James W Stark wrote:
> on 6/25/01 9:10 AM, george murphy at email@example.com wrote:
> > Tim Ikeda wrote:
> >> ...............................................
> >> I purposefully skipped the series proofs because I have an irrational fear
> >> that prevents me from even considering infinite series (Well, that and a
> >> problem I have getting the signs of the odd and even terms correct).
> > ................................................
> > This is unfortunate because it probably keeps you from accepting a
> > brilliant proof of the doctrine of _creatio ex nihilo_:
> > 1 - 1 + 1 - 1 + ....
> > = (1-1) + (1-1) + ....
> > = 0 + 0 + 0 ....
> > = 0
> > 1 - 1 + 1 - 1 + ....
> > = 1 - (1-1) - (1-1) - ....
> > = 1 - 0 - 0 ....
> > = 1
> > Therefore 0 = 1.
> > Shalom, George
> This "proof" does not appear so brilliant to me.
> In the above proof one has to assume that inserting ( ) and adjusting signs
> does not alter the equivalence of the initial infinite series and that an
> equivalence between infinite series is the same as equal for finite numbers.
> Inserting the ( ) maintains a one to one correspondence needed for
> equivalence. However, you are interpreting 0 = 1 as equals. Isn't this
> misleading? It should not be interpreted as an equality.
> Something out of nothing with finite numbers is often shown from 1 - 1 = 0
> Here we "create" two somethings out of nothing.
> To me _creatio ex nihilo_ is only a convenient assumption for a story.
Because Tim's original post was obviously parody, I thought that it would
be clear that this should be read in the same spirit. Alas, it seems that one
has to label such things with :) or something similar in order for them not to be
Yes, the apparently innocuous procedure of group terms in an infinite
series in the ways required by the above "proof" is not valid unless the series
in question have the required sorts of convergence properties, which of course
the manifestly divergent (because its sequence of partial sums alternate between
1 & 0 & thus have no limit) series 1 - 1 + 1 - .... doesn't have.
Before modern ideas about convergence were well developed, however, quite
competent mathematicians handled divergent series in ways that would earn a
calculus student today an F, & even today there are consistent ways of _defining_
sums for divergent series. E.g., if the sum is
defined as the limit of the _mean_ of the sequence of partial sums (Cesaro
summation) then the above series has the value 1/2, which is also the value of
the function 1/(1 + x), which equals 1 - x + x^2 - x^3 + .... when x < 1. When x
= 1 then 1/(1+x) = 1/2 and 1 - x + x^2 - x^3 + .... = 1 - 1 + 1 - 1 + .... .
The 17th-18th century Italian mathematician Guido Grandi in fact used
such an argument with this series to suggest a parallel with _creatio ex nihilo_
- a dubious procedure theologically even if it could be justified
mathematically. _Creatio ex nihilo_ is neither a mathematical theorem nor "only
a convenient assumption for a story" but the claim that the universe depends
ultimately on God alone.
George L. Murphy
"The Science-Theology Dialogue"
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