Volume 05 of W. Ross Ashby's Journal
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1941



Volume 05



0970+03	0970+04



0970+05	0970+06

Summary: A higher level must usually change more slowly than a lower level, in order that the lower level may be given time to catch its neutral point. Dominance and velocity Independence and velocity Parameter if not known Probability of parameter
Summary: If, from a given system, we remove knowledge of a variable, we must introduce probability to replace it. (But see next paragraph)
0971	0972

Stimulus probability distribution of
Summary: If initial conditions are unknown we must replace them with probability. Problems mine Statistical mechanics
0973	0974

Memory potential memory Organisation and memory
Summary: The idea is suggested that the old memories, as organisations, may be present implicitly rather than explicitly. Environment in parts
0975	0976


Summary: The lower animals, at any rate, with their environment may be much simplified for our purpose by noting that one animal may be considered to be split into several, or many, parts, each of which has its own environment. So animal and environment = several machines, not one. Dominance and velocity Equilibrium of organisations Independence and velocity Levels mechanism of Organisation stable
0977	0978


Dominance chain of Organisation chain of organisations
0979	0980

Organisation number of parameters Parameter number of
Summary: We have discussed the situation: p's dominate x's, and x's dominate y's. Under these conditions we can get a stability of organisation. Also we can get y-point in y-space moving twice through the same point in different directions. If the x's react rapidly they will tend to disappear functionally. A succession of such gives transmission through a series of organisations. If one level has only a few, or a single, variable this introduces an essential simplicity into all subsequent levels. A large organisation may be 'simple' because it depends on only one or a few parameters. Equilibrium of organisations Organisation exploring organisation Oddments [11]: Stability of organisation 0982, examples 0701, 0606.
0981	0982

Summary: Details are given showing that it is possible to explore, experimentally, a given field or organisation. To do this parameters are necessary, and it may be necessary to introduce new ones not mentioned before. Field (of substitution) exploring
Organisation number of parameters Society [12]: Organisation has two complexities: number of variables and number of parameters, 0984. In man-made machines, 1054.
0983	0984

Summary: An organisation with n variables and m parameters has two separate complexities. Subject to conditions, m describes the number of coordinates in the space in which the neutral point moves, when m=n we have a 'transative' state. Neutral point (of equilibrium - including 'cycle', 'region' etc.) control of
Dependence test for Dominance test for Independence test for Substitution (mathematical) reducibility
0985	0986

Summary: A (better) restatement of the theorem of 680.
Neutral point (of equilibrium - including 'cycle', 'region' etc.) in differential equation Substitution (mathematical) as differential equation
0987	0988

Summary: Mathematical definition and test is given for 'neutral point' and 'neutral cycle' when the substitution is given as a differentil equation. (Actual example next paragraph).
Summary: An example of neutral cycle in a differential equation is given. Neutral point (of equilibrium - including 'cycle', 'region' etc.) examples
0989	0990

Break and step-function Break example Organisation break of Substitution (mathematical) break and

0991	0992



0993	0994

Summary: A break may be treated as a mere incident in the development (in time) of one machine. Also one machine may be considered as split into two parts with a break between if one of the variables is a step-function of the time (see next paragraph). A break is a change of organisation. Changes of organisation have two causes: (1) Due to conditions outside the machine, which are arbitary parameter changes, and are my doing. (2) Due to conditions inside the machine - a break if we ignore the cause. Step function defined

0995	0996


Substitution (mathematical) step-function in
0997	0998


Environment step-functions in Personal notes [23]: Re-read my notes completely from page 1000 on, 4606. Personal notes [29]: Notes re-read from p. 1000 in Feb '57 5536.
0999	1000


Break definition Organisation break Parameter and break
1001	1002

Break surface Critical surface if not related to essential variables
Summary: "Step-function" is defined. An analytic formula given for one. If a function in a substitution is a step function of the variables, the corresponding variable in the solved equations is a step-function of the time. The effect in a field of a step-function is discussed, The essential conditions for a break are a cloud of dots, each of which has a number associated with it saying "change one of the step-functions to this new value" and not a surface as suggested on 898.
1003	1004

