On 6/1/2022 11:00 AM, Ross A. Finlayson wrote:
> On Tuesday, May 31, 2022 at 9:40:03 PM UTC-7, sergi o wrote:
>> On 5/31/2022 3:15 PM, mitchr...@gmail.com wrote:
>>> On Monday, May 30, 2022 at 9:54:09 PM UTC-7, zelos...@gmail.com wrote:
>>>> fredag 27 maj 2022 kl. 19:38:33 UTC+2 skrev mitchr...@gmail.com:
>>>>> On Friday, May 27, 2022 at 10:23:29 AM UTC-7, Ross A. Finlayson wrote:
>>>>>> On Thursday, May 26, 2022 at 2:17:50 PM UTC-7, sergi o wrote:
>>>>>>> On 5/26/2022 3:47 PM, mitchr...@gmail.com wrote:
>>>>>>>> On Thursday, May 26, 2022 at 1:37:42 PM UTC-7, Chris M. Thomasson wrote:
>>>>>>>>> On 5/26/2022 1:25 PM, Michael Moroney wrote:
>>>>>>>>>> On 5/25/2022 11:49 PM, Chris M. Thomasson wrote:
>>>>>>>>>>> On 5/25/2022 7:21 PM, Dan joyce wrote:
>>>>>>>>>>>> On Monday, May 23, 2022 at 7:11:22 PM UTC-4, Chris M. Thomasson wrote:
>>>>>>>>>>>>> On 5/23/2022 1:25 PM, Michael Moroney wrote:
>>>>>>>>>>>>>> Just because in theory an infinite number of steps is required doesn't
>>>>>>>>>>>>>> mean the limit cannot be reached. Consider Zeno's Paradox where
>>>>>>>>>>>>>> Achilles
>>>>>>>>>>>>>> races a tortoise with a head start. Each time Achilles reaches a point
>>>>>>>>>>>>>> where the tortoise was, the tortoise advances somewhat. When Achilles
>>>>>>>>>>>>>> reaches that point, the tortoise advances more. And so on for an
>>>>>>>>>>>>>> infinite number of steps. Yet Achilles catches up to the tortoise and
>>>>>>>>>>>>>> passes it and wins the race, despite taking an infinite number of
>>>>>>>>>>>>>> steps
>>>>>>>>>>>>>> to catch up to the tortoise.
>>>>>>>>>>>>> If Achilles strictly plays by the tortoises rules on a step-by-step
>>>>>>>>>>>>> basis:
>>>>>>>>>>
>>>>>>>>>> What are "the tortoise's rules"? The only rules are the tortoise gets a
>>>>>>>>>> head start and both it and Achilles run as fast as they can to the
>>>>>>>>>> finish line, and whoever does so first, wins.
>>>>>>>>>>>>>
>>>>>>>>>>>>> step 1: tortoise moves one meter; Achilles moves one meter. The
>>>>>>>>>>>>> tortoise
>>>>>>>>>>>>> is ahead because of the head start.
>>>>>>>>>>>>>
>>>>>>>>>>>>> step 2: tortoise moves one meter; Achilles moves one meter. The
>>>>>>>>>>>>> tortoise
>>>>>>>>>>>>> is still ahead because of the head start.
>>>>>>>>>>>>>
>>>>>>>>>>>>> on and on. The turtle will cross the finish line before Achilles.
>>>>>>>>>>>>
>>>>>>>>>>>> The turtle will never cross the finish line but will always be ahead
>>>>>>>>>>>> of Achilles.
>>>>>>>>>>>
>>>>>>>>>>> Yes. True. It gets infinitely closer and closer to the finish line.
>>>>>>>>>>
>>>>>>>>>> That is not Zeno's Paradox. The tortoise gets a head start, at point
>>>>>>>>>> A[1]. The race starts. When Achilles reaches A[1], the tortoise has
>>>>>>>>>> moved ahead somewhat, to what we call A[2]. When Achilles reaches A[2].
>>>>>>>>>> the tortoise has reached A[3], at A[3] the tortoise is at A[4] and so
>>>>>>>>>> forth.
>>>>>>>>>>
>>>>>>>>>> Since Achilles is faster than the tortoise, the distances A[1], A[2],
>>>>>>>>>> [A3], ... get smaller and smaller, since the time it takes Achilles to
>>>>>>>>>> run from the start to A[1] equals the time it takes the slower tortoise
>>>>>>>>>> to run from A[1] to A[2], and so on.
