My point is that you need to have the concepts of "angles" (so that they can be equal), "directions" (so that you can make the points on your lines aligned, i.e. colinear), "distances" apriori, before resorting to analytic geometry. (And formally, we would call that an affine space today. Although the terminological designation would not be present historically, the ideas would be the same.) It would be backwards thinking if we started with pairs of numbers, declared them to be the Cartesian coordinate system for implicit space of entities and finally tried to infer a sensible explanation of the nature of the metrics of those entities. — simeonz

I guess I don't follow your point. The historical evolution is well known, from Euclid to Descartes. And in modern math we start with a 2-dimensional coordinate system and define the Euclidean distance. Either way works. What exactly is the question or issue?

The whole point is that the 2D Cartesian coordinate system is not a picture. It is ascription of coordinates to some plane of points, which points correspond in pairs to vectors, which vectors individually correspond to lengths and in pairs correspond to angles. Ok, the points are angles-apples, but the remaining properties are not automatic. They are not provided by the Cartesian coordinate system for you, mathematically. They are provided by you, originally, so that you can justify the use of Cartesian coordinate system. Otherwise, what you have are just pairs of numbers corresponding to points, and the rest is as real as Tolkien's world. — simeonz

Sure, but I'm not following your point. The Pythagorean theorem goes back over a couple of thousand years, and Cartesian coordinates go back only to Descartes. We don't need an orthogonal coordinate system to have the theorem. Everyone agrees with that I'm sure. I am not understanding the point you're making. A right angle (if I remember my high school geometry) is when you have a line intersecting another line and making equal angles on each side. No coordinates or numbers needed.

I'd like to see that! — SophistiCat

Thanks, maybe I'll get started on it.

Likewise if you have a proof that space is discrete. They are two different mathematical assumptions. — aletheist

I've made no such claim. You did. And they're physical, not mathematical assumptions.

What do you think? — Shawn W

There's a theory of proofs over uncountable alphabets but I don't know anything about it. But as I say, by complexity I think of complexity theory, and there's an incompleteness theorem that can be proved that way. That's literally all I know about it.

Since space is continuous, it has infinitely many potential parts, but its only actual parts are those that we create by marking them off. — aletheist

If you have a proof that space is continuous that would be a discovery on a par with the revolutions of Newton and Einstein.

That is because there is no basic unit intrinsic to space itself, — aletheist

I just pointed out that Max Planck introduced the Planck length as something aliens would discover. It's intrinsic to space itself as I understand it. I'm no expert but this is my understanding. It's not like inches or centimeters. It's a fundamental length that a physicist anywhere in the universe would discover.

Chaitin has used complexity theory to prove a version of Gödel's incompleteness theorem. But I wonder if you're using complexity in a different way. Here are a couple of links.

https://math.stackexchange.com/questions/1001948/is-it-possible-to-deduce-godels-first-incompleteness-theorem-from-chaitins-inc

https://www.cs.ox.ac.uk/activities/ieg/e-library/sources/georgia.pdf

However, on my other account "Shawn" I have surmised that a growing alphabet can be able to determine the complexity of the proof of the theorem if logic comes next to mathematics. — Shawn W

This makes me want to make a joke about my other account @Metaphysician Undercover.

Something discrete. Yet "discrete space" is impossible if it is to remain space. — Gregory

The integers are discrete. And if you have any space whatsoever that's made of points, you can define the distance between two points to be 1 if the points are different, and 0 if they're the same. This is called the discrete metric. It makes any space into a discrete space; even an uncountably infinite space.

https://en.wikipedia.org/wiki/Discrete_space

But you didn't really answer the question. You said you're troubled that the unit interval [0,1] can be partitioned into infinitely many intervals of the form [0,1/2), [1/2, 3/4), [3/4, 7/8), and so forth. I asked you what is your objection and I am still unclear as to why you are troubled by this example. In fact it seems to relate to your original question about dividing a block. This is the one-dimensional version of the same question.

This is a commonly held sophistry. — Gregory

The Planck length is a fundamental aspect of modern physics. And by modern I mean since 1899, when Planck came up with the idea. He noted that it's defined only in terms of the speed of light, Newton's gravitational constant, and Planck's constant. His idea was that the Planck length was universal, in the sense that aliens would come up with it.

Here's Sabine Hossenfelder discussing the Planck length.

http://backreaction.blogspot.com/2020/02/does-nature-have-minimal-length.html

As i demonstrated on this thread, everything in this world is made of infinite parts and I BELIEVE the conclusion is that everything is finite and infinite in the exact same respect. That last part is what I was trying to explore — Gregory

You claimed it. You haven't demonstrated it. Such a demonstration, that the world is infinitely divisible, would be a breakthrough on the order of Newton or Einstein.

I don't see how anyone with a brain wouldn't want to know how to get two objects out of one without referring to infinities. — Gregory

The Banach-Tarski theorem does depend on the axiom of infinity. You do have to be able to treat infinite collections as sets. No getting around that. And at one point in the proof you have to invoke the axiom of choice to pick a set consisting of one point from each of an uncountably infinite collection of sets. So there's no question that infinity is involved.

Such a theorem is incredible and I hope you do codify it into a thesis that others will read and appreciate. — Gregory

One of these days ...

I for one am having trouble with it because it's of such a nature which I do not think I will understand it by READING it, as opposed to having it explained in person where I can cross examine every step. Reading it is just to much for me — Gregory

Should I come over with a blackboard and chalk?

