## 2042: "Rolle's Theorem"

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Flumble
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### Re: 2042: "Rolle's Theorem"

Soupspoon wrote:Makes me wonder about the problem I considered when I was maybe five or six and playing Crazy Golf... Given an initial impetus of a ball rolling over an (arbitrarily profiled, even multi-humped but smooth) hill, with zero rolling/air resistance or other dampening forms of friction, up to a certain initial velocity there's a point up the rise (or maybe one of several rises) subsequent to which the ball will reach. This will never be a/the local maxima (because if it rolls onto it, it will continue to roll over it) or horizontal inflection. It will also never be a downwards-inclined slope beyond a maxima, or indeed any subsequent slope not higher than the prior maximal point. (There's also a velocity beyond which the ball departs the undulating surface, which we either decide is an upper limit of its own or else treat the last concurrently rolled-over point as the solution to "where the ball gets to".)

However simple and pure the hill curve is, the derivative y'=g(x') - distance for attained for each velocity - derived from the y=f(x) - the height of the slope at any given distance (horizontal or slope-hugging, to taste) is discontinuous and not even like a tangent's vertical asymptote to infinity but a distinct up-to-but-not-including limit.

I'm unconvinced there's no nice hill where the ball ends up (in a lim t→∞) stationary. And I mean rolling towards a point where dy/dx=0, so y=-e^-x is disqualified.
If you've already disproven its existence with the second paragraph, can you please elaborate? My geometry/calculus understanding is too rudimentary to even imagine how to properly map time to ball position on a constant slope. (that is, by a derivation that works for general curves, instead of just using y=v₀t-½at²) And if not, can you prove/disprove its existence?

Oh right, it's pretty trivial that a ball with exactly enough speed to reach the top of any continuous hill (at height h) will take forever to get there. It can't move horizontally at y=h (any other direction is fine, however the hill is perfectly flat at h, so it can't be at the top) and it can never stop moving because y<h except at the top means it has kinetic energy (and it can't turn around because that requires a non-zero slope at h).

I'm still very much interested in how you construct the motion equations or parametrization for the ball.
Last edited by Flumble on Fri Sep 07, 2018 7:39 pm UTC, edited 2 times in total.

Sableagle
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### Re: 2042: "Rolle's Theorem"

rmsgrey wrote:
Heimhenge wrote:
Soupspoon wrote:
... And I'm still not quite sure what the proper mathematical description for such a discontinuity would be. Except that there'd be various output (distance reached) values that are undefined for given inputs (initial speed), and likewise certain initial speeds that remain undefined as the solution to the inverse solution applied to given distances reached. It doesn't even rightfully go to imaginary values (maybe they appear in the integration/differentiation, but given discontinuousnesses there are already other issues needing to be resolved).

That a great puzzle. Just had to say that, if I understand your posed undulating hill scenario, I suspect for whatever output you observe, there's probably more than one input that would map to it. If you just provide the ball the negative of its exit velocity at the exit point, it would trace back only a single trajectory, sure. But my intuition says you could get that same output with other inputs simply because of the number of degrees of freedom. I just don't have the balls to prove it.

In the absence of friction, a given initial kinetic energy converts to a given maximum potential energy, and a given position has a given potential energy, so a different starting speed must produce a different highest point. Different initial velocities with the same speed may or may not come to the same point, but will come to the same contour (possibly a disconnected component of the same one).
Given a 3D course with hill curvature gentle relative to ball velocity, yes, but if ball speed is too high for the hill curvature and the ball departs from the hill surface, you get some differing results.

In the last example, the windmill can probably be considered a non-obstacle.
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### Re: 2042: "Rolle's Theorem"

This reminds me of some experiments we did in GCSE science. The idea was to explore the equation of motion (s=ut+1/2at2, type of thing) for various conditions, by rolling a ball down a slope, and the conditions were (a) constant speed and (b) constant acceleration. I was flummoxed as to how you could achieve constant speed, but in the next lesson, the teacher just said you needed to adjust the gradient of the slope so that the friction cancelled out the gravitational acceleration. What really bugged me was how we would know we had just the right slope to get a constant speed. Of course you could show that the ball covers equal distance in equal times, but the whole experiment was presented as an exercise in deriving experimentally the laws of motion, whereas any plea to the laws of motion to prove that you had the right conditions would make the entire exercise a circular argument.
xtifr wrote:... and orthogon merely sounds undecided.

Kit.
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### Re: 2042: "Rolle's Theorem"

Soupspoon wrote:The only Real World application I can currently identify as "matrices are 'useful' in this context" is 3D graphical transforms/rendering,

Matrix multiplication is essential for everything that involves derivatives of a vector function. That includes a huge amount of optimization problems, including most of modern machine learning.

DanD
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### Re: 2042: "Rolle's Theorem"

erejnion wrote:So Randall just has to pay several people to start citing something as "Munroe's theorem". Maybe high-profile professors, so that there's a higher chance for adoption.

He already has an effect. https://en.wikipedia.org/wiki/Shaped_charge#Munroe

cellocgw
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### Re: 2042: "Rolle's Theorem"

In response to all the posts about the ball rolling to the top of a hill (Local maximum), here's why it can't happen.

Gedankenerfahrung: [with zero friction, of course]
Since all classical mechanics are fully reversible, consider a ball placed at the top of the hill with zero velocity. It doesn't move. The only way to get it to fall is to push it , i.e. add kinetic energy to an unstable equilibrium point. Thus not reversible, thus the ball can't roll to a stop.

