From alans@mailhost.sva.edu Mon Jun 10 17:37:37 2002
Date: Fri, 7 Jun 2002 18:41:17 -0400 (EDT)
From: Alan Sondheim
To: sondheim@panix.com
Subject: abacus
The Laying-Out (infinite abacus)
Consider an abacus with infinitely long columns; this is similar to a num-
ber system with base infinity. (See below for finite approaches.) Now to
add, simply move the requisite number of beads X to the already-calculated
Y; you have X + Y. To subtract, do the reverse. To multiply, three columns
are used. The first column is infinitely deep; the second is set to zero;
the third to Y. Set X on the first; move one bead on the second; set a
second X on the first; move a second bead on the second, and do this until
the second and third columns are equivalent.
One might also use two measure-strings and two columns. Set one measure-
string to X, the second to Y; one column measure out to X, the second set
at 1; measure a second X on the first column, add another 1 on the second
and repeat until the second column is the length of the Y measure-string.
One could use as well just one measure-string and a marker on the side of
column Y; when the beads reach the marker, the calculation is finished.
Division is a reversal of this process; set X and set Y on a measure-
string; subtract the measure-string from X; add one to the second empty
column; repeat until no more subtractings are possible; the length of the
beads in the second column represents the integral quotient; what remains
in the first is the remainder.
The measure-strings in both instances are placed next to the requisite
beads on the first and second columns.
What is unique in this system is that there is no necessity whatsoever to
name the numbers of beads, i.e. assign them to particular integral values.
Instead, one has greater or lesser numbers of beads in the first column;
after the operation, the rough length of the beads is the result. In this
manner, a king may count his horses, a queen her subjects, without further
need of specific tallying. When the column gets especially low, judge as
"more or less"; that is all that is necessary.
This is to some extent the mathematics of the heap or pile, a mathematics
with an inner exactitude, but a fuzzy reading of both givens and results.
Of course one can also consider an abacus with one bead per column and an
infinite number of columns. In this manner, addition and subtraction are
again to the base infinity, simply the moving up and down of beads at the
leading edge of the quantity. But if one is insistent on multiplication, a
second tally is necessary, and if one is insistent on division, one must
look for the same.
Moreover, it is unnecessary to specify an infinite length or number of
beads or positions. An inaccessibly high finite number will do - or even a
finite number practically greater than any conceivable calculations might
warrant. Of course such a number could be arbitrarily extended or retract-
ed by convention or convenience. In any case, problems of platonism or in-
finity are bypassed in this fashion; the systems are both functional and
phenomenologically interesting.
Addition with columns or rows
1oooooooooooooooooooooooooooooo
2oooooooo
1oooooooooooooooooooooooooooooooooooooo
2
Division with columns or rows and tallies (2ooo can be measure-string)
1ooooooooooo
2ooo
3
1oooooooo
2ooo
3o
1ooooo
2ooo
3oo
1oo (remainder)
2ooo
3ooo
I'm fascinated by these simple systems of primitive measurings and tabula-
tions of exactitude, of quantities precisely calculated but unknown, of
the measurings of kingdoms without largesse and the origins of bureaucra-
cies. For nothing more is needed than the laying-out of rows upon the
ground, exalting at the beads disappearing in the distance, and worrying
when the line becomes shorter, almost starved and measurable.
