Volume 06 of W. Ross Ashby's Journal
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1942



Volume 06



1176+03	1176+04

Summary: A field can be explored easily, but break-surfaces are destroyed by their discovery. This may involve curious philosophical properties. Break surface exploring Organisation exploring organisation

1177	1178


Summary: Dynamic systems are, in general, fundamentally irreversible. Dynamic system irreversible Reversible process in dynamic system
1179	1180

1943

Summary: The concept of "break" does not need that of "irreversibility". Break reversible Reversible process breaks
Summary: Theory has been submitted for publication for the third time. Meanings of words in my form inhibition, reflex
1181	1182



1183	1184



1185	1186

Summary: The concept of "a reflex" is translated into my organisational terminology. Reflex in dynamic system
Break surface controlled Dominance over breaks Levels controlling breaks
1187	1188

Summary: A dominating system can control the position of break-surfaces of a second system. Dominance two types

1189	1190



1191	1192

Dominance and velocity Independence and velocity Order of velocity

1193	1194



1195	1196


Summary: It has been shown that a representative point, staying within a region bounded by a layer of break-surfaces, can act as a "variable" in a substitution composed of n such points provided the representative points move with a velocity of a higher "order" than that of the substitution. "Order" is defined and explained. The ordinary substitution can be considered as the limit of this type. Break surface controlling substitution
1197	1198

Reflex in dynamic system
Summary: A discussion of a simple reflex along my lines. Meanings of words in my form inhibition, reflex
1199	1200

Summary: "Adaptation" is more properly divided into: the adapted state after this has been reached, and the process of finding this state. Adaptation defined
Mathematics 1+1?2
1201	1202


Adaptation growing Equilibrium graduation of
1203	1204


Organisation splitting into parts
1205	1206

Part-function defined

1207	1208

Summary: We study how adaptation can increase qualitatively, and are led to define and examine "part-function" and "part-surface". (Continued 1219)
Organisation number of types of
1209	1210



1211	1212



1213	1214



1215	1216

Summary: The number of possible ways of organising n variables is at last answered. It is of the order of \|n Operator example Substitution (mathematical) example
Summary: An interesting elementary substitution is described. It demonstrates paths going to infinity and neutral cycles. (Better 3776) Neutral point (of equilibrium - including 'cycle', 'region' etc.) example of cycle
1217	1218

Summary: A property of step-functions. Step function (not)

1219	1220



1221	1222


Threshold significance
1223	1224

Holism excessive
Summary: A method is described by which a machine can show increasing adaptation, by one part after another getting into equilibrium. A clear explanation of "threshold" and "summation" in the Central Nervous System follows. It is concluded that between a sense organ and the adaptive part a "distributor" must occur. 5345 Adaptation non-adaptation Survival failure of
1225	1226

Break surface peculiarities of

1227	1228



1229	1230



1231	1232

Summary: An attempt is made to classify and exhaust the causes of non-adaptation; but it seems that non- adaptation must be taken as fundamental, adaptation occuring only if there is some special reason for it.
Book sketch of
1233	1234



1235	1236



1237	1238



1239	1240



1241	1242


Summary: Arrangement and collected materials for my book.
1243	1244

Invariant essence of Substitution (mathematical) invariants of
Field (of substitution) invariants of
1245	1246

Summary: Discusses the application of the concept of the "invariant" of a substitution.
Organisation number of Organisation number of types of
1247	1248


Adaptation after previous adaptation
1249	1250


Complex (Freudian) as internal environment
1251	1252

Environment "internal" environment
Summary: Notes on adaptation to "internal" environment; and an example of how a set of adaptations can collapse. Adaptation and mutations Adaptation chains of Evolution as law and chance Holism in mutations Natural Selection [13]: In adaptation by heredity there is the recombination effect 1254. Quotation 2163.
1253	1254


Break mostly harmful Organisation selection of Selection of organisations
1255	1256

Summary: Huxley's book reviewed, and proof that a holistic set must be altered by infinitesimal steps. Adaptation growing Organisation must change only infinitesimally
Equilibrium tends to "normal" type
1257	1258

Summary: We have a right to expect that normal equilibrium will be commoner than other sorts
Part-function further discussion Step function in substitution Substitution (mathematical) part-function in
1259	1260

Break as path property Field (of substitution) special types Organisation and special fields

1261	1262



1263	1264

Summary: Part-functions and step-functions should be defined as special types of path in a field.

1265	1266

Summary: Whittaker defines "equilibrium" and also a "neutral cycle".