That is arbitrary, as is the Plank length — Gregory

The Planck length is not arbitrary, Planck noted that an alien would discover it. A mathematical unit of length, formed by choosing two points on a line and calling one 0 and the other 1, is indeed arbitrary. But any choice at all will do, since any choice of points can be transformed into any other by a translation and a linear scale factor.

Something I need to consider more, thanks. — Gregory

Are you familiar with the fact that the harmonic series diverges? This is the basis of the proof that the area under 1/x from 1 to infinity diverges. That is, 1 + 1/2 + 13 + 1/4 + 1/5 + 1/6 + ... diverges to infinity. Very cool proof if you haven't seen it.

https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)


"Sir Henry Savile remarked in 1621, there were only two blemishes in Euclid, the theory of parallels and the theory of proportion. It is now known that these are almost the only points in which Euclid is free from blemish. Countless errors are involved in his first eight propositions. That is to say, not only is it doubtful whether his axioms are true, which is a comparatively trivial matter, but it is certain that his propositions do not follow from the axioms which he enunciates."
Bertrand Russell[/quote]

That's why Tarski and others have cleaned up Euclidean geometry in the 20th century.

It becomes very confusing, which is why I was trying to find something basic about space that I could use as "first principles" in a Cartesian fashion — Gregory

Not sure what you're after. To me, Cartesian means two copies of the real numbers at right angles to each other, with the point of intersection labeled (0,0), and all the analytic geometry following from that by labeling points in the plane as pairs (x,y) of real numbers. You can do quite a lot with that, especially if you put in a third axis so you can do 3-D, and even a 4th axis with a funny relationship between it and the other 3 so you can do special relativity. You can go pretty far using nothing more than mutually perpendicular copies of the real number line. That's Cartesian. To me, anyway.

Then what were you talking about? — TheMadFool

You mentioned sobriety checkpoints, and I went off-topic for a moment to express my ambivalence about them in my dual societal role as both a civil libertarian and a driver. Like the scene in Full Metal Jacket where Joker is confronted by a Colonel for wearing a peace sign and having "Born to Kill" written on his helmet, and he responds, "The duality of man. The Jungian thing Sir!" That was another little off-topic excursion.

https://rolandscivilwar.wordpress.com/2017/06/24/full-metal-jacket-pvt-jokers-born-to-killpeace-and-the-jungian-duality-of-man/

And now that you mention it ... perhaps that's what Gabriel's horn is ultimately about. A shape that reflects aspects of both the finite and the infinite. I think Jung would approve.

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". — Metaphysician Undercover

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

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. — Metaphysician Undercover

That's now how limits are defined. What's literally true here is that as $a$ gets arbitrarily large, the volume of the solid between 1 and $a$ gets arbitrarily close to $\pi$. Since that is the case, we say that the limit is $\pi$. That's the definition of a limit.

https://en.wikipedia.org/wiki/Limit_(mathematics)

Then what were you talking about? — TheMadFool

Do you have a cat? Because someone in control of your account typed:

I feel like a drunk driver being asked to conduct a sobriety test on himself. — TheMadFool

Not so? If it wasn't you, perhaps your cat used your keyboard while you were otherwise occupied.

By the way, what's "normal programming"? Do you have one yourself? — TheMadFool

Whatever my vatkeepers have scheduled for me today.

And also, you haven't gotten round to pointing out the error, if there's one, in my argument. Please focus on the issue. — TheMadFool

I scrolled back and did not find the argument you're referring to, can you please repeat it?

Get to the point if you don't mind me saying. Either tell me where I'm wrong or stop wasting your time. :smile: — TheMadFool

We now return you to your normal programming.

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. — Metaphysician Undercover

$\frac{1}{\infty} = 0$ in the extended real number system, which is always the implicit domain of integration problems. The Numberphile guy didn't mention it since it's taken for granted: either glossed over, in the case of freshman calculus; or explicitly formalized, in the case of a more rigorous class in real analysis. Either way it's perfectly rigorous. We could live without it by substituting the phrase, "increases without bound" instead of infinity, and plugging in limits as needed. But that's far more messy and confusing than simply defining the extended real numbers as a notational shorthand.

https://en.wikipedia.org/wiki/Extended_real_number_line

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.

Sorry timwood because right now I feel like a drunk driver being asked to conduct a sobriety test on himself. — TheMadFool

As a civil libertarian I am always conflicted. On the one hand, sobriety checkpoints are unconstitutional as they are a search without probable cause. On the other hand as a driver I'm perfectly happy to get some drunks off the road.

Modern mathematicians seem to have forgotten that Aristotle covered up this problem with a sophistry and that Kant presented this problem in one of his antimonies. If I want to know how many parts an oven has or a loaf of bread baked in it, I simply have to ask how many times I can mentally divide it. And it turns out I can do this infinitely, yet the bread and the oven are finite. Mathematicians now longer see this as a problem or as even strange, and I don't know why — Gregory

If I'm understanding you, you're concerned that the unit interval [0,1] contains infinitely many points. Is that correct? If so, how many points do you think it should have?

Yes, but this is contradicted by infinite divisibility, which all space must have. — Gregory

You are confusing math with physics. And modern physics does not posit infinite divisibility. In fact in physics, space is divisible down to the Planck length, equal to around $1.616255... \times10^{-35}$ meters. Below this distance, our physics breaks down and we cannot sensibly speak of what goes on or how space is. There's a Planck time as well, a minimum time interval below which our physics breaks down and can't be applied.

