[continued from previous message]   
      
   >>>>>>>>> A note about Kosmanson's emphasis on what's often truncated in an   
   >>>>>>>>> infinite series. A year or so back I was forming baby problems   
   >>>>>>>>> in a   
   >>>>>>>>> blog   
   >>>>>>>>> for a Linux newsgroup frequenters to solve, and in one of them one   
   >>>>>>>>> would   
   >>>>>>>>> begin with a correct equation, would make correct changes in it,   
   >>>>>>>>> but   
   >>>>>>>>> would end up in an obviously wrong equation :) Nobody solved it of   
   >>>>>>>>> course (audience were mostly morons). But I now wonder if that   
   >>>>>>>>> problem   
   >>>>>>>>> had something about Kosmanson's concerns about handling   
   >>>>>>>>> infinities.   
   >>>>>>>>>   
   >>>>>>>>> Here I quote the part of the blog that contained that problem:   
   >>>>>>>>>   
   >>>>>>>>> (beginning of the quote)   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>>    "Then, swoooooooshhshsh!.... and Jesus and all that intense   
   >>>>>>>>> light   
   >>>>>>>>> went   
   >>>>>>>>> back up and out of there. Physfit looked up and there wasn't   
   >>>>>>>>> even an   
   >>>>>>>>> opening in the ceiling anymore. But now for some reason he was   
   >>>>>>>>> horizontally on the floor, in his bed. Right in the living room!   
   >>>>>>>>>   
   >>>>>>>>> He thought a bit about what was happening, when he found himself   
   >>>>>>>>> quite   
   >>>>>>>>> hungry. Last time he had eaten anything was the night before he   
   >>>>>>>>> had   
   >>>>>>>>> waken up on the summit of the magic mountain in an urban Dallas   
   >>>>>>>>> area.   
   >>>>>>>>>   
   >>>>>>>>> He thought to himself, "I'm going to assume that more than 48   
   >>>>>>>>> hours has   
   >>>>>>>>> passed since. So got up and walked to the kitchen and took a look   
   >>>>>>>>> inside   
   >>>>>>>>> refrigerator. There was nothing there but the cat food he had   
   >>>>>>>>> cooked on   
   >>>>>>>>> the day he first saw the magic mountain. He got on the computer to   
   >>>>>>>>> order   
   >>>>>>>>> something zesty from HelloFresh. After choosing the closest to a   
   >>>>>>>>> healthy   
   >>>>>>>>> nice pre-agricultural food kit, he clicked, "Go to checkout"   
   >>>>>>>>> button,   
   >>>>>>>>> after which the computer waited for a few seconds but instead of   
   >>>>>>>>> getting   
   >>>>>>>>> to the check out screen, a screen came up to make sure Physfit was   
   >>>>>>>>> not a   
   >>>>>>>>> robot. It had a simple question that he had to give it the correct   
   >>>>>>>>> answer, otherwise food nommo.   
   >>>>>>>>>   
   >>>>>>>>> The question went like this:   
   >>>>>>>>>   
   >>>>>>>>>      "In math, is there a difference between the two numbers   
   >>>>>>>>> 0.999999...   
   >>>>>>>>> and 1 ?"   
   >>>>>>>>>   
   >>>>>>>>> The digits of "9" continued forever to the right of the radix   
   >>>>>>>>> point. So   
   >>>>>>>>> of course, Physfit clicked on the "yes" button. If there was not a   
   >>>>>>>>> difference, then one wouldn't even bother to write 1 in that funky   
   >>>>>>>>> form,   
   >>>>>>>>> using an infinite series of digit 9.   
   >>>>>>>>>   
   >>>>>>>>> But the screen disappeared, and a message said, "You're a robot.   
   >>>>>>>>> Bye!"   
   >>>>>>>>>   
   >>>>>>>>> Physfit said, "Fuck!" (first of the fix number of curses Jesus had   
   >>>>>>>>> allowed him for that day). So he took a pen and paper and started   
   >>>>>>>>> jotting down:   
   >>>>>>>>>   
   >>>>>>>>>      x = 0.99999....   
   >>>>>>>>>   
   >>>>>>>>> Therefore:   
   >>>>>>>>>   
   >>>>>>>>>      10x = 9.99999....   
   >>>>>>>>>   
   >>>>>>>>> Now he subtracted the former from the latter:   
   >>>>>>>>>   
   >>>>>>>>>      10x - x = 9.99999... - 0.99999...   
   >>>>>>>>>   
   >>>>>>>>> Which simplifies to:   
   >>>>>>>>>   
   >>>>>>>>>      9x = 9   
   >>>>>>>>>   
   >>>>>>>>> And therefore:   
   >>>>>>>>>   
   >>>>>>>>>      x = 1   
   >>>>>>>>>   
   >>>>>>>>> "What the fuck??", said Physfit (his 2nd curse of the day).   
