netzweltler explained on 10/2/2017 : > Am Sonntag, 1. Oktober 2017 17:45:39 UTC+2 schrieb FromTheRafters: >> netzweltler formulated the question : >>> Am Sonntag, 1. Oktober 2017 15:20:16 UTC+2 schrieb FromTheRafters: >>>> After serious thinking netzweltler wrote : >>>>> Am Sonntag, 1. Oktober 2017 13:56:01 UTC+2 schrieb FromTheRafters: >>>>>> >>>>>> It seems counterintuitive when a number is viewed (or represented) as >>>>>> an infinite unending 'process' of achieving better and better >>>>>> approximations, and that we can never actually reach the destination >>>>>> number. In my view, this sequence and/or infinite sum is a >>>>>> representation of the destination number "as if" we could have gotten >>>>>> there by that process. >>>>> If the process doesn't get us there then we don't get there. Where do you >>>>> get your "as if" from? >>>> >>>> If you had sufficient time, then you would get there. >>> Show how time is involved in our process. >> >> If you have to add a next number (like one quarter) to a previous >> result of adding such previous numbers (like one plus one half) then >> you have introduced time. Thee is a 'previous' calculation needed as >> input to the next calculation. The idea that you 'never' get there (to >> two) introduces time also. I'm with you, I don't think time has any >> place in this. >> >>>>>> IOW "*After* infinitely many 'better' >>>>>> approximations" we reach the destination number *exactly* even if we >>>>>> cannot 'pinpoint' that number on the number line. >>>>> Please define "*After* infinitely many 'better' approximations". All >>>>> we've got is infinitely many approximations - each approximation telling >>>>> us that we get closer to 1 but don't reach 1. There is no *after* >>>>> specified in this process. >>>> >>>> There is also no "time" mentioned, so why is there an assumption of a >>>> process which takes time to complete? It is already completed (pi >>>> exists as a number despite our inability to pinpoint it on the number >>>> line by using an infinite alternating sum or any of the other infinite >>>> processes) we just can't pinpoint it because we exist in a time >>>> constrained universe with processes which take time to complete. >>> If you insist on introducing time to our process, try this: >> >> You misunderstand me. I'm not insisting that, in fact I insist the >> opposite. I take the infinite sequence or series representation to be >> just that, a represenation of a number -- not a process at all. This >> avoids the idea that time is a constraint against a number being exact. >> >> When it come to application, then you may have to consider the >> indicated process and get as close an approximation as you desire. The >> representations 0.999... and the infinite series or the sequences >> related to it, are all just different representations of the number >> one, just as our current representation are all representations of the >> number two. Time has nothing at all to do with it, hence there is no >> 'almost, but not quite there' to worry about. > > Correct. Time is of no concern. So, let me modify the list: > > t = 0: write 0.9 > t = 0.9: append another 9 > t = 0.99: append another 9 > ... > > to > > 1. write 0.9 > 2. append another 9 > 3. append another 9 > ... > > Do you still agree that this is a _complete_ list of all the actions needed > to write 0.999... (already present - in no time)? It is a list of additions > as well. All the additions it takes to sum up to 0.999... Again the question: > If your claim is, that we reach point 1, you need to show which step on this > list of infinitely many steps accomplishes that.
Why would I need to do that?
e raised to the pi times i equals -1 exactly without my needing to explain what the ellipses means in the 3.1415... or 2.71828... decimal representations. Those representations are of numbers not strictly procedures for approximations of numbers.
Were they only meant to be procedures then I could substitute division by zero with division by 'e to the pi times i plus one' and avoid ever dividing by zero again since pi and e could *only* be approximated. You see then that everything in calculus would be an approximation under this scenario.
Of course, some people think that this is already the case.