1267	1268

Observable equals projection

1269	1270



1271	1272

Dominance in field

1273	1274

Field (of substitution) dominance in

1275	1276



Hover mouse here to display note
1277	1278



1279	1280

Part-function defined Step function defined
Summary: We have got a grip of "part-function", finding that it depends simply on zero values of dx_i/dt.
1281	1282

Summary: Some points from a book.

1283	1283+01



1283+02	1284



1285	1286



1287	1288



1289	1290


Summary: A description is given of relations between differential equations and solutions when certain variables are not present in some of the equations. Two matrices \|f\| and \|F\| are defined. Particularly it is shown that the "independence" test of p applies to either.
1291	1292



1293	1294

Summary: A view of Levy's book. He specifically notices that breaks are an essential feature of matter and not a trivial one.

1295	1296

Summary: The concept of "dominance" involves an inverted way of looking at things, and is better replaced by the same variables being "independent of the others" in a system.

1297	1297+01

Summary: We may not write arbitrary functions in the solutions x_i=F_i(x^o;t), for the f's are to be free from t. This means that there are restrictions on the F's, and it is shown that suitable F's will satisfy certain equations. (Cf. 1315)(and 1341)

1298	1298+01

Summary: Definition of the First and Second Jacobian matrices of a dynamic system, with a note that "completion" applies to the Second and not the First.

1299	1300

Homeostasis Carrel's list

1301	1302


Summary: A review of Carrel's "Man, the unknown".
1303	1304


Summary: The concept of "parameter" should be replaced, (except in simple cases), by the idea of a variable having some special properties, These are given. The fundamental is [x^-_k=0]. (But see 1324)
1305	1306



1307	1308



1309	1310

Oddments [16]: Removing excess variables 1311.

1311	1312



1313	1314


Summary: Exploring the interaction of a given set of variables means finding the F's in x_i=F_i(x^o;t). (Assembling a machine gives us the [x^o_i=f_i(x)] equations). By the independence test on the Second Jacobian Matrix applied in one stroke we eliminate what is not wanted. That its behaviour is reproducible is equivalent to the requirement that t is explicitly absent from the f's. This restricts possible F's. An equation is given which they must satisfy. It is proved that under these conditions the F's are always completed.
1315	1316


Summary: "Step-function" in practice is not usually so restricted as on 1279.
1317	1318


Summary: At last an exact meaning can be given to the idea of whether one variable does, or does not, affect another. It can only be tested when the complete system containing the affected one is obtained. A set, independant of the others, contained in a complete set, must itself be complete.
1319	1320

Summary: Nil.

1321	1322

Summary: A definition of a complete system, and some elementary properties.
Summary: Parameters which are regarded as constant "variables" thereby lose some freedom, perhaps too much sometimes.
1323	1324


Summary: A single permanent zero in [f] introduces a slight, permanent restriction in the field.
1325	1326

Summary: The non-zero elements in [f] correspond, in a sense, to dendrons.

1327	1328



1329	1330



1331	1332


Summary: The chance that n variables should all independently be in equilibrium is discussed and this gives an estimate of the time required to reach equilibrium. The fastest method of getting equilibrium will be the one found in practice, for the system selects the fastest. And this suggests that the brain will automatically manifest an "analysing" tendency.
1333	1334



1335	1336



1337	1338


Summary: The environment (probably) consists of many small complete systems contained in larger complete systems, etc slow time changes upsetting all. Two more ways of graduating adaptation are noted. The dynamic form of "whole" and "part" is clarified.
1339	1340



1341	1342


Summary: The solutions of a complete system form a finite continuous group of order one.
1343	1344


Summary: Notes from Bieberbach on finite continuous groups.
1345	1346

Dominance and velocity Independence and velocity

1347	1348


Summary: Variables changing at different orders of velocity hardly interact. A study of interaction must therefore assume the variables are of the same order of velocity (Now turn to 1474!)
1349	1350



1351	1352

Summary: The relations of "complete sets which contain complete sets which ..." can be shown accurately by an isomorphic diagram.

1353	1354



1355	1356

Summary: Assuming each variable has a fixed chance of getting equilibrium, it is shown that a system of n₁, variables dominating n₂ will in 1-p^n₂ cases get equilibrium by getting it in the n₁ and then in the n₂, while in p^n₂ cases it will get the whole simultaneously, the latter proportion being vanishingly small. Experiment will therefore demonstrate the equilibrium appearing in stages.
Hour glass system defined
1357	1358

Summary: An unsolved problem in organisation. (Now see 1420)

1359	1360



1361	1362



1363	1364



1365	1366

Summary: If a complete system has n variables and r parameters [x^-_i=f_i(x;λ)], then the λ's can, from given starting point, control the movement of the x-point within an r-dimensional space which moves with time through the n-space, but the λ's cannot control the movement of the r-space. (Now see 1376)

1367	1368


Summary: A Permanent zero in the 1st. Jacobian Matrix, i.e. incomplete joining, means that a sudden change of the variables does not immediately alter the path as projected on to the other variable's axis. (Continued 1372)
1369	1370

Summary: The 1st Jacobian Matrix (1) cannot be filled in arbitrarily (2) does not accurately specify a dynamic system.

