Resolving Zeno's Paradox without calculus

[Dan Christensen]

The paradox:

"In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise."
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise


Why it seemed paradoxical:

That this would have seemed paradoxical to ancient Greek philosophers can be explained by the fact that they had no precise notion of what speed was. It wasn't until Galileo in the 16th century first measured speed by considering the distance covered and the time it takes.
http://en.wikipedia.org/wiki/Speed#Definition


Resolving the paradox by algebraic methods:

Assuming constant speeds S_A and S_T (m/s) for Achilles and the Tortoise respectively, we can show that, in this example, Achilles would have caught up to the Tortoise in 100/(S_A - S_T) seconds. A trivial application of the s=d/t formula.

With only a vague notion of speed, ancient Greek philosophers were perplexed by the fact, in that time interval, both racers would have passed through infinitely many points in space, the arrival at each point being an "event".

As Aristotle wrote, "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." (see first link above)

(It's hard to be believe ancient Greek engineers could have been as deluded on such mechanical issues, but that is probably another story.) In modern modern mathematics and physics, we have no problem with infinitely many such events occurring in a finite time interval.

Comments?

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
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[Roland Franzius]

Ahistorical nonsense. The question was central in a scinetific war
between two philosophical schools.

In that war the "physical faction" lost to Parmenides over the question
if in the world of pure ideas, thought and logical resoning is always
the same or changes with time.

Zenos paradox served to exclude the very thought of continuity of time
and continous development of states in the framework of rational
arithmetics.

So the notion of continuity of all physical processes in time became
questionable on logical reasons as it is still today. This paradox is
partially resolved only now by quantum physics with an excluded observer
of evolution processes and a notion of obervable effects as wheighted
interference sums over all possible wave histories.

While the triumphant development of academic geometry in ancient Athens
and later on in Alexandria, that cumulated in Euclids work, gave the
logicians a strong point over the ancient greek physicists and
engineers, these busy ingenious people nevertheless used "analysis" as a
somehow logical dirty digging method to excavate mathematical theorems.
Sometimes they could be proved to be logically true "more geometrico", a
method still used in the times of Gau� at the birthtimes of modern
mathematics.

The usefulness of analysis has been demonstrated eg by Hipparchos (his
quadratrix was the idea of a limes machine that made Zenon so angry),
Archimedes (would probably have invented modern mathematics if those
mathematically and numerically handicapped Romans with their much to
small language and awfully bad number system had build him an academia
instead of killing him).

--

Roland Franzius

[Chris M. Thomasson]

> "Dan Christensen" wrote in message
> news:412c14f0-23ce-48aa...@googlegroups.com...
> The paradox:
> [...]

Sorry if this comes across as moronic ramblings, but:


1. Draw a line L between points p_0 and p_1.

2. Update p_0 to be equal to the mid-point of L.

3. If p_0 = p_1 then halt.

4. Goto step 1.


AFAICT, p_0 will _never_ be equal to p_1, unless they
were equal from the _first_ iteration...



Now, if you had an infinity border, then I guess
you can say that after N iterations, we are close
enough to say that p_0 is close enough to p_1?

Humm...

[Dan Christensen]

> method still used in the times of Gau� at the birthtimes of modern

>
> mathematics.
>
>
>
> The usefulness of analysis has been demonstrated eg by Hipparchos (his
>
> quadratrix was the idea of a limes machine that made Zenon so angry),
>
> Archimedes (would probably have invented modern mathematics if those
>
> mathematically and numerically handicapped Romans with their much to
>
> small language and awfully bad number system had build him an academia
>
> instead of killing him).
>

Thanks for that bit of history. But when you accept that speed equals distance over time (from Galileo), the resolution seems so obvious. Even if you postulate as certain quantum "graininess" to space-time, if that is what you are getting at, the relation would surely hold on the macroscopic level. There is really no need for limits and infinitesimals to resolve this problem. Simple algebra is all you need.

[Dan Christensen]

On Monday, May 12, 2014 1:09:33 AM UTC-4, Chris M. Thomasson wrote:
> > "Dan Christensen" wrote in message
>
> > news:412c14f0-23ce-48aa...@googlegroups.com...
>
> > The paradox:
>
> > [...]
>
>
>
> Sorry if this comes across as moronic ramblings, but:
>

Not at all!

>
>
>
>
> 1. Draw a line L between points p_0 and p_1.
>
>
>
> 2. Update p_0 to be equal to the mid-point of L.
>
>
>
> 3. If p_0 = p_1 then halt.
>
>
>
> 4. Goto step 1.
>
>
>
>
>
> AFAICT, p_0 will _never_ be equal to p_1,

This turns out not be a very useful way to look at space-time. The fact is, motion does occur. Contrary to Aristotle, faster runners do overtake slower runners. If they are running at a constant speed along the same straight line, it is a simple matter to calculate when and where that event will occur.

[Martin Shobe]

>> method still used in the times of Gau� at the birthtimes of modern

>> mathematics.

>> The usefulness of analysis has been demonstrated eg by Hipparchos (his
>> quadratrix was the idea of a limes machine that made Zenon so angry),
>> Archimedes (would probably have invented modern mathematics if those
>> mathematically and numerically handicapped Romans with their much to
>> small language and awfully bad number system had build him an academia
>> instead of killing him).

> Thanks for that bit of history. But when you accept that speed equals distance over time (from Galileo), the resolution seems so obvious. Even if you postulate as certain quantum "graininess" to space-time, if that is what you are getting at, the relation would surely hold on the macroscopic level. There is really no need for limits and infinitesimals to resolve this problem. Simple algebra is all you need.

You solved the wrong problem. The problem isn't "when will Achilles
overtake the tortoise?" The problem is "how can Achilles overtake the
tortoise when he has to complete an infinite number of subtasks to do so?".

Martin Shobe

[Dan Christensen]

On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:

>
> You solved the wrong problem. The problem isn't "when will Achilles
>
> overtake the tortoise?" The problem is "how can Achilles overtake the
>
> tortoise when he has to complete an infinite number of subtasks to do so?".
>

Do you really dispute that speed equals distance over time?

Looking at it as an infinite number of subtasks to be completed complicates the problem unnecessarily, as shown by the fact that you can actually calculate when and where the event will occur using a simple formula. What more do we need to know?

[Martin Shobe]

On 5/12/2014 9:34 AM, Dan Christensen wrote:
> On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>> You solved the wrong problem. The problem isn't "when will Achilles
>> overtake the tortoise?" The problem is "how can Achilles overtake the
>> tortoise when he has to complete an infinite number of subtasks to do so?".

> Do you really dispute that speed equals distance over time?

Of course not. What makes you think I do?

> Looking at it as an infinite number of subtasks to be completed complicates the problem unnecessarily, as shown by the fact that you can actually calculate when and where the event will occur using a simple formula. What more do we need to know?

The existence of that decomposition creates the paradox. It's necessity
isn't relevant to the paradox, so you're not resolving the paradox by
saying, "Use a different decomposition."

Martin Shobe

[Dan Christensen]

On Monday, May 12, 2014 12:00:12 PM UTC-4, Martin Shobe wrote:
> On 5/12/2014 9:34 AM, Dan Christensen wrote:
>
> > On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>
> >> You solved the wrong problem. The problem isn't "when will Achilles
>
> >> overtake the tortoise?" The problem is "how can Achilles overtake the
>
> >> tortoise when he has to complete an infinite number of subtasks to do so?".
>
>
>
> > Do you really dispute that speed equals distance over time?
>
>
>
> Of course not.

Good. Now, suppose Achilles runs at a constant speed of 11 m/s, and the Tortoise at 1 m/s. Where would they be in 10 seconds? Both would be at 110 m from the start line. An instant late, Achilles will have passed the Tortoise.

How did I figure that out? (Hint: No need to decompose the race into infinite, ever decreasing intervals. Just use S = d/t.)

> > Looking at it as an infinite number of subtasks to be completed complicates the problem unnecessarily, as shown by the fact that you can actually calculate when and where the event will occur using a simple formula. What more do we need to know?
>
>
>
> The existence of that decomposition creates the paradox.

It's a dead-end. Dump it. Use S = d/t instead, and no more paradox.

[Martin Shobe]

On 5/12/2014 11:20 AM, Dan Christensen wrote:
> On Monday, May 12, 2014 12:00:12 PM UTC-4, Martin Shobe wrote:
>> On 5/12/2014 9:34 AM, Dan Christensen wrote:
>>> On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>>>> You solved the wrong problem. The problem isn't "when will Achilles
>>>> overtake the tortoise?" The problem is "how can Achilles overtake the
>>>> tortoise when he has to complete an infinite number of subtasks to do so?".

>>> Do you really dispute that speed equals distance over time?

>> Of course not.

> Good. Now, suppose Achilles runs at a constant speed of 11 m/s, and the Tortoise at 1 m/s. Where would they be in 10 seconds? Both would be at 110 m from the start line. An instant late, Achilles will have passed the Tortoise.

> How did I figure that out? (Hint: No need to decompose the race into infinite, ever decreasing intervals. Just use S = d/t.)

What makes you think that was the problem that needed resolving? (Hint:
It wasn't.)

>>> Looking at it as an infinite number of subtasks to be completed complicates the problem unnecessarily, as shown by the fact that you can actually calculate when and where the event will occur using a simple formula. What more do we need to know?

>> The existence of that decomposition creates the paradox.

> It's a dead-end. Dump it. Use S = d/t instead, and no more paradox.

Which doesn't help.

Martin Shobe

[Dan Christensen]

On Monday, May 12, 2014 12:35:18 PM UTC-4, Martin Shobe wrote:
> On 5/12/2014 11:20 AM, Dan Christensen wrote:
>
> > On Monday, May 12, 2014 12:00:12 PM UTC-4, Martin Shobe wrote:
>
> >> On 5/12/2014 9:34 AM, Dan Christensen wrote:
>
> >>> On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>
> >>>> You solved the wrong problem. The problem isn't "when will Achilles
>
> >>>> overtake the tortoise?" The problem is "how can Achilles overtake the
>
> >>>> tortoise when he has to complete an infinite number of subtasks to do so?".
>
>
>
> >>> Do you really dispute that speed equals distance over time?
>
>
>
> >> Of course not.
>
>
>
> > Good. Now, suppose Achilles runs at a constant speed of 11 m/s, and the Tortoise at 1 m/s. Where would they be in 10 seconds? Both would be at 110 m from the start line. An instant late, Achilles will have passed the Tortoise.
>
>
>
> > How did I figure that out? (Hint: No need to decompose the race into infinite, ever decreasing intervals. Just use S = d/t.)
>
>
>
> What makes you think that was the problem that needed resolving? (Hint:
>
> It wasn't.)
>

It trashes the bizarre, Aristotelian notion that faster runners cannot overtake slower ones. It shows that there is no paradox -- just a ridiculous analysis of the situation. All is exactly as expected.

[Martin Shobe]

What premise of the argument does it argue against? What step in the
argument does it show to be invalid? Since the answers to those
questions are quite obviously none and none, you quite clearly have
resolved nothing.

Martin Shobe

[Dan Christensen]

On Monday, May 12, 2014 1:17:35 PM UTC-4, Martin Shobe wrote:
> On 5/12/2014 11:45 AM, Dan Christensen wrote:
>
> > On Monday, May 12, 2014 12:35:18 PM UTC-4, Martin Shobe wrote:
>
> >> On 5/12/2014 11:20 AM, Dan Christensen wrote:
>
> >>> On Monday, May 12, 2014 12:00:12 PM UTC-4, Martin Shobe wrote:
>
> >>>> On 5/12/2014 9:34 AM, Dan Christensen wrote:
>
> >>>>> On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>
> >>>>>> You solved the wrong problem. The problem isn't "when will Achilles
>
> >>>>>> overtake the tortoise?" The problem is "how can Achilles overtake the
>
> >>>>>> tortoise when he has to complete an infinite number of subtasks to do so?".
>
>
>
> >>>>> Do you really dispute that speed equals distance over time?
>
>
>
> >>>> Of course not.
>
>
>
> >>> Good. Now, suppose Achilles runs at a constant speed of 11 m/s, and the Tortoise at 1 m/s. Where would they be in 10 seconds? Both would be at 110 m from the start line. An instant late, Achilles will have passed the Tortoise.
>
>
>
> >>> How did I figure that out? (Hint: No need to decompose the race into infinite, ever decreasing intervals. Just use S = d/t.)
>
>
>
> >> What makes you think that was the problem that needed resolving? (Hint:
>
> >> It wasn't.)
>
>
>
> > It trashes the bizarre, Aristotelian notion that faster runners cannot overtake slower ones. It shows that there is no paradox -- just a ridiculous analysis of the situation. All is exactly as expected.
>
>
>
> What premise of the argument does it argue against?

