Re: The spinoza papers: design of the extra-precision number object 2
- From: spinoza1111 <spinoza1111@xxxxxxxxx>
- Date: Fri, 23 May 2008 18:17:28 -0700 (PDT)
On May 23, 11:39 pm, Richard Heathfield <r...@xxxxxxxxxxxxxxx> wrote:
spinoza1111said:
On May 23, 4:30 pm, "kwikius" <a...@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
<snip>
I'm interested in the rationale behind representing integers and reals
together within one type.
To me the useage of the 2 types is distinct. A real is a digital
approximation to an analogue quantity, IOW a physical quantity or a
ratio of 2 quantities. This has an interesting consequence in that the
representation should be constant and it should be accepted that the
approximation will be truncated to some arbitrary point (either a fixed
constant in the language, or supplied parametrically). The alternative
is that we get the distinct possibility to get an approximation that
doesnt terminate, which cause practical problems, e.g in representation
of pi.
But many "real" values are also exact. I can make a block of wood
exactly 1.23 inches long, can't I?
I doubt very much whether even a highly skilled carpenter could do so,
except by a stupendous fluke. The "real" world is not kind to those who
like exact measurements. When we're counting, we can be reasonably sure
(if we count carefully) that we have an exact number. But when we're
measuring, there comes a point (which varies depending on what we're
measuring and what we're using to measure it) at which we must admit that
we're guessing. It might be a good guess. It might even be a very good
guess. But a guess it remains.
Nonetheless, the block has a length (perhaps unknowable in Kant's
sense) at any given time,, acknowledging that it is actually in motion
by way of warping, shrinking and expanding. Only the "Copenhagen
interpretation" of quantum theory would deny this, and say that the
real number is within some small interval.
It's irrational to do so. First of all, the interval upper and lower
bounds could also be inexact. Furthermore, a limited, bounded
precision representation still is a claim that "this is the length".
Limiting the precision doesn't change this claim.
Our first lab in my Mechanics class was measuring a stack of punched
cards...the prof made us use two digits only.
You raise an interesting point. If physically we can't really ensure
that the block is 1.23 inches long, then sure, a bounded
representation at least tells us, indirectly to be certain, not to
imagine we have the "exact" value.
That's correct. In fact, we can go a little further than that, and say that
the exact value lies within a range of values. How wide that range is will
depend, again, on what we're measuring and what we're using to measure it
(and to no small extent on who is doing the measuring - for a low
tolerance, obviously we'd prefer a skilled operator with good health, a
good eye, a steady hand, and no distractions).
IF we cannot know the exact length, we cannot know the upper or lower
bounds of the range by the same argument. Doesn't the Copenhagen
interpretation of quantum theory mean that it's possible an elephant
could be suspended by a thread?
I believe that was what Einstein believed, and he was a smart guy,
probably smarter than Bohr: IF we cannot measure exactly, THEN we
cannot measure anything, including quantum uncertainty represented as
ordered pairs of numbers {upperbound, lowerbound}.
<snip>
But wouldn't it be best not to have to do numerical analysis and have
the computer spin out the mantissa and exponent to unlimited
"lengths", perhaps by systematically modifying the base such that
hyperlarge numbers are represented say by offsets to powers of two, or
powers of powers?
There comes a point (and for many real world situations it was reached long
ago) when the precision offered by the computer exceeds the accuracy of
the data. Take your wooden block, for example. A PC can easily record the
length of the block as 1.2300000000 inches without our even having to
invent an extra-precision type - but we would be unwise to trust that the
block has been measured to an accuracy of +/- 5/100000000000 inches.
Your arguments have great merit. But they imply, I'm afraid, that we
shouldn't represent real numbers as scalars at all. Instead, an object
oriented representation containing the best guess AND an uncertainty
plus or minus is needed.
This is interesting, because it would be a radical rethink of my
number object: I am most obliged to you and Kwikius.
Since INTEGERS are primarily social values and thence exact (one's
bank balance is always considered to be exact to the penny: the number
of widgets produced by a worker in a day can always be known exactly,
since a widget is a commodity in a Marxist ontology).
Leibniz and Newton may have believed that at some future date, we
could inventory every particle in the universe and thus calculate its
future direction precisely. But given the Copenhagen interpretation,
the universe isn't some sort of shop in which we can count things
exactly, nor a counting-house, nor a go-down.
But, computers have no business, in my opinion, in imposing artificial
limits. Far from keeping us honest, they merely cause more errors.
Therefore my number object seems to need to represent reals as a
triplet: {m, e, u} where m is the unlimited precision mantissa, e is
the unlimited precision exponent, and u is the uncertainty.
u could be a single real number in turn, normally with its own u' of
zero.
Which takes us to "modal logic": it is puzzling to nest "modal"
qualifiers like "necessary", "possible", "impossible": if God exists
by necessity, is it or is it not necessarily that She necessarily
exists? What would it mean to have an uncertainty of zero with an
uncertainty of +-.0000005?
[And, the uncertainty could have different positive or negative
values, meaning that it needs to be represented as the union of a
scalar representation and an ordered pair. Unfortunately, C Sharp
don't got unions, probably so programmers wouldn't form syndicalist
ideas.]
[Another possibility: the number has a scattershot uncertainty and is
one of a possible set of values and/or intervals.]
[You'd have to redefine equals, greater, and less with an overload to
handle relations of these numbers.]
[If quantum calculations are done with ordinary numbers, errors could
be introduced when the main number is "uncertain" but the uncertainty
isn't certain.]
The most advanced treatment of this might be that of Saul Kripke, a
Princeton philosopher who subsequent to his magnum opus developed a
set of models for this issue, which unfortunately I haven't mastered.
A wicked-smart colleague at work told me about it the other day, and I
rushed over to the library to get a book about Kripke. I knew his
former secretary when at Princeton, and had learned all sorts of
things about what he's like. I would have been better off reading more
of his work.
Limiting precision is done, IMO, to complicate hardware and far from
preventing errors, causes more errors. Computers shouldn't do the
basic ops in hardware using fixed length values. They should in a RISC
fashion, support fast single-digit operations.
.
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