Aristotle's wheel paradox

[Jon Letendre]

4 hours ago, Ellen Stuttle said:

Granted that developments in mathematics were essential for later working out of "the physical mechanics of revolution and rotation."  However,  I've been thinking that even with the limited math available in classical times, the "paradox" came from ignoring reality and thinking in a sloppy fashion.

As I wrote yesterday - here:

"Seems to me that this wheel problem differs from the special-relativity paradoxes in that those come from reasonable assumptions whereas the wheel problem doesn't."

The question raised by the "paradox" in effect asks, "Why doesn't something which can't move independently move independently?"

Answer:  It doesn't because it can't.

Ellen

 

Well stated, Ellen.

It would have been a simple matter in Aristotle's time to stand still and hold one's forearm out flat, to make a "road" for the inner hub of a large wheel, and the slipping/skidding would be quite apparent.

I ride an 1870's style highwheel, or penny farthing, with a mere 48" front wheel, and it is quite apparent as you stand in place and rock it back and forth.

It would have been a quick matter to make a four foot wheel, mark it, roll it, and soon you could draw my 1/8th rotation illustration, and, you're done.

I think the rate of confoundedness over this one is probably about the same as it was back then. I think it comes down to some people not noticing certain things and to some people not well equipped to cognate certain things.

I don't believe Aristotle could ever have fallen for it for long.

Maybe he was intentionally spurring inquiry into better mathematics.

Me on my highwheel...

[Brant Gaede]

13 hours ago, Ellen Stuttle said:

LOL.  You do like to play the fool.

In what year did precise thinking become possible?

Ellen

I suspect between the ages of 3 and 5.

--Brant

and 7?

[BaalChatzaf]

18 hours ago, Ellen Stuttle said:

LOL.  You do like to play the fool.

In what year did precise thinking become possible?

Ellen

On my 6 th birthday.  It became possible for me.

 

[Brant Gaede]

7 hours ago, BaalChatzaf said:

On my 6 th birthday.  It became possible for me.

Musta been quite a party!

--Brant

[BaalChatzaf]

41 minutes ago, Brant Gaede said:

Musta been quite a party!

--Brant

Yes it was.  I taught myself algebra.  

 

[Brant Gaede]

1 hour ago, BaalChatzaf said:

Yes it was.  I taught myself algebra. 

On the very day?

--Brant

wowser!

[BaalChatzaf]

4 hours ago, Brant Gaede said:

On the very day?

--Brant

wowser!

starting about that time.  I was mostly an autodidact and I taught myself things that were not generally taught in the public schools at that time.  I remember while being in my tenth grade geometry class teaching myself  non-euclidean geometry and a bit of fundamental differential geometry.   When I brought these things up in class my geometry teacher nearly had a fit.  When I demonstrate a Moebius Strip in class  the geometry (Miss Talmage)  threatened to  send me to the Principle (that was like the Ultimate Punishment)  and to flunk me. The only people who appreciated my extraciricula learning were my science teachers   Joe Louderstein and Gabriel Helman.. To this day this pair of supportive science teachers have a permanent  place in my  heart and memory..  You might say I had an interesting childhood.

[Peter]

I would walk to school around August and get old beat up text books that were going to be used in the coming semester and read them cover to cover. I must have seemed like Horshak from the show, "Mr. Kotter" 'Ooh ooh pick me! I know!' Oddly, no one ever asked me what I was doing walking around the school and taking books out. 

[Ellen Stuttle]

On October 14, 2017 at 2:25 PM, BaalChatzaf said:

On my 6 th birthday.  It [precise thinking] became possible for me.

The question I was asking isn't at what age you believe you became capable of precise thought but in what year you think precise thought became possible for anyone.

You earlier wrote, in response to my saying that I think that clear thinking could have seen through the supposed paradox even with the limited math the Greeks had available:

 

On October 13, 2017 at 7:24 PM, BaalChatzaf said:

Math Think and Symbolic Logic Think are the only truly precise methods of thinking I am certain of. 

Symbolic Logic wasn't developed until the second half of the 19th century, and calculus wasn't developed until the mid-17th century, so are you claiming that precise thought wasn't possible prior to the first, or to both, of those developments?

Ellen

[Ellen Stuttle]

On October 14, 2017 at 12:16 AM, Jon Letendre said:

Well stated, Ellen.

It would have been a simple matter in Aristotle's time to stand still and hold one's forearm out flat, to make a "road" for the inner hub of a large wheel, and the slipping/skidding would be quite apparent.

I ride an 1870's style highwheel, or penny farthing, with a mere 48" front wheel, and it is quite apparent as you stand in place and rock it back and forth.

It would have been a quick matter to make a four foot wheel, mark it, roll it, and soon you could draw my 1/8th rotation illustration, and, you're done.

I think the rate of confoundedness over this one is probably about the same as it was back then. I think it comes down to some people not noticing certain things and to some people not well equipped to cognate certain things.