Implicit was that infinitely many events (e.g. arrival at point) cannot occur in a finite time interval. This is no longer accepted.

[Martin Shobe]

Very good. Since you've chosen that route, two things should now be
obvious. First, your argument about speed doesn't address the issue of
whether or not there are infinitely many tasks occurring in a finite
time period one way or the other, and is therefore not a resolution to
the paradox. Second, you need a way to handle the infinite sub-tasks
that occurred in the finite time that is consistent with the treatment
of the task as a whole. Your speed argument doesn't address the infinite
sub-tasks, so you will need to look elsewhere.

Martin Shobe

[Dan Christensen]

On Monday, May 12, 2014 2:06:51 PM UTC-4, Martin Shobe wrote:

>
> > Implicit was that infinitely many events (e.g. arrival at point) cannot occur in a finite time interval. This is no longer accepted.
>
>
>
> Very good. Since you've chosen that route, two things should now be
>
> obvious. First, your argument about speed doesn't address the issue of
>
> whether or not there are infinitely many tasks occurring in a finite
>
> time period one way or the other, and is therefore not a resolution to
>
> the paradox. Second, you need a way to handle the infinite sub-tasks
>
> that occurred in the finite time that is consistent with the treatment
>
> of the task as a whole. Your speed argument doesn't address the infinite
>
> sub-tasks, so you will need to look elsewhere.
>

I think it does. Of course, nothing is formally stated here (this is physics, not math), but it seems to me that at the heart of the speed argument is the notion that a moving object passes through infinitely many points every second with an event associated with the arrival at any one of those points.

[Martin Shobe]

On 5/12/2014 1:32 PM, Dan Christensen wrote:
> On Monday, May 12, 2014 2:06:51 PM UTC-4, Martin Shobe wrote:
>
>>
>>> Implicit was that infinitely many events (e.g. arrival at point) cannot occur in a finite time interval. This is no longer accepted.
>>
>>
>>
>> Very good. Since you've chosen that route, two things should now be
>>
>> obvious. First, your argument about speed doesn't address the issue of
>>
>> whether or not there are infinitely many tasks occurring in a finite
>>
>> time period one way or the other, and is therefore not a resolution to
>>
>> the paradox. Second, you need a way to handle the infinite sub-tasks
>>
>> that occurred in the finite time that is consistent with the treatment
>>
>> of the task as a whole. Your speed argument doesn't address the infinite
>>
>> sub-tasks, so you will need to look elsewhere.
>>
>
> I think it does. Of course, nothing is formally stated here (this is physics, not math), but it seems to me that at the heart of the speed argument is the notion that a moving object passes through infinitely many points every second with an event associated with the arrival at any one of those points.

The heart of your speed argument was to treat it as a whole and ignore
the fact that it could be broken up into infinitely many sub-tasks. You
said, "Looking at it as an infinite number of sub-tasks to be completed
complicates the problem unnecessarily."

Martin Shobe

[Dan Christensen]

On Monday, May 12, 2014 2:40:00 PM UTC-4, Martin Shobe wrote:

>
> The heart of your speed argument was to treat it as a whole and ignore
>
> the fact that it could be broken up into infinitely many sub-tasks. You
>
> said, "Looking at it as an infinite number of sub-tasks to be completed
>
> complicates the problem unnecessarily."
>

Well said. But so what? The precise notion of speed, however it may have been arrived at historically, allows you to mathematically solve the problem of a faster runner overtaking a slower one in the above scenario. You can calculate when and where the event will happen. What more is to be said?

I suppose there was never really any paradox, just a poorly formulated, needlessly complicated analysis -- a dead-end as it turned out.

[Chris M. Thomasson]

> "Dan Christensen" wrote in message

> news:fbee0d2c-80b3-4892...@googlegroups.com...

> >On Monday, May 12, 2014 1:09:33 AM UTC-4, Chris M. Thomasson wrote:
> > > "Dan Christensen" wrote in message
> >
> > > news:412c14f0-23ce-48aa...@googlegroups.com...
> >
> > > The paradox:
> >
> > > [...]
> >
> >
> >
> > Sorry if this comes across as moronic ramblings, but:
> >

> Not at all!
>
> > 1. Draw a line L between points p_0 and p_1.
> > 2. Update p_0 to be equal to the mid-point of L.
> > 3. If p_0 = p_1 then halt.
> > 4. Goto step 1.
> >
> > AFAICT, p_0 will _never_ be equal to p_1,

> This turns out not be a very useful way to look at
> space-time. The fact is, motion does occur.

Motion always occurs in step 2 where the point
p_0 moves 1/2 across the line L created in step 1.

After a number of iterations, p_0 eventually gets
extremely close to p_1, but never equal. IMHO, if
you zoom in far enough, you will see that p_0 has
some more "traveling" to do...

I guess the question is how close does p_0 have to
get to p_1 before we can assume they are equal and
halt the algorithm in step 3? Once this occurs we can
say that the action of moving p_0 to p_1 is complete
because we found that p_0 = p_1 during the iteration
process.

> Contrary to Aristotle, faster runners do overtake
> slower runners. If they are running at a constant
> speed along the same straight line, it is a simple
> matter to calculate when and where that event
> will occur.

Agreed.


Imagine you have 2 runners r_0, r_1 starting at p_0.

At each step of the iteration, r_0 moves 1/2 across
L, and r_1 goes 1/3. Therefore, r_0 will always reach
p_1 before r_1 because 1/3 is less than 1/2.

However, if r_1 decides to go much faster by moving
9/10's across L at each iteration, then it will overtake
the now slower moving r_0 because 9/10 is greater
than 1/2.

[Dan Christensen]

p_0 will never be equal to, and never be greater than p_1. As such, this is not a good model for a faster runner overtaking a slower one.

[Chris M. Thomasson]

> "Dan Christensen" wrote in message

> news:c10bb922-3856-4836...@googlegroups.com...

> On Monday, May 12, 2014 4:53:18 PM UTC-4, Chris M. Thomasson wrote:

> [...]

> p_0 will never be equal to, and never be
> greater than p_1. As such, this is not a
> good model for a faster runner overtaking
> a slower one.

Humm... How about:


Let line l_0 connect points p_0 and p_1.

Let point p_2 be equal to p_0.

Let point p_3 be 1/2 across l_0.



The algorithm:

1. Draw a line l_1 connecting p_2 and p_1.

2. Draw a line l_2 connecting p_3 and p_1.

3. Update p_2 to be 1/2 across l_1.

4. Update p_3 to be 1/4 across l_2.

5. if distance(p_1, p_2) < distance(p_1, p_3) then halt.

6. Go to step 1.



AFAICT, this means that p_2 will beat p_3
to p_1 even though p_3 has a 1/2 head start
down l_0. The algorithm will halt and not get
into an infinite loop.

Is this algorithm a better model for showing how
a runner with a head start can be overtaken by a
faster runner?

[Martin Shobe]

On 5/12/2014 2:51 PM, Dan Christensen wrote:
> On Monday, May 12, 2014 2:40:00 PM UTC-4, Martin Shobe wrote:
>
>>
>> The heart of your speed argument was to treat it as a whole and ignore
>>
>> the fact that it could be broken up into infinitely many sub-tasks. You
>>
>> said, "Looking at it as an infinite number of sub-tasks to be completed
>>
>> complicates the problem unnecessarily."
>>
>
> Well said. But so what? The precise notion of speed, however it may have been arrived at historically, allows you to mathematically solve the problem of a faster runner overtaking a slower one in the above scenario. You can calculate when and where the event will happen. What more is to be said?

Quite a bit. You need to be able to handle varying speeds (which your
current analysis fails to handle). You need to be able to handle
arbitrary partitions of the path (which your current analysis fails to
handle). Etc.

> I suppose there was never really any paradox, just a poorly formulated, needlessly complicated analysis -- a dead-end as it turned out.

Of course there wasn't really a paradox. The fact that there's a
resolution to it shows that. However, that "needlessly overly
complicated analysis" has to be made to work since all the steps before
the last one are valid. You still have some work to do to resolve it.

Martin Shobe

[Dan Christensen]

On Monday, May 12, 2014 8:36:36 PM UTC-4, Martin Shobe wrote:

> > Well said. But so what? The precise notion of speed, however it may have been arrived at historically, allows you to mathematically solve the problem of a faster runner overtaking a slower one in the above scenario. You can calculate when and where the event will happen. What more is to be said?
>
>
>
> Quite a bit. You need to be able to handle varying speeds (which your
>
> current analysis fails to handle). You need to be able to handle
>
> arbitrary partitions of the path (which your current analysis fails to
>
> handle). Etc.
>

Again, in the above scenario with constant speeds in a straight line, algebraic methods are best approach. It makes the resolution of Zeno's "paradox" accessible to students in a pre-calculus course. Typically, it is resolved -- even in this simplified scenario -- with a converging series.

>
>
> > I suppose there was never really any paradox, just a poorly formulated, needlessly complicated analysis -- a dead-end as it turned out.
>
>
>
> Of course there wasn't really a paradox. The fact that there's a
>
> resolution to it shows that. However, that "needlessly overly
>
> complicated analysis" has to be made to work since all the steps before
>
> the last one are valid.

No, it doesn't. This approach is a dead-end. The very fact that this analysis concludes that a faster runner cannot possibly overtake a slower one should tip you off to that. It's like some kind of prank on a gullible public.

[Dan Christensen]

I think you are missing the point. We want to determine the point at which the faster runner catches up to the slower one using simple algebra. There is no need to decompose the race into infinite, ever decreasing intervals as seems to be the usual approach.

[Chris M. Thomasson]

> "Dan Christensen" wrote in message

> news:0baaa820-35a9-4b4e...@googlegroups.com...

> On Monday, May 12, 2014 6:08:35 PM UTC-4, Chris M. Thomasson wrote:
> > > "Dan Christensen" wrote in message

> > Is this algorithm a better model for showing how
> >
> > a runner with a head start can be overtaken by a
> >
> > > faster runner?

> I think you are missing the point. We want to determine
> the point at which the faster runner catches up to the
> slower one using simple algebra.

Okay. So an equation that will tell exactly what iteration
number will be responsible for rendering a frame in which
the the faster runner will be equal to, or overcome the
slower runner. Right?

> There is no need to
> decompose the race into infinite, ever decreasing
> intervals as seems to be the usual approach.

Well, this setup does allows for an infinitely long run
on a track with finite area...

[Dan Christensen]

On Monday, May 12, 2014 11:09:28 PM UTC-4, Chris M. Thomasson wrote:
> > "Dan Christensen" wrote in message
>
> > news:0baaa820-35a9-4b4e...@googlegroups.com...
>
> > On Monday, May 12, 2014 6:08:35 PM UTC-4, Chris M. Thomasson wrote:
>
> > > > "Dan Christensen" wrote in message
>
> > > Is this algorithm a better model for showing how
>
> > >
>
> > > a runner with a head start can be overtaken by a
>
> > >
>
> > > > faster runner?
>
>
>
> > I think you are missing the point. We want to determine
>
> > the point at which the faster runner catches up to the
>
> > slower one using simple algebra.
>
>
>
> Okay. So an equation that will tell exactly what iteration
>
> number will be responsible for rendering a frame in which
>
> the the faster runner will be equal to, or overcome the
>
> slower runner. Right?
>

Not really. Here is the derivation of formula in my original posting here:

Let S_A and S_T be the speeds (m/s) of Achilles and the Tortoise respectively.

The distance (m) covered by Achilles at time t (s) is S_A * t.

The distance (m) covered by the Tortoise at time t (s) is S_T * t + 100. Recall that the Tortoise has a 100 m head start.

When Achilles catches up to the Tortoise, their distances traveled will be the same. At that point, we will have:

S_A * t = S_T * t + 100.

Solving for t:

t = 100 / (S_A - S_T)

Example: If S_A = 11 m/s and S_T = 1 m/s, then Achilles will catch up to the Tortoise in 10 s.

I hope this helps.

[Martin Shobe]

On 5/12/2014 9:36 PM, Dan Christensen wrote:
> On Monday, May 12, 2014 8:36:36 PM UTC-4, Martin Shobe wrote:
>
>>> Well said. But so what? The precise notion of speed, however it may have been arrived at historically, allows you to mathematically solve the problem of a faster runner overtaking a slower one in the above scenario. You can calculate when and where the event will happen. What more is to be said?
>>
>>
>>
>> Quite a bit. You need to be able to handle varying speeds (which your
>>
>> current analysis fails to handle). You need to be able to handle
>>
>> arbitrary partitions of the path (which your current analysis fails to
>>
>> handle). Etc.
>>
>
> Again, in the above scenario with constant speeds in a straight line, algebraic methods are best approach. It makes the resolution of Zeno's "paradox" accessible to students in a pre-calculus course. Typically, it is resolved -- even in this simplified scenario -- with a converging series.

