From : Annales of the New York Academy of Sciences, 138(2), 1967, p

Transcription

1 Fro : Annales of the New York Acadey of Sciences, 38(2), 967, p Panel Discussion TIME AND MEMORY HEIINZ VON FOERSSTER (University of Illinois, Urbana, Ill.): I a sorry that Dr. Whitrow is absent this orning, because I hoped he would elaborate upon soe of his delightful rearks of yesterday evening on the relationship between tie and eory. Since I believe that this relationship ay be of interest to soe participants in this conference, I wish to add just a few words. Dr. Whitrow said yesterday that we know little about "eory." I whole-heartly agree but I would like to add that we know even less about "tie". The cause for this deficiency I see in the superior survival value for all perceptive and cognitive living organiss if they succeed to eliinate quickly all teporal aspects in a sequence of events or, in other words, if "tie" is abandoned as early as possible in the chain of cognitive processes. I believe that I can give at least two plausible arguents to support this proposition. The first arguent is purely nuerical and attepts to show the infinitely superior econoy of a "tie-less" eory copared to a record which is isoorphic to the teporal flow of events. Consider a finite "universe" which ay assue at any particular instant precisely one of n possible states, S ; S 2 ; S 3 ;... S n. Let be the length of a sequence of states: e.g.: The nuber, N, of distinguishable sequences of length is equivalent to the nuber of cobinations of n distinct objects taken at a tie, ultiplied with the nuber of perutation of distinct objects. This is because the sequence S a S b is, of course, different fro the sequence S b S a. Hence n n! N =! = ( n )! If both n and are large nubers, but n is uch larger than, one ay approxiate N n The nuber of binary relays to hold this nuber - or the "aount of inforation" to be stored - is approxiately H = log 2 N = log 2 n = H 0 bits, where H 0 is the aount of inforation necessary to specify one state of the universe. As an exaple, let us consider the retinal osaic of various excited states of its rods and cones as the states of our "visual universe". Conservative estiates suggest 32 distinguishable states for each receptor of which there are about two hundred illions in both eyes. Hence, the visual universe has distinguishable states and S 5 S 23 S 4 S S 22 n = ( 32) H 0 = log 2 n = 0 9 bits

2 If only ten states per second are processed by the retina - a regular ovie projector presents 24 pictures per second - then during one second a sequence of length = 0 is to be stored, i.e., H ( second) = 0 0 bits However, the entire brain has "only" 0 0 neurons at its disposal. Let us be optiistic and assue one thousand bits stored within each neuron, then in one thousand seconds - or slightly over a quarter of an hour - the whole brain is flooded with inforation, ost of which ay be copletely worthless. On the other hand, if it were possible to integrate the sequence of a events into a single "operator" that perits reconstruction of the sequence, say, "a galloping elephant," "a flight over the Atlantic," etc., the nuber of distinguishable "acro-states" becoes n N * = and H * = H 0 log 2 The copression ratio between these two ethods of "storage" is siply * H H = log 2 log2 n If the length of the sequence is extended, this reduction in uncertainty ay take on values of considerable agnitude. FIGURE_. Scheatic representation of a unicellular anial oving fro one spot to another by extending a tubular pseudopod and pulling itself up through this extended capillary. I hasten to deystify these "operators" which I have just entioned. Indeed, they are perpetually coputed in our perceptive syste and their abstracting powers becoe apparent in their linguistic representation, usually in for of naes for spatial abstracts and of verbs for teporal abstracts. In fact, without these abstracting operators we could not conceive of otion or of change, and - as an 2

