Re: [Phys-L] irrationals

[John Denker <jsd@av8n.com>]

On 09/11/2017 10:45 AM, Bob Sciamanda via Phys-l wrote:

> To me, this is a weirdity 
>
> Somebody please straighten me out!

Observation:  You are reliving one of the epochal events in the
history of human thought.  The Pythagoreans worshiped integers
and tolerated rationals;  they were shocked to discover irrationals.

At a higher level, they were shocked to discover the power of
the idea of *proof* ... the idea that you could prove things
that were totally unexpected and highly counterintuitive.

Gradually they pivoted from concentrating on integers to
concentrating on proofs.  This issue reverberates to the
present day.  It is a lot easier to come to terms with the
idea of irrationals than to come to terms with the idea
of formal proof.  (Try asking a lawyer to /quantify/ how
much circumstantial evidence constitutes a proof.  This
is a question that can be answered, but verrrry few
people have any idea how to do it.)

If you think irrationals such as √2 are weird, wait until
you get to real numbers, which include transcendentals such
as π and e.  These are steps along the road whereby we
generalize the concept of "number".  And they are just
the first steps.  You can complexify things by adjoining
i to the integers, or the rationals, or the reals.  You
can form vectors, bivectors, and higher-grade objects.
Sometimes multiplication is non-commutative.  And so on.

> a weirdity that rivals (perhaps exceeds) the weirdities of Quantum Mechanics

I doubt it.  Quantum mechanics in its usual form involves
non-commutative operators acting on infinite-dimensional
vectors over the complex numbers, so it includes the
rational weirdity as a verrrry small subset of the overall
weirdity.  Even in the simplest discrete system, such as
an isolated two-state spin system, factors of √2 show up
all over the place.

There have been some efforts to formulate quantum mechanics
in terms of integers.  In fact, Feynman Volume III uses a
semiconductor lattice as a stepping-stone toward understanding
the Schrödinger equation -- not vice versa!
  http://feynmanlectures.caltech.edu/III_13.html
There's a lot more you can do with this, and there is some
insight to be gained thereby;  see e.g. the book by Feynman
and Hibbs.  Ken Wilson ran this ball downfield quite a ways:
  https://en.wikipedia.org/wiki/Lattice_gauge_theory

However, most mortals find the continuum methods easier
to use for most purposes.

OTOH there is still a lot of discrete mathematics, including
Diophantine analysis, finite or countable groups, et cetera.
This has real-world applications including almost all of
computer science, cryptography, data storage and communication,
et cetera ... although spinoffs from quantum cryptography
have caused people to take renewed interest in analog computing
and continuum coding.

Maybe Mother Nature is one big digital computer, or maybe
analog.  There's no proof either way ... and it is not at
all clear which would be weirder.

My advice is don't worry about rational numbers.  There are
plenty of far weirder things to worry about.