Once again, you insist on solving the wrong problem. The problem was
never trying to determine when Achilles caught the tortoise. The Greeks
knew that Achilles would catch the tortoise. The problem is that they
also had an argument that shows Achilles not catching the tortoise. To
resolve the problem, you have to explain what goes wrong with the
reasoning that ends in Achilles not catching the tortoise, and coming up
with valid methods of handling that argument. You should know enough
logic to realize that you can't do that by showing alternate methods of
proving Achilles catches the tortoise.

>>> I suppose there was never really any paradox, just a poorly formulated, needlessly complicated analysis -- a dead-end as it turned out.
>>
>>
>>
>> Of course there wasn't really a paradox. The fact that there's a
>>
>> resolution to it shows that. However, that "needlessly overly
>>
>> complicated analysis" has to be made to work since all the steps before
>>
>> the last one are valid.
>
> No, it doesn't. This approach is a dead-end. The very fact that this analysis concludes that a faster runner cannot possibly overtake a slower one should tip you off to that. It's like some kind of prank on a gullible public.

You should know enough logic by now to know that it doesn't matter how
complex things get as long as every step remains valid. If a complex but
valid process gets a different answer than a valid simple process, you
still have a paradox that needs resolving.

Martin Shobe

[Dan Christensen]

On Tuesday, May 13, 2014 1:15:11 AM UTC-4, Martin Shobe wrote:
> On 5/12/2014 9:36 PM, Dan Christensen wrote:
>
> > On Monday, May 12, 2014 8:36:36 PM UTC-4, Martin Shobe wrote:
>
> >
>
> >>> Well said. But so what? The precise notion of speed, however it may have been arrived at historically, allows you to mathematically solve the problem of a faster runner overtaking a slower one in the above scenario. You can calculate when and where the event will happen. What more is to be said?
>
> >>
>
> >>
>
> >>
>
> >> Quite a bit. You need to be able to handle varying speeds (which your
>
> >>
>
> >> current analysis fails to handle). You need to be able to handle
>
> >>
>
> >> arbitrary partitions of the path (which your current analysis fails to
>
> >>
>
> >> handle). Etc.
>
> >>
>
> >
>
> > Again, in the above scenario with constant speeds in a straight line, algebraic methods are best approach. It makes the resolution of Zeno's "paradox" accessible to students in a pre-calculus course. Typically, it is resolved -- even in this simplified scenario -- with a converging series.
>
>
>
> Once again, you insist on solving the wrong problem. The problem was
>
> never trying to determine when Achilles caught the tortoise.

Actually that is precisely the problem. From my original posting here:

As Aristotle wrote, "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."

What a bizarre world view, but there you have it.

> The Greeks
>
> knew that Achilles would catch the tortoise.

Not the professional navel-gazers among them (Aristotle et al).

> The problem is that they
>
> also had an argument that shows Achilles not catching the tortoise. To
>
> resolve the problem, you have to explain what goes wrong with the
>
> reasoning that ends in Achilles not catching the tortoise, and coming up
>
> with valid methods of handling that argument.

No, the problem was to first determine IF Achilles catches up to the Tortoise. The navel-gazers denied it. Then to determine WHEN and WHERE.

[Roland Franzius]

Not for Aristotle.

He is famous for putting Platonic idealism back on the floor of physical
kwnowledge.

In teaching philosophy of course one has to refer all fruitful ideas,
open questions, controversies and fallacies from the past. This does not
say that the author (who was defnitely not Aristotele but all of his
school) is a follower of a certain school, especially here the Eleates.

>
>> The problem is that they
>>
>> also had an argument that shows Achilles not catching the tortoise.
>> To
>>
>> resolve the problem, you have to explain what goes wrong with the
>>
>> reasoning that ends in Achilles not catching the tortoise, and
>> coming up
>>
>> with valid methods of handling that argument.
>
>
> No, the problem was to first determine IF Achilles catches up to the
> Tortoise. The navel-gazers denied it. Then to determine WHEN and
> WHERE.

Arises the question what your dowonload software does.

Aristoteles software deduced from a set of axioms an apparent paradox
with respect to physics that cautioned the students to not to believe to
much into logical production if the physical world shows the opposite to
be true.

This scientific principle works all the times exept in strongly
religiously predefined educational systems.

Even in the times of Aristoteles the Akademia was full of
Kounoupiachorions (M�ckenheims) with their half baked knowlegde pudding
of math and logic with a chocolate cover of logical absurdities.

The fun with this kind of logics was always something to make hard
mathematical work interesting for learners during the coffee break.

Just read the problems book of Aristotle (2 Volumes, full with questions
like "why men have balls?", "why is breathe hot but a blow cold?") and
have more fun.

--

Roland Franzius

[Peter Percival]

Dan Christensen wrote:

> This turns out not be a very useful way to look at space-time. The
> fact is, motion does occur. Contrary to Aristotle, faster runners do
> overtake slower runners.

Surely Harry Stotle never doubted it. The question is not, we know it
happens so where and when does it happen? Rather the question is we
know it happens but how is that possible? You have answered the first
question but not the second.

> If they are running at a constant speed
> along the same straight line, it is a simple matter to calculate when
> and where that event will occur.

--
[...] They listened at his heart.
Little-less-nothing!-and that ended it.
No more to build on there. And they, since they
Were not the one dead, turned to their affairs.
"Out, Out-", Robert Frost, 1916.

[scattered]

Dan Christensen has some weird ideas about the world. He actually wrote

"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."

What a bizarre world view, but there you have it.

I wonder why Dan Christensen doesn't understand that in the real
world the quickest runner will in fact overtake the slowest. Maybe he
doesn't know elementary algebra.

One could object that I am ascribing to Dan an argument which he disagrees
with but is merely quoting. However, I am following Dan's hermeneutic.
After all, Dan himself ascribes to Aristotle an argument of Zeno which
Aristotle is merely quoting in order to refute.

Oh dang -- I guess that now *I* must believe in the impossibility of motion as
well since I just quoted it.

More seriously: why not *study* the history of philosophy before commenting on
it? Something as simple as reading the Wikipedia article on Zeno's paradox
would be sufficient to discover that Aristotle did not endorse Zeno's views.

[Dan Christensen]

It seems I owe Aristotle et al an apology. The quote from Aristotle at Wiki is a bit misleading:

"'In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.' - as recounted by Aristotle, Physics VI:9, 239b15"
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise

This is not, as you might think, a statement of Aristotle's view on the matter. Rather it is a recounting of Zeno's ideas. From a translation of his works (with my emphasis), we see that he actually disagrees with Zeno's notion that a faster runner cannot overtake a slower runner:

"Zeno's arguments about motion, which cause so much disquietude to
those who try to solve the problems that they present, are four in
number. The first asserts the non-existence of motion on the ground
that that which is in locomotion must arrive at the half-way stage
before it arrives at the goal. This we have discussed above.

"The second is the so-called 'Achilles', and it amounts to this, that
in a race the quickest runner can never overtake the slowest, since

the pursuer must first reach the point whence the pursued started,

so that the slower must always hold a lead. This argument is the same
in principle as that which depends on bisection, though it differs
from it in that the spaces with which we successively have to deal
are not divided into halves. The result of the argument is that the
slower is not overtaken: but it proceeds along the same lines as the
bisection-argument (for in both a division of the space in a certain
way leads to the result that the goal is not reached, though the 'Achilles'
goes further in that it affirms that even the quickest runner in legendary
tradition must fail in his pursuit of the slowest), so that the solution
must be the same. AND THE AXIOM THAT THAT WHICH HOLDS A LEAD IS NEVER OVERTAKEN IS FALSE: it is not overtaken, it is true, while it holds
a lead: but it is overtaken nevertheless if it is granted that it
traverses the finite distance prescribed."
http://classics.mit.edu/Aristotle/physics.mb.txt

[Dan Christensen]

On Tuesday, May 13, 2014 6:58:41 AM UTC-4, scattered wrote:
> Dan Christensen has some weird ideas about the world. He actually wrote
>
>
>
> "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."
>
>
>
> What a bizarre world view, but there you have it.
>
>
>
> I wonder why Dan Christensen doesn't understand that in the real
>
> world the quickest runner will in fact overtake the slowest. Maybe he
>
> doesn't know elementary algebra.
>

This is not my world view, but that of the Greek philosopher, Zeno (which I mistakenly attributed to Artistotle as well). See my reply to Roland.

[Dan Christensen]

On Tuesday, May 13, 2014 6:37:55 AM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
>
>
>
> > This turns out not be a very useful way to look at space-time. The
>
> > fact is, motion does occur. Contrary to Aristotle, faster runners do
>
> > overtake slower runners.
>
>
>
> Surely Harry Stotle never doubted it.

I mistakenly attributed the ideas of Zeno to Aristotle. Aristotle actually rejected Zeno's conclusion. See my reply just now to Roland.

> The question is not, we know it
>
> happens so where and when does it happen? Rather the question is we
>
> know it happens but how is that possible?

It was Zeno's bizarre conclusion that a faster runner can never overtaken a slower one. (Aristotle rejected that "axiom" as he called it.)

> You have answered the first
>
> question but not the second.
>

The notion of speed (v = d/t), which was formulated by Galileo some 2,000 years after Zeno, is the answer. If the speeds of the runners are constant along a straight line, it is a matter of simple algebra to prove that it is possible. The decomposition of the race into infinite, ever decreasing intervals is completely unnecessary -- a distraction and a dead-end to complete understanding.

[Chris M. Thomasson]

> "Dan Christensen" wrote in message

> news:2432e912-33f7-44d0...@googlegroups.com...

> On Monday, May 12, 2014 11:09:28 PM UTC-4, Chris M. Thomasson wrote:
> > > "Dan Christensen" wrote in message
> >
> > > news:0baaa820-35a9-4b4e...@googlegroups.com...
> >
> > > On Monday, May 12, 2014 6:08:35 PM UTC-4, Chris M. Thomasson wrote:

> [...]

> > Okay. So an equation that will tell exactly what iteration
> >
> > number will be responsible for rendering a frame in which
> >
> > the the faster runner will be equal to, or overcome the
> >
> > slower runner. Right?
> >

> Not really. Here is the derivation of formula in my
> original posting here:

> [...]

> Example: If S_A = 11 m/s and S_T = 1 m/s, then
> Achilles will catch up to the Tortoise in 10 s.

> I hope this helps.

It makes perfect sense to me.

[netzweltler]

Am Montag, 12. Mai 2014 22:53:18 UTC+2 schrieb Chris M. Thomasson:
>
> I guess the question is how close does p_0 have to
> get to p_1 before we can assume they are equal and
> halt the algorithm in step 3? Once this occurs we can
> say that the action of moving p_0 to p_1 is complete
> because we found that p_0 = p_1 during the iteration
> process.

"After infinitely many steps" p_0 is close enough to p_1,
because there is no space between p_0 and p_1 anymore.
Nevertheless, p_0 does not equal p_1, even "after
infinitely many steps".

[dull...@sprynet.com]

On Mon, 12 May 2014 07:34:31 -0700 (PDT), Dan Christensen
<Dan_Chr...@sympatico.ca> wrote:

>On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>
>>
>> You solved the wrong problem. The problem isn't "when will Achilles
>>
>> overtake the tortoise?" The problem is "how can Achilles overtake the
>>
>> tortoise when he has to complete an infinite number of subtasks to do so?".
>>
>
>Do you really dispute that speed equals distance over time?

For heaven's sake, do you really think he's disputing that?

The point is that you're not "resolving" the "paradox",
you're simply ignoring the paradox.


>
>Looking at it as an infinite number of subtasks to be completed complicates the problem unnecessarily, as shown by the fact that you can actually calculate when and where the event will occur using a simple formula. What more do we need to know?