Yes, a ball moving up will come to a zero - Y-velocity, but there is nothing that impedes its X-velocity. This is why a launched ball (in vacuum, yada yada) follows a parabola. The ball rolling up the hill will run out of y-velocity, but not x-velocity, at the peak. Sysiphus ensues.
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### Re: 2042: "Rolle's Theorem"

On the other hand, with perfect conditions and setup, you can get the ball to stop and turn around arbitrarily close to the top of the hill. In the smooth hill most people would imagine, close to the top it is very flat. By getting arbitrarily close to the flat top before stopping, you can have the ball creep slowly enough near the peak that it appears to be stopped at the top of the hill. At least long enough that there will be no witnesses to it rolling back down.

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### Re: 2042: "Rolle's Theorem"

Sableagle wrote:
rmsgrey wrote:
Heimhenge wrote:
Soupspoon wrote:
... And I'm still not quite sure what the proper mathematical description for such a discontinuity would be. Except that there'd be various output (distance reached) values that are undefined for given inputs (initial speed), and likewise certain initial speeds that remain undefined as the solution to the inverse solution applied to given distances reached. It doesn't even rightfully go to imaginary values (maybe they appear in the integration/differentiation, but given discontinuousnesses there are already other issues needing to be resolved).

That a great puzzle. Just had to say that, if I understand your posed undulating hill scenario, I suspect for whatever output you observe, there's probably more than one input that would map to it. If you just provide the ball the negative of its exit velocity at the exit point, it would trace back only a single trajectory, sure. But my intuition says you could get that same output with other inputs simply because of the number of degrees of freedom. I just don't have the balls to prove it.

In the absence of friction, a given initial kinetic energy converts to a given maximum potential energy, and a given position has a given potential energy, so a different starting speed must produce a different highest point. Different initial velocities with the same speed may or may not come to the same point, but will come to the same contour (possibly a disconnected component of the same one).
Given a 3D course with hill curvature gentle relative to ball velocity, yes, but if ball speed is too high for the hill curvature and the ball departs from the hill surface, you get some differing results.

CrazyGolf.png

In the last example, the windmill can probably be considered a non-obstacle.

There's still a maximum height the ball can reach for a given starting speed - for a high enough speed, that height is somewhere above the entire course.

It is possible for the ball to never become stationary, but if it does, it will be on a contour defined by its initial speed. And, yes, this is slightly different than what I said previously.

cellocgw wrote:In response to all the posts about the ball rolling to the top of a hill (Local maximum), here's why it can't happen.

Gedankenerfahrung: [with zero friction, of course]
Since all classical mechanics are fully reversible, consider a ball placed at the top of the hill with zero velocity. It doesn't move. The only way to get it to fall is to push it , i.e. add kinetic energy to an unstable equilibrium point. Thus not reversible, thus the ball can't roll to a stop.

Yes, a ball moving up will come to a zero - Y-velocity, but there is nothing that impedes its X-velocity. This is why a launched ball (in vacuum, yada yada) follows a parabola. The ball rolling up the hill will run out of y-velocity, but not x-velocity, at the peak. Sysiphus ensues.

The ball will never be stationary at a local maximum after finite time, but you have two cases - one where the ball doesn't quite make it to the peak, and rolls back; and one where the ball only just crosses the peak and rolls down the other side. As the latter case starts with lower and lower speeds, the time taken to reach the peak becomes greater and greater, until it becomes infinite; in the former case, as the starting speed increases, the highest point reached increases until it reaches the peak. There's a boundary case where the ball takes an infinite time to reach the peak, and never falls back. That corresponds to the reverse case where the ball is stationary on top of the hill, and takes an infinite time to achieve a finite speed.

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### Re: 2042: "Rolle's Theorem"

@maniexx wrote:Interesting to see that Rolle didn't even belive in differential calculus when he proved this.
https://en.wikipedia.org/wiki/Rolle%27s_theorem#History

Interesting. I assumed that it was a corollary to the Mean Value Theorem. Note: I only remember the Mean Value Theorem as it was an equally blindingly obvious statement that required an entire lecture for the professor to prove (and was also necessary to prove the fundamental theorem of calculus or similar critical proof). And I suspect that there were a few shortcuts in the "one lecture" proof to make it fit.

cellocgw wrote:
Showsni wrote:
rmsgrey wrote:I like to tell people that when I studied maths at a prestigious university, it took 4 weeks to get to the proof that 2+2=4. It's not that the proof itself is terribly difficult, but that it starts from ZF axioms, and the conceptual groundwork had to be laid for those first.

We didn't get that far until year four of the course.

How could you get to year four without proving that? Did you start by proving 1+2 = 3 in your third year?

In Whitehead and Russel's Principia Mathematica, it takes at least 360* pages of dense mathematical foundation before addition is defined. I wouldn't be at all surprised if an even more modern attempt to build up an axiomatic mathematics takes even longer.

Velo Steve wrote:I'll pay \$100 to have that theorem named after me. And \$5000 if you don't tell anyone that I got it by paying for it.

L'Hôpital's rule is the most well known case of this happening. Presumably L'Hôpital didn't pay Johann Bernoulli enough money to keep quiet.

* the infallible wiki gives these numbers and also mentions that addition is "proved" on page 86 which a note that "this is occasionally useful". I suspect I don't want to see the edit war around these comments.