-----
Appendix of language etiquette:
oooooooooooooooooooooooooooooo File: ww
1oooooooooooooooooooooooooooooo
oooooooo File: ww
1oooooooooooooooooooooooooooooo
2oooooooo
oooooooooooooooooooooooooooooooooooooo File: ww
1oooooooooooooooooooooooooooooooooooooo
0: oooooooooooooooooooooooooooooo oooooooo
1: oooooooooooooooooooooooooooooo-oooooooo
2: oooooooo oooooooooooooooooooooooooooooo
3: oooooooo-oooooooooooooooooooooooooooooo
ooo File: ww
Division with columns or rows and tallies (2ooo can be measure-string)
00: boo 01: coo 02: foo 03: goo 04: loo 05: moo 06: Ofo 07: oho 08: Oto
09: too 10: woo 11: zoo
ooooooooooo File: ww
1ooooooooooo
0: oooooooo ooo
1: oooooooo-ooo
2: ooo oooooooo
3: ooo-oooooooo
ooooo File: ww
1ooooo
oo File: ww
2ooo
3oo
00: Ao 16: moo 32: Oto 01: bo 17: no 33: ow 02: boo 18: o 34: ox 03: coo
19: od 35: Oz 04: do 20: oe 36: po 05: Fo 21: of 37: Ro 06: foo 22: Ofo
38: so 07: go 23: Og 39: to 08: goo 24: oh 40: too 09: ho 25: oho 41: wo
10: io 26: Ok 42: woo 11: jo 27: om 43: yo 12: ko 28: on 44: zo 13: lo 29:
ooo 45: zoo 14: loo 30: or 15: mo 31: os
=====
Plain and Heap
No longer the line, the geometry, only the gatherings of beads, tokens,
markers, units, only the rubble of accountancy:
At the level of the plain, the abacus behaves differently. Consider an in-
finite (or inaccessibly high finite etc.) flat surface of beads; it be-
comes necessary to isolate portions among them, for example a 1-portion
and a 2-portion, in order to carry out any operations.
The portions have to be specified by closed, i.e. Jordan, curves which do
not cross themselves. For addition and subtraction, either channels must
be opened between the portions (in order to join or separate them), or an
operation may be carried out, such that removing 1 bead from a 2-portion
is accompanied by adding 1 bead to a 1-portion:
2-portion - 1 => 1-portion + 1
until 2-portion is empty, i.e. 2-portion = 0 (in which case the 2-portion
may no longer exist).
Subtraction is in the opposite direction, but in order to subtract a spec-
ified amount note that we dispense with channels, so that
2-portion - 1 => 1-portion - 1,
with the bead from 1 portion returning to the undifferentiated plain, and
this continues until 2-portion = 0 (or becomes non-existent).
Note that a method is necessary to keep 1-portion and 2-portion separate
and labeled.
Multiplication and division also work by repeated additions and subtrac-
tions, using a 3-portion, if not 4-portion and 5-portion for tallying -
see heaps, below.
And how are portions diminished or incremented in the midst of a solid sea
of beads? Moats must be constructed, portions piled in grounds cleared of
any interferences.
So that from the plain, one gathers beads into heaps. The heaps are separ-
ated by blanked space, the ground-state; this state G = 0. G is always
existent; it is a territorialization, a boundary, a marker by virtue of
lack of demarcation. If G = 0 it is 0 anywhere, everywhere that G is.
Think of it as the Basin in Bon religion, or Kristevan chora. Nothing is
ever placed within G; G surrounds heaps. New heaps may be created, but G
is not among them, within them. In this sense, G is not a natural number,
but a numberless state, not even an emptying of number. In this sense as
well, the integers move from 1 upwards; negative integers might be indi-
cated by heaps labeled as negative, but in fact all heaps are of the form:
HN = | N | where N > 0.
To add: combine heaps. To subtract: Either remove any number of beads from
one heap and place them in a second heap or:
Begin with H1 = X and H2 = Y. To reach X - Y:
Subtract a bead from both; place these beads in a third heap H3; carry
this operation out until H2 is empty. When H2 is empty, H2 = 0 = G, the
undifferentiated state, H2 no longer existent. But see below, labeling.
This depends of course on Y < or = X.
To multiply X * Y. Create H3 and H4. Move H1 bead by bead to H3; each
time, add one bead to H4. When H1 = 0 (i.e. non-existent), subtract one
bead from H2 and place it in H5. Then move H3 bead by bead to H1 (which
must be created anew - H3 and H1 oscillating in this fashion); each time,
add one bead to H4. When H3 = 0 (i.e. non-existent), subtract another bead
from H2 and place it in H5. We then have
H1 = X
H3 = 0 (i.e. non-existent)
H2 = Y - 2
H4 = X * 2
continue until
H2 = 0 (i.e. non-existent) at which point H4 = X * Y.