[Dan Christensen]

On Wednesday, May 14, 2014 12:54:31 PM UTC-4, dull...@sprynet.com wrote:
> On Mon, 12 May 2014 07:34:31 -0700 (PDT), Dan Christensen
>
> <Dan_Chr...@sympatico.ca> wrote:
>
>
>
> >On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>
> >
>
> >>
>
> >> You solved the wrong problem. The problem isn't "when will Achilles
>
> >>
>
> >> overtake the tortoise?" The problem is "how can Achilles overtake the
>
> >>
>
> >> tortoise when he has to complete an infinite number of subtasks to do so?".
>
> >>
>
> >
>
> >Do you really dispute that speed equals distance over time?
>
>
>
> For heaven's sake, do you really think he's disputing that?
>
>
>
> The point is that you're not "resolving" the "paradox",
>
> you're simply ignoring the paradox.
>

Using the speed formula and simple algebra, you can prove that not only does Achilles pass the Tortoise (actually disputed by Zeno and others), but you can determine precisely when and where this event will occur. What more do you want? The decomposition of the race into infinitely many, ever decreasing intervals was completely unnecessary in this case.

Perhaps I was being unfair, calling it a dead-end. With the development of a whole lot of extra machinery, the paradox can also be resolved using this very composition and the methods of modern calculus.

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message
> news:lkuku4$26s$1...@speranza.aioe.org...

> > "Dan Christensen" wrote in message

> [...]

> > I hope this helps.

> It makes perfect sense to me.

WRT the runners on an infinite track in a finite
plane algorithm, I could use the iteration count
to define time. This turns it into a parametric
equation where each iteration, or frame, represents
a derivative of movement.

[Chris M. Thomasson]

> "netzweltler" wrote in message
> news:3b45cf4f-5eb4-4daf...@googlegroups.com...

> Am Montag, 12. Mai 2014 22:53:18 UTC+2 schrieb Chris M. Thomasson:
> >
> > I guess the question is how close does p_0 have to
> > get to p_1 before we can assume they are equal and
> > halt the algorithm in step 3? Once this occurs we can
> > say that the action of moving p_0 to p_1 is complete
> > because we found that p_0 = p_1 during the iteration
> > process.

> "After infinitely many steps" p_0 is close enough to p_1,
> because there is no space between p_0 and p_1 anymore.

Well, there might be "some space" if you zoom in enough to
observe it?

> Nevertheless, p_0 does not equal p_1, even "after
> infinitely many steps.

Right. So, p_1 - p_0 is the distance of space between
them, and will never be zero. IMHO, if you zoom in and
visually plot this distance, it will not seem so small
anymore?

[djoyce099]

Here is a novel idea.

Using the sqrt(2) to make my point!

Turtle shmertle.

You have two runners, the first runner is at 1.5 km from the
starting line and the other runner is at 1.4 km from the starting
line. The first runner was given a starting handicap. ;-)

At smaller smaller time intervals you have --

First runner --------- Second runner.

1.5 km------------------- 1.4 km. 2nd runner 1/10 km behind ----stop watch sum.
1.42 km------------------ 1.41 km. " " 1/100 km behind ---- 1/10 sec.+
1.415 km----------------- 1.414 km. " " 1/1000 km behind ----1/100 sec.+
1.4143 km---------------- 1.4142 km. " " 1/10000 km behind ---1/1000 sec.+
1.41422 km. ------------- 1.41421 km. " " 1/100000 km behind --1/10000 sec.+
etc.

The second runner will keep advancing on the first runner but the smaller
and smaller time interval along with the shorter and shorter distances covered
will never overtake the first runner.

Just a thought.

Cheers,

Dan

[Spac...@hotmail.com]

the mini-M-sets are artefacts of "zooming around
in hte complex floatingpoint specification (IEEE-754,
-854 ;the first is an article in a magazine,
teh second I have not seen

[Dan Christensen]

So, we have two competing models of movement through physical space: One says it is impossible for a faster runner to overtake a slower one. The other says that, not only is it possible, but you can calculate precisely where and when this event will occur. Which one shall we choose? Hmmmmmmm....

[Chris M. Thomasson]

> "djoyce099" wrote in message
> news:d7eac4f9-9c9f-495c...@googlegroups.com...

> Here is a novel idea.
> Using the sqrt(2) to make my point!

;^)

How about something "crazy" like:


Let the 2d point p_0 be a type that contains its
x and y components respectively, where:

p_0 = (0, 0)


Let the 2d point p_1 be a type that contains its
x and y components respectively, where:

p_1 = (100, 0)



If the y component of p_0 and p_1 are equal, they
can be omitted from the process.

0 = 0, therefore we will only focus on the x
component:

Let the value p_1_x be equal to the x component
of p_1.



Imagine we have two runners declared as the values
r0 and r1 having a race from p_0 to p_1. These values
represent where the racers are along the x-axis of the
line connecting p_0 and p_1.



Let the iterative equation of the first runner r0 be:
_______________________________________
r0_[0] = 0

r0_[n+1] = r0_[n] + (p_1_x - r0_[n]) / 2
_______________________________________


Let the iterative equation of the second runner
r1 be:
_______________________________________
r1_[0] = p_1_x / 2

r1_[n+1] = r1_[n] + (p_1_x - r1_[n]) / 4
_______________________________________



Here is output generated by the iterative process
of runner r0:
_______________________________________
r0_[0] = 0
r0_[1] = 0 + (100 - 0) / 2 = 50
r0_[2] = 50 + (100 - 50) / 2 = 75
r0_[3] = 75 + (100 - 75) / 2 = 87.5
r0_[4] = 87.5 + (100 - 87.5) / 2 = 93.75
_______________________________________



And here is output generated by the iterative process
of runner r1:
_______________________________________
r1_[0] = 60
r1_[1] = 60 + (100 - 60) / 4 = 70
r1_[2] = 70 + (100 - 70) / 4 = 77.5
r1_[3] = 77.5 + (100 - 77.5) / 4 = 83.125
r1_[4] = 83.125 + (100 - 83.125) / 4 = 87.34375
_______________________________________


Therefore, I can say the the faster runner r0 passes
the slower runner r1 along the x-axis, even though
r1 had a 60% head start handicap! This event occurs
when n = 3.

[0...3] is the interval when r0 will catch up to r_1, which
means the r0 overcomes r1 in 4 steps of iteration where:

r1_[3] is less than r0_[3], because 83.125 < 87.5, which
means that the faster runner overcomes the slower one
when n = 3.



Does this make any sense?

;^o


Is it a _completely_ shitty base for any sort of formal
proof?


Sorry!!!

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll1aa2$fsl$1...@speranza.aioe.org...

> Let the iterative equation of the second runner
> r1 be:
> _______________________________________
> r1_[0] = p_1_x / 2
> r1_[n+1] = r1_[n] + (p_1_x - r1_[n]) / 4
> _______________________________________

UMMMM! Of course r1_[0] should equal 60 to equal
the 60% of 100 head start handicap of runner r1!

Damn bugs!


Grrrrrrr.....


Actually, in the iterative equation for r0 and r1, I really
do not need to initialize the n = 0 state of the runners.

I can do that later when I define concrete values to define
the race.

Humm...


Any advise?

[djoyce099]

On Wednesday, May 14, 2014 9:00:41 PM UTC-4, djoyce099 wrote:

Ok, can anyone find out what the handicap at the starting line
(so many meters off the line for the first runner) given the distance and time
shown above?
There probably is an answer but I don't have it.

[netzweltler]

Am Donnerstag, 15. Mai 2014 01:32:51 UTC+2 schrieb Chris M. Thomasson:
>> "netzweltler" wrote in message
>> news:3b45cf4f-5eb4-4daf...@googlegroups.com...
>
>> Am Montag, 12. Mai 2014 22:53:18 UTC+2 schrieb Chris M. Thomasson:
>>>
>>> I guess the question is how close does p_0 have to
>>> get to p_1 before we can assume they are equal and
>>> halt the algorithm in step 3? Once this occurs we can
>>> say that the action of moving p_0 to p_1 is complete
>>> because we found that p_0 = p_1 during the iteration
>>> process.
>
>> "After infinitely many steps" p_0 is close enough to p_1,
>> because there is no space between p_0 and p_1 anymore.
>
> Well, there might be "some space" if you zoom in enough to
> observe it?

If there is space left you haven't done infinitely many
steps.

>> Nevertheless, p_0 does not equal p_1, even "after
>> infinitely many steps.
>
> Right. So, p_1 - p_0 is the distance of space between
> them, and will never be zero. IMHO, if you zoom in and
> visually plot this distance, it will not seem so small
> anymore?

You are drawing a line from p_0 to p_1 in infinitely
many steps. If we are creating a list of these steps,
then this is the complete list (p_0 = 0, p_1 = 1):

t = 0 s: draw a line from 0 to 0.5 in 0.5 s
t = 0.5 s: draw a line from 0.5 to 0.75 in 0.25 s
t = 0.75 s: draw a line from 0.75 to 0.875 in 0.125 s
...

At t = 1 second we have done infinitely many steps.
There is no step on this list like

t = x s: drawing the line from x to 1.0 in (1-x) s

So, p_1 will not be covered, but every point to the
left of p_1.

No space left, no matter how much you zoom in.

[Chris M. Thomasson]

> the mini-M-sets are artefacts of "zooming around
> in hte complex floatingpoint specification (IEEE-754,
> -854 ;the first is an article in a magazine,
> teh second I have not seen

I always thought that the M-Sets are generated by
a "fixed" iteration of a formula (z_[n+1] = z_[n]^2 + c)
where each c represents each pixel, and stays constant
during iteration.

When the real and imaginary parts of z "trip" the
following condition under iteration:
_______________________________________
if (z.real^2 + z.imaginary^2 > 4)
{
// inset
halt;
}
_______________________________________

we can "assume" that the point is "outside" of the
radius trap.

The border of points that escape and are outside, vs.
the ones that do not and are inside.

Inside vs. outside defines the infinite boundary of the
M-Set under the given level of iteration...

[Ross A. Finlayson]

Or, it's that they start in an
order, defined by the initial
distance.

Then classically no matter what
order of motions the most proximal
or likely initial winner has to get
there, all the other runners get the
same order of motions.

Ways to look at it.

Here the "fixed" velocities of the
runners forms an effective ratio.

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll1eo0$pdl$1...@speranza.aioe.org... [...]

> When the real and imaginary parts of z "trip" the
> following condition under iteration:
> ___________________________________

> if (z.real^2 + z.imaginary^2 > 4)
> {
> // inset
> halt;
> }
> ___________________________________

YIKES!!! I meant _outside_ in the line before
the halt;

if (z.real^2 + z.imaginary^2 > 4)
{

// the pixel is out of the set!!!

halt;
}


damn it!

[dull...@sprynet.com]

On Wed, 14 May 2014 10:26:51 -0700 (PDT), Dan Christensen

<Dan_Chr...@sympatico.ca> wrote:

>On Wednesday, May 14, 2014 12:54:31 PM UTC-4, dull...@sprynet.com wrote:
>> On Mon, 12 May 2014 07:34:31 -0700 (PDT), Dan Christensen
>>
>> <Dan_Chr...@sympatico.ca> wrote:
>>
>>
>>
>> >On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>>
>> >
>>
>> >>
>>
>> >> You solved the wrong problem. The problem isn't "when will Achilles
>>
>> >>
>>
>> >> overtake the tortoise?" The problem is "how can Achilles overtake the
>>
>> >>
>>
>> >> tortoise when he has to complete an infinite number of subtasks to do so?".
>>
>> >>
>>
>> >
>>
>> >Do you really dispute that speed equals distance over time?
>>
>>
>>
>> For heaven's sake, do you really think he's disputing that?
>>
>>
>>
>> The point is that you're not "resolving" the "paradox",
>>
>> you're simply ignoring the paradox.
>>
>
>Using the speed formula and simple algebra, you can prove that not only does Achilles pass the Tortoise
> (actually disputed by Zeno and others), but you can determine precisely when and where this event will occur.

Do you really think that this is not completely obvious to everyone?

To "resolve" a paradox is not to explain what actually happens; to
resolve a paradox is to explain away the seemingly paradoxical
nature of whatever.

Jeez. Never mind. Congratuations - you're the first person in
history to realize that distance is speed times time. Wow.