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### Re: 2042: "Rolle's Theorem"

rmsgrey wrote:There's a boundary case where the ball takes an infinite time to reach the peak, and never falls back. That corresponds to the reverse case where the ball is stationary on top of the hill, and takes an infinite time to achieve a finite speed.

Indeed, like me thinking that it would be 'elegant' that the amount of energy applied in the Big Bang be tuned exactly to the mass in it¹ that it would hover exactly between the devil and the deep blue sea.
Soupspoon wrote:(And yet I was also quite convinced that the Universe was tuned to be expanding at the rate exactly between the lower speeds that would lead to a Big Crunch and the higher speeds that led to an eternal expansion, despite the obvious parallels with my crazy-golf ball problem.)

(And the inverse being a practically Steady State non-colappsing universe because everything is just too mind-bogglingly spread out so that there's no noted middle to collapse to, everywhere is maybe locally collapsing but everywhere non-local is disinclined to collapse towards the locality because of the amount of beyond-non-local stuff pulling it away. Obv. needs an infinite brane, or something, but weirder things have been (and are being) imagined.)

But as a theoretical outcome, it makes me wonder what order of infinity this unique solution has us waiting for. Less than the forever that it would be forever expanding into, given any additional expansion?

¹ After mass/energy equivalencies eventually settle down, I suppose. This was a childhood thought, before I knew more about E=mc², and much longer before irregular distribution/chaos/quantum foam ever factored into my worldcosmos view. Oh, and the whole thing about it not exploding into anything, etc.

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### Re: 2042: "Rolle's Theorem"

Heh, nothing like crazed, convoluted proofs that take a class period or more for concepts that look blazingly obvious to the student.

I remember a doopy one from Linear Algebra, wherein the professor spent the whole class period proving on the board that if you had n orthogonal vectors in an n-dimensional space, you had a basis for that space and no more orthogonal vectors could be drawn. I'm sure it's useful for proving other things with, but at the time I was thinking "Yes, duh, that's how dimensionality works. 'Got n orthogonal vectors in an n-space? I'll bet you a very large sum you have a basis for it, which yes, can be linearly combined to make any vector pointing to any point in space, because that's what a basis is for."
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### Re: 2042: "Rolle's Theorem"

So how many centuries does one have to spend meditating in a cave on a mountaintop before one can prove that 1 = 1, or have we still not accomplished that?
Oh, Willie McBride, it was all done in vain.

rmsgrey
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### Re: 2042: "Rolle's Theorem"

Sableagle wrote:So how many centuries does one have to spend meditating in a cave on a mountaintop before one can prove that 1 = 1, or have we still not accomplished that?

That one's pure definition - a thing always equals itself. You could get into the concept of equivalence relations if you want (an equivalence relation is a relation such that: a thing always relates to itself; if x relates to y, then y relates to x; and if x relates to y and y relates to z, then x relates to z) but all you need to know is that, by definition, x = x for any x.

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### Re: 2042: "Rolle's Theorem"

rmsgrey wrote:
Sableagle wrote:So how many centuries does one have to spend meditating in a cave on a mountaintop before one can prove that 1 = 1, or have we still not accomplished that?

That one's pure definition - a thing always equals itself. You could get into the concept of equivalence relations if you want (an equivalence relation is a relation such that: a thing always relates to itself; if x relates to y, then y relates to x; and if x relates to y and y relates to z, then x relates to z) but all you need to know is that, by definition, x = x for any x.

That's just as well for the kind of proof that my A-level maths teacher derided, where the last step goes
"1=1 ∴ true"
xtifr wrote:... and orthogon merely sounds undecided.

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### Re: 2042: "Rolle's Theorem"

wumpus wrote:In Whitehead and Russel's Principia Mathematica, it takes at least 360* pages of dense mathematical foundation before addition is defined. I wouldn't be at all surprised if an even more modern attempt to build up an axiomatic mathematics takes even longer.

Modern attempts to build up mathematics (say, homotopy type theory or some other new age woo) tend not to have arithmetic as their aim, as it's known to be a consequence of other things. Of course, everyone knows about incompleteness, and so nobody really sets out to provide a 'rigorous' foundation of arithmetic anymore.

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### Re: 2042: "Rolle's Theorem"

bobleboffon3 wrote:Am I a clueless art person if I look at the "Orange, Red, Yellow" painting and genuinely believe my kid could make that?

Could s/he?

Like, maybe if someone told hir "put this particular color here" s/he could reproduce it as a technician if someone had already figured out what to do, although we would still lack a language to specify in clear, unambiguous terms the specific manner of brush stroke, paint mixing, etc. Rothko used.

But coming up with the particular combination and arrangement of colors to produce the specific visual effect requires an intimate, artistic, almost intuitive understanding of how those colors interact. Not to mention the recommendations Rothko had for viewing it, etc. And then there's the choice of medium (not just paint on canvas, but what kind of paint? how is it to be mixed?) that itself requires similar intimate knowledge, how to use the brush in the exact way, etc.

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### Re: 2042: "Rolle's Theorem"

Are you sure of that?

Could Rothko have been spouting utter bollocks?

I doubt it would be hard to find a hundred webcomic artists who could do convincing parody explanations for a "work of art" that was actually just a canvas square near which I had shot several full pots of various paints with Matt Carriker's various rifles.