3 ultiate abstraction - of the flow of tie. Conteplate for a oent the soewhat foralized representation of a unicellular anial oving fro one spot to another by extending a tubular pseudopod and pulling itself up through this extended capillary (See FIGURE, Stages -6). However, what we see in fact is a sequence of six apparitions of quite distinct shapes of which it is difficult-i believe even ipossible-to assert that they represent the "sae object" unless, of course, the set of transforations is specified under which the properties of this "object" reain invariant. These transforations ay accoodate spatial as well as teporal aspects of the object under consideration, as can be seen by the linguistic representation of the totality of events depicted in FIGURE : "a unicellular anial oving fro one spot to another by extending a tubular pseudopod and pulling itself up through this extended capillary." The invariance of the sequence, 2, 3, 4, 5, 6 is suggested by a feeling of "inappropriateness" if any perutation of this sequence, say, is proposed as an alternate possibility. It is clear that it is due to eory that teporal abstracts can be coputed and stored. Although eories ay have soe charing aspects, their crucial test lies in their efficacy to anticipate sequences of events, in other words, to perit inductive inferences. The conceptual construct of "tie" is, so far as I see it, just a by-product of our eory, which in soe instances ay use "tie" as a convenient paraeter - a tertiu coparatu, so to say - to indicate synchronis of events belonging to two or ore spatially separated sequences. Of course, there is no need to refer to tie in such coparison, for it is always sufficient to take one sequence as "standard" and to associate with standard events the events of another sequence as, for instance, the anticipation of the sequence of events regarding Peter: [] " Verily I say unto thee, that this day, even in this night, before the cock crows twice, thou shalt deny e thrice." In parenthesis it ay be interesting to note that this prediction would iediately lose its punch if it would use teporal reference, for instance: " even in this night, before 6:30 a.., thou shalt deny e thrice." The intellectual slup one suffers in this version stes, I believe, fro the fact that idealized absolute, or Newtonian, tie carries no inforation: H 0. Unfortunately, in spite of considerable efforts by clock designers to build a perfect clock, this ideal goal has not yet been attained. There is still a sall residue of uncertainty H = ε > 0, which is due to sall inaccuracies of even the best clocks and, ultiately, there will be quantu noise which will set an absolute liit to this enterprise. Since these assertions which represent y second arguent ay, at first glance, sound surprising, let e first deonstrate that Newtonian tie is a useful but unnecessary paraeter in a coplete description of the universe; second, let e briefly state what I ean by an accurate clock. In practice an approxiation to Newtonian absolute tie, called "epheeris tie," is obtained in two steps as Dr. McVittie pointed out earlier this orning. First the equations of otion in Newtonian echanics are solved for various celestial bodies, in particular for the different planets P i. These solutions usually express the positions r i of these bodies in ters of a linear paraeter, t, called tie: r i = r (t) Second, the scale of this paraeter is adjusted so as to give the best fit between observation and the theoretical solutions. In fact, this established scale fits the i 3

4 observations so well that there is a residual error of only one unit in about 0 units. [2] With this estiate one ay calculate the uncertainty H of reading this astronoical clock. With the probability p of aking an error p = 0 - and with the definition of H for a binary choice: H = - p log 2 p ( - p) log 2 ( - p), one finds the uncertainty to be H = bits/ unit. It ay be interesting to deterine whether this sall residue of uncertainty is due to "noise" in the observed syste, that is the planets refuse to obey Newton by occasionally perforing sall extravagancies in their otherwise predictable behavior, or whether this noise is introduced by the transission channel, that is by inaccuracies of observation due to rando fluctuations in the atosphere, in the optical equipent or in the evaluation of data. If the latter should be the case, then this uncertainty can be ade arbitrarily sall on the basis of Shannon's [3] far reaching theore which states that if the sae sequence of signals is repeated again and again the effect of errors (within certain liits) that are introduced during transission can be ade arbitrarily sall. This principle of "Reliability through Redundancy" is ost effectively applied in all periodic or repetitive phenoena. All "clocks," for instance, are based on periodic event sequences, and environental periodicities can be recognized, or absorbed, by the genetic code of reproducing and utating living organis by prograing these periodicities into the organis in the for of circadian rhyths: "Even if you read e poorly, if you read e often enough, you will get y essage." If I should ake a guess as to the causes of the sall residue of uncertainty left in establishing epheeris tie, I would venture to say that they coe indeed fro a certain capriciousness of the planets, for the channel noise has ost probably been eliinated by the prolonged observation of these periodic events. After having assured ourselves that this paraeter "tie" is universal *) and does not change scale fro planet to planet, we can now eliinate it fro the set of equations which represent the positions of the planets as a function of this paraeter by selecting the positions of one planet r 0 as reference for the positions of all others: R (r ) ri = i 0 In other words, one would, for instance, tell the position of Mars in reference to the position of Venus, etc. Adhering to this schee, appointents would be ade in this for: " we shall eet after the sun has risen twice." Of course, this is precisely the ethod outlined earlier in which siultaneity of events belonging to different sequences are used to establish the "when" of an event of one sequence by reference to a particular event of a standard sequence. *) That is within the fraework of Newton's equations of otion. 4

