Re: [Phys-L] Irrationals

[John Denker <jsd@av8n.com>]

On 11/12/2017 10:07 AM, Bob Sciamanda via Phys-l wrote:

> > An interesting QUORA discussion  ==>
> >
> > https://www.quora.com/What-is-the-importance-of-irrational-numbers

Uncle Albert famously said:

> As far as the laws of mathematics refer to reality, they are not
> certain; and as far as they are certain, they do not refer to
> reality.

The mathematicians in the quora discussion seem to
have lost sight of that.

Mathematicians are free to construct their own little
worlds in which exactitude is required.  However, when
they try to justify their constructions in terms of
real-world activities, such as throwing things in the
air, they are seriously out-of-bounds.

A child can throw a ball without irrational numbers.
Indeed, humans can build rockets and fly to the moon
without rational numbers.  Hint:  floating-point
arithmetic suffices for designing sports equipment,
spacecraft, et cetera, and does not use irrational
numbers.

The claim that irrationals are required for calculus
is wrong even by mathematical standards.  You can take
the limit as b goes to zero by writing b as a rational
and letting the denominator become very large.

Irrationals are important to the /history/ of human
thought, because a bunch of ancient Greeks made a
big fuss over integers, rationals, exactitude, and
completeness.  They even built a religious cult and
a political party around that.  There was a big stink
when it was proved that the rationals did not suffice
to represent exact, complete solutions to certain
equations.  This was an early example of the power
of rigorous /proof/ to prove unexpected results.
Spectacularly unexpected.