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.
>> 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. > >> A 'limit' is not an >> approximation, it is the destination number (if there is one in that >> field) implied by the sequence or series in question.
You could define a sequence or series by progressing from zero, to zero plus one, to zero plus one plus one half, to zero plus one plus one half plus one quarter, etcetera. This looks like it goes on forever getting closer and closer to some number without actually ever getting there.
You could also define the same sequence or series by starting from two and pulling something from one toward you by half the remaining distance each time. In this second case, you already know the destination even though the other representation of the same sequence looks like it never gets there. Using the concept of infinity as in,"infinitly many steps" you relieve yourself of the neccessity of calculating the infinite sum approximations since you already know the destination number (known as a limit). Despite the fact that the process itself is neverending, these two 'things' are both representations of the same number - namely two.