5 For practical purposes, however, it is convenient to have a highly redundant signal generator - a reliable clock - which facilitates the estiates of the siultaneity of events in a large nuber of sequences. Clearly, such a device should not inject into the universe of observation unwanted uncertainties, i.e., each subsequent state should be well deterined by its predecessor. This is ost easily accoplished if this device goes at a constant rate, which gives the additional advantage that such a device ay read epheeris tie which is a useful paraeter in Newton's equations of otion. This brings e to y second point, naely, what do I ean when I speak of a "reliable," of an "accurate," clock that goes at a "unifor rate". As Dr. McVittie has already pointed out, coparison of one clock with another ay never establish which one goes ahead and which one falls back. Since cross-correlation between two clocks leads us nowhere, I propose to consider for a oent auto-correlation of one clock with itself. FIGURE 2 shows such an arrangeent where an illuinated diagonal slit in a circular rotating disk, representing the clock, is located at the center of curvature R of a convex irror. In this arrangeent, the clock is optically apped onto itself. +) FIGURE 2. Deterination of the accuracy of a clock through auto-correlation. If the disc rotates with angular velocity ω, the angular position φ of the slit and φ 2 of its iage are given in paraetric for (tie t is paraeter): φ = ω t φ 2 = ω(t 2R/c) where c is the velocity of light. Eliinating t fro this pair of equations expresses the position of the slit's iage as a function of the position of the slit: φ 2 = φ - 2Rω/c +) The inversion is copensated by aking the hand a diagonal slit rather than a radial one. 5

6 This eans that the slit will be trailed by its iage at an angle of φ 2 - φ = - 2Rω/c If the curvature of the irror, its distance fro the slit and the velocity of light are for the oent assued to be constant, then it appears that the angular displaceent between slit and iage will reflect all variations in the rate by which this clock proceeds. If ω increases, so will and conversely, a reduced w will result in a saller displaceent. One is tepted to say that a constant indeed ascertains an accurate clock that proceeds at a constant rate. Nevertheless, this naive interpretation has been subjected to a severe criticis by E. A. Milne ++) who rightly points out that a constant displaceent eans only that its variation vanishes. With = - 2Rω/c this eans that or δ dr δr δ + δω δ = 0 δ dω + dc = 0 δc Calculating the partial derivatives suggested above, this expression ay be rewritten to read dω dr dc + = ω R c if in spite of constancy of the displaceent a dependency of ω with the position φ of the slot is suspected, one ay write: dω dr dc + = ω dφ R dφ c φ d This expression suggests that indeed a variation of rate at which the clock proceeds dω 0 dφ ay go unnoticed by relying on an invariable displaceent, if the irror flaps or wiggles, or if the velocity of light jerks back and forth precisely in such a anner as to copensate for the variation of ω. One ay iagine a "law of nature" that couples the three quantities R, c and φ so that the relation obtains dc dr = Ω( φ) c dφ R dφ where the function on the right hand side represents the expression dω( φ) Ω( φ) = ω( φ ) dφ ) ++) Milne's criticis is addressed against a "Gedanken experient" which incorporates entirely different physical devices. However, the basic features in both experients are equivalent. 6

7 There are, of course, an infinite nuber of functions c(φ ) or R(φ ) or couplings between R and c, which will satisfy the above differential equation. However, the ausing upshot of this side issue is that whatever these laws ay be, they ust transit with great accuracy the fluctuations in the clock ω(φ ) to the irror and force it to wiggle or to flap, or to the velocity of light to jiggle in precise copensation of the fluctuation of the clock so as to ake the deviation to appear unchanged or, at least, to change so little that epheeris tie can still be deterined with the great precision entioned earlier. Hence, these processes - iaginary or real - are of such redundancy that they do not interfere with our use of the paraeter "tie" as a tertiu coparatu which facilitates highly accurate deterinations of the siultaneity of events belonging to different cognitive sequences. References. ST. MARK: G. M. McVITTIE. 96. Fact and Theory in Cosology : 25. Eyre & Spottiswoode. London. (96). 3. C. E. SHANNON & W. WEAVER The Matheatical Theory of Counication. : 39. University of Illinois Press. Urbana, Ill. 7