Environment for survival Probability of survival Survival by-product
Death achieved Intelligence is blind
1005	1006



1007	1008

Summary: (1) Brain activity will sometimes conduct an animal, with great ingenuity, to its death. (2) Survival is a by-product of brain activity. Summary: It is agreed, with 928, that a reversible system is of no interest from our point of view and does not exist in nature anyway. Neutral point (of equilibrium - including 'cycle', 'region' etc.) effect of change of parameter Organisation irreversible Parameter and neutral point
Summary: We show how to calculate the shift of a neutral point for a small change of parameter when the substitution is given as differential equations, (if finite substitution 927) (if several parameters, 1023) Parameter changes of state of equilibrium
1009	1010

Operator and movement

1011	1012



1013	1014


Summary: The general principle of "pressures", that difference means movement, suggests a method of combining sustitutions, or stimuli, to form a "product". If the number of parameters is greater than the number of variables, this product exists always, and powers are associative. The inverse in not unique. But the whole suggests a way in which groups might get in.
1015	1016

Break surface interaction of
Summary: In general, after a break has occurred due to the x- point touching a break point, not only the field changes but also the break points.
1017	1018

Step function all linear functions of one
Summary: All step-functions can be expressed as a linear function of one basic step-function, stp (x), "step-x", here defined. (Not true)
1019	1020

Critical surface if not related to essential variables Neutral point (of equilibrium - including 'cycle', 'region' etc.) examples
Summary: The behaviour of break-surfaces.
1021	1022

Summary: Another example of the conclusion of 1006. Brain fundamental function Death achieved Neutral point (of equilibrium - including 'cycle', 'region' etc.) effect of change of parameter Parameter and neutral point Survival by-product
Summary: Equations are given for determining the shift in a neutral point if several parameters are altered a little. The change in each coordinate is a linear function of the changes of parameters. Neutral point (of equilibrium - including 'cycle', 'region' etc.) examples
1023	1024


Summary: A carefully calculated field is given, with four neutral points. Useful for experimenting. (Others are on 817, 828, 839, 885, 941, 990, 1021) Break example Break surface interaction of
1025	1026

Critical surface if not related to essential variables Critical surface interactions
Summary: An example is given, in all detail, of a substitution with two step-functions. It confirms the theorum of 1021. The existence of "false neutral points" is noted. Neutral point (of equilibrium - including 'cycle', 'region' etc.) "false"
1027	1028

Step function collected properties

1029	1030



1031	1032



1033	1034



1035	1036


Summary: A much better statement is given of the idea of varying patterns of dominance etc in a system. Summary: "Break" does not involve "irreversibility". Break definition Dominance varying Reversible process breaks
1037	1038

Break equations for

1039	1040

Summary: In the specification of a system with step-functions present, the latter cannot be specified by differential equation form. It seems that our equations for the system must be in form { dx_i/dt = f_i(x;y), y'_i = a_i+b_istp{V_i(x;y)} } or { x_i = F_i(x⁰;y;t), y'_i = a_i+b_istp{V_i(x;y)} }. And as these define the future behaviour of the x's, and as in any case they can usually be solved only numerically, we might as well leave them in this state. (Compare 1048) (Better 1086) Step function in differential equations

1041	1042

Summary: Later we shall have to show how we can break down the minute rigidity of our dynamic systems, where the minutest change has to be put in and may lead to something profoundly different. Suggested way of doing it. Break surface no free edge Critical surface has no free edge
Summary: The V-surface of a step-function cannot have a free edge. Group (mathematical) finite continuous
1043	1044

Summary: Substitutions may, perhaps, define an infinite continuous group.
Summary: "Simplicity", "wholeness", etc are perhaps clarified by the discussion above. Simplicity meaning of
1045	1046

Summary: The idea that "orderliness" or "intelligence" spreads like crystallisation is probably covered more correctly by the more precise idea that it is "reaching neutral point and stopping still" which spreads along a chain of dominance. Break equations for Dominance chain of Equilibrium spread of Organisation spread of Step function in differential equations

