[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   
   >>>>>>> function in functional analysis have their usual methods about it,   
   >>>>>>> it's also a pdf and CDF of the natural integers at uniform random,   
   >>>>>>> of which there are others, because there are at least three laws   
   >>>>>>> of large numbers, at least three Cantor spaces, at least three   
      
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