Re: Infinite precision floating-point
- From: "josephoswaldgg@xxxxxxxxxxx" <josephoswald@xxxxxxxxx>
- Date: 26 May 2005 14:00:37 -0700
Raffael Cavallaro wrote:
> On 2005-05-26 14:15:47 -0400, tar@xxxxxxxxxxxxx (Thomas A. Russ) said:
>
> > Well, I would say that there are a fair number of irrational numbers in
> > the analog world. Pi and the square root of 2 for starters.
>
> I believe you're missing the point. pi as defined relies on the
> existence of *perfect* circles. No perfect circles in the real world
> (and the quantum nature of the physical world means all "circles" are
> actually slightly bumpy) means no pi.
>
> Similarly, square roots are a mathematical construct. Basically this is
> an argument between quantum mechanics and mathematical platonism.
> Pysicists hold that the platonic view of the world is not borne out by
> physical experimentation. If all physical quantities are discrete, not
> continuous, then there are no real world irrational quantities - every
> real world quantity can be reduced to an integer number of the
> fundamental unit for that quantity (charge, length, time, etc.).
This discussion is veering off into amateur metaphysics, but that won't
stop me from responding with my own. :-)
"Physical quantities" includes all sorts of things that are not
quantized under the most widely accepted theories of physics. Electric
and magnetic fields, for instance, have a well-developed quantum
theory, but the resulting fields are still continuous in general. [In
trendy-popularized-science-land, Wolfram, of course, believes that
there is some way an underlying discrete phenomenon could explain all
this, but I believe he is simply hand-waving, and no practical method
for describing reality can come of his ideas, much less an experimental
proof of such discreteness.]
Pi does not depend on a physical circle for its definition. It also
arises from such things as phases of wave functions. Saying pi doesn't
exist physically is sort of like saying -1 doesn't exist physically.
cos(pi), e^pi*i, and all that. Many things are hard to "point at" but
still exist in physical theories.
I remember reading long ago in one of Asimov's popularized science
essays about a person who objected to the reality of imaginary numbers.
The response was, "well, how about fractions? are those real?" "But, of
course!" "Well, then, why don't you show me what half a piece of chalk
is." The skeptic breaks a piece of chalk in two, and hands it back.
"Well, neither of those is 1/2 a piece of chalk; both are just
*smaller* pieces of chalk. If you aren't even clear on fractions, how
can I possibly explain imaginary numbers to you?"
Analyzing anything in the physical world while studiously restricting
one's calculations to be completely contained within the field of
rationals is pretty much impossible. You can't use trig functions,
exponentials, or logarithms. Calculus is completely out of bounds.
Even the mathematics of prime numbers, which seems like the most
discrete kind of mathematics, is intimately connected to the Riemann
zeta function, which has pi and continuity written all over it.
Of course, in some sense, all physical theories are just as imaginary
as pure mathematics, just mental constructs we humans amuse ourselves
with; theories do not determine what the world *is*, just how we
describe it. Nonetheless, quantum physics by no means allows us to
escape continuous mathematics.
.
- Follow-Ups:
- Re: Infinite precision floating-point
- From: Raffael Cavallaro
- Re: Infinite precision floating-point
- From: Barry Jones
- Re: Infinite precision floating-point
- References:
- Re: Infinite precision floating-point
- From: Thomas A. Russ
- Re: Infinite precision floating-point
- From: Raffael Cavallaro
- Re: Infinite precision floating-point
- Prev by Date: Re: Infinite precision floating-point
- Next by Date: Re: Infinite precision floating-point
- Previous by thread: Re: Infinite precision floating-point
- Next by thread: Re: Infinite precision floating-point
- Index(es):
Relevant Pages
|