[Dan Christensen]

On Thursday, May 15, 2014 1:50:24 PM UTC-4, dull...@sprynet.com wrote:
> On Wed, 14 May 2014 10:26:51 -0700 (PDT), Dan Christensen
>
> <Dan_Chr...@sympatico.ca> wrote:
>
>
>
> >On Wednesday, May 14, 2014 12:54:31 PM UTC-4, dull...@sprynet.com wrote:
>
> >> On Mon, 12 May 2014 07:34:31 -0700 (PDT), Dan Christensen
>
> >>
>
> >> <Dan_Chr...@sympatico.ca> wrote:
>
> >>
>
> >>
>
> >>
>
> >> >On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>
> >>
>
> >> >
>
> >>
>
> >> >>
>
> >>
>
> >> >> You solved the wrong problem. The problem isn't "when will Achilles
>
> >>
>
> >> >>
>
> >>
>
> >> >> overtake the tortoise?" The problem is "how can Achilles overtake the
>
> >>
>
> >> >>
>
> >>
>
> >> >> tortoise when he has to complete an infinite number of subtasks to do so?".
>
> >>
>
> >> >>
>
> >>
>
> >> >
>
> >>
>
> >> >Do you really dispute that speed equals distance over time?
>
> >>
>
> >>
>
> >>
>
> >> For heaven's sake, do you really think he's disputing that?
>
> >>
>
> >>
>
> >>
>
> >> The point is that you're not "resolving" the "paradox",
>
> >>
>
> >> you're simply ignoring the paradox.
>
> >>
>
> >
>
> >Using the speed formula and simple algebra, you can prove that not only does Achilles pass the Tortoise
>
> > (actually disputed by Zeno and others), but you can determine precisely when and where this event will occur.
>

> Do you really think that this is not completely obvious to everyone?
>

It wasn't obvious to Zeno and his followers.

>
>
> To "resolve" a paradox is not to explain what actually happens; to
>
> resolve a paradox is to explain away the seemingly paradoxical
>
> nature of whatever.
>

We have identified the faulty assumption that an infinite number of events cannot occur in a finite time interval. And we don't need calculus to get to the bottom of it. Simple algebra will do.

[Peter Percival]

Dan Christensen wrote:

> We have identified the faulty assumption that an infinite number of
> events cannot occur in a finite time interval.

Zeno surely knew that Achilles would overtake the tortoise. The problem
is that A. must pass though every point that T. has already passed
through. How is that possible? That it _is_ possible no one doubts,
but an explanation of _why_ is not, it seems to me, just a matter of
'simple algebra'. You might like to read
http://www.mathpages.com/rr/s3-07/3-07.htm .

> And we don't need
> calculus to get to the bottom of it. Simple algebra will do.

[Dan Christensen]

On Thursday, May 15, 2014 2:33:07 PM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
>
>
>
> > We have identified the faulty assumption that an infinite number of
>
> > events cannot occur in a finite time interval.
>
>
>
> Zeno surely knew that Achilles would overtake the tortoise. The problem
>
> is that A. must pass though every point that T. has already passed
>
> through. How is that possible? That it _is_ possible no one doubts,
>
> but an explanation of _why_ is not, it seems to me, just a matter of
>
> 'simple algebra'.

The explanation is indeed just a simple application of the Galileo's speed formula v=d/t. Prior to Galileo's breakthrough, people didn't have the tools to measure speed and to predict where and when a faster moving object might overtake a slower moving one. Now, we almost can't imagine that people ever had a problem with the idea.

[Roland Franzius]

Am 15.05.2014 20:16, schrieb Dan Christensen:
>
> We have identified the faulty assumption that an infinite number of
> events cannot occur in a finite time interval. And we don't need
> calculus to get to the bottom of it. Simple algebra will do.

Define "occur".

You will never reach the intellectual level of Demokritos, Parmenides,
Zenon, Diogenes, Aristotle just by introducing some more undefined terms
of modern English.

And we definitely need calculus as a proof of existence that defining
the mechanical or geometrical algebra is not an empty game.

--

Roland Franzius

[Dan Christensen]

On Thursday, May 15, 2014 2:59:51 PM UTC-4, Dan Christensen wrote:

>
> The explanation is indeed just a simple application of the Galileo's speed formula v=d/t. Prior to Galileo's breakthrough, people didn't have the tools to measure speed and to predict where and when a faster moving object might overtake a slower moving one. Now, we almost can't imagine that people ever had a problem with the idea.
>

The ancient Greek philosophers also believed a heavier object would fall faster than a lighter one. And that an javelin thrown by a soldier would travel not in a curve, but in a straight line, suddenly stop and fall straight to the ground. It didn't occur them to test their ideas, to take measurements, etc. That was beneath them! They believed they could, by introspection alone, answer all such questions. Ultimately, I suppose, that was the faulty assumption at the heart of Zeno's Paradox, as well.

[Dan Christensen]

On Thursday, May 15, 2014 3:14:04 PM UTC-4, Roland Franzius wrote:
> Am 15.05.2014 20:16, schrieb Dan Christensen:
>
> >
>
> > We have identified the faulty assumption that an infinite number of
>
> > events cannot occur in a finite time interval. And we don't need
>
> > calculus to get to the bottom of it. Simple algebra will do.
>
>
>
> Define "occur".
>

I think my meaning is clear enough.

>
>
> You will never reach the intellectual level of Demokritos, Parmenides,
>
> Zenon, Diogenes, Aristotle just by introducing some more undefined terms
>
> of modern English.
>
>
>
> And we definitely need calculus as a proof of existence that defining
>
> the mechanical or geometrical algebra is not an empty game.
>

For non-constant velocities along curving paths, you will need calculus. But simple algebra will suffice in this case.

[Dan Christensen]

On Thursday, May 15, 2014 3:20:28 PM UTC-4, Dan Christensen wrote:

>
> The ancient Greek philosophers also believed a heavier object would fall faster than a lighter one. And that an javelin thrown by a soldier would travel not in a curve, but in a straight line, suddenly stop and fall straight to the ground.

Actually, it's worse than that! According to Thomas Grissom in "The Physicist's World: The Story of Motion and the Limits to Knowledge" p. 64, Aristotle held that the javelin would go on forever, propelled forward by air rushing in to fill the void created at the end of the javelin!!!

http://books.google.ca/books?id=9FQc7quzSeEC&pg=PA64&lpg=PA64&dq=aristotle+javelin&source=bl&ots=48E0QvX89x&sig=l0udH3eizKWryZ6jrBZNbiZt-IU&hl=en&sa=X&ei=6xx1U8z0O9WeqAaWgoLAAQ&ved=0CEUQ6AEwBA#v=onepage&q=aristotle%20javelin&f=false

[Chris M. Thomasson]

> "djoyce099" wrote in message
> news:6f36bf7d-c4e3-4321...@googlegroups.com...

> Ok, can anyone find out what the handicap at the starting line
> (so many meters off the line for the first runner) given the distance and
> time
> shown above?
> There probably is an answer but I don't have it.

wrt the iterative process, run it backwards...

First of all, get a formula that can guess at a
iteration count to start at, then work backwards
from there?

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll1aa2$fsl$1...@speranza.aioe.org...

> > "djoyce099" wrote in message
> > news:d7eac4f9-9c9f-495c...@googlegroups.com...
> > Here is a novel idea.
> > Using the sqrt(2) to make my point!
>

> How about something "crazy" like:

[...]

Wrt each iteration of the iterative equation
of the faster runner r0:

r0_[n+1] = r0_[n] + (p_1_x - r0_[n]) / 2

Where r0_[0] is equal to the faster runners
initial starting point:

AFAICT, the following stupid simple equation
can be used to gain level of movement per-
iteration:

r0_[n]' = r0_[n + 1] - r0_[n]

Which means each frame r0_[n]' is the derivative
of movement from p_0 to p_1.

The sum of r0_[n]' would be the integral of the
current number of iterations performed.

Is that a valid assertion?

[Chris M. Thomasson]

> "netzweltler" wrote in message news:

> d1bede46-4b68-4417...@googlegroups.com...

> Am Donnerstag, 15. Mai 2014 01:32:51 UTC+2 schrieb Chris M. Thomasson:
> >> "netzweltler" wrote in message
> >> news:3b45cf4f-5eb4-4daf...@googlegroups.com...
> >
> >> Am Montag, 12. Mai 2014 22:53:18 UTC+2 schrieb Chris M. Thomasson:
> >>>
> >>> I guess the question is how close does p_0 have to
> >>> get to p_1 before we can assume they are equal and
> >>> halt the algorithm in step 3? Once this occurs we can
> >>> say that the action of moving p_0 to p_1 is complete
> >>> because we found that p_0 = p_1 during the iteration
> >>> process.
> >
> >> "After infinitely many steps" p_0 is close enough to p_1,
> >> because there is no space between p_0 and p_1 anymore.
> >
> > Well, there might be "some space" if you zoom in enough to
> > observe it?

> If there is space left you haven't done infinitely many
> steps.

Okay. I agree with you here. This would be akin to an
infinite amount of iterations in my algorithm posted
in this thread. So if you observe infinity, you have
obviously stopped which means you have not completed
infinity many steps!

Is this a decent way to look at infinity long iterations
of running on a finite track?

> >> Nevertheless, p_0 does not equal p_1, even "after
> >> infinitely many steps.

This sounds like a bit of a paradox...

> > Right. So, p_1 - p_0 is the distance of space between
> > them, and will never be zero. IMHO, if you zoom in and
> > visually plot this distance, it will not seem so small
> > anymore?

> You are drawing a line from p_0 to p_1 in infinitely
> many steps. If we are creating a list of these steps,
> then this is the complete list (p_0 = 0, p_1 = 1):

> t = 0 s: draw a line from 0 to 0.5 in 0.5 s
> t = 0.5 s: draw a line from 0.5 to 0.75 in 0.25 s
> t = 0.75 s: draw a line from 0.75 to 0.875 in 0.125 s
> ...

> At t = 1 second we have done infinitely many steps.
> There is no step on this list like

> t = x s: drawing the line from x to 1.0 in (1-x) s

> So, p_1 will not be covered, but every point to the
> left of p_1.

> No space left, no matter how much you zoom in.

So t = 1 is akin to t = Positive Infinity? If so, AFAICT, that
means there can be no space left, however, p_0 does
not equal p_1. So, p_1 – p_0 is non-zero which means
that there is some “logical” space left per iteration of
the infinite process. The act of examining an iteration
frame means that infinite precision is lost. The act of
observing a frame logically stops Infinity from existing
wrt the problem at hand at that moment in the iteration
frame?

IMVVVVHO, there is a paradox that p_1 – p_0[Infinity]
does not equal zero!

;^o

[Martin Shobe]

On 5/15/2014 3:38 PM, Dan Christensen wrote:
> On Thursday, May 15, 2014 3:20:28 PM UTC-4, Dan Christensen wrote:
>
>>
>> The ancient Greek philosophers also believed a heavier object would fall faster than a lighter one. And that an javelin thrown by a soldier would travel not in a curve, but in a straight line, suddenly stop and fall straight to the ground.
>
> Actually, it's worse than that! According to Thomas Grissom in "The Physicist's World: The Story of Motion and the Limits to Knowledge" p. 64, Aristotle held that the javelin would go on forever, propelled forward by air rushing in to fill the void created at the end of the javelin!!!
>
> http://books.google.ca/books?id=9FQc7quzSeEC&pg=PA64&lpg=PA64&dq=aristotle+javelin&source=bl&ots=48E0QvX89x&sig=l0udH3eizKWryZ6jrBZNbiZt-IU&hl=en&sa=X&ei=6xx1U8z0O9WeqAaWgoLAAQ&ved=0CEUQ6AEwBA#v=onepage&q=aristotle%20javelin&f=false

Reading comprehension is not your forte, is it? Your source does not say
that Aristotle held that belief, just that his explanation of motion
*could* result in that happening.

Martin Shobe

[djoyce099]

On Thursday, May 15, 2014 5:34:07 PM UTC-4, Chris M. Thomasson wrote:
> > "djoyce099" wrote in message
>
> > news:6f36bf7d-c4e3-4321...@googlegroups.com...
>

> First of all, get a formula that can guess at a
>
> iteration count to start at, then work backwards
>
> from there?

I will give it a try. The time element for the second runner
could be an important part of how to get the starting point
handicap for the first runner.

[Jim Burns]

It's not really clear to me how we have resolved Zeno's
paradox, though it's clear that we have. I wonder whether we
have done that by modeling a continuum, a continuum of space,
of time, of speed, of what-have-you -- by using the reals,
which is to say a Dedekind-complete, totally-ordered field.

If that is the case, then Dan is right -- in a sense --
except that all the resolving was done in the re-modeling of
the situation that gives rise to the paradox, and ends
just before Dan begins his part of the problem.

I am reminded of a course in advanced calculus I took,
possibly the most difficult math course I've had. It
covered one theorem: the modern, generalized version of
Stokes' Theorem, and it took us an entire semester of hard
work to acquire the background we would need to prove it.
However, when it came time to prove Stokes itself, it was
just handed off as an exercise, plug-and-chug. The
hard-thinking part was woven into the machinery. I think
this may be similar.

I recently had a discussion with WM about the Intermediate
Value Theorem, and its equivalence to Dedekind completeness
in this case (something I'd just learned or re-learned). One
way (of many ways) to state the theorem, it seems to me, is
"Two continuous curves that cross meet at one or more points."
Isn't that the sort of property we want included in our
description of the continuum?

