netzweltler
Posts:
473
From:
Germany
Registered:
8/6/10


Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 2, 2017 1:54 PM


Am Montag, 2. Oktober 2017 16:15:47 UTC+2 schrieb FromTheRafters: > on 10/2/2017, netzweltler supposed : > > Am Montag, 2. Oktober 2017 13:12:21 UTC+2 schrieb FromTheRafters: > >> 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? > > > > If there is no such step, then there is no reason to assume that we reach > > point 1. > > If you assume that it is a stepwise process to approach a number, then > of course it is a stepwise process to approach a number. I can't argue > against a stipulation like that from within the system which you insist > I use, which in turn stipulates that assumption. > > The number pi can be represented as a process (actually many equivalent > ones) like you stipulate, and the 'process' can never be completed > because of the 'infinite steps' aspect. Nevertheless pi is an exact > number. Only when you leave out the 'ad infinitum' part does it become > a only a rational approximation. > > Algebraically, different 'infinite stepwise process' forms of pi's > representation can exactly cancel with other representations, or > interact with other numbers (like e) *exactly* without any need > whatsoever to calculate using the 'infinite stepwise processes' you > seem to be insisting on. > > The number pi is already there, you don't have to approach it in > discreet steps. Same with 0.999... and 1.000...  they are just > different representations of the number one.
Yes. pi is already there and we can exactly locate its position on the number line, but you cannot locate a point on the number line representing pi if this point would be the result of a stepwise process  neither a finite process nor an infinite.