Presenting the problem in terms of numbers instead of space obscures the issue — Gregory

Ok. So in the unit interval, consider the spacial intervals [0,1/2), [1/2, 3/4), [3/4, 7/8), and so forth, where the square bracket on the left means that the endpoint is included, and the paren on the right means that the endpoint is excluded. Aren't there infinitely many of those spacial intervals making up the 1-unit long segment? What do you make of that? Why does it trouble you?

I am trying to comprehend the first few axoims of all geometry, and i'm not sure the specifics of B/T relate. I only was talking about B/T in terms of taking an infinite of points out of another infinity of points. — Gregory

Oh darn, I sandbagged myself again. People always bring up Banach-Tarski, and I say, "B-T is at heart a simply syntactic phenomenon that I could describe in a page of exposition if anyone was interested," and they invariably have no interest. One of these days someone's going to say, "I'd like to see that" and I'll do it. But I see once again that you name-checked B-T but don't actually have an interest in it. And I got hopeful, only to be disappointed again. I am telling you that the heart of B-T is simple and surprising and perfectly clear, but nobody wants to hear about it. I pointed you at the references but you had no questions. One of these days ...

Sure. If we have two 12 inch rulers, they are equal 1 to 1. However with numbers half of 1 is also a number, so if we apply to this the ruler we have 2 six inches on one side and 2 six inches on the other, hence instead of 1 and 1 being compared, it's 2 and 2. — Gregory

Well yes, instead of 1 whole plus 1 whole we have 2 halves plus 2 halves. Surely it's sophistry to claim that this means anything or that it's some kind of paradox or contradiction. If I drive to the store a mile away, I did indeed cover half a mile twice, or a third of a mile three times, or a quarter mile four times. You don't mean for me to take this commonplace fact as some kind of antinomy, do you?

The reason is that in arithmetic you have to have basic numbers that are understood as not divided. In geometry, all space is divisible and its impossible to find the basic unit. — Gregory

In physics you can take the Planck length as the basic unit. In math, given a line, you can pick any two points, label one point 0 and the other point 1, and that length is your basic unit.

Not precisely. I was good at pre-calculus in high school but in college I only did geometry and that was over ten years ago. — Gregory

Well, if you take the graph of y = 1/x from 1 to infinity, the area under it is infinite. But the area of 1/x^2 from 1 to infinity is finite. That falls directly out of calculus. And as @andrewk noted in the Gabriel's horn thread, it's analogous to the fact that the infinite series 1/2 + 1/3 + 1/4 + 1/5 ... sums to infinity, yet the series 1/4 + 1/9 + 1/16 + 1/25 + ... has a finite sum. Just a mathematical fact that takes a bit of getting used to, but is undeniably true. The paint business is just a way of making it more confusing.

I am coming at this from a more basic fundamental level and perhaps I can't avoid highwe mathematical ideas but I had wanted to find the first few axioms of geometry and am confused why it's become to problematic — Gregory

Well Euclid's axioms are a fine set of basic axioms. And if you drop the parallel postulate and replace it with either zero or many parallels through a point parallel to a given line, you get various flavors of non-Euclidean geometry. In modern times, Tarski's axiomitization of Euclidean geometry is of interest.

https://en.wikipedia.org/wiki/Tarski%27s_axioms

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. — Metaphysician Undercover

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. I could be wrong, I only looked at a couple of seconds of the vid. Either way though, I already explained that he's implicitly using the extended real number system as a shorthand for the cumbersome use of limits.

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. — Metaphysician Undercover

That was already clear two years ago. But I'm surprised to see you still complaining about something I already explained to you -- that we're working in the extended reals -- and that you're rejecting freshman calculus. I must admit you're making progress ... from being confused about 2 + 2 = 4 to being confused about integrating 1/x.

I suggest that the proper representation is that the volume is necessarily indefinite, rather than finite, — Metaphysician Undercover

Nonsense.

and there is no paradox. — Metaphysician Undercover

There is already no paradox, only a veridical paradox.

This means that the amount of paint required to fill the horn cannot be determined. — Metaphysician Undercover

So you don't agree with the determination of the volumes of solids of revolution. Ok. Whatever. An intellectual nihilist. Throw out all the engineering, all the physics, all the physical science simply because you aren't willing to take the time to understand it.

Therefore no act of pouring a determined amount of paint into the horn will fill it — Metaphysician Undercover

Yeah yeah.

ps -- You know, if you said, "Modern calculus does get the right answers and it's useful for physics and engineering; but the mystery of the ultimate nature of infinitesimal quantities, whether mathematical or physical, is not satisfactorily addressed by the formalism," that would be an intelligent criticism.

But to deny the mathematical result of computing the volume of the solid of revolution ... that's just ignorance for its own sake.

Why sausages and not pizza, you fail to ask? — unenlightened

Exactly. You need to put the sausages on the pizza. Only then may satori be achieved by the devoted supplicant.

The vsauce video was where I first encountered B\T. His supertask video also showed me that I was not alone in thinking about "Zenonian cubes". I know that mathematicians look at Banach-Tarski with many equations in mind, but I've always looked at it from the angle of Zeno's paradox alone. So my series of questions has been — Gregory

You should look at it from the paradoxical decomposition of the free group on two generators. There are Wiki pages on the subject. It's simple, it's non-geometric, and it's the heart of the paradox. Regrettably Wiki doesn't have a clear writeup of the phenomenon. I'd like to write it up sometime but I don't know if this site is the right place for expository math of any length. Meanwhile these two pages will have to do.

https://en.wikipedia.org/wiki/Free_group
https://en.wikipedia.org/wiki/Paradoxical_set

1) if space is infinitely divisible than it has infinite parts despite the fact that we experience geometric things as finite — Gregory

Banach-Tarski (B-T from now on) pertains only to mathematical Euclidean space, and most definitely not to the real world. It's a mistake to confuse the two. In particular, B-T involves partitioning 3-spaces into sets that are so jagged and discontinuous that they could not possibly exist in reality. They're non-measurable sets, meaning sets of points that can not possibly be assigned any sensible measure of volume.