My conception of how we get from a properly described
continuum to the finessing of the infinite actions is
more than a little blurry for me.

(Speculating here:)
One of the many points that Virgil has made to WM is
that any real number interval, such as [0, 1], necessarily
contains uncountably many undefined reals, even if the
endpoints are always defined or referenced in some fashion.

I wonder if The Secret of Resolving Zeno's Paradox lies
in treating the unresolved, unreferenced greater mass
of points properly (and what is "proper"? Dedekind-
completeness, because crossing lines should meet).

Once we figure that out, all those infinite actions
or space-points or time-events sink back into the
anonymity of the unresolved, unreferenced greater
mass of points. And no more paradox.

The amusing thing about my speculation would be if the
problem of the infinite points to which Achilles had
to go to were resolved by introducing a larger infinity
of points to the problem.

I wonder -- if WM or David Petry somehow got their
way, hard as that is to imagine, would Zeno's Paradox
return from its grave to trouble us? Wouldn't that
be a fine howdy-do, given that they are always going
on (and on and on) about how Mathematics Should Model
Reality, if their "improvements" broke a pretty basic
capability of mathematics to do that very thing.

[Chris M. Thomasson]

> "djoyce099" wrote in message
> news:89821121-8b12-478d...@googlegroups.com...

Humm...

I think one can caculate where a runner is at a given frame.


Let d be the total distance from p_0 to p_1.

Let s be the scale value for a runner. So, the faster runner
r0 will have an s equal to 2.

Let r be the distance from r to p_1.

For a given n:

r_[n] = d - d / s^n


Lets try some values...

d = 10
s = 2


r_[0] = 10 - 10 / 2^0 = 0
r_[1] = 10 - 10 / 2^1 = 5
r_[2] = 10 - 10 / 2^2 = 7.5
r_[3] = 10 - 10 / 2^3 = 8.75
r_[4] = 10 - 10 / 2^4 = 9.375


For the following table posted up thread:

r0_[0] = 0

r0_[n+1] = r0_[n] + (p_1_x - r0_[n]) / 2

_______________________________________
r0_[0] = 0
r0_[1] = 0 + (100 - 0) / 2 = 50
r0_[2] = 50 + (100 - 50) / 2 = 75
r0_[3] = 75 + (100 - 75) / 2 = 87.5
r0_[4] = 87.5 + (100 - 87.5) / 2 = 93.75
_______________________________________

The formula r_[n] = d - d / s^n holds when you multiply
r_[n] by 10, because 10 * 10 = 100.


Now we an interpretation between the actual iteration
count and the iterated equation internals.

[Chris M. Thomasson]

>"Chris M. Thomasson" wrote in message

>news:ll3qae$jg9$1...@speranza.aioe.org... [...]

> r_[n] = d - d / s^n

Never mind, it does not work!

[Virgil]

In article <537562BA...@osu.edu>, Jim Burns <burn...@osu.edu>
wrote:

I was by no means the first to make that point.
It was well known well before I was born.
That WM still rejects it is a measure of the dominance of his ego over
his intellect.

>
> I wonder if The Secret of Resolving Zeno's Paradox lies
> in treating the unresolved, unreferenced greater mass
> of points properly (and what is "proper"? Dedekind-
> completeness, because crossing lines should meet).
>
> Once we figure that out, all those infinite actions
> or space-points or time-events sink back into the
> anonymity of the unresolved, unreferenced greater
> mass of points. And no more paradox.
>
> The amusing thing about my speculation would be if the
> problem of the infinite points to which Achilles had
> to go to were resolved by introducing a larger infinity
> of points to the problem.
>
> I wonder -- if WM or David Petry somehow got their
> way, hard as that is to imagine, would Zeno's Paradox
> return from its grave to trouble us? Wouldn't that
> be a fine howdy-do, given that they are always going
> on (and on and on) about how Mathematics Should Model
> Reality, if their "improvements" broke a pretty basic
> capability of mathematics to do that very thing.

--

[Dan Christensen]

Maybe you missed (ignored?) this part:

"The air flowing in behind the javelin, he [Aristotle] argued, is what continues to push it forward.... A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."

[netzweltler]

> not equal p_1. So, p_1 - p_0 is non-zero which means

> that there is some "logical" space left per iteration of
> the infinite process. The act of examining an iteration
> frame means that infinite precision is lost. The act of
> observing a frame logically stops Infinity from existing
> wrt the problem at hand at that moment in the iteration
> frame?
>

> IMVVVVHO, there is a paradox that p_1 - p_0[Infinity]

> does not equal zero!
>
> ;^o

If we are looking for a process that enables us to cover
p_1, then countably infinitely many steps are not useful.

We better use finitely many steps like

t = 0 s: draw a line from 0 to 0.5 in 0.5 s

t = 0.5 s: draw a line from 0.5 to 1 in 0.5 s

and stop at t = 1.

The only way I can think of, how to stop an infinite
process, is to end up in an infinite loop of zero-size
steps, like

t = 0 s: draw a line from 0 to 0.5 in 0.5 s
t = 0.5 s: draw a line from 0.5 to 0.75 in 0.25 s
t = 0.75 s: draw a line from 0.75 to 0.875 in 0.125 s
...

t = 1s: draw a line from 1 to 1 in 0 s
t = 1s: draw a line from 1 to 1 in 0 s
t = 1s: draw a line from 1 to 1 in 0 s
...

Here we end up in an infinite loop that causes no
change in time and in space anymore. This can be
treated as "stopping an infinite process".

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll3rqq$me5$1...@speranza.aioe.org...

Ahhhh, now this is a direct formula:

n = iteration count
d = distance
s = scale

r_[n] = (d / s^n) * (s^n - (s-1)^n)


just might work for finding the total distance
traveled at a given iteration count of the following
iterated equation:

r_[n+1] = r_[n] + (d - r_[n]) / s



Here is the sequence for d = 10 and s = 4 using the
iterative formula:
__________________________________
r_[0] = 0
r_[1] = 0 + (10 - 0) / 4 = 2.5
r_[2] = 2.5 + (10 - 2.5) / 4 = 4.375
r_[3] = 4.375 + (10 - 4.375) / 4 = 5.78125
r_[4] = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
__________________________________


And here is the sequence for d = 10 and s = 4 using
the direct formula:
__________________________________
r_[0] = 10 / 1 * 0 = 0
r_[1] = 10 / 4 * 1 = 2.5
r_[2] = 10 / 16 * 7 = 4.375
r_[3] = 10 / 64 * 37 = 5.78125
r_[4] = 10 / 256 * 175 = 6.8359375
__________________________________


As you can see, they are identical!

Humm...

[scattered]

[snip]

Dan said:

The explanation is indeed just a simple application of the Galileo's speed formula v=d/t. Prior to Galileo's breakthrough, people didn't have the tools to measure speed and to predict where and when a faster moving object might overtake a slower moving one. Now, we almost can't imagine that people ever had a problem with the idea.

Response: The Greeks didn't have the notion of speed as a quantity in the
modern sense partly because they didn't have a modern notion of rational
numbers, but instead thought in terms of ratios conceived as relationships between whole numbers. But -- they were quite adept at dealing with ratios.
I find it implausible that the Ancient Greeks (who produced the mathematics
described in Euclid's Elements) could neither understand nor solve a problem
like "If Achilles can run 10 meters in the time it takes the tortoise to run 1
meter and the tortoise has a 20 meter head start, how far must Achilles run in
order to catch the tortoise?" Just because they didn't have Galileo's clear
formulation doesn't mean that they didn't have any grasp on
the relevant background algebra.

[Martin Shobe]

No, I saw it. It doesn't say, "Aristotle held that the javelin would go

on forever, propelled forward by air rushing in to fill the void created
at the end of the javelin!!!"

Martin Shobe

[Dan Christensen]

It seems you are in a state of denial, Martin. Oh, well.

[djoyce099]

Now you have to add (t)for time in there somewhere. ;-)

[Ross A. Finlayson]

It's a wonder that there could be
many approaches, then for the
classical courses.

And calculus is a general approach,
here to the tenuous definition of
the line integral, calculus is a
suitable approach for generally
addressing these, and for the
continuous spaces under their
support, these "finite differenceing
or series methods" or what-have-you
transform to and from the integral
calculus or real analysis.

Then here the point is that D. Joyce
can, about the structures under the
primes, make a point, and anyone can
make a point about Newton-Cotes, or
use geometry under various
constructions, building about the
relevant structures of support of a
model here of relative motion.
Then, for the _effect_, the most is
known about the general transform in
the calculus then as for requirement
where it is general. For the
effect, of course it is built from
all the usual combinatorics, and
then also from the space that
results. This is the space of all
things, for any things. So, then as
begun here building models of models
of systems, advises from their
natural costs in construction what
they are. Then, in our
framework(s), still there is that
the classical limits would as well
be the sums, for what holds and what
lets.

My uncle's a calculus professor, I
described a resolution of Zeno's
paradox in the development once and
he noted "ah, now you've solved
light speed". Point being I was
flattered but also of note, all the
classical concerns then are for what
are of interest to the research
physicist the _effect_.

And it's the classical because it's
pretty much everywhere true -
classical physics.

This, for mechanics.

[Dan Christensen]

That may well be true, but the philosophers of ancient Greece seemed to think they could do physics by introspection alone . It seemed to be a point of honour NOT to experimentally verify their theories. If Aristole, for example, had simply gone out to observe people throwing javelins, he would have immediately trashed his silliness about the path they took when thrown (see my posting yesterday). Instead, it was passed on as received knowledge in academia for centuries.

I'm sure of the issue of motion through space had long been resolved by men of world like the builders of the Parthenon, but Zeno and company seem to have been fixated on what was for them a dead-end to a trivial problem. They would probably have done well to bring in a guest-speaker from the architects across town.

[Martin Shobe]

Any evidence that Aristotle actually believed that a javelin would
travel on forever as opposed to it being an unintended consequence of
his theory? It seems to be missing.

Martin Shobe

[Ross A. Finlayson]

The question is,
do _you_ believe
it will travel on forever?

Aristotle would entertain
that it did and didn't,
in entertaining that it does.

This, what it does.

[dull...@sprynet.com]

On Thu, 15 May 2014 11:16:17 -0700 (PDT), Dan Christensen
<Dan_Chr...@sympatico.ca> wrote:

>On Thursday, May 15, 2014 1:50:24 PM UTC-4, dull...@sprynet.com wrote:
>> On Wed, 14 May 2014 10:26:51 -0700 (PDT), Dan Christensen
>>
>> <Dan_Chr...@sympatico.ca> wrote:
>>
>>
>>
>> >On Wednesday, May 14, 2014 12:54:31 PM UTC-4, dull...@sprynet.com wrote:
>>
>> >> On Mon, 12 May 2014 07:34:31 -0700 (PDT), Dan Christensen
>>
>> >>
>>
>> >> <Dan_Chr...@sympatico.ca> wrote:
>>
>> >>
>>
>> >>
>>
>> >>
>>
>> >> >On Monday, May 12, 2014 10:03:06 AM UTC-4, Martin Shobe wrote:
>>
>> >>
>>
>> >> >
>>
>> >>
>>
>> >> >>
>>
>> >>
>>
>> >> >> You solved the wrong problem. The problem isn't "when will Achilles
>>
>> >>
>>
>> >> >>
>>
>> >>
>>
>> >> >> overtake the tortoise?" The problem is "how can Achilles overtake the
>>
>> >>
>>
>> >> >>
>>
>> >>
>>
>> >> >> tortoise when he has to complete an infinite number of subtasks to do so?".
>>
>> >>
>>
>> >> >>
>>
>> >>
>>
>> >> >
>>
>> >>
>>
>> >> >Do you really dispute that speed equals distance over time?
>>
>> >>
>>
>> >>
>>
>> >>
>>
>> >> For heaven's sake, do you really think he's disputing that?
>>
>> >>
>>
>> >>
>>
>> >>
>>
>> >> The point is that you're not "resolving" the "paradox",
>>
>> >>
>>
>> >> you're simply ignoring the paradox.
>>
>> >>
>>
>> >
>>
>> >Using the speed formula and simple algebra, you can prove that not only does Achilles pass the Tortoise
>>
>> > (actually disputed by Zeno and others), but you can determine precisely when and where this event will occur.
>>
>
>> Do you really think that this is not completely obvious to everyone?
>>
>
>It wasn't obvious to Zeno and his followers.

How could you possibly know that?

>
>>
>>
>> To "resolve" a paradox is not to explain what actually happens; to
>>
>> resolve a paradox is to explain away the seemingly paradoxical
>>
>> nature of whatever.

