In order to lay the foundations for an hierarchical analysis
of the Ideal Gas Law (which will show that though Volume,
Pressure and Temperature look to be very different physical
entities, they are essentially the same thing under different
aliases) I can't do better than to reproduce a post I sent
to Ing.Saviour's Yahoo Group last January.
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Hi Savvy,
I'll try to deal with another of your points.
You wrote:-
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A line is 1m long, a plane is 1m^2, and a
volume is 1m^3. Now, you can eliminate the
metre unit and say that length is 1 in 1D,
area is 1 in 2D, volume is 1 in 3D, and
you would not loose any information, you
would be 100% correct in both value and
description. But... if one does not believe
in higher than 1 dimensions, and tries to
do the same job, he will say that length =1,
area=1x1=1 and volume =1x1x1=1, numerically
correct, but he lost all information.
-------------------------------------------
I rather think that we may be more or less agreement on this one but
to make sure I will give my view of the situation. The central
problem is what one means by dimension. For instance, if one
takes the x, y and z Cartesian dimensions to be unlimited in
magnitude to the indefinitely large and the indefinitely small and to
include all the interleaved number sets, integer, real, irrational,
transcendental and any others which I've forgotten or haven't yet
been heard of at "Harvard", ;-) then space has only three Cartesian
dimensions and to talk of more is nonsense. However, I take a
different view of spatial dimension, a view more in tune with an
engineering viewpoint which thinks in terms of upper and lower
bounds, a view more in tune with Ross-Ashby's concept of requisite
variety, a view more conformable with the realisation that any
experimental data is unavoidably finite in extent, represents a
limited a limited number of information, and therefor requires only a
finite number set to represent it; a finite dimension in other words.
Now many people treat the Cartesian axes as if they are all equal.
They are not. No matter how far you extend the x axis you are never
going to get anywhere along the y axis. The x, y, and z axes are
separated by unbridgeable gulfs. The trouble is that the hierarchical
order in which they are used is arbitrary and not prescribed. Though
conventionally one uses x for increasing length, y for increasing
area and z for increasing volume, one is not forced to do so. This
means that the necessary hierarchical relationship between length,
area and volume is hidden.
Another conventional feature which hides the necessary hierarchical
relation is the discordance between the convention governing the
presentation of Cartesian co-ordinate and the convention governing
the decimal numbering system.
I will take a particular example to illustrate what I mean.
Suppose we have three Cartesian axes, x, y and z, with each axis
having finite intervals of 0 to 999. In effect we have a 1000 by
1000 by 1000 array of volumes. Think of them as solid pixels, eh!. 8-)
Now suppose we want to represent the pixel x = 674, y = 392 and z =
518. The conventional way of doing this is to represent the co-
ordinate in the order, x, y, z. i.e. as 674,392,518 which is
unconformable with the decimal number system. If we put the co-
ordinates in the reverse order, l.e. z, x, y ¬ 518,392,674 the co-
ordinate can then be viewed as a decimal number. The 674 represents
where the pixel is along the finger of pixels axis. The 392
represents where the pixel is in the slice of pixel fingers. And the
518 represents where the pixel is in the sliced loaf. You can see
that the order, largest first, smallest last, is now the same for the
pixel co-ordinates and the number system.
Unfortunately, the concordance in the above example, though improved
by reversing the conventional order of presenting the pixel co-
ordinates, is not as good as it might be. The independent number
orders corresponding to the loaf, slice and finger, are millions,
thousand and units. The independence of these orders is symbolised by
the commas between them which are totally conformable with the commas
between the z, x and y co-ordinates. However, in the number system
there is a sub-order of hundreds, tens and ones, in each of what
might be termed the "comma orders" of millions, thousands and units.
If we wanted to line up the decimal number system and the coordinate
system even closer we will have to either change the example to a 10
x 10 x 10 array which will give us the number 537 with each digit
representing an independent dimension - OR - we can change our number
system to make it conformable with the example by going to a number
system with nine hundred and ninety nine separate characters, in
which case we will have a number which looks like this say # -
which is not a very practical proposition - me thinks. 8-)
There are two interesting examples where people have recognised the
unconformity of different systems and tried to bring them into
conformity. A contemporary example is the way that certain
programmers represent a date such as the 11th of September 2001
as 20010911 (convenient for sorting dates in order) rather than the
British convention of 11/9/2001 or even more irrational and confusing,
the US way of 9/11/2001 which as far as I can see has nothing
whatsoever to recommend it. <g>
An older example is the attempt by the French revolutionaries to de-
Christianize the calendar by dividing each month into three decades
of 10 days, of which the final day was a rest day. A terribly
unpopular move, as you might imagine, since it increased the number
of working days between the rest days by 50%.