2) calculus says that a infinite number can be subsumed by a finite measurement. But in spatial terms how is this possible? — Gregory

I don't know what you mean by subsumed in this context, nor do I relate that to anything in calculus. Can you be more specific so that I can understand your concern?

3) how can something be spatially finite and infinite is what appears to be "the same respect"? — Gregory

Can't understand the question, can you please give an example of what you mean? If something is spatially finite it's finite, not infinite. Do you mean how can a finite length, like the unit interval [0,1], contain infinitely many points? That's pretty simple, if nothing else 1/2, 1/4, 1/8, 1/16, ... are infinitely many points contained within the interval. Is that what you mean?

4) if an object has infinite parts we can take infinite parts out and have a new object, hence Banach Tarski. But isn't this entirely counter intuitive? — Gregory

Yes it's a very counterintuitive theorem. A veridical paradox. Counterintuitive but not actually a logical contradiction, on the contrary a provable theorem. But there's much more too it than just removing points from an infinite set. The theorem says that you can partition the unit ball in 3-space into as few as five parts, and move them around rigidly -- that is, preserving all distances -- and end up with two balls. That's the real puzzler, that the motions are rigid.

For example if we take the counting numbers 1, 2, 3, ... we can partition them into two disjoint subsets, the odds and the evens, and each subset is bijectively equivalent to the original set. That's a puzzler in itself. But in B-T we partition the unit ball into five pieces and rigidly rotate the pieces to put together two balls, each the size of the original. That's definitely counterintuitive.

5) this is all paradoxical to because of the way I think of objects as finite. What is the way forward? — Gregory

The way forward is the Wikipedia outline of the proof. (1) The free group on two letters has a paradoxical decomposition; (2) The group of rigid transformation of Euclidean 3-space contains a copy of the free group on two letters; (3) Apply the paradoxical decomposition of the free group to the ball in 3-space; (4) Fix up a few dangling anomalies. There are a lot of buzzwords in there but the Wiki proof is pretty decent.

I wanted to explore the non-Euclidean stuff with more care because it is also counter intuitive and might give me a clue on how to find the fundamental principle of all geometry and space. — Gregory

That's pretty ambitious but go for it.

But to be honest, earlier you claimed that 1 + 1 might be 4, and you didn't respond when I asked for clarification. May I suggest nailing that down first. Also, if you seek to understand the true nature of geometry, you need to study Riemann and also Klein, who pointed out that geometry is really group theory. And in fact Banach-Tarski is essentially group-theoretic.

https://en.wikipedia.org/wiki/Bernhard_Riemann
https://en.wikipedia.org/wiki/Felix_Klein
https://en.wikipedia.org/wiki/Erlangen_program

In particular, the modern view of geometry is that a geometry is determined by the collection of transformations that preserve its properties. Euclidean space is defined by Euclidean transformations, etc. Again a little buzzwordy but things to look at.

I'm not trying to prove anything to other people, but trying to find an understanding that satisfies myself. Some are ok with Gabriel 's horn. I don't have peace with it — Gregory

Do you follow the calculus in Gabriel's horn? The integral of 1/x from 1 to infinity is infinite, and the integral of 1/x^2 is finite. It's just how it is and the proofs are perfectly straightforward.

I think the real issue is that it's cumbersome to talk about limits when the subject is infinite, because it's contradictory. — Metaphysician Undercover

No it's not. We have a logically rigorous theory of the calculus used in the example

Is the horn closed (limited), or is it infinite (unlimited). — Metaphysician Undercover

It's infinite. But let me ask you this. Are you familiar with the graph of the function $y = \frac{1}{x}$? Isn't it infinite on the right? You can go out as far as you like, right? Did you need me to copy a picture from the web? No matter how far you go, like x = a zillion, there's a point on the graph at y = 1/zillion. Right? It's the cross-section of Gabriel's horn. We can do the integration to see that the area is infinite. And if you don't like calculus, there's a simple visual demonstration that I could provide.

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?

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. — Metaphysician Undercover

You're equivocating the word limit, as in "limited," versus the mathematical theory of limits to infinity. A cheap rhetorical trick. How can you accept confusing yourself like this? Surely you know better. Or you could just ask.

And that's why the appearance of a paradox arises. — Metaphysician Undercover

The apparent paradox arises because 1/x has an infinite integral and 1/x^2 has a finite one.

Are there many female philosophers or is it more of a "good old boys" club? — TiredThinker

Penelope Maddy is the foremost philosopher of set theory. Patrica Churchland is another name I happen to know. Luce Irigaray is a famous feminist philosopher. There are many others, these are just three off the top of my head.

When we do arithmetic, any number can have a half, so 1 plus 1 can really equal 4 in that case,[/quote}

How do you conclude that? It's a bit much for me.
— Gregory

I hope to become bolder and use my ambition to solve the paradox of Banach and Tarski — Gregory

It's much simpler than people imagine. It comes up from time to time on this forum. The proof outline on Wiki is very good.

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

At heart it's only a syntactic paradox that's very easy to explain, thought it takes a little work. It's a long sequence of steps, but each step is very understandable. I say it's syntactic because it's based on the fact that the free group on two letters has a paradoxical decomposition. That is, we can do the paradox by only talking about the collection of words that can be made out of two letters and their inverses, without reference to geometry. But perhaps if you have some questions about the paradox I could respond to them without going into the technicalities.