>>>>>>>>>>
>>>>>>>>>> The paradox is, no matter how big n gets, A[n] (Achilles' position) is
>>>>>>>>>> always behind A[n+1] (the tortoise's position), even as n approaches
>>>>>>>>>> infinity. So Achilles can never beat the tortoise, right? But, as long
>>>>>>>>>> as the head start isn't _too_ large, in real life, Achilles passes the
>>>>>>>>>> tortoise and wins, just as you'd expect. So what's wrong with this?
>>>>>>>>>>
>>>>>>>>>> As I said, just because there's an infinite limit, it doesn't mean the
>>>>>>>>>> limit is absolute. In this case, the total time passed also reaches a
>>>>>>>>>> limit (at n=infinity) but that time limit isn't infinite, so what
>>>>>>>>>> happens after the "limit" on time passes? As always, time marches on...
>>>>>>>>>> At that point Achilles passes the tortoise and remains ahead for the
>>>>>>>>>> rest of the race, and the infinite series no longer applies.
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>>> Now, if Achilles tells the tortoise to f-off and just starts
>>>>>>>>>>>>> running, he
>>>>>>>>>>>>> will quickly pass the tortoise...
>>>>>>>>>>
>>>>>>>>>> In real life, yes, but in Zeno's Paradox, no.
>>>>>>>>>>>>>
>>>>>>>>>>>>> ;^)
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Extra credit: Given the speeds of Achilles S1 and the tortoise S2
>>>>>>>>>> (S1>S2), and the head start distance A1, how long does it take for
>>>>>>>>>> Achilles to pass the tortoise? :-)
>>>>>>>>>>
>>>>>>>>> I did some equations on this a while back:
>>>>>>>>>
>>>>>>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/6tr-_qY-3DgJ
>>>>>>>>>
>>>>>>>>> Here are my comments:
>>>>>>>>>
>>>>>>>>> Iirc, scale was speed:
>>>>>>>>> ____________________________
>>>>>>>>> [...]
>>>>>>>>> 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...
>>>>>>>>> ____________________________
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Here is another post:
>>>>>>>>>
>>>>>>>>> https://groups.google.com/g/sci.math/c/UKBgW2IOZkI/m/ysjxQWu9URMJ
>>>>>>>>> ____________________________
>>>>>>>>> I think I found a way to find the handicap of a
>>>>>>>>> runner in an infinite race on a finite track...
>>>>>>>>>
>>>>>>>>> How about something like:
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Let:
>>>>>>>>>
>>>>>>>>> d = total distance in track
>>>>>>>>> s = scale, which relates to speed
>>>>>>>>> n = integer iteration count, which relates to time
>>>>>>>>> r_h = a runners starting handicap
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Here is the iterative equation for finding the
>>>>>>>>> distance a runner is down the track that I posted
>>>>>>>>> up thread:
>>>>>>>>>
>>>>>>>>> r_[n + 1] = r_[n] + (d - r_[n]) / s
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> The handicap of the runner is equal to r_[0]
>>>>>>>>> because n = 0 is the starting position of every
>>>>>>>>> runner.
>>>>>>>>>
>>>>>>>>> The goal is to find the handicap of a runner with
>>>>>>>>> a given distance, iteration count, total distance
>>>>>>>>> of the track, and a scale or speed. AFAICT, the
>>>>>>>>> following formula solves for the handicap of a
>>>>>>>>> runner using that information:
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> r_h = ((s-1) / s)^(-n) * ( (d * (s-1)^n * s^(-n) - d + r)
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Here is output of a racer using the iterative equation
>>>>>>>>> with the following attributes:
>>>>>>>>>
>>>>>>>>> d = 10
>>>>>>>>> s = 4
>>>>>>>>> r_h = 6.8
>>>>>>>>> _______________________________________
>>>>>>>>> r_[0] = 6.8
>>>>>>>>> r_[1] = 6.8 + (10 - 6.8) / 4 = 7.6
>>>>>>>>> r_[2] = 7.6 + (10 - 7.6) / 4 = 8.2
>>>>>>>>> r_[3] = 8.2 + (10 - 8.2) / 4 = 8.65
>>>>>>>>> r_[4] = 8.65 + (10 - 8.65) / 4 = 8.9875
>>>>>>>>> _______________________________________
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> As we can see this runner has a head start of 6.8 out
>>>>>>>>> of 10. Also, in the third frame, the runner r_[2] has
>>>>>>>>> traveled 8.2 out of a possible 10.0.
