On 6/1/2022 2:00 PM, mitchr...@gmail.com wrote:
> On Wednesday, June 1, 2022 at 9:52:27 AM UTC-7, Ross A. Finlayson wrote:
>> On Wednesday, June 1, 2022 at 9:43:27 AM UTC-7, sergi o wrote:
>>> 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.
>> By the time we had formal real analysis after a theory of limits,
>> the other courses included atomism, particle/wave duality, ....
>>
>> Avogadro's number is a stoichiometric constant relating
>> abstractly indistinguishable atoms in count, to mass, kinetic.
>>
>> It's a finite number.
>>
>> Anyways if there are "finitesimals" or not, if not, then "infinitesimals".
>>
>> Here of course "finitesimals isn't a word", but, it just means smallest
>> quantities in some fixed-point arithmetic, for example, then that in
>> the unbounded, those are arbitrarily small.
>
> We know 1 divided by infinity is just as real as any other quantity.

Who are "we", Mitch? You and Roy Masters?