There's nothing to solve, by the way. It's a theorem, not a paradox. It's a veridical paradox, meaning that it's counterintuitive but not actually a true logical paradox. It's entirely Euclidean, it takes place in standard 3-space.

ps -- This is a very good video on Banach-Tarski.

https://www.youtube.com/watch?v=s86-Z-CbaHA

Does it go up forever into space or does it stop at a limit? — Gregory

Suppose the cuts are horizontal, parallel to the surface on which the block rests. Then we can think of all the infinitely many cuts as already there. Whether you see the block as having height 1, or having height 1/2 + 1/4 + 1/8 + 1/16 + ... = 1, it's exactly the same block either way. If you see a block in front of you, you can imagine it having the cuts already there.

There is no "next to last" or "last" cut, for the same reason there's no last element to the infinite series.

ps Wiki has a 2D picture. Not with strictly horizontal cuts but the idea is the same. The area (or volume in the 3D case) doesn't change just because we make a lot of cuts, any more than carving a turkey changes the amount of turkey.

https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF

Clearly you are using a different calculation than the one in the video then. — Metaphysician Undercover

Try Wiki.

https://en.wikipedia.org/wiki/Gabriel%27s_Horn#Mathematical_definition

I can't speak for Numberphile, which I generally don't watch because the guy annoys me. Perhaps they were trying to simplify the fact that $\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$ and you were confused by their simplification. In which case it's on them and not on you. But now that I've explained it to you, it's up to you. Truly if they said that on Numberphile they shouldn't have exactly for this reason. It confuses people.

When we see the expression 1/infinity it's a shorthand for the limit and people generally know that. But if the video is intended (as it apparently is) for people who don't know that, they shouldn't have done it.

ps -- Ok I watched that section of the original video. That's a shorthand they do in integration problems, formally you'd replace it with a limit. I guess now I'd be in a position of trying to defend calculus pedagogy, that's hopeless. In general 1/infinity isn't defined but when it comes up in problems like this you can take it to be zero. I can't argue with you that there's a bit of flimflam to the whole enterprise. I can only say that the procedure can be made rigorous, but at the level of calculus problems, it rarely is. I don't expect that to be a satisfying answer, it's not to me either.

pps -- For integration problems we can consider ourselves working in the extended real numbers. These are the standard reals with special symbols $\pm \infty$.

One of the rules for these symbols is that if $a$ is a positive real number, the symbol $\frac{a}{\infty} = 0$.

That's actually the formally correct answer to your question and I probably should have thought of it earlier. The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity.

One divided by infinity is not zero, it is indefinite. — Metaphysician Undercover

Absolutely correct. Also absolutely irrelevant, since nothing in this problem involves dividing one by infinity. I'm afraid this looks like yet another case of your misunderstanding the math, making up a story about what the math is saying, then arguing with your own story. A classic strawman argument.

But you know, you did say something mathematically correct, that one divided by infinity is not defined. I give you credit for that.

If you assume that one divided by infinity equals zero, — Metaphysician Undercover

Which it doesn't. So if we assume a falsehood we can prove anything by Ex falso quodlibet". If 2 + 2 = 5 then I am the Pope.

you assume that the value for y reaches zero, — Metaphysician Undercover

and I am the Pope, since your antecedent is false. Go in peace and sin no more. You can cos as much as you like.

therefore closure. — Metaphysician Undercover

Likewise.

The cone or horn is NEVER closed. Pick any point on the real line and the cone is still open at the bottom. There is no "point at infinity" in this problem nor is there one in the real numbers. Again you misunderstand the math, make up a story, then argue against your own story. Your strawman has by now had all the straw beaten right out of it.

It's very clearly stated in the YouTube video, he says we're taking the value of y to zero. — Metaphysician Undercover

Ok to be fair, as I've mentioned, I've seen Gabriel's horn before but didn't watch the video, so I don't know if perhaps he said something misleading. But it's true that as x gets arbitrarily large, y gets arbitrarily close to zero. That's what's meant by "going to zero." It's a technical phrase meaning that as x increases without bound, y gets as close as you like to zero. But x never "becomes infinite" nor does y ever become zero. The mathematical phrasing is a clever and subtle way of talking about these things WITHOUT saying that x becomes infinite or that y becomes zero. It's your mathematical ignorance of this terminology that's leading you into error. And since you repeatedly do this, when Wikipedia and other online sources could easily explain these things to you, I must assume at some point you choose not to learn the math, but rather to flail at strawmen of your own creation. I don't mean to sound uncharitable but if you have a better explanation I'm open to hearing it.

The tl;dr here is that as x increases without bound, but remains at all times finite; y gets as close as you like -- "arbitrarily close" as they say -- to zero. x does NOT "become infinite" nor does y ever become zero. That's the math. It's well-known. They teach it to college freshman. These days they even teach it to high school students. Not that it's taught well at that level or that anyone understands it. I'll stipulate that nobody comes out of calculus class with a clear understanding of the logical fine points. But it's all on Wiki.

However, this clearly contradicts the premise that the horn continues infinitely. — Metaphysician Undercover

Since your beliefs are false, you haven't got a contradiction; only confusion caused by your lack of mathematical knowledge. But now that I've explained it to you, you can no longer claim to be ignorant.