1371	1372

Summary: If each break (a) depends only on one variable, (b) affects, or appears in only that variables' f, then each variable will become stabilised almost independently of the others. Under these conditions the time taken by n is of the order of log n.

1373	1374


Summary: As first approximation, the "largest of a sample of n" tends to increase as log n.
1375	1376

Summary: If r parameters controlling a complete system are arbitrarily under our control, then we can, by controlling the parameters, force an arbitrarily selected set of r variables to behave as we chose. The detailed control can, so to speak, be transmitted through the many other variables without any loss of control! Input control possible Parameter degrees of freedom
Summary: Note from Eddington.
1377	1378



1379	1380

Reducibility defined Oddments [15]: A "Reducible" complete system defined 1381.

1381	1382



1383	1384


Oddments [6]: "Almost absolute" systems, defined 1409, some properties 1386.
1385	1386



1387	1388

Summary: The problem of several complete systems joining into an interacting system without losing (entirely) their completeness is discussed and partially solved.

1389	1389+01



1389+02	1390


Summary: The solutions are given of the problems of: Given the f's (or the F's), to find the F's (or the f's).
1391	1392



1393	1394


Summary: A proof, with modern technique, of the old problem, showing that two stable machines can be joined to form an unstable one.
1395	1396



1397	1398


Summary: A test to see whether a neutral point is stable or unstable. (Test for neutral cycle, 1494)
1399	1400



1401	1402



1403	1404



1405	1406

Summary: The old case of several variables affecting one another chain-fashion is re-examined. It is shown that if an "increase" leads back to a "decrease" the system will be stable, though probably with oscillations (of decreasing amplitude). If it leads to an "increase" the system may still be stable.

1407	1408

Summary: Contrary to p.____ [0599], the concept of equilibrium does not depend on a circuit. Oddments [6]: "Almost absolute" systems, defined 1409, some properties 1386.
Summary: Definition of an "almost" complete system.
1409	1410

Memory and unobservables Unobservables replacement by other observations

1411	1412

Higher geometry of fields and matrix theory [5]: If some variables in a complete system are unobservable, a path fixed by n t,x combinations can be accertained empirically, by using substitute variables, 1413, numerical example 1470.
Summary: If the study of a complete system of n variables is restricted to some of the variables only, the others being hidden, the behaviour of the visible variables can be predicted correctly when we know any n coordinate-time combinations. A machine may appear to show imagination. (Restated 1424)
1413	1414


Summary: The (real) environment may be absolutely anything. But we can devise theoretical systems to which a given brain could and would adapt, and we then examine the real world to see if such sorts exist.
1415	1416



1417	1418


Summary: The idea of a "constraint" added to a dynamic system may have meaning with Newtonian dynamics but it has no general meaning. And the idea of thereby losing a "degree of freedom" is also of restricted applicability. Hour glass system properties
1419	1420



1421	1422


Summary: It is shown that the "hour-glass" type of organisation will differ little from others in its properties of adaptation.
1423	1424

Summary: If, in a system of n variables complete or not, we are given n coordinate-time pairs, the particular path is fixed. Higher geometry of fields and matrix theory [4]: A path is fixed (in a given field) by any n combinations of t and x, 1425.

1425	1426

Summary: Notes from Eisenhart. (Ref. 476)

1427	1428



1429	1430

Summary: Six definitions of a "complete" system are given and are all proved equivalent.

1431	1432

Summary: Some references to amoeboid activity in nerve cells.

1433	1434

Summary: The brain is an equilibrium-trap. And if the equilibrium can only occur on certain conditions then the brain will trap those conditions too! 1487.

1435	1436



1437	1438

Summary: Stabilising some variables almost certainly stabilises those other variables connected with them.

1439	1440

Summary: More notes on the "hour-glass" case.
Summary: A definition of transient and permanent equilibria.
1441	1442

Summary: The projections of a path, and the solutions x_i=F_i(x^o,t) are two forms of the same thing.

1443	1444



1445	1446



1446+01	1446+02

These images are reproduced courtesy of The Estate of W. Ross Ashby. Copyright 1972, 2008 © The Estate of W. Ross Ashby