In order to divide, a similar process is used, tracking beads removed from
H1 in quantities of H2, until H2 > H1 (as divided), in which case what
remains in H1 is the remainder.
We have numbers to no base or perhaps to base infinity or to base 1. At
the level of the heap, infinity cycles back to 1; at the level of the
heap, there is no positionality. With the infinite or inaccessibly high
finite column abacus, an infinite base might be theoretically employed,
each integer individuated; with the infinite or inaccessibly high finite
row abacus, a base of 1 might be theoretically employed, each integer an
extension.
Throughout all of this, the heaps must be labeled (just as the portions
had to be labeled); think of a interior beads in label quantities o oo ooo
oooo ooooo surrounding an emptied center, with rays connecting the center
through the label quantities to the heaps beyond. If there are no heaps
beyond certain label quantities, those spaces are reserved for the crea-
tion of heaps, i.e. as in the oscillation of H1 and H3 above, which is
based on the constant recreation of heaps in order to carry out multiplic-
ation and division.
Naturally the ground must be kept clear of any rolling beads, false beads,
embezzlement beads, stolen beads, extra beads, and masqueraded bead-like
objects. And naturally the ground must be level to avoid such rolling
beads as well as emptied, flat, in order to create maximum conditions of
visibility. One might think of Bentham's Panopticon as described by Fou-
cault in this regard - while beads remain undifferentiated, except by
virtue of belonging as a member of a set (heap), they must nonetheless
leave a trail or trace; nothing must remain unaccounted or unaccounted-for
and nothing must be unaccountable.
=====
Infinitely Small
What happens when the size S of beads grows infinitely (or inaccessibly
low finite) small?
Of course one does not consider the size of abacus beads. But, given the
heap, one might want to...
With the abacus of infinitely high (or deep) columns, or the abacus of
infinitely wide rows, measurement becomes an issue. Thus to add length X
and length Y, measure X + Y, move beads. To subtract X from Y, measure X
and within X from the far end, remove beads.
To divide, repeatedly subtract X from Y until X < Y which is the remain-
der. But without quantity, who knows how many subtractions? Keep track of
subtractions with finite beads! (Note the need for a second measurement,
second instrumentation.)
However, to multiply is impossible; without scale, without beads, there is
no way of telling how many beads are necessary. This is fascinating - in
this instance it's easier to divide than multiply!
One might measure X against Y by moving string, and in that fashion create
an accurate multiplication - but only if X/Y or Y/X = an integer. Other-
wise one is lost without further calculation, fractions, and so forth. An
exercise: What calculations are necessary? What additional instruments?
Given the plain, it's easy to add portions and subtract portions - the
latter by laying the flat-measure of one portion on top of the other, or,
given the two-dimensional invariant, removing portions from X and placing
Y within them, until the roughly the same shape occurs (providing one has
geometry at work); one might also place Y next to X, and remove portions
of X until it appears that Y and the removed portions are equal.
Appearances, appearances. Multiplication and division become much more
difficult. An exercise! (And an exercise in perception. Again, we're run-
ning into external instrumentation, tallying.)
Consider the heap: To add, add portions. TO subtract, take away. To sub-
tract a given portion, take portions away until they appear equivalent to
the given portion.
Or weigh the results. Or displace, measure water. Think of Archimedes.
All of these bring quantity back by other means. Suppose there are no mea-
sure strings, no way of judging equivalences, no waters, no scales. Then
one might add by bringing portion to portion, and one might subtract - but
only ikonically, by removing the subtracted portion. (This is quite impor-
tant; if the abacus is indexical in Peirce's terms, the heap of substance
is ikonic; representation is peripheral and obtuse at best. One is mucking
in the real, not fucking in the symbolic. Literally!)
One could not, for example, say, remove 1/3 of that heap, or remove this
amount Y from that heap. One would begin with heap X and remove Y, both
unquantified, and then one would be left with:
X - Y
and Y.
How to remove a second portion Z such that Z = Y? This is impossible with-
out external measuring.
One begins with heap or heaps, one separates, one combines. It is a kind
of concatenation without labeling. One can't say synchronically, spatial-
ly, that X = Y + Z, but one can say diachronically, temporally, that X
_did equal_ Y + Z. It is always a question of process.