The real issue is that integrals are approximations, — Metaphysician Undercover

Integrals are exact. The Riemann sums that define them are approximations, but the integrals are the limits of the Riemann sums and they are perfectly exact. The volume of the horn is exactly pi and the surface area is "infinite," which has a technical meaning: It's greater than any real number you could name. It's not the metaphysical infinite. I think that's a point of confusion in these conversations.

Integrals are perfectly exact, that's the takeaway.

and infinity has no place in an approximation. — Metaphysician Undercover

Well they're not approximations. And secondly, "infinity" is not a magic thing that you should allow to cloud your mind. Infinity in this context is nothing more than a shorthand for "increases without bound." We say "x goes to infinity" meaning that x increases without bound, but is at every point finite. Mathematical infinity is not the metaphysical infinity. Perhaps that needs to be said more often. It's more of a technical gadget or shorthand for a quantity that increases without bound, or is larger than any finite number.

So that method of integration is simply not applicable to an infinitely long cylinder. — Metaphysician Undercover

Of course it is. It's freshman calculus. It's been working well since the days of Archimedes, formalized into a systematic scientific tool in the late 17th century, and put on a logically rigorous footing since the late 19th century.

But, as clearly evidenced by the many needless care-home resident Covid-19 deaths, big business does not always know or practice what's best for its consumers. — FrankGSterleJr

It was politicians, not corporations that caused the nursing home deaths. There's a major scandal about Cuomo right now. He's the governor of NY state, not a corporate CEO. He ordered covid patients into nursing homes, where they died and infected others. Then he lied and covered up the numbers of deaths.

If something is moving at infinite speed - the speed of light — Paul S

Just a quibble. The speed of light is finite. That's the entire point. It's infinite in Newtonian physics. In special relativity the speed of light is finite. That's what makes relativity strange.

For time 0 to 1, the particle starts at position (0, 2), — andrewk

You meant (0, -2) here.

traverses the infinite lengths of the two hyperbolic arms — andrewk

"I just flew in from infinity and boy are my hyperbolic arms tired!"

Nice example. Finite time, infinite length traversed.

We can have thoughts of objects we have seen. Are those thoughts emergent properties of the brain? — Don Wade

Other way 'round. ALL we have is the thoughts of those objects. Who says there need to be objects at all? Maybe it's the objects that are emergent properties of the thoughts. Or perhaps there aren't any objects at all, there are ONLY the thoughts. That's Bishop Berkeley.

https://en.wikipedia.org/wiki/Subjective_idealism

After all, could there be better method to disguise the true intention of of any system then to package it in mathematical mumbo-jumbo and then tell people, "You must follow the science." — synthesis

Not for nothin' do they call it techofascism. Coming soon to a bankrupt empire near you.

Absolutely, but everyone else's as well, including those of those who are coming after you. — tim wood

How do I get to one of these better worlds?

↪fishfry By "any amount" I do not mean zero, of course - I thought that would be obvious. — SophistiCat

In the example I gave, the speed is unbounded over "any amount of time" for any interval containing zero as the intervals get small. I see your point but at the time it wasn't obvious. And when someone says "any amount" I take them at their word, zero being an amount. I'm not being cute, that's exactly the case that popped into my head. My pedantic turn of mind. If you'd said, "any nonzero amount of time" I wouldn't have needed to post. Perhaps I should have realized that's what you meant. In some other universe I did. Does the Many Worlds theory absolve us of our sins?

It is not possible for anything with a starting point to move infinitely fast per finite amount of time. — elucid

0 is finite. Did you mean to exclude that?

What would you say is the velocity of my function f at the point 0? Well in calculus we would say it doesn't have a velocity because the function's not defined at 0 and it's not differentiable there. But if you take any tiny interval including zero, the particle moves 1 unit up in arbitrarily small intervals of time. So you can make its velocity as large as you want, without bound. That's what infinite means. So I would say it has infinite velocity at zero by that definition. But I admit it could be argued otherwise.

Not if you are moving infinitely fast. If your speed is infinite over any amount of time, then you will have moved an infinite distance. — SophistiCat

I already explained that this is false. Did you read my post? Would you be good enough to do so now? You could move a billionth of an inch in zero time and that would be infinite speed but you've only moved a billionth of an inch.

But in cases like .999... and the volume of the horn, the infinite is rounded off and given closure. — Metaphysician Undercover

Bullpucky. There is no "closure at the bottom of the horn." You just make stuff up and claim mathematicians said it when they didn't. That's called a strawman argument.

like the closure at the end of the infinitely long horn — Metaphysician Undercover

Bullpucky bullpucky bullpucky. YOU said that, not any mathematician, ever. As always you take your own mathematical ignorance and project it onto mathematics itself. You wield your ignorance like a weapon.

ps -- Let me not be so ill mannered. Perhaps you could explain to me your version of calculus in which the area of 1/x from 1 to infinity isn't infinite, and the area under 1/x^2 from 1 to infinity isn't finite. If you could elucidate your version of calculus then I'd be enlightened. I seek your wisdom. Or perhaps you reject calculus entirely. I'd like to hear your perspective on this.

Do you happen to understand that there is no "closure at the bottom" of the cone? That this is NOT anything that any mathematician says or thinks? That is is entirely something you made up and then claim to mock? A strawman, as they say. Do you apprehend this point?

Because no matter where it's at after travelling, it has not travelled infinite spaces. — elucid

Not a good argument. You need not travel an infinite distance to reach infinite speed. Consider any jump discontinuity. For example suppose a particle is moving in the plane according to the rule

f(x) = 0 if x <= 1
f(x) = 1 if x > 1.