If "the art world" accepted one of their paragraphs of tripe and my bespattered old sail as a masterpiece of world-shaking enormity, would that mean it was an incredible work of art that, while you might say your six-year-old could have made, couldn't actually be reproduced without an amazing amonut of analysis to figure out the positions of the paint cans, the impact points on them, the calibres used, the sequence of the shots and so on and so forth, and even if that could be reenacted your six-year-old couldn't produce the recommendations for viewing it that another webcomic artist produced, because your six-year-old wouldn't understand the profound spiritual and emotional impact of viewing such a work in a particular way?
Oh, Willie McBride, it was all done in vain.

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### Re: 2042: "Rolle's Theorem"

Sableagle wrote:Could Rothko have been spouting utter bollocks?

My feelings exactly. While he may be sincere in what he's saying, I reckon that artistic analysis has gone entirely too far up its own rectum by now.

Kind of reminds me of an incident in high school where I'd worked a little Discworld shout-out1 in a French assignment, that launched our teacher in full-on analysis mode, asserting that I was illustrating the inherent meaninglessness of words, or something like that. When I told her it was just a random funny quote, she retorted that her analysis is what I subconsciously meant. Yeah, right...

Point is, not everything has a profound commentary on society and the inevitability of death, no matter how much the analytical-minded might be convinced of the contrary. If that's what they see in it, more power to them, but could they please stop talking our ears off with that nonsense?

1That scene in... I think it's the Light Fantastic, where Death is sharpening His scythe and it's described as so sharp that it even separates speech into ribbons of vowels and consonants
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### Re: 2042: "Rolle's Theorem"

Farabor
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### Re: 2042: "Rolle's Theorem"

Eebster the Great wrote:A far more egregious example of this is DeMorgan's Laws. Not only are these obvious, they are trivial to prove and have been known forever. It's like naming the exponent property (ab)c = abc "Eebster's Law."

You'd think so, but it turns out there's a subtle use of the law of the excluded middle in 1/4th of DeMorgan's laws, so you only get 3/4's of them in constructive mathematics.

ijuin
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### Re: 2042: "Rolle's Theorem"

The “Death’s scythe so sharp that it cuts words” scene is from “Reaper Man”, where Death is preparing to face his potential replacement the following midnight.

As to the “what is Great Art” argument, at least half of the praise heaped upon works by famed artists is because of the perceived greatness of the artist rather than that of the individual piece.

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### Re: 2042: "Rolle's Theorem"

I don't know much about painting or modern art and I've never seen the painting in person, but when I look at high quality scans of "Orange, Red, Yellow," I don't get it at all. I guess my strongest reaction is that it looks uncomfortable, basically just ugly, and it makes me want to look away. Past that, idk. I don't even see any yellow, just muddy green. That doesn't make the painting "bad," but it does make it very much worthless and confusing to me.

Farabor wrote:
Eebster the Great wrote:A far more egregious example of this is DeMorgan's Laws. Not only are these obvious, they are trivial to prove and have been known forever. It's like naming the exponent property (ab)c = abc "Eebster's Law."

You'd think so, but it turns out there's a subtle use of the law of the excluded middle in 1/4th of DeMorgan's laws, so you only get 3/4's of them in constructive mathematics.

How do you get any of the laws without excluded middle?

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### Re: 2042: "Rolle's Theorem"

ijuin wrote:The “Death’s scythe so sharp that it cuts words” scene is from “Reaper Man”, where Death is preparing to face his potential replacement the following midnight.

That happens in Reaper Man (at a quick scan, pp.150x3, 172, and 174, in this Corgi paperback I have that is undamaged and yet bound badly such that it starts on p.49, with Windle already not-dead-already and - if memory serves - on his way back from a crossroads), but the consonant/vowel separation thing is a direct quote from TCOM (p.189 in this 1989 2nd ed. hardback, as I apparently have leant my contemporary paperback of it out eons ago to whoknowswho…) when Rincewind and Twoflower are starting to try to escape Krull (finding an 'astrolabe room') and it jump-cuts for a small scene where Fate goes to visit Death in His garden with "a task" for Him that he thought He would be pleased to have.

In this garden-scene, it is implied to be a standard quality, after a 'standard' whetstoning, to Death's scythe (also the cutting of a flame in three with a quick double-swipe at one), whereas in RM it is a scythe that has been used in agriculture that the more earthly discly Bill Door now sharpens up to mythical levels by rather more esoteric sharpening methods (and, when that's done, henceforth needed to be further processed to make it useful to Him in the upcoming Plot). But bear in mind that Death in TCOM was still not quite the Death we know and love in the later and more mature books. No real consistency of The Duty, etc, so, for example, He takes out His annoyance on summarily 'killing' a random swarm of flies(?) with a click of His fingers when he misses out on dealing with Rincewind. And, of course, Bill Door was less Him and more merely him, at least in the depths of his 'absence'.

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### Re: 2042: "Rolle's Theorem"

Euphonium wrote:
bobleboffon3 wrote:Am I a clueless art person if I look at the "Orange, Red, Yellow" painting and genuinely believe my kid could make that?

Could s/he?

Like, maybe if someone told hir "put this particular color here" s/he could reproduce it as a technician if someone had already figured out what to do, although we would still lack a language to specify in clear, unambiguous terms the specific manner of brush stroke, paint mixing, etc. Rothko used.