   >>>>>>>>>   
   >>>>>>>>> Why x which was 0.99999... and not 1, turned out to be 1? ... "   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>> (end of quote)   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>> So, is this problem pointing to what Kosmanson has been so keen   
   >>>>>>>>> about? :)   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>>   
   >>>>>>>>   
   >>>>>>>> Once I was reading a book or article,   
   >>>>>>>> and was introduced the introduction of .999 (...),   
   >>>>>>>> vis-a-vis, 1. A cohort of subjects was surveyed   
   >>>>>>>> their opinion and belief whether .999, dot dot dot,   
   >>>>>>>> was equal to, or less than, one. About half said   
   >>>>>>>> same and about half said different.   
   >>>>>>>>   
   >>>>>>>>   
   >>>>>>>> It's two different natural notations that happen   
   >>>>>>>> to collide and thus result being ambiguous.   
   >>>>>>>>   
   >>>>>>>> So, then these days we have the laws of arithmetic   
   >>>>>>>> introduced in primary school, usually kindergarten,   
   >>>>>>>> about the operations on numbers, and also inequalities,   
   >>>>>>>> and the order in numbers.   
   >>>>>>>>   
   >>>>>>>> Yet, even the usual account of addition and its   
   >>>>>>>> inverse and its recursion and that's inverse,   
   >>>>>>>> as operators, of whole numbers, has a different   
   >>>>>>>> account, of increment on the one side, and, division   
   >>>>>>>> on the other, sort of like the Egyptians only had   
   >>>>>>>> division or fractions and Egyptian fractions,   
   >>>>>>>> and tally marks are only increment, that though   
   >>>>>>>> it was the Egyptian fractions that gave them a   
   >>>>>>>> mathematics, beyond the simplest sort of conflation   
   >>>>>>>> of "numbering" and "counting".   
   >>>>>>>>   
   >>>>>>>> So, where ".999 vis-a-vis 1" has a deconstructive account,   
   >>>>>>>> to eliminate its ambiguities with respect to what it's   
   >>>>>>>> to model, or the clock-arithmetic and field-arithmetic,   
   >>>>>>>> even arithmetic has a deconstructive account, then,   
   >>>>>>>> even numbering versus counting has a deconstructive account,   
   >>>>>>>> to help eliminate what are the usually ignored ambiguities.   
   >>>>>>>>   
   >>>>>>>>   
   >>>>>>>> So, pre-calculus, the course, goes to eliminate or talk   
   >>>>>>>> away the case .999, dot dot dot, different 1. Yet,   
   >>>>>>>> it can be reconstrued and reconstructed, on its own   
   >>>>>>>> constructive account. So, it's a convention.   
   >>>>>>>>   
   >>>>>>>>   
   >>>>>>>> It's "multiplicity theory", see, that any, "singularity   
   >>>>>>>> theory", which results as of admitting only the principal   
   >>>>>>>> branch of otherwise a "bifurcation" or "opening" or "catastrophe"   
   >>>>>>>> or "perestroika (opening)", as they are called in mathematics,   
   >>>>>>>> branches, that singularity theory is a multiplicity theory,   
   >>>>>>>> yet the usual account has that it's just nothing,   
   >>>>>>>> or that it's apeiron and asymptotic.   
   >>>>>>>>   
   >>>>>>>>   
   >>>>>>>> So, there's a clock arithmetic where there's a reason why   
   >>>>>>>> that there's a .999, dot dot dot, _before_ 1.0, in the   
   >>>>>>>> course of passage of values from 0, to 1, and, it's also   
   >>>>>>>> rather particularly only between 0 and 1, as what results   
   >>>>>>>> thusly a whole, with regards to relating it to the modularity   
   >>>>>>>> of integers, the integral moduli.   
   >>>>>>>>   
   >>>>>>>> Thusly, real infinity has itself correctly and constructively   
   >>>>>>>> back in numbers for "standard infinitesimals" here called   
   >>>>>>>> "iota-values".   
   >>>>>>>>   
   >>>>>>>> Then, this is totally simple and looks like f(n) = n/d,   
   >>>>>>>> for n goes from zero to d and d goes to infinity, this   
   >>>>>>>> is a limit of functions for this function which is not-   
   >>>>>>>> a- real- function yet is a nonstandard function and that   
   >>>>>>>> has real analytical character, it's a discrete function   
   >>>>>>>> that's integrable and whose integral equals 1, it illustrates   
   >>>>>>>> a doubling-space according to measure theory in the measure   
   >>>>>>>> problem,   
   >>>>>>>> it's its own anti-derivative so all the tricks about the   
   >>>>>>>> exponential   
      
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