When were real numbers born? (was for / while else doesn't make sense)

[Rustom Mody]

On Tuesday, May 24, 2016 at 4:10:59 AM UTC+5:30, Ian wrote:
> On Mon, May 23, 2016 at 9:30 AM, Rustom Mody wrote:
> > Yes the point is being missed but in a different direction:
> > The SET (as a completed whole) of real numbers (ℝ) is no more than a 100 years
> > old.
> > People may have used fractions earlier
> >
> > And even here the first line of Steven's http://nrich.maths.org/2515 says
> > "Did you know that fractions as we use them today didn't exist in Europe until the 17th century?"
> >
> > Egypt and Babylon (and India for that matter) are really only of archaeological
> > interest in the sense that there is almost complete loss of continuity
> > from then to now
> 
> So 13th century European merchants would have been entirely incapable
> of cutting a cheese wheel in half in order to accommodate a customer
> who didn't the whole thing?

That people could compute with fractions does not mean they had reified
ℚ as a set.
Cantor did something which was a complete No-No in math until that point
-- assume that completed infinities are meaningful.
A programming example would be the question: What does this program do after it finishes printing?
i = 0
while True:
  print i
  i += 1




> 
> > That the set ℝ legitimately exists was a minority view -- Cantor,Dedekind,
> >  Weierstrass...
> 
> I'm not sure where ℝ comes into this in the first place. Existing
> Python numeric types only represent various subsets of ℚ (in the case
> of fractions.Fraction, the entirety of ℚ).
> 
> > On the other side Kronecker belligerently declared:
> > "The good Lord made the natural numbers (Zahlen in German)
> > All the rest is the work of man"
> >
> > This was the MAINSTREAM view in the 1880s.
> >
> > As late as 1918 Weyl and Polya took a bet that math concepts such as
> > real numbers, sets, countability etc would be relegated to history as a bad
> > dream and the pristine purity of constructive math would be firmly established
> > -- where "constructive math" basically means ℕ is the only reasonable infinite set and that ℝ is anything but real!
> >
> > https://en.wikipedia.org/wiki/Hermann_Weyl#Foundations_of_mathematics
> 
> I'm rather skeptical that this bet would have extended to fractions.

Yes... Its hard to guess what Kronecker/Cantor/Hilbert/Brouwer etc believed
other than the records we have.
But we can make some guesses...
Insofar as the fractions are enumerable and computable they would be said to
exist (by everyone)
Insofar as ℝ is non-denumerable its existence is suspect (by the constructivists)

Now float is a ghastly approximation to ℝ doesnt preclude better computable
approximations eg continued fractions
 (by the constructivists)