I don't believe Aristotle could ever have fallen for it for long.

Maybe he was intentionally spurring inquiry into better mathematics.

Thanks.

Re Aristotle falling for it:

For one thing, we have no extant writings of Aristotle's.  What we have is lecture notes, and I think there's some doubt about the "work" in which the wheel paradox appears coming from Aristotle's lectures.  I've been meaning to look into the issue but haven't had time yet.

Ellen

[BaalChatzaf]

6 hours ago, Ellen Stuttle said:

The question I was asking isn't at what age you believe you became capable of precise thought but in what year you think precise thought became possible for anyone.

You earlier wrote, in response to my saying that I think that clear thinking could have seen through the supposed paradox even with the limited math the Greeks had available:

 

Symbolic Logic wasn't developed until the second half of the 19th century, and calculus wasn't developed until the mid-17th century, so are you claiming that precise thought wasn't possible prior to the first, or to both, of those developments?

Ellen

I was born in 1936.  All the math and logic I needed was already invented.

 

[Brant Gaede]

12 hours ago, BaalChatzaf said:

I was born in 1936.  All the math and logic I needed was already invented.

And when did you realize all you could do was use that math?

--Brant

 

[BaalChatzaf]

3 hours ago, Brant Gaede said:

And when did you realize all you could do was use that math?

--Brant

 

When I was 25

 

[Ellen Stuttle]

On October 18, 2017 at 8:26 PM, BaalChatzaf said:

I was born in 1936.  All the math and logic I needed was already invented.

 

Your inability to read - or failure to read - is amazing.

Repeating the question, though I don't expect a straight answer:

"Symbolic Logic wasn't developed until the second half of the 19th century, and calculus wasn't developed until the mid-17th century, so are you claiming that precise thought wasn't possible prior to the first, or to both, of those developments?"

Ellen

[BaalChatzaf]

1 hour ago, Ellen Stuttle said:

Your inability to read - or failure to read - is amazing.

Repeating the question, though I don't expect a straight answer:

"Symbolic Logic wasn't developed until the second half of the 19th century, and calculus wasn't developed until the mid-17th century, so are you claiming that precise thought wasn't possible prior to the first, or to both, of those developments?"

Ellen

Precise?  To what degree.  To what degree of rigor?  The calculus invented by Newton and Leibniz does not meet modern standards of rigor.  Prior to the invention of calculus  the intricacies of motion simply were not understood.  How about rigor.  Euclid does not meet modern standards of rigor.  This defect was cured by Hilbert in 1899 when he gave a complete and rigorous axiom system for Euclidean spaces. So to answer your question as you asked it   no,  prior to the invention of calculus thinking pertaining to physical quantities was NOT precise.  Are you satisfied with the answer now?

[Ellen Stuttle]

4 minutes ago, BaalChatzaf said:

Are you satisfied with the answer now?

I'm satisfied that you answered.  Your reply is what I thought is your view.  But here's the problem I see with it:   How could modern methods have been developed without someone's thinking precisely in order to develop them?  Modern methods certainly help in understanding physics, and there's much we wouldn't understand without them, but I think their development rested on previous precision within the available context of knowledge.  Maybe you think so, too, and just weren't saying that clearly.

Ellen

[Brant Gaede]

On 10/21/2017 at 6:41 PM, Ellen Stuttle said:

I'm satisfied that you answered.  Your reply is what I thought is your view.  But here's the problem I see with it:   How could modern methods have been developed without someone's thinking precisely in order to develop them?  Modern methods certainly help in understanding physics, and there's much we wouldn't understand without them, but I think their development rested on previous precision within the available context of knowledge.  Maybe you think so, too, and just weren't saying that clearly.

Ellen

Failure is vastly underrated.

--Brant

[BaalChatzaf]

20 hours ago, Ellen Stuttle said:

I'm satisfied that you answered.  Your reply is what I thought is your view.  But here's the problem I see with it:   How could modern methods have been developed without someone's thinking precisely in order to develop them?  Modern methods certainly help in understanding physics, and there's much we wouldn't understand without them, but I think their development rested on previous precision within the available context of knowledge.  Maybe you think so, too, and just weren't saying that clearly.

Ellen

Sadi Carnot invented thermodynamics  and got the second law of thermodynamics right even though he started with the wrong theory of heat (coloric theory).  Clausius got to entropy by trying to rescue Carnot's law  from the caloric theory of heat.  Sometimes wrong leads to right.  It happens all the time in science.  

[merjet]

On 10/13/2017 at 9:28 AM, Jonathan said:

Actually, curved paths do ignore the circles moving. You eliminate the circles from the setup, and replace them with three points on a line segment which undergoes rotation and translation. The "paradox" setup is about circles/wheels traveling on lines/surfaces, not about a line/stick with points/dots on it flipping through the air ...