>>
>
>We have identified the faulty assumption that an infinite number of events cannot occur in a finite time interval. And we don't need calculus to get to the bottom of it. Simple algebra will do.
>

[Dan Christensen]

On Friday, May 16, 2014 12:07:33 PM UTC-4, Martin Shobe wrote:

> >>> "The air flowing in behind the javelin, he [Aristotle] argued, is what continues to push it forward.... A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>

> >> No, I saw it. It doesn't say, "Aristotle held that the javelin would go
>
> >>
>
> >> on forever, propelled forward by air rushing in to fill the void created
>
> >>
>
> >> at the end of the javelin!!!"

>
> > It seems you are in a state of denial, Martin. Oh, well.
>
>
>
> Any evidence that Aristotle actually believed that a javelin would
>
> travel on forever as opposed to it being an unintended consequence of
>
> his theory? It seems to be missing.
>

Quoting from Grissom again, "A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."

Do you deny this?

[Dan Christensen]

On Friday, May 16, 2014 12:33:54 PM UTC-4, dull...@sprynet.com wrote:

> >> Do you really think that this is not completely obvious to everyone?
>
> >>
>
> >
>
> >It wasn't obvious to Zeno and his followers.
>
>
>
> How could you possibly know that?
>

Again, from Aristotle himself. See http://classics.mit.edu/Aristotle/physics.mb.txt

[Peter Percival]

Dan Christensen wrote:
> On Friday, May 16, 2014 12:07:33 PM UTC-4, Martin Shobe wrote:
>
>>>>> "The air flowing in behind the javelin, he [Aristotle] argued, is what continues to push it forward.... A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>>
>
>>>> No, I saw it. It doesn't say, "Aristotle held that the javelin would go
>>
>>>>
>>
>>>> on forever, propelled forward by air rushing in to fill the void created
>>
>>>>
>>
>>>> at the end of the javelin!!!"
>
>>
>>> It seems you are in a state of denial, Martin. Oh, well.
>>
>>
>>
>> Any evidence that Aristotle actually believed that a javelin would
>>
>> travel on forever as opposed to it being an unintended consequence of
>>
>> his theory? It seems to be missing.
>>
>
> Quoting from Grissom again, "A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>
> Do you deny this?

To show that Aristotle believed something you should quote Aristotle not
Grissom. If you believe B and I claim that B entails C, does that mean
that you believe C? Even if my claim is true, does it mean that?


--
[...] They listened at his heart.
Little-less-nothing!-and that ended it.
No more to build on there. And they, since they
Were not the one dead, turned to their affairs.
"Out, Out-", Robert Frost, 1916.

[Martin Shobe]

On 5/16/2014 11:50 AM, Dan Christensen wrote:
> On Friday, May 16, 2014 12:07:33 PM UTC-4, Martin Shobe wrote:
>
>>>>> "The air flowing in behind the javelin, he [Aristotle] argued, is what continues to push it forward.... A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>>
>
>>>> No, I saw it. It doesn't say, "Aristotle held that the javelin would go
>>
>>>>
>>
>>>> on forever, propelled forward by air rushing in to fill the void created
>>
>>>>
>>
>>>> at the end of the javelin!!!"
>
>>
>>> It seems you are in a state of denial, Martin. Oh, well.
>>
>>
>>
>> Any evidence that Aristotle actually believed that a javelin would
>>
>> travel on forever as opposed to it being an unintended consequence of
>>
>> his theory? It seems to be missing.
>>
>
> Quoting from Grissom again, "A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>
> Do you deny this?

That Grissom said that? No. Now where is the evidence that Aristotle
believed it as opposed to it being an unintended consequence of his theory?

Martin Shobe

[Dan Christensen]

On Friday, May 16, 2014 1:07:42 PM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
>
> > On Friday, May 16, 2014 12:07:33 PM UTC-4, Martin Shobe wrote:
>
> >
>
> >>>>> "The air flowing in behind the javelin, he [Aristotle] argued, is what continues to push it forward.... A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>
> >>
>
> >
>
> >>>> No, I saw it. It doesn't say, "Aristotle held that the javelin would go
>
> >>
>
> >>>>
>
> >>
>
> >>>> on forever, propelled forward by air rushing in to fill the void created
>
> >>
>
> >>>>
>
> >>
>
> >>>> at the end of the javelin!!!"
>
> >
>
> >>
>
> >>> It seems you are in a state of denial, Martin. Oh, well.
>
> >>
>
> >>
>
> >>
>
> >> Any evidence that Aristotle actually believed that a javelin would
>
> >>
>
> >> travel on forever as opposed to it being an unintended consequence of
>
> >>
>
> >> his theory? It seems to be missing.
>
> >>
>
> >
>
> > Quoting from Grissom again, "A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>
> >
>
> > Do you deny this?
>
>
>
> To show that Aristotle believed something you should quote Aristotle not
>
> Grissom.

Ideally. I am not motivated to plow through the collected works of Aristole, so I will rely on the assessments of others. Apart from a handful of internet kooks, I am not even aware that this is a controversy.

[Peter Percival]

Dan Christensen wrote:
> On Friday, May 16, 2014 1:07:42 PM UTC-4, Peter Percival wrote:
>> Dan Christensen wrote:
>>
>>> On Friday, May 16, 2014 12:07:33 PM UTC-4, Martin Shobe wrote:

>>>> Any evidence that Aristotle actually believed that a javelin would
>>>> travel on forever as opposed to it being an unintended consequence of
>>>> his theory? It seems to be missing.
>>
>>> Quoting from Grissom again, "A javelin propelled in this fashion could continue to move forward forever since there is nothing to stop the process that Aristotle invokes."
>>
>>> Do you deny this?
>>
>> To show that Aristotle believed something you should quote Aristotle not
>> Grissom.
>
> Ideally. I am not motivated to plow through the collected works of Aristole, so I will rely on the assessments of others. Apart from a handful of internet kooks, I am not even aware that this is a controversy.

So those who disagree with you are kooks are they? Well, that's one way
of dealing with the matter.

Because Grissom draws a particular conclusion from a theory of
Aristotle, that does not mean that Aristotle believed that conclusion.

[Dan Christensen]

On Friday, May 16, 2014 2:01:45 PM UTC-4, Peter Percival wrote:

>
> Because Grissom draws a particular conclusion from a theory of
>
> Aristotle, that does not mean that Aristotle believed that conclusion.
>

From another source:

"Unfortunately, there are many things in the [Aristotle's] Physics that seem either strange or factually wrong. He claims, for example, that there can be no
such thing as an empty void, that an isolated point cannot move, that all
motion (in the sense of change-of-place) is either in a straight line or a
circle with circular motion restricted to the heavenly bodies and straight
line motion restricted to earthly things. He claims that an object tossed in
the air becomes stationary before changing direction and that a projectile,
such as a javelin, moves after it leaves the thrower's hand only because it is pushed along by the air."

A. W. Stetz, "Beginning with Aristotle ," 1999
http://www.physics.oregonstate.edu/~stetza/ph407H/Aristotle.pdf

Further evidence of an anti-Aristotelian conspiracy?

[Martin Shobe]

On 5/16/2014 2:01 PM, Dan Christensen wrote:
> On Friday, May 16, 2014 2:01:45 PM UTC-4, Peter Percival wrote:
>
>>
>> Because Grissom draws a particular conclusion from a theory of
>>
>> Aristotle, that does not mean that Aristotle believed that conclusion.
>>
>
> From another source:
>
> "Unfortunately, there are many things in the [Aristotle's] Physics that seem either strange or factually wrong. He claims, for example, that there can be no
> such thing as an empty void, that an isolated point cannot move, that all
> motion (in the sense of change-of-place) is either in a straight line or a
> circle with circular motion restricted to the heavenly bodies and straight
> line motion restricted to earthly things. He claims that an object tossed in
> the air becomes stationary before changing direction and that a projectile,
> such as a javelin, moves after it leaves the thrower's hand only because it is pushed along by the air."
>
> A. W. Stetz, "Beginning with Aristotle ," 1999
> http://www.physics.oregonstate.edu/~stetza/ph407H/Aristotle.pdf
>
> Further evidence of an anti-Aristotelian conspiracy?

No. Just further evidence that you don't understand the objections to
your claims.

Martin Shobe

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll0tj8$kp8$1...@speranza.aioe.org...

> > "Chris M. Thomasson" wrote in message

> > news:lkuku4$26s$1...@speranza.aioe.org...
> > > "Dan Christensen" wrote in message
> > [...]
> > > I hope this helps.

> > It makes perfect sense to me.

> WRT the runners on an infinite track in a finite
> plane algorithm, I could use the iteration count
> to define time. This turns it into a parametric
> equation where each iteration, or frame, represents
> a derivative of movement.

Hope this is not total crap, but:



Each iteration count can be used for a time t because
the count is a monotonically increasing scale in the
form of an unsigned integer...

So very quickly:

ts = 1/2
t_[n] = n * ts


A possible sequence for time linking iteration counts:

ts = 0.5
t_[0] = 0 * 0.5 = 0
t_[1] = 1 * 0.5 = 0.5
t_[2] = 2 * 0.5 = 1.0
t_[3] = 3 * 0.5 = 1.5
...


Each iteration count represents 0.5 seconds of time.



Is this legit? Any advise would be greatly appreciated...


Thank you all!

:^)

[Dan Christensen]

On Friday, May 16, 2014 3:27:33 PM UTC-4, Martin Shobe wrote:

>
> > Further evidence of an anti-Aristotelian conspiracy?
>
>
>
> No. Just further evidence that you don't understand the objections to
>
> your claims.
>

We have two independent experts that agree with me. Get back to us when when you have even one that disagrees. Or plow through Aristotle's "Physics" yourself and show where I am wrong.

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll5psh$d7o$1...@speranza.aioe.org...

> > "Chris M. Thomasson" wrote in message
> > news:ll0tj8$kp8$1...@speranza.aioe.org...
> > > "Chris M. Thomasson" wrote in message
> > > news:lkuku4$26s$1...@speranza.aioe.org...
> > > > "Dan Christensen" wrote in message
> > > [...]
> > > > I hope this helps.

> > > It makes perfect sense to me.

[...]

> Each iteration count represents 0.5 seconds of time.

[...]

I could scale the time t_[n] with the same scale as the
iterated equation of a runner I laid out up thread.

I am trying to come up with a set of functions that call
upon equations that return relevant per-frame/iteration
information but only require an iteration count as a
base parameter.

IMVVVHO, these functions can be beneficial for a
computer, because it means that it can reap information
without actually physically executing n iterations. Instead,
it can call the direct formulas and gain relevant information
using n in a "single step"?

Humm...

[Chris M. Thomasson]

AHHH!

http://www.endlesspools.com

Can be "possibly" used as an contrived
example of an endless race held on a finite
area?

You reach the back of the pool when
you cannot fight the current any longer,
or the power goes out...

;^)

[Chris M. Thomasson]

"djoyce099" wrote in message
news:7acf3c01-76de-4e64...@googlegroups.com...

> Ahhhh, now this is a direct formula:

[...]

> Now you have to add (t)for time in there somewhere. ;-)

I am trying to come up with a set of functions that call
upon equations that return relevant per-frame/iteration
information but only require an iteration count as a
base parameter.

Hope this is not total crap, but:

Could each iteration count possibly be used for a time t

because the count is a monotonically increasing scale
in the form of an unsigned integer...

So very quickly:

ts = 1/2
t_[n] = n * ts


A possible sequence for time linking iteration counts:

ts = 0.5
t_[0] = 0 * 0.5 = 0
t_[1] = 1 * 0.5 = 0.5
t_[2] = 2 * 0.5 = 1.0
t_[3] = 3 * 0.5 = 1.5

...


Each iteration count represents 0.5 seconds of time.

I can scale the time t_[n] with the same scale as the

iterated equation of a runner I laid out up thread.

[Martin Shobe]

On 5/16/2014 3:17 PM, Dan Christensen wrote:
> On Friday, May 16, 2014 3:27:33 PM UTC-4, Martin Shobe wrote:
>
>>
>>> Further evidence of an anti-Aristotelian conspiracy?
>>
>>
>>
>> No. Just further evidence that you don't understand the objections to
>>
>> your claims.
>>
>
> We have two independent experts that agree with me. Get back to us when when you have even one that disagrees. Or plow through Aristotle's "Physics" yourself and show where I am wrong.

You have two independent experts who haven't agreed with you or
disagreed with you. Get back to us when you find one that agrees with
you. (BTW, I already showed you where you went wrong. The statement you
think says Aristotle held a particular belief doesn't say that Aristotle
held that belief. You're just in denial.)

