Re: Infinite precision floating-point

josephoswaldgg@xxxxxxxxxxx wrote:

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.


That's why I brought up the issue of di/dt, as in v=C di/dt. Calculus falls apart without continuity.

It's interesting that digital electronics deals with discontinuities as a matter of course, but when things start to fail, your attention seems to be drawn to those curved corners of the square wave signals. They are caused by analog issues like stray capacitance, and bandwidth, and put you right back in the analog world.

As long as we're opining about quantum physics, I wonder if we look hard enough, whether we might find some "rounded corners" there as well. :^) The full story is not yet written. When an electron goes from one discrete energy state to another, is there no room within a quantum level for some small variation, implying an analog world beneath the quantum one? Maybe we're just not looking close enough. Even Heisenberg seems to be telling us, "you never know."

Sorry to have hijacked the floating point representation issue. Funny how that one word, "fantasy" set me off.

Barry
.

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