You reject our (original, unborrowed) presentations with the claim that they are, or could be, optical illusions. ...

Show it, or else your claims about curved paths are just hearsay based on optical illusions.

I enjoyed a two week hiatus from OL. I see the obnoxious, self-deluded, mathematically incompetent (a line integral, what's that?), logic-challenged, dishonest ignoranus added more ad hominem, butchering of the truth, double-standards, and other nonsense. Yap, yap, yap, yap, yap. All bark and no bite. He's as pleasant as Annie Wilkes or preparing for a colonoscopy. 

His first and second sentences are plainly false and stupid. The third sentence is about as stupid as saying that measuring the height of a fence eliminates the fence. Any such point is never separate from the circle it is part of and traverses the entire circle with one rotation, like how a pencil in a compass draws a circle, except its center moves. Your dishonesty, stupidity, and incompetence exceed what they were only two weeks ago.

Duh. I reject your con games. Original and unborrowed? That’s due to nobody else being such a self-deluded con artist. Unborrowed?  You borrowed somebody’s software.

Reverse your double standard and prove the slipping in your videos is not based on an optical illusion.

Toiletathan is OL’s intellectually pretentious con man, who revels in name-calling and posting his ignorant nonsense.

[Brant Gaede]

Merlin,

The "paradox" is in your head. The reality is on the ground. Circles (epistemological) are what you draw on wheels (metaphysical). Simply replace all circles with actual wheels and the paradox ceases to exist.

That's my reading, anyway.

You'll not get a simpler criticism--why not a simple refutation which you've not yet proffered?

It's so easy a caveman can do it (post-wheel caveman)

--Brant

what's in your head trumps what's out there?

[Jonathan]

5 hours ago, merjet said:

Any such point is never separate from the circle it is part of and traverses the entire circle with one rotation, like how a pencil in a compass draws a circle, except its center moves.

Are you capable of operating a pencil, compass and straightedge? If so, use them to show that the points travel the paths that your online sources claim.

 

Quote

Duh. I reject your con games. Original and unborrowed? That’s due to nobody else being such a self-deluded con artist. Unborrowed?  You borrowed somebody’s software.

Okay, so, go out and mine the ore to make a compass...

 

Quote

Reverse your double standard and prove the slipping in your videos is not based on an optical illusion.

 

What would you accept as proof? Let me guess: only your cycloid method qualifies as proof, right?

The actual proof is much simpler. The smaller circle has a circumference whose length is shorter than the line segment which represents the length of the larger circle's circumference. Therefore, by definition, the smaller slips as it rolls. QED.

Yeah, I know that your objection is that you can't grasp that proof, so it doesn't count as real proof.

[Jonathan]

1 hour ago, Brant Gaede said:

Merlin,

The "paradox" is in your head. The reality is on the ground. Circles (epistemological) are what you draw on wheels (metaphysical). Simply replace all circles with actual wheels and the paradox ceases to exist.

That's my reading, anyway.

You'll not get a simpler criticism--why not a simple refutation which you've not yet proffered?

It's so easy a caveman can do it (post-wheel caveman)

--Brant

what's in your head trumps what's out there?

It's more like what's NOT in Merlin's head trumps what's in eveypryone else's. He can't see it, envision it, or grasp the logic of it, so therefore he needs to believe that no one else can either. People who report grasping what he can't must be con men and scam artists. Merlin is the cognitive limit of all mankind. It's a very common mindset among Rand's followers.

[Max]

The crux of the paradox is the implied -and false- suggestion that both wheels can turn without slipping on their respective supports (rail or road etc.). Both (concentric) wheels are part of a rigid body, so they have the same rotational velocity and the same translational velocity (of their common center). When the larger wheel makes one rotation without slipping, it travels over a distance of 2 π R. So does the smaller wheel, but if this wheel wouldn't slip, it would only travel over a distance 2 π r (r < R). However, it has to travel over a distance of 2 π R, so apart from its rotation it must also slip with respect to its support, to keep up with the larger wheel. Mutatis mutandis if it is the smaller wheel that rotates without slipping. It's all so very simple and trivial, so why should we have a discussion that now covers already 25 pages? It isn’t that difficult!

[Jonathan]

6 minutes ago, Max said:

It's all so very simple and trivial, so why should we have a discussion that now covers already 25 pages? It isn’t that difficult!

It IS that difficult to Merlin. Plus he's stubborn. He doesn't grasp it, plus he doesn't want to grasp it.

 

[Jonathan]

Where's Merlin?

Where's he been?

I envision him experimenting in his garage with an actual wheel made of wood or cardboard, with a smaller wheel glued to it, just like in my rock wheel video. He sets up the wheels and ledges, and is stupefied to discover that the "scam" has infiltrated his garage! The small wheel skids as it rolls! How did that happen? Why, Jon and Jonathan must have snuck into his garage and done some illusion tampering with his wood or cardboard when he wasn't there!

J