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Alan Sondheim on 27 Jan 2001 21:54:32 -0000 |
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<nettime> The Laying-Out (infinite 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 ===== # distributed via <nettime>: no commercial use without permission # <nettime> is a moderated mailing list for net criticism, # collaborative text filtering and cultural politics of the nets # more info: majordomo {AT} bbs.thing.net and "info nettime-l" in the msg body # archive: http://www.nettime.org contact: nettime {AT} bbs.thing.net