But coming up with the particular combination and arrangement of colors to produce the specific visual effect requires an intimate, artistic, almost intuitive understanding of how those colors interact. Not to mention the recommendations Rothko had for viewing it, etc. And then there's the choice of medium (not just paint on canvas, but what kind of paint? how is it to be mixed?) that itself requires similar intimate knowledge, how to use the brush in the exact way, etc.

I think when people say "my six-year-old could do that", they don't mean specifically that, as a faithful reproduction; they mean that what the artist has produced requires only the artistic abilities of a six-year-old, and are therefore disappointed.

The problem with this so-called art is this: People go to look at things they couldn't do themselves. I can put rounded-off squares of paint on a canvas. I can pick colors that look like they go together. I can even, given some time, come up with some convincing bullshit about what it means. This latter is exactly what I was learning to do for three semesters of architecture school before I decided that faking my way through four years of college by waxing enthusiastically poetic about buildings I detested was not an enticing career prospect. It's not that hard.

We don't go to the circus to watch people ride a bike and have some announcer tell us what it means. We want to see someone on a unicycle balancing a five-foot stack of dishes and tossing them to other unicycle riders. We don't go to baseball games to watch some dad-bod schlub whiff badly at a softball and huff his way around the bases. We want to see the elite, hitting a pitch we can barely see over a fence a very long ways away. Likewise, most people don't want to go to art museums to see finger-painting. I look at the work of true artists and know there's no way in ten thousand years I could produce that. If I could produce that, and the only difference between what's on the wall and what I could make is that the wall thing was produced by someone who has been Accepted as a genuine member of the Art Community and therefore their tripe is taken seriously, I'm not interested.

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### Re: 2042: "Rolle's Theorem"

There is a great deal of fantastic modern and abstract art, but Rothko is well and truly useless.
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### Re: 2042: "Rolle's Theorem"

sonar1313 wrote:I think when people say "my six-year-old could do that", they don't mean specifically that, as a faithful reproduction; they mean that what the artist has produced requires only the artistic abilities of a six-year-old, and are therefore disappointed.

The problem with this so-called art is this: People go to look at things they couldn't do themselves. I can put rounded-off squares of paint on a canvas. I can pick colors that look like they go together. I can even, given some time, come up with some convincing bullshit about what it means. This latter is exactly what I was learning to do for three semesters of architecture school before I decided that faking my way through four years of college by waxing enthusiastically poetic about buildings I detested was not an enticing career prospect. It's not that hard.

We don't go to the circus to watch people ride a bike and have some announcer tell us what it means. We want to see someone on a unicycle balancing a five-foot stack of dishes and tossing them to other unicycle riders. We don't go to baseball games to watch some dad-bod schlub whiff badly at a softball and huff his way around the bases. We want to see the elite, hitting a pitch we can barely see over a fence a very long ways away. Likewise, most people don't want to go to art museums to see finger-painting. I look at the work of true artists and know there's no way in ten thousand years I could produce that. If I could produce that, and the only difference between what's on the wall and what I could make is that the wall thing was produced by someone who has been Accepted as a genuine member of the Art Community and therefore their tripe is taken seriously, I'm not interested.

Basically this, though I do swing back and forth in my opinion.

First, there is usually something clever about these kinds of artworks - not in the execution, but in the idea. Granted, most of us probably need to be told what this cleverness is, but I can believe that people who know a lot about art would see it immediately. Now, my issue with this is that I'm not convinced that it needs to be a talented artist who does this. Anybody could potentially come up with the idea: it could be an art connoisseur or better still an art critic or art historian. For me, the idea is the work, and the actual execution of it is almost irrelevant.

However, the artists who produce these works always are extremely talented and skilled. Your interpretation, which I tend to agree with, is that you don't get the opportunity to exhibit a blank canvas or a shark cut in two unless you've previously exhibited a repertoire of evocative landscape or a sensitive portrait. Why do these artists choose to do this kind of thing? Don't they risk devaluing their own skills by redefining the boundaries of art to include technically unimpressive works, however thought-inspiring? This is what tends to undermine my own certainty that this stuff is just nonsense. I have the same issue with music: it's difficult to get why Coltrane would have produced the atonal squeaks, infantile motifs and repetitive dirges of his later career given his absolute proven mastery of the previous jazz genres and his creation of whole new original idioms of his own --- unless he really was genuinely pushing back the frontiers in what he felt was an important and interesting way. In other words, it's kind of the Emperor's New Clothes, except that rather than a pair of con-men, they're provided by tailors who actually have a long-established record of making really top-notch robes. Why risk it?

Then there's the layers of subversion. The recent excellent David Hockney exhibition at the Tate Britain showed several canvases which looked initially like the kind of simple geometric patterns that abstract (con-)artists were producing, but on closer inspection turned out to be highly realistic paintings of real objects that happened to be geometric in shape. In particular, one showed a lifebuoy in a swimming pool in plan view. Apparently it was intended as a tongue-in-cheek criticism of that type of abstract art.
xtifr wrote:... and orthogon merely sounds undecided.

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### Re: 2042: "Rolle's Theorem"

I am ambivalent about these things. "Fountain" was art because nobody had considered it art-worthy before to present a perpendicular urinal bowl as such (the "R Mutt" signature just pressed the point home, whatever that point might be perceived to be), and that made it art-worthy to do. Maybe. There were then several more "Fountain"s, which touched upon the later Warhol "duplication" process, which I can see would irk some people.