In other words it's going left to right along the x-axis, and at x = 0 it jumps up to a height of 1. It's moved a distance of 1 unit up in arbitrarily small time, in other words it has infinite speed. But it only need to hop up one unit. In fact the same example would work if it only hopped up 1/zillionth of a unit. As long as the movement is discontinuous, the "speed" is effectively infinite if you think about it that way.

Note that I'm not commenting on your first assertion, that infinite speed isn't possible. I'm only pointing out that your argument doesn't support that conclusion. You need a better argument.

As far as whether such a thing could happen in the physical world, see quantum tunneling. I googled around to see if quantum tunnelling is regarded as going at infinite speed or exceeding the speed of light, and it seems there's some discussion of this but I didn't have the patience to drill down to a conclusion. Regardless, discontinuous motion does seem to be possible in the physical world and I don't see why this wouldn't count as infinite speed, if only for a moment. I could be wrong about the physics though.

The video, and others, tells me the volume in the horn is finite, is in fact π in appropriate units. — tim wood

Yes. It's a finite volume with infinite surface area. It's a veridical paradox: "A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless." In other words it's a true fact (provable in freshman calculus) that's so counterintuitive it seems impossible. But there is no actual paradox. There's no statement that's both true and false. Math is filled with these kinds of things.

https://en.wikipedia.org/wiki/Paradox

That it is fillable, with π "mathematical paint" - whatever that means. — tim wood

Whatever that means. It's a shape with finite volume and infinite surface area. That's all you can say. Yes it's "fillable with paint" if you want to view it that way. But if you push too hard on that visualization it gets confusing.

I take it to mean analogously the same as when a paint can is filled. If a paint can is filled with paint, then it seems fair to say that the inner surface of the can is painted. — tim wood

I agree with that, about actual paint cans. But I don't agree with that about Gabriel's horn, because we're not filling anything with paint. And there is no inner surface. I went at this in detail in my previous post. You understand that for example a circle in the plane has no "inner surface." The horn is a two-dimensional surface in 3-space. It's not like a paint can. That's another source of bad intuition. A paint can has thickness, with an inner and outer surface. But the horn has no thickness and no inner or outer surfaces.

Analogously if the horn can be filled, then whatever it has that passes for a surface is "mathematically" painted. Any problem with this so far? — tim wood

Well, to the extent that I accept that, and I sort of do, it's a veridical paradox. We have a finite volume that has infinite surface area, and it's a seeming common-sense paradox without being an actual logical contradiction.

The proposition of the paradox, as I get it from the video, is that the amount of "paint" is not enough to cover the outside of the horn because the area to be covered is infinite. — tim wood

I didn't actually watch this particular video, I've seen this example in the past. But I disagree with that statement because we could paint the outside of the horn by using pi/2 gallons of paint on the segment between 1 and 2; and pi/4 gallons between 2 and 3; and pi/8 gallons between 3 and 4; and so forth. We'd use pi gallons of paint to cover the entire infinite surface area. Of course we can't paint it uniformly, unless we imagine paint of zero thickness, but then we can't sensibly have any meaningful volume of paint at all.

But the question retains its edge: — tim wood

I agree with that. I saw this example years ago and it hasn't lost its force. It's a real puzzler. The best you can say is that it's a veridical paradox. There is no actual logical contradiction; only a violation of common sense. But the math is clear. The horn has finite volume and infinite surface area.

if the inside is covered by the "paint" inside, then why cannot the same volume of paint cover the outside? Is the area of the "outside" somehow different from the area "inside"? — tim wood

I don't know the answer. It's a veridical paradox. How can an infinite surface area enclose a finite volume, or a finite volume have an infinite surface area? I don't know. It just does. I don't suppose that's satisfactory. That's why Numberphile got everyone talking about it. For what it's worth, Wiki has my solution and nothing better:

Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its inner surface. The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it simply needs to get thinner at a fast enough rate (much like the series 1/2^N gets smaller fast enough that its sum is finite). In the case where the horn is filled with paint, this thinning is accomplished by the increasing reduction in diameter of the throat of the horn.

https://en.wikipedia.org/wiki/Gabriel%27s_Horn

If you didn't find my explanation compelling you won't be satisfied by Wiki's identical explanation (except by virtue of Wiki's authority). And as I say, nobody on Wikipedia could come up with a better explanation. It's just one of those things in math that, as von Neumann said, you don't understand; you just get used to it.

ps -- Here's a tl;dr:

* It's just one of those things. It's counterintuitive but not a true paradox.

* The horn is two-dimensional whereas a paint can has thickness. That's throwing off our intuition. Paint cans have an inside and outside surface. The horn doesn't.

Are we at the point yet where we - as a Nation - could be openly governed by one, or more, computers? — Don Wade

What does that even mean?

Why? Why cannot it be a consistent thickness, mathematically thin? — tim wood

Because if the thickness is constant its volume must be infinite. Think of the infinitely long real line and a rectangle above it with height 1/10. Width x height is infinite. Say the height is 1/million. Same thing. 1/zillion. Same thing. As long as the height is nonzero and uniform, the total area across an infinite surface must be infinite.

If I wish to paint my living room, it does not occur to me that I must fill the living room with paint or that the paint must vary in thickness. — tim wood

Nothing to do with the previous sentence. And the whole idea of painting your living room is a red herring, since this is a mathematical and not a physical situation. It's like people complaining about Hilbert's hotel that there can't be any such hotel, or how would the maid make up all those beds, and so forth. The attempt to put the puzzle in "practical" terms ends up confusing the issue. But the point still stands that if the surface to be painted is infinite and the paint layer is uniform and of positive thickness, no matter how small, the volume of paint must be infinite.

