Working with the set of real numbers

[Rustom Mody]

On Wednesday, March 5, 2014 10:07:44 AM UTC+5:30, Ben Finney wrote:
> Roy Smith writes:

> > I stopped paying attention to mathematicians when they tried to convince 
> > me that the sum of all natural numbers is -1/12.

> I stopped paying attention to a particular person when they said "I
> stopped paying attention to an entire field of study because one
> position expressed by some practicioners was disagreeable to me".

In general this is a correct response
In this particular case (apart from Roy speaking tongue-in-cheek)
it (Roy's viewpoint) is more appropriate and central to our field than you
perhaps realize:

Nonsensical results believed in by a small minority (Cantor's time)
became full scale war between platonists (Hilbert) and constructivists
(Brouwer) a generation later.

Gödel staunchly in Hilbert camp made his incompleteness theorem to rebut the
constructivists

Turing unable to disagree with Gödel's result but disagreeing with platonic
philosophy made his 'machine'. The negative result that he did not like but
had to admit was uncomputability/undecidability. However he trumped Gödel in
making a 'universal' machine

And so we are here :-)