I am sure I am making errors throughout. Nonetheless: With infinitely
small beads, with the reign of _substance,_ digital meets analog; the
raster is now infinitely fine, and the modeling of the real has become
equivalent to the real itself. As above, the ikonic becomes identity, a
movement from the equivalences constituting the digital, a movement from
the _adjudicated_ (and hence the origins of culture) raster back to
relative undifferentiation.
(If the digital inhabits eternity, the analog is worn, worn down; who is
to say that erosion doesn't enter into the mess of the heaps, that with
the division into Y + Z noise enters as well - Serres' Parasite - a bit of
spillage, entropy, ultimately bringing down the constituted house? We are
close to the _nerves of the real_ here, dissipated signals within which
even the digital must reside. Pass the thresher, pass the threshold...)
=====
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
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ooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
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ooooooooooooooo
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=====
infinite abacus, accountancy, culture
meandering paths among heaps and stones
slow streamings, quietude
murmurs of pebble movement
who is counting here
who knows the names
=====
no hands, names, dullness of flame, spark, & name. & knew there was some-
thing i had dreamed, of names, of the numbers of names,, who knows the
names i tried,, spreading words & wounds, numberless states, places,
unemptied, handless, ashes & charnalhouses, sparks & nameless flames,,,
=====
what remains after the deluge
maya maya maya maya maya maya maya maya maya maya maya maya maya maya maya
maya &a; &b; &c; &d; &e; < > & &lb; maya maya maya & maya & " maya maya
maya maya maya maya maya maya maya < maya maya maya maya maya maya maya
maya maya maya maya ^? maya ^? maya maya maya maya maya
maya-prayer-extensions maya maya maya maya maya maya maya maya maya maya
maya maya maya maya maya maya maya maya maya maya maya maya maya maya maya
maya maya maya maya maya maya maya maya maya maya maya maya maya maya maya
maya maya " maya maya maya maya maya < > maya > maya maya maya
after the filtering and after the protocols
after the enumerations, after the denumerations
what remains after the orbit
after the rockets thrust into datagrams and routings
across softwares and hardwares
what remains as higher ascii particles are cleaned up, the text
rendered taut / distraught, nothing but ashes, it's just like
any work, all work at a _loess_
=====
implementation of (infinite) abacus
balls in grooves & if you slant the rock they roll dependent
on the tilt and depth of cut so they do not fall out,
so they are there or not there
anyone can tell they are there or not there
bring the balls back, lower the tilt
so they lower the tilt, the balls remain
let ones move from groove to groove
later on it will remain for all time: let ones remain from groove
to groove
count or capitulate balls
later on, the grooves worn, balls flailing against all accountancy
we must move the tallying
later on, museums of grooves, a few remaining balls, weakened, witness,
positionless
lift the grooves from the surface of the rock, breaths of different
exhalations furrowing the air
creatures scurrying around the elder place of tallying
office lamps, illuminations, you can hear the clacking of the balls
even on the lower floors
=====
living within our means
of the abacus as if we're haunted by beadwork
and its mobility, as if ends were always elsewhere
beyond the framework
really nothing more than a domain or groundwork
where you might find someone sweeping the garden
of stones into patterns always forgetting
every one of them moving, about to be moved,
about to move, every one of them quiet, in place
beyond the framework,
invisible, as we're
borne in streams, as we're borne streams, muting
along worn and barely visible furrows, in earth
or stone, or in water or among membranes, most
comforting, surrounding us, placenta feeding us,
or feeding gardens, or earthwork,
or groundwork, with furrows, as we begin,
to constitute, or begin, to cancel and as we begin,
to calculate or begin,
the smallest particle of speech, already bead,
already speechwork, already labor, and
the measurement of the hole, where nearly, the bead
might move, in placework, might be placed,
might place, among our conscious, within, withal,
our perplexity, that we may comprehend, mindwork,
furrows and garden, work and beadwork, softly
already
=====
on abacus onon on
on on abacus onon
onon abacus on on
=====