I'll attempt a more rigorous description of the paint and painting. And for that purpose I'll borrow from the argument that the cardinality of the points on the number line between zero and one is the same as the cardinality of the points in a cube measuring one mile on a side. — tim wood

Since that's true, that's a hint that a cardinality argument will be of no use here.

Let's consider the cardinality of the points that make up the inner and outer surfaces of the horn respectively. — tim wood

There's a big problem, which is that there is no such thing as the "inner or outer surface" of a line. For example, what is the "upper surface" of the real line? There clearly isn't one because if the surface is any positive height above or below the line, it's not the "surface," and if it's directly on the line, it's the line.

Same argument with a circle. There is no inner and outer surface of a circle. Do you follow this point? It's crucial, since the "inner and outer surface" of a two-dimensional surface in 3-space is being bandied about in this thread and there is no such thing.

What are the inner and outer surfaces of a sphere? Can you name a point on the inner surface? No. Because if the point is zero distance from the sphere, it's on the sphere. And if it's a positive distance inside the sphere, it's not the "inner surface" because there are many points strictly between that point and the sphere.

In fact the "surface of a sphere" can only be the sphere itself. Like FDR said! The only surface of the sphere is sphere itself. Ok nevermind.

It seems to me it must be the same for both. — tim wood

Bearing in mind that there is no such thing as the inner or outer surfaces of the horn, which is a tw- dimensional object living in 3-space, I'll play along. But remember the premise is false already.

Now, if we may, the cardinality of the points that make up the paint itself. By painting is meant an assignment of one point of paint to each point of surface. — tim wood

Ok. But think of a circle in the plane. Can you paint it's "inner" or "outer" surfaces? No, because there's no such thing. If you paint the interior of a circle you either include the points on the circle or you don't. If you do, you've painted the circle itself. But if you don't, then there's always a space between your paint and the circle into which you could have put more paint; that is, gotten a little closer to the circile.

I hope you see this, if not please say so, because this point is essential. There are no inner or outer surfaces to a 1-D object living in 2-D, or a 2-D object living in 3-D.

It seems to me the cardinality of the paint must be greater than or equal to that of one of the surfaces. — tim wood

Why? The cardinality of any circle concentric to a given circle is the same. Ancient proof, just draw rays from the center, there's your bijection.

And greater because for any cross section of the horn the inner surfaces never meet, and consequently there is always more paint in the cross section when the horn is filled than is needed to just paint the surface. — tim wood

This doesn't make sense (because there are no inner and outer surfaces) and doesn't follow logically even if I granted you that there are such things. It's your own cardinality argument. Any continguous chunk of 3-space has the same cardinality as the unit interval on the line.

Thus on the assumption that the horn is filled with a finite amount of paint, understood to be proved, then the inner surface has been painted. Because the inner surface thus paintable, and the outer surface the same area, the outer surface must be paintable. — tim wood

There are no inner or outer surfaces. But you haven't got an argument here even if there were. After all the interval []0,1] has the same cardinality as [0,2] which is twice as long. You already made that point earlier. I don't see that you've made an argument.

Is there an error? — tim wood

At least half a dozen so far. But more to the point you haven't made an argument, only claims that you yourself refuted at the beginning by noting that every contiguous subset of 3-space has the same cardinality as the real line or as a finite segment of the real line.

Where is it? — tim wood

Which of the many errors I pointed out do you disagree with? Starting from the false concept of inner and outer surfaces of 2-D shapes living in 3-space? Just think a sphere in 3-space or (easier) a circle in the plane. Name a point on the "inner" or "outer" surface? You can't. If you pick a point on the circle that's on the circle. But if the point is any positive distance whatsoever off the circle, it's not the inner or outer surface because there are points strictly between your point and the circle.

If it depends on an if, then the if is that the horn can be filled in the first place and the rest "flows" from that, so it seems. — tim wood

This I didn't understand. But imagine filling the inside of a circle in the plane. If you tell me you're including the circle in your painted region, then the points on the circle are painted. But if not, then there's some positive distance between painted region and the circle, so that's not the "inner or outer" surface after all.

In the end perhaps casting this problem in terms of paint causes more problems than it solves. But just saying, "1/x is square-integrable but not integrable" (a point made earlier by @andrewk) doesn't have quite the same ring to it.

the series has to be either eternal or it came out of nothing. — Gregory

If you run the known motion of the matter in the universe backward, you get the big bang. That was the initial oomph. And what caused the big bang, and what came before it, and do we know it's even true? Nobody knows but everyone has an opinion. Lawrence Krauss says that In the Beginning was the quantum soup and the laws of physics. Genesis says that In the Beginning God created the heavens and the earth. Can anyone clearly distinguish science from theology here? Of course the simulationists say that In the Beginning was the great computer in the sky, and we're all programs. I'm Microsoft Word, and you're Tetris. Science? Or theology? The Many Worlds folks insist that while in this universe I wrote this paragraph, in some other universe I thought better of it and didn't. Science? Or theology? And why is it exactly that so much of our science lately is indistinguishable from theology?

It would seem, then, that if we want to paint the interior surface, we need only pour in π amount of paint — tim wood

Correct, but the paint can't be uniform since it must be spread over an infinite surface area. The thickness of the paint has to decrease as some convergent infinite series: 1/2 gallon for the area between x = 1 and x = 2, 1/2 gallon for the next chunk, and so forth. So nothing you said is in conflict with the story, which is simply that 1/x has an infinite area but its solid of revolution has finite volume.