Martin Shobe

[Dan Christensen]

Someone here is denial. And it ain't me.

[John Gabriel]

On Monday, 12 May 2014 05:14:16 UTC+2, Dan Christensen wrote:

> http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise

It's not a paradox and does not even seem like a paradox to the astute mind. Only stupid people like you might ever believe that it is a paradox.

> That this would have seemed paradoxical to ancient Greek philosophers can be explained by the fact that they had no precise notion of what speed was.

Thanks for confirming that you are an ignorant moron. Archimedes knew exactly what is speed. If you had studied any of the works of Archimedes, then you would have known which propositions discuss uniform speed and lead to the formation of the spiral. But how would an imbecile like you know...

The exact definition of speed is given in the propositions on spirals.

One look at your crap on Peano's rot, and there is no need to look further.


> Assuming constant speeds S_A and S_T (m/s) for Achilles and the Tortoise respectively, we can show that, in this example, Achilles would have caught up to the Tortoise in 100/(S_A - S_T) seconds. A trivial application of the s=d/t formula.

How you arrived at 100/(S_A - S_T) seconds is strange.

We know that Achilles and the Tortoise will meet at the same time, but each will have covered different distances.

So, you have D_a / S_a = D_t / S_t

where D_a is the distance covered by Achilles and D_t is the distance covered by tortoise. Likewise, S_a is Achilles' speed.

You idiotically assumed their distances are the same and then did something very odd in your brain to arrive at the bullshit you wrote down.

You can't even do simple high school math properly. And here you are talking about axioms. What a monkey!

> With only a vague notion of speed, ancient Greek philosophers were perplexed by the fact, in that time interval, both racers would have passed through infinitely many points in space, the arrival at each point being an "event".

The only one who is perplexed, is YOU, O moron!

> (It's hard to be believe ancient Greek engineers could have been as deluded on such mechanical issues, but that is probably another story.) In modern modern mathematics and physics, we have no problem with infinitely many such events occurring in a finite time interval.

They were so 'deluded' that they constructed the first computer 2000 years before a modern version was built. Fucking moron.

> Comments?

Yes. Give up on mathematics. You have no aptitude for it.

[Dan Christensen]

On Friday, May 16, 2014 7:52:43 PM UTC-4, John Gabriel wrote:
> On Monday, 12 May 2014 05:14:16 UTC+2, Dan Christensen wrote:
>
>
>
> > http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise
>
>
>
> It's not a paradox and does not even seem like a paradox to the astute mind.

Yes, if you understand the speed formula and basic algebra, it is easy to not only show that Achilles does overtake the Tortoise, but also to calculate precisely when and where that event will occur. It seems, the ancient Greek philosophers were incapable of this. I can't imagine that their contemporaries who were building the Parthenon did not have something like our own understanding of mechanics. The philosophers probably should have talked to them.

A.W. Stetz outlines 3 fundamental problems with Aristotle's world view, in particular:

"1. Motion should be treated quantitatively and algebraically. This is seen especially clearly in the argument that a point cannot move. In Book VI, Chapter 2, Aristotle discusses what we would call speed or velocity. In modern notation, v = x/ t, where x is the distance traveled by the point in some very short interval of time t. He cannot say this, however, because Physics lacks the concept of space as we understand it mathematically. We would say that the particle moves from x1 to x2 in time t = t2 - t1, so that x = x2 - x1. The idea of assigning a variable to describe position is foreign to Aristotle's way of thinking. As a consequence the reasoning in this section is very labored and obscure. When he comes around to the motion of a point in Chapter 10 of the same book, he says in effect that v = x/ t, but x is the size of the point. That is zero of course, so v = 0 as well. Put this way the error is immediately obvious, but without a quantitative notion of space it is extremely obscure.

"2. Despite his remarks about the logical structure of scientific arguments in the Posterior Analytics, Aristotle does not use syllogisms in the Physics but rather another kind of argument that can best be called the process of elimination. In response to a question or "impasse" he will formulate three (or more) explanations, say A, B, and C. A and B can be ruled out with some simple arguments, and so, it is claimed, the correct answer is C. This is a treacherous argument, because it is usually impossible to prove that A, B, and C are the only possibilities. Perhaps the right answer is Z, which cannot even be formulated with the language and concepts at hand.

"3. [Take special note, John Gabriel!] Aristotle's definition of motion is difficult to make sense of. I will quote at length from Sachs's translation of the crucial passage from Book III, Chapter 2.

"'Therefore, motion is the being-at-work-staying-itself of the movable, and happens to it by contact with what is moving, so that the latter too is acted upon. And what moves will always bear a form, whether a this or an of-this-kind or a this much, which will be the source and cause of its motion whenever it moves.'

"The elaborate hyphenated noun, being-at-work-staying-itself, is Sach's attempt to translate entelecheia, a word that Aristotle has invented by combining and punning on several different words. In his commentary on this passage the translator makes the strange remark that this word has been misunderstood by "almost everybody" for the last thousand years. (I certainly don't understand it!) The point is that science is, first of all, a community enterprise. A useful scientific idea must be understandable to all the practitioners in the field, and it must be possible to reformulate it in many ways without losing it's content. A concept that has proved incomprehensible to a thousand years of serious scholarship hardly fits into that category."

http://www.physics.oregonstate.edu/~stetza/ph407H/Aristotle.pdf

>
> > Assuming constant speeds S_A and S_T (m/s) for Achilles and the Tortoise respectively, we can show that, in this example, Achilles would have caught up to the Tortoise in 100/(S_A - S_T) seconds. A trivial application of the s=d/t formula.
>
>
>
> How you arrived at 100/(S_A - S_T) seconds is strange.
>

[snip]

I suggest you go back and review your 9th grade math notes, John Gabriel.

>
>
> We know that Achilles and the Tortoise will meet at the same time, but each will have covered different distances.
>

[snip]

You might brush up on your reading skills while you are at. It seems you missed the part about the Tortoise being given a head start.

> > Comments?
>
>
>
> Yes. Give up on mathematics. You have no aptitude for it.

This from a kook who touts some ancient, informal axioms of arithmetic to replace Peano's axioms (not making this up, folks!) but is unable to establish even simplest properties of numbers like the associativity of addition. He makes WM and AP look like a geniuses!

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll49gu$fh2$1...@speranza.aioe.org...

> > "Chris M. Thomasson" wrote in message

> > news:ll3rqq$me5$1...@speranza.aioe.org...

> > >"Chris M. Thomasson" wrote in message

> > >news:ll3qae$jg9$1...@speranza.aioe.org... [...]
> > > r_[n] = d - d / s^n
> > Never mind, it does not work!

> Ahhhh, now this is a direct formula:

> n = iteration count
> d = distance
> s = scale
> r_[n] = (d / s^n) * (s^n - (s-1)^n)

Well, AFAICT the following equation holds when
n is an unsigned integer. Avoiding all of the nasty
divide by zero conditions of course:

(d / s^n) * (s^n - (s-1)^n) = r_[n] + (d - r_[n]) / s

r_[n] is the distance of the runner at the n'th
iteration count.

I was thinking about the following sequence of
numbers generated by executing the left hand
side of the equation above when d = 10 and
s = 4:

__________________________________
r_[0] = 10 / 1 * 0 = 0
r_[1] = 10 / 4 * 1 = 2.5
r_[2] = 10 / 16 * 7 = 4.375
r_[3] = 10 / 64 * 37 = 5.78125
r_[4] = 10 / 256 * 175 = 6.8359375
__________________________________

The set [0, 1, 7, 37, 175] looked interesting to be, so
I typed into OEIS, and had a hit:

https://oeis.org/search?q=0%2C+1%2C+7%2C+37%2C+175

The (s^n - (s-1)^n) portion of the equation seems to be
the key element wrt relating the iteration count with the
correct distance traveled in one step without actually
iterating the equation.


I should probably turn this into a function:

F(n, s) = s^n - (s-1)^n

Where:

(d / s^n) * F(n, s) = r_[n + 1] = r_[n] + (d - r_[n]) / s

To save some typing...

;^)


I think I can relate time to each iteration of the equation
being a frame of time. Where each iteration of the equation
means time is passing, and is related to the iteration, or
frame count if you will...

Time = Iteration Count in a high level general sense?

Humm. Am I doing something very terribly wrong here?

:^o

[John Gabriel]

On Saturday, 17 May 2014 05:27:13 UTC+2, Dan Christensen wrote:

> Yes, if you understand the speed formula and basic algebra, it is easy to not only show that Achilles does overtake the Tortoise, but also to calculate precisely when and where that event will occur. It seems, the ancient Greek philosophers were incapable of this. I can't imagine that their contemporaries who were building the Parthenon did not have something like our own understanding of mechanics. The philosophers probably should have talked to them.

Ha, ha. Greek philosophers were incapable?! :-) You moron! If they were incapable, you would not be sitting behind a keyboard masturbating your low IQ brain.

Those who built the Parthenon were Greeks. Idiot!

A very common misconception is that an infinite sum is actually possible. Well, common sense will tell us that no infinite process is theoretically or practically possible. For example, Xeno's faux paradox (it is really not a paradox at all, rather it is an exercise in mathematical illusion by means of flawed and deceptive arguments) led misguided mathematicians to all the wrong inferences.

Xeno's argument is purely theoretical and there is nothing paradoxical about it. The argument involves an infinite process. Reinforced with incorrect inferences that arise from deceptively producing a link to the physical world, it misleads a thinker to conclude that something is not quite right. Of course, if one has to complete a distance x (by walking) and one divides the distance by half each time, theoretically one will never complete the distance. Now back to reality: each time one takes another step, the remaining distance is less than x, hence, when one reaches the halfway mark of the remaining distance, eventually one will be able to take the final step to complete the distance.

There is nothing magical or paradoxical about this. To use juvenile American lingo, it's not even cool. Cantorian theory and incorrect ideas about discrete and continuous distances, are based on this rot. There is no such thing as a discrete or continuous distance.

> <snip>
> A load of wrong inferences from Aristotle.

> > How you arrived at 100/(S_A - S_T) seconds is strange.

> I suggest you go back and review your 9th grade math notes, John Gabriel.

That's hilarious. 100/(S_A - S_T) seconds is bullshit and you're the one who needs to learn mathematics.

A_______T____E____

The race begins at time t=0. At that time, Achilles and Tort. are in the positions shown. At E is where they meet. Now pay attention you imbecile!

When they meet at E, they will both have covered different distances in the same time. So how did you arrive at your nonsense 100/(S_A - S_T)?

> You might brush up on your reading skills while you are at. It seems you missed the part about the Tortoise being given a head start.

Tsk, tsk. Stupid can't be fixed. No matter what.

> This from a kook who touts some ancient, informal axioms of arithmetic to replace Peano's axioms (not making this up, folks!) but is unable to establish even simplest properties of numbers like the associativity of addition. He makes WM and AP look like a geniuses!

The crank is calling me a 'kook'. That's too funny! If Peano's rot were debunked to you 1000 times, you will still not get it, because you are not intellectually capable.

You sound just like AP and speaking of dimwits, where is your fellow moron Port563? I am glad Dimwit-of-Oz has finally quit spewing out his BS. Now follow his example and get lost also.

[Chris M. Thomasson]

> "Chris M. Thomasson" wrote in message

> news:ll6vor$pul$1...@speranza.aioe.org... [...]

> I was thinking about the following sequence of
> numbers generated by executing the left hand
> side of the equation above when d = 10 and
> s = 4:
> __________________________________
> r_[0] = 10 / 1 * 0 = 0
> r_[1] = 10 / 4 * 1 = 2.5
> r_[2] = 10 / 16 * 7 = 4.375
> r_[3] = 10 / 64 * 37 = 5.78125
> r_[4] = 10 / 256 * 175 = 6.8359375
> __________________________________

Sorry for all of the posts, but yet another table of
relationships when d = 10 and s = 4, n is the current
iteration count as a monotonically increasing unsigned
integer:

(d / s^n) * (s^n - (s-1)^n) = r_[n] + (d - r_[n]) / s

______________________________________________
r_[0] = 10 / 1 * 0 = 0 = 0
r_[1] = 10 / 4 * 1 = 0 + (10 - 0) / 4 = 2.5
r_[2] = 10 / 16 * 7 = 2.5 + (10 - 2.5) / 4 = 4.375
r_[3] = 10 / 64 * 37 = 4.375 + (10 - 4.375) / 4 = 5.78125
r_[4] = 10 / 256 * 175 = 5.78125 + (10 - 5.78125) / 4 = 6.8359375
______________________________________________


I am liking the results more and more.

:^)