On Wednesday, June 1, 2022 at 2:10:24 PM UTC-7, Michael Moroney wrote:
> 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?

It really only works to define infinitesimals uniformly from zero through one,
then those elements named "iota-values" are contiguous, that "EF is a function"
and "EF(1) > EF(0)", but it's not really the usual attachment to division, where
it's written according to the existence of the limit that 1/oo = 0.

Also "iota-sums" and "iota-mutiples" are not the same.

In something like "surreal numbers", there is something like "1/w".

Given its rank or order, or "in omega", the infinitely many regions
[0/w, 1/w], [1/w, 2/w], ..., are, like infinitely many regions and have
trichotomy and are well-ordered and countably many and so on.

https://en.wikipedia.org/wiki/Surreal_number

Of course Conway's one of the most famous mathematicians of our time.