Many pieces of porcelain are 'works of art' as a design, before and after, but it would be no longer be a revelatory piece of art to be so presented, because if the art is in the novelty then that no longer applies as it did then.

A Rube Goldberg (or, for us Brits, Heath Robinson) construct of wheels and pulleys and frames meant to represent something like The Suffering Of Middle-Aged Women (<- spontaneous example by me, though I have no doubt something like it exists, perhaps dragging an iron continuously across frilly knickers, whilst opening and closing and oven door and various other tasks, or whatever) might be deemed an artwork, and leads the way thence into the realm of skill, not pure conceptualising, yet even more considered engineering designs are at work that arise from the pens of engineers and might claim no art to them, necessarily (though I do also see things like The Falkirk Wheel as artistic, melded with the more practical).

And then how many "Lady Madonna With Child"s are there? The novelty factor is gone, yet (some of!) these are art-of-proficiency, in the various styles of their time. Some selfies may well find themselves composed artistically, by good luck or practiced eye, despite the near effortlessness of their creation.

And in looking for "the most painted subject" I found this, not what I intended to find, but reveals art transiently/sequentially, maybe with aberrations and no-art gaps between, even by the most liberal of art-appreciators.

I think talent and skill make art, but not exclusively, and it's even possible for art to arise through no skill, though it may require a declaration by an acknowledged 'art finder' (different from a 'found art' ist!), which may be an 'art' in itself, and just as arguable against/ignorable for those with different thoughts about these things.

Art?
Spoiler:
Maybe to someone, once. And maybe to some people now. But there will probably be critical critics in both eras.

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### Re: 2042: "Rolle's Theorem"

What makes something art in the first place is just its being presented to an audience. What makes something good art is if it evokes the right mental response in the audience. "Right" might mean according to the artist, according to the audience, according to some objective standard of what ought to be evoked in people, whatever -- if it evokes the "right" response according to whatever, then it's "good art" according to whatever. The state of mind evoked may be a pure abstract feeling devoid of propositional content (e.g. instrument music that just feels sad, but not about anything in particular), or it may have propositional content, which may be either of a descriptive or prescriptive nature, and may be either assertive (pushing that proposition on you) or inquisitive (asking you to evaluate the proposition yourself).

Things like Fountain and (moving away from visual arts) 4'33" seem intended to evoke a question, and a very meta question: "what is art?" They appear to be very successful at evoking that question, which would make them, in the eyes of their creators and anyone else who approves of that question being asked, good art.
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### Re: 2042: "Rolle's Theorem"

Pfhorrest wrote:What makes something art in the first place is just its being presented to an audience. What makes something good art is if it evokes the right mental response in the audience. "Right" might mean according to the artist, according to the audience, according to some objective standard of what ought to be evoked in people, whatever -- if it evokes the "right" response according to whatever, then it's "good art" according to whatever. The state of mind evoked may be a pure abstract feeling devoid of propositional content (e.g. instrument music that just feels sad, but not about anything in particular), or it may have propositional content, which may be either of a descriptive or prescriptive nature, and may be either assertive (pushing that proposition on you) or inquisitive (asking you to evaluate the proposition yourself).

Things like Fountain and (moving away from visual arts) 4'33" seem intended to evoke a question, and a very meta question: "what is art?" They appear to be very successful at evoking that question, which would make them, in the eyes of their creators and anyone else who approves of that question being asked, good art.

By that definition, blowing up the World Trade Center was art. A drunk streaker on the football field is art. Both pretty much manage to evoke the "right", even the intended, mental response in the audience.

Practically everything that happens in society evokes a mental response, and then if we allow the definition of the "right" response to be that subjective, everything is art, from painting to jaywalking to emitting a particularly loud and smelly fart. Indeed, I wouldn't put it past anyone at all to start jaywalking around in the middle of New York City traffic and calling him or herself a performance artist, and the only difference between that person and a certified nutjob is that they declare their intent to create art. Which to "art critics" might be a bold groundbreaking performance, but to the rest of the world, that difference is such a fine line as to be invisible.

Art, first and foremost, requires skill and beauty. The latter is still highly subjective, of course, but on the fringes it is objective enough to work. There are enough people that think a dog turd on the sidewalk is not beautiful, that you can declare it objectively so. There are enough people that think a fiery pink and orange sunset is beautiful, that you can declare it objectively so. Silence may be beautiful, but it takes no skill to create it. A urinal is neither beautiful nor skillful; Dadaism is just running around going "look how different I am!" (Actually, absent mass-production techniques, it may take some level of skill to produce a porcelain pisser, but Duchamp bought the thing.)

Ultimately, a great deal of things that are called art these days are only so because their creators came out and said THIS IS ART. If people can't tell the difference between your "art" and everyday nonsense without you bashing them over the head with it, you're probably doing art wrong.

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### Re: 2042: "Rolle's Theorem"

orthogon wrote:
sonar1313 wrote:I think when people say "my six-year-old could do that", they don't mean specifically that, as a faithful reproduction; they mean that what the artist has produced requires only the artistic abilities of a six-year-old, and are therefore disappointed.

The problem with this so-called art is this: People go to look at things they couldn't do themselves. I can put rounded-off squares of paint on a canvas. I can pick colors that look like they go together. I can even, given some time, come up with some convincing bullshit about what it means. This latter is exactly what I was learning to do for three semesters of architecture school before I decided that faking my way through four years of college by waxing enthusiastically poetic about buildings I detested was not an enticing career prospect. It's not that hard.

