[continued from previous message]   
      
   >>>>>>>>>>> Yeah, I imagine if you let Kosmanson go on then there'd   
   >>>>>>>>>>> be quite more to it.   
   >>>>>>>>>>>   
   >>>>>>>>>>>   
   >>>>>>>>>>>   
   >>>>>>>>>>>   
   >>>>>>>>>>>   
   >>>>>>>>>>   
   >>>>>>>>>>   
   >>>>>>>>>> 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-   
      
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