Re: Real numbers , what are they good for ?

On Dec 21, 4:53 pm, Ray Vickson <RGVick...@xxxxxxx> wrote:
On Dec 21, 2:11 pm, mjc <mjco...@xxxxxxx> wrote:



On Dec 21, 1:06 am, Ray Vickson <RGVick...@xxxxxxx> wrote:

On Dec 20, 4:27 pm, "FredJeffr...@xxxxxxxxx" <FredJeffr...@xxxxxxxxx>
wrote:

On Dec 16, 2:37 pm, jane <jane1...@xxxxxxxxxx> wrote:

A bit shaky question :

A friend of mine was asked the following question:

What are real numbers good for ?

How would you answer this question, stressing properties which, differ exactly real numbers and which are useful in analysis ?

I can give my own answer as well, but i would like to see other opinions, perhaps written shorter and having more information.

Thanks.

I came across the following which may be of interest: From Alexandre
Borovik's bloghttp://dialinf.wordpress.com/2007/12/17/commented-out/
referring to Vladimir Arnold's "What is Mathematics"

Vladimir Arnold forcefully stated in one of his books that it is wrong
to think about finite difference equations as approximations of
differential equations. It is the differential equation which
approximates finite difference laws of physics;

As far as we know, the laws of physics are described /exactly/ by
differential equations, and so finite-difference approaches are
approximations. Arnold seems to have it wrong, if he said what you
claim. I have not read his blog.

it is the result of
taking an asymptotic limit at zero. Being an approximation, it is
easier to solve and study.

It is not an approximation, as far as we know.

In support to his thesis, Arnold refers to a scene almost everyone has
seen: old tires hanging on sea piers to protect boats from bumps. If
you control a boat by measuring its speed and distance from the pier
and select the acceleration of the boat as a continuous function of
the speed and distance, you can come to the complete stop precisely at
the wall of the pier, but only after infinite time: this is an
immediate consequence of the uniqueness theorem for solutions of
differential equations. To complete the task in sensible time, you
have to allow your boat to gently bump into the pier. The asymptotic
at zero is not always an ideal solution in the real world. But it is
easier to analyze!

OK, so in a situation of human (or machine) control we are limited in
accuracy, and may need to be satisfied with a discrete approximation
of the true continuous-time, continuous-space situation. However, what
about those cases where human or machine control does not enter, such
as the orbits of planets, the behaviour of electrons within atoms,
etc.? Are you (or Arnold) claiming that "Nature" is discrete?

R.G. Vickosn

Yes, but look at how these differential equations are derived:

Equations are derived for finite step sizes,

Not so. APPROXIMATIONS are derived for finite step sizes, but one
obtains exact results only in the limit.

and the step sizes are
allowed to approach zero. In other words, the differential equations
are limits of difference equations. So the difference equations come
first!

I don't know what this means. Yes, you start with finite differences,
but No, you don't stay with them.

R.G. Vickson

In the "real"world, h cannot get arbitrarily small. The reason, imho,
that we let h -> 0 and replace the difference equations with
differential equations (and sums with integrals) is that they are much
easier to work with. They fit well because h in the real world is
quite small.
.

Relevant Pages

  • Re: Real numbers , what are they good for ?
    ... to think about finite difference equations as approximations of ... differential equations. ... you control a boat by measuring its speed and distance from the pier ...
    (sci.math)
  • Re: Real numbers , what are they good for ?
    ... to think about finite difference equations as approximations of ... approximates finite difference laws of physics; ... differential equations, ... you control a boat by measuring its speed and distance from the pier ...
    (sci.math)
  • Re: Real numbers , what are they good for ?
    ... approximates finite difference laws of physics; ... differential equations, ... It is not an approximation, ... you control a boat by measuring its speed and distance from the pier ...
    (sci.math)
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