>>>>>>>>>
>>>>>>>>> Given that information alone, we can plug it all into
>>>>>>>>> the formula for finding the handicap, and get:
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> r_h = ((4-1) / 4)^(-2) * ((10 * (4-1)^2 * 4^(-2) - 10 + 8.2) = 6.8
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Bingo! We now know that the handicap for the runner
>>>>>>>>> is 6.8 at n = 0 by information reaped in a later moment
>>>>>>>>> in time when n = 2... Three frames later.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Is this Kosher?!?!
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> :^o
>>>>>>>>>
>>>>>>>>> ____________________________
>>>>>>>>
>>>>>>>> If you add zero to .999 repeating you still get .999 repeating.
>>>>>>>> Add the infinitely small and you get 1 instead.
>>>>>>> .999 repeating = 1.000 repeating anyway
>>>>>> Mitch, for that ".999... is add infinitesimal", just first
>>>>>> have it that "1 minus infinitesimal, is, .999..., lesser".
>>>>> .999 is lesser than one by the infinitely small not zero.
>>>>>
>>>>> Mitchell Raemsch
>>>>>>
>>>>>> Then though it's always that "the .999..., lesser, is
>>>>>> only on its way to zero, least or none", because there
>>>>>> are two kinds of relations: related motion and lattice
>>>>>> relations, that the field defines lattice relations while
>>>>>> the infinitesimals is only part of a "range" or "course".
>>>>>>
>>>>>> I.e., the infinitesimal changes between 1.0 and 0.0,
>>>>>> going through each .aaa... as far as it could be measured,
>>>>>> are instead of that "this .333... times 3 = .999... = 1", that
>>>>>> this "1 minus .000...1" is writing out a notation, where
>>>>>> the ...1's "sum their differences, to zero", while the numbers,
>>>>>> "round up".
>>>>>>
>>>>>> So, when someone writes ".999, ..., repeating", is mostly
>>>>>> reflecting the notion that the notation after numbers introducing
>>>>>> the "..." or over-bar or the usual way of indicating the
>>>>>> repeating part for any rational number, basically works from
>>>>>> the field of course that _all_ and _only_ rational numbers,
>>>>>> end with a repeating terminus.
>>>>>>
>>>>>> Then there's only that
>>>>>>
>>>>>> 000... <- 0
>>>>>> 000...
>>>>>>
>>>>>> 011...
>>>>>> 011... <- 1/2
>>>>>> 100...
>>>>>>
>>>>>> 111...
>>>>>> 111... <- 1
>>>>>>
>>>>>> Notice the bounds are only at the ends,
>>>>>> and each column is half 1's and half 0's.
>>>>>>
>>>>>> It's easier to reduce the discussion to [0,1] instead of
>>>>>> involving all the real numbers.
>>>> There are no infinitesimals in real numbers. The real numbers are archimedian. I have told you this
>>>
>>> How do you know they are more real than the Calculus fundamental infinitesimal?
>>> .999 repeating is not the same quantity as the first integer.
>>> Add zero to .999 repeating and you get .999 repeating.
>>>
>>> Mitchell Raemsch
>> how do you know you actually have an infinitesimal ?
>
> Deduction: "continuous exists? could not be not infinitesimal".
>
> It's more that you know that you _don't_ have an infinitesimal, but,
> that according to the existence of some analog process like the
> procedure in time, that "effectively" that given any specific frequency
> of otherwise finite events, there's another of not-necessarily finite,
> "effectively", events. (That includes them.)
>
> Basically that time goes on forever and never stops.
>
> Or a mathematical model of same, ....
>
> Deduction, that's how. Deductive inference is what's seated under
> inference, anyways. (This) ...after complementary terms, and
> complementarity of course is of greatest grounds for deduction.
>
> "Infinite" is a qualia, if it's the numbers, not ours.
>
> "Atomism" is probably a most familiar theory for
> "effectively, ..., infinitesimal atoms exist". Beyond that,
> then, there's superstring theory, "atoms' infinitesimal
> superstrings exist". That's about it, with atomic scale about
> 25 orders of magnitude and superstring scale about 50,
> orders of magnitude smaller than 1.0 meter.
>
> In theory, ....
>
>
>
>
>
>
>

you could be covered in infinitesimals, and not know it, they do itch though.