We don't go to the circus to watch people ride a bike and have some announcer tell us what it means. We want to see someone on a unicycle balancing a five-foot stack of dishes and tossing them to other unicycle riders. We don't go to baseball games to watch some dad-bod schlub whiff badly at a softball and huff his way around the bases. We want to see the elite, hitting a pitch we can barely see over a fence a very long ways away. Likewise, most people don't want to go to art museums to see finger-painting. I look at the work of true artists and know there's no way in ten thousand years I could produce that. If I could produce that, and the only difference between what's on the wall and what I could make is that the wall thing was produced by someone who has been Accepted as a genuine member of the Art Community and therefore their tripe is taken seriously, I'm not interested.

Basically this, though I do swing back and forth in my opinion.

First, there is usually something clever about these kinds of artworks - not in the execution, but in the idea. Granted, most of us probably need to be told what this cleverness is, but I can believe that people who know a lot about art would see it immediately. Now, my issue with this is that I'm not convinced that it needs to be a talented artist who does this. Anybody could potentially come up with the idea: it could be an art connoisseur or better still an art critic or art historian. For me, the idea is the work, and the actual execution of it is almost irrelevant.

However, the artists who produce these works always are extremely talented and skilled. Your interpretation, which I tend to agree with, is that you don't get the opportunity to exhibit a blank canvas or a shark cut in two unless you've previously exhibited a repertoire of evocative landscape or a sensitive portrait. Why do these artists choose to do this kind of thing? Don't they risk devaluing their own skills by redefining the boundaries of art to include technically unimpressive works, however thought-inspiring? This is what tends to undermine my own certainty that this stuff is just nonsense. I have the same issue with music: it's difficult to get why Coltrane would have produced the atonal squeaks, infantile motifs and repetitive dirges of his later career given his absolute proven mastery of the previous jazz genres and his creation of whole new original idioms of his own --- unless he really was genuinely pushing back the frontiers in what he felt was an important and interesting way. In other words, it's kind of the Emperor's New Clothes, except that rather than a pair of con-men, they're provided by tailors who actually have a long-established record of making really top-notch robes. Why risk it?

Then there's the layers of subversion. The recent excellent David Hockney exhibition at the Tate Britain showed several canvases which looked initially like the kind of simple geometric patterns that abstract (con-)artists were producing, but on closer inspection turned out to be highly realistic paintings of real objects that happened to be geometric in shape. In particular, one showed a lifebuoy in a swimming pool in plan view. Apparently it was intended as a tongue-in-cheek criticism of that type of abstract art.

So, it completely makes sense to me that once you've shown the ability to produce some great artistic works by the traditional definition, you might be encouraged to push the boundaries a little. Constantly doing the same thing would get boring for an artist. Duchamp was a rather talented painter before he put a urinal on a stand. Le Corbusier designed some pretty traditional (and quite beautiful) buildings before moving on into the kind of thing that made him famous.

If the worlds of art and architecture at large are anything like my brief upbringing in it, however, aspiring artists are skipping all the parts where you prove you have some skill in the discipline. At a minimum, they're certainly hiding them from the public. Architecture school, as it was when I went through it, tried to start you off immediately in the world of creating something completely brand new and groundbreaking and get you used to explaining it to a puzzled audience. They wanted you to be a freshman Frank Gehry before you proved your buildings could be appealing, follow rules of form and function, or even stand on their own.

I don't think it does the art/architecture world any favors. Classical art for centuries had rules, within which you had to operate, and it forced artists to prove they could produce something within the rules. You would never, for example, have an odd number of Greek columns. Modern art has no rules and thus no way to prove you're good at it. I remember reading some free-form "poem" and seeing the poet declare that what he had to say could not be contained within a traditional meter - and later realizing, no, the ability to say exactly what you want and fit it into the strictures of a meter and have it scan, that's what takes talent. Any fool can spit words onto a page. I totally get that someone who's proven themselves might want to - and should - branch off and do something different.... but that first step of proving yourself is too often being skipped.

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### Re: 2042: "Rolle's Theorem"

On the other hand, a toilet spurting into the air is still a toilet spurting into the air even if created by Pablo Picasso or Vincent van Gogh.

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### Re: 2042: "Rolle's Theorem"

(In architecture, at a minimum buildings¹ should stand on their own. )

¹ Yes, even mid-terrace rebuilds, because what if each next-door needs to be rebuilt!

sonar1313
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### Re: 2042: "Rolle's Theorem"

Soupspoon wrote:(In architecture, at a minimum buildings¹ should stand on their own. )

¹ Yes, even mid-terrace rebuilds, because what if each next-door needs to be rebuilt!

I actually did mean "stand on their own" in the most literal sense possible, as in "can physically bear its own weight without spontaneously collapsing into a materials pile." That wasn't part of the curriculum until much later (how much later, I dunno - I was in a different major by then), and I have no idea how much it actually mattered. They felt it more important for us to do assignments like reaching into a closed box, feeling the object inside, and drawing it. (It was a teapot. We were encouraged to imagine the pattern by feel.) It was also a reason I got quickly disillusioned. I essentially did not realize I was going to art school. I guess the "make it stay building-shaped after it's built" stuff is for structural engineers.