>From the above exposition, you can see that anyone who uses the
decimal number system implicitly believes in a multi-dimensional
system since the decimal system is multi-dimensional with ten
divisions in each dimension. A non-dimensional number system would
have separate characters for each cardinal number and no way of
ordering them. It would in fact be a naming system since the names
Smith, Brown and Jones are merely identifies and have no ordinal
(hierarchical) content. If some perverse person argues that they do
have ordinal characteristics since you can order the names
alphabetically, I reply that this characteristic relates to the names
and not their owners. One must distinguish between the two since the
connections are arbitrary, just as the connections between the
characters { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } and the ordinal numbers,
{zero, one, two, three, four, five, six, seven, eight, nine} are
likewise arbitrary. You can demonstrate this quite easily by asking
someone who has been brought up with different characters for 0 to 9,
to arrange the european characters in their ordinal sequence. Their
arrangement will be largely intelligent guesswork and highly
dependent on the graphics with which the numbers are represented.
To summarise then, I see the integer decimal number system which
practical scientists and engineers [as opposed to mathematical
metaphysicians ;-) ] use to represent the physical world, as
multidimensional. What one might call the main triangulation points
are represented by the use of commas, between groups of three figure.
However, the total number of dimensions is best seen by representing
the number as ten to the power. The number of dimensions is then
simply the power, i.e. the power to which 10 is raised.
The number of dimensions is therefore dependent on the number base
which the user chooses. If the user adds six more characters, A to F
say, and chooses a number system with base 16 then the number of
dimensions will be decreased correspondingly. The maximum number of
dimensions that can be obtained is, of course that given by a number
system with only two characters, zero and one, i.e. the binary
system. This has many advantages, but is not much used outside
computing circles because our education in the decimal system makes
instant association of the symbolism of such binary numbers as
1110001111, 1010011010 and 111001 impossible. In base 10 however, the
associations are instant, viz. terrorism, the antichrist and beans. I
should point out to any executive officer of H.J.Heinz Company that I
do not wish to imply that there is any connection between
terrorism, the antichrist and their nutritious comestible. <g>
==========================================
Going somewhat off topic I have often mused that a good way to
disguise one's pin number would be to choose a memorable octal
number, 1776 say if you're a colonial, and use the decimal equivalent
of that number, 1022 as your pin number. Even better, write 1776 on
the back of your card and then any thief finding it will think your a
complete and utter moron and never dream you have coded in octal. If
you do ever forget your number then all you will need is a pencil and
paper to reconstruct it.
But surely, not even colonials could be so stupid as to choose such
an obvious number as a pin number. ;-)
You'd be surprised. I'm ashamed to say that one of my own daughters
was using 1066 until I warned her that the date of the Battle of
Hastings was one of the first dates that any English thief would try -
precisely because it is so memorable to English cardholders.
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Re-reading my post I'm not sure I have really conformed my view to
yours.
I'll take just the last point [1x1x1=1] to see if I can engineer a
meeting of minds.
Now my example for the loss of information that you recognise would
be as follows.
Suppose we have a lorry full of cigarettes. The cigs are 20 to a
packet, 10 packets to a box, 12 boxes to a case and 8 cases to a
lorry load.
The number of cigs is 20x10x12x8 = 19200.
So, the multiplication string has lost us quite a few items of
information.
We do not know how many cigs in a packet, how many packets in a case,
how many cases in the lorry. But surely, any multiplication string
loses information. There is nothing mysterious about that. The whole
of physics is built on losing information.
The cigs example is also useful in showing how one can multiply up
dimensions at will. In fact we have 5 dimensions, cigs, packets,
boxes, crates and lorries - and you will see that there is nothing at
all mysterious about the fourth and fifth dimensions. The reason we
can have multiple dimensions is that we have done away with the
Cartesian tyranny of infinite axes and recognised that for real
objects there must be upper and lower bounds. The trouble with the
quantum and 'c' is that these bounds are seen as absolute, whereas,
like all other bounds they can only be relative to the particular
physical objects they are measuring.
Cheers
Frank
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The above sets the scene for recognising the hierarchical nature
of PV = RT
Frank Grimer
