and the results of that act, thoughts.

If so how has that thinking been provoked? Might it not have been provoked by something exterior like a tree or a traffic cop. Or provoked by an earlier thought coming from either of the two minds or from the newly discovered default mind?

How does "directed" come into it? — Ken Edwards

What I tried to explain in the last post, is that i do not think that thinking is provoked, it just goes on and on, somewhat automatically. However, it is directed, guided. This is why it is not totally random like dreams can be, though dreams are generally guided to an extent anyway, and so are not completely random. We can know that it is guided rather than provoked, because we sense many different things at the very same time, and we only direct our thinking toward particular things. You would say that these are the things which provoke us, but if we ask why did this provoke, rather than that, we must turn to an internal reason and explain it by saying that the thought was directed toward this rather than that, for whatever reason.

So, consider your example, "a tree or a cop". What makes you think about the tree rather than the cop, or the cop rather than the tree? Think of all the visual stimulus around you, the aural stimulus, and things like smells. Of all those things, why do you direct your attention to this or that thing instead of the many other things? You would say that these are the things which provoke you, but it's not the thing which is responsible here, it is you who is seeing the thing as interesting. The thing is doing something, which for some reason interests you, so I turn to you, and ask why does this thing interest you. Why are you directing your thought toward this thing and not something else?

The sense of touch provides a very good example because it is very high in the hierarchy of capacity to attract one's attention. If something, or someone touches you physically, it's almost impossible not to direct your attention, and therefore think about that thing which touches you. Now if you think about the sense of sight, there is at any time many things within your field of sight, which you see, but you will not think about. So I think I can conclude that something touching me physically is much more important to me than something simply being in my field of vision. That's why when something touches me I direct my attention to it.

Banno thinks that if two distinct numbers are made compatible through an equation, then they become one — Banno

You expect me to believe that? Two distinct things become one and the same thing, if they are assumed to be equal. No, equal things are not the same thing. Doesn't anyone have any respect for the law of identity anymore?

Philosophers are trying to save the human race from their greatest enemy, themselves. They haven't been able to do it, therefore philosophy is a failure. Wait a minute, human beings haven't annihilated themselves yet, so on what basis is "failure" claimed.

The point being that we judge a process as a success or failure in relation to whether it achieves a stated goal. Snakes states the goal of philosophy as to "unlock general features of the universe", and concludes that since features remain locked, philosophy has failed. But Snakes seems completely ignorant of the role which philosophy has played in unlocking those features which have already been unlocked.

Banno thinks that if two distinct frames of reference are made compatible through a transformation theory, then they become one. What Banno doesn't recognize is that this is just a matter of using theory to reconcile the differences between two distinct perspectives. And so it is just a matter of Banno saying that they are one, when they are treated by the theory as two, and that's hypocrisy.