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### Re: 2042: "Rolle's Theorem"

Sonar, that "declaing it art" step is almost but not quite the defining factor I was spreaking of. The "presenting it to an audience" part. You might also call it something like "framing". Consider for example a beautiful landscape found in nature. If you happen across that and snap a picture of it, is that not art? Photography is widely recognized as an art form. But all you did was take something you found and frame it. Okay, you might say that the choice of camera settings and lenses constitutes the art of it, but never mind that you might just snap a pic with whatever smart phone you happen to have and it could still be art, consider if instead you just bought a frame and some easle-like stand for it from a store, walked back to the trail where you saw the nice view, and literally just put a frame around it. Is that not an art installation now? It seems uncontroversially so.

It was putting the urinal in an art museum that made Fountain art. If you don't like the response it evoked in you, then you're free to call it BAD art on that account. But art doesn't have to be good just to be art in the first place. And beauty is certainly not the only quality of good art, though since of course most people like the feelings that beauty invokes in them (pretty much by definition), beauty will always be a popular quality of art.
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### Re: 2042: "Rolle's Theorem"

Soupspoon wrote:(In architecture, at a minimum buildings¹ should stand on their own. )

¹ Yes, even mid-terrace rebuilds, because what if each next-door needs to be rebuilt!

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### Re: 2042: "Rolle's Theorem"

Pfhorrest wrote:It was putting the urinal in an art museum that made Fountain art. If you don't like the response it evoked in you, then you're free to call it BAD art on that account. But art doesn't have to be good just to be art in the first place. And beauty is certainly not the only quality of good art, though since of course most people like the feelings that beauty invokes in them (pretty much by definition), beauty will always be a popular quality of art.

It occurs to me also that, consistent with my view on what constitutes art and good art, you (Sonar specifically) could argue for the position that those good feelings people like when they see beauty are the only kinds of feelings that art objectively ought to be evoking, and that evoking things like the question "what is art?" is not what art ought to be doing, and that something like Fountain is thus objectively bad art, not just art that you don't like, because it fails to do what art ought to do, even if it does what its creator meant it to do. But it is still nevertheless art, even if it's objectively bad art.
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### Re: 2042: "Rolle's Theorem"

rmsgrey wrote:
Soupspoon wrote:(In architecture, at a minimum buildings¹ should stand on their own. )

¹ Yes, even mid-terrace rebuilds, because what if each next-door needs to be rebuilt!

How much do you want me banging on them because you're too loud and I wasn't invited?

(Wall, mumblemumble, wall. A good old two-sided cavity-wall fully load-bearing on either side and centrally insulating Also means we don't leach off of each other's central-heating bills.)

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### Re: 2042: "Rolle's Theorem"

"Rolle's" theorem?

Now there's a name to on-the-nose for Legend of Everfree.

sonar1313
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### Re: 2042: "Rolle's Theorem"

Pfhorrest wrote:
Pfhorrest wrote:It was putting the urinal in an art museum that made Fountain art. If you don't like the response it evoked in you, then you're free to call it BAD art on that account. But art doesn't have to be good just to be art in the first place. And beauty is certainly not the only quality of good art, though since of course most people like the feelings that beauty invokes in them (pretty much by definition), beauty will always be a popular quality of art.

It occurs to me also that, consistent with my view on what constitutes art and good art, you (Sonar specifically) could argue for the position that those good feelings people like when they see beauty are the only kinds of feelings that art objectively ought to be evoking, and that evoking things like the question "what is art?" is not what art ought to be doing, and that something like Fountain is thus objectively bad art, not just art that you don't like, because it fails to do what art ought to do, even if it does what its creator meant it to do. But it is still nevertheless art, even if it's objectively bad art.

So I still disagree with the original idea, that being, that anything is art as long as you call it art. An old riddle, attributed to Abraham Lincoln (like every other quote in the world) has him asking, "If you call a cow's tail a leg, how many legs does it have? Four - calling a tail a leg doesn't make it so."

I think there's a difference between the definition of art and the definition of good art. A great many people (including me) would certainly dispute that setting up an empty frame in front of a pretty landscape isn't really art. I would define art a lot more narrowly than "anything the creator says it is" - I would say it requires skill; beauty; it must enrich humanity in some way, even if minuscule; or at least, it must have the potential for the above if executed correctly. Enriching humanity matters. Humanity would certainly be impoverished if, in something I called performance art, I burned the Mona Lisa in front of a shocked and horrified audience. I could call that art all I wanted; the rest of the world would objectively disagree.

There are countless rules of composition for painters, poets, architects, and other artists, by which we can objectively judge the quality of the art and without which there is no way to say art is objectively bad or good. A thing like "Fountain" has no rules to judge it by, so how can we call it objectively good or bad art? Simple: it is neither, as it is not art. What is the so-called creator? Painter, sculptor, actor, poet, author, playwright, composer?** None - there is no word for it. William McGonagall was an objectively horrible poet, but he was a poet.

I maintain that if you have to explain that what you've done is art, it probably isn't. A urinal was never and will never be considered a piece of art unless sitting in an art museum. A painting laying on the street will always be considered a work of art. Once we've defined what art is and what it isn't, and once we've decided what sort of art it is, then we can start calling it objectively bad or good.

**Duchamp was a painter, but certainly not acting in that capacity when installing Dada objects into museums.

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### Re: 2042: "Rolle's Theorem"

Spoiler for furry webcomic. Yes, really!

Spoiler:

I don't actually like the proportions on that piece, but I respect the talent that went into making it, and nobody can deny it is what it's clearly meant to be.
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