1047	1048



1049	1050

Summary: Differential equations with step-functions are fundamentally unsolvable. Adaptation by break Break and adaptation

1051	1052

Summary: The concept of "breaks" by itself is not sufficient to cause any emergence of adaptation or intelligence. Brain, i.e. a machine of particular type, is necessary. (See 1063) Adaptation brain necessary Brain necessary Intelligence brain necessary
Break in machine Organisation in machinery, examples Society [12]: Organisation has two complexities: number of variables and number of parameters, 0984. In man-made machines, 1054.
1053	1054

Summary: Examples are given in ordinary machinery of "change of organisation" and "break". Both are rare.
Summary: Our definition of "dominance" of 960 is correct. See 1077 for a fuller survey. Dominance definition
1055	1056

Summary: The idea of a system, like the brain, altering its own organisation necessarily implies the presence of step-functions and breaks. Break to change organisation Organisation self change = break

1057	1058

Summary: One stage in our long journey is finished and solved: the 'exact' case, i.e. an organisation where we are given full and exact information about every little detail. Differential equation and linear partial differential equations
Group (mathematical) and isomorphism Isomorphism ? group necessary
1059	1060

Summary: It is shown conclusively that "isomorphism" does not necessarily imply "group". Organisation properties different from parts Oddments [9]: Properties of organisation may be quite different from those of the parts: 1061 (wheel rolling, temperature of gas, etc).
Summary: Some examples are given showing how a statement may be quite true about the whole and yet quite untrue of all the parts.
1061	1062

Adaptation by break Brain essentials of Break and adaptation
Summary: Although a general system has no tendency to survival by adaptive behaviour, yet a "brain" has. Details are given. (see 1068) Organisation two meanings united
1063	1064

Summary: A definition of 'organisation' is given which covers both dynamic, machine, organisations, and static, pattern ones. Organisation definition
Summary: An "organisation", by the definition of the previous page, need not be a group. Group (mathematical) and organisation Organisation in group
1065	1066

Summary: Formulae are given in the special case where one variable always moves towards some function of the other variables. Operator special
Adaptation by break Break and adaptation
1067	1068



1069	1070



1071	1072

1942


Summary: Actual equations are constructed giving the theoretical views of the nervous system in mathematical form. (See 1092) Adaptation References Break and change of neutral point Equilibrium range of Neutral point (of equilibrium - including 'cycle', 'region' etc.) choice of several
1073	1074



1075	1076

Summary: A discussion is given of the meaning of the "change of organisation" (if any) which occurs when a system settles at a new neutral point without change of the field. i.e. a variable, without change of field, going outside the "range of stability" of one neutral point. A complete clarification is given, together with its relation to my previous ideas of "breaks". Dominance definition Organisation change of neutral point Substitution (mathematical) dominance in
Substitution (mathematical) 'dependance' in
1077	1078


Matrix of organisation Organisation matrix of organisation
1079	1080

Parameter definition

1081	1082

Summary: The question of "dominance" is still further clarified. I define "immediate", "distant" and "ultimate" dependance. Also "completed matrix of an organisation". "Dominance" (two equivalent definitions). "Parameter" is defined as "dominant and constant". It is proved that if a dominates b, and b dominates c, then a dominates c.
Break continuous approximation Step function eqivalent continuous form Substitution (mathematical) change to differantial equation form
1083	1084


Summary: A method is given for changing the abrupt h'=... method of defining a break to an equivalent dh/dt method. This puts the whole system into ordinary differential equation form. The equations are in "normal" form.
1085	1086

Break example Organisation break

1087	1088


Summary: An example of a break is given in substitution form, like 991.
1089	1090

Congruence Equilibrium defines a 'thing' Equilibrium essential definition Field (of substitution) nomenclature
Summary: "Equilibrium" means not moving out of a given region. (But see 1143)
1091	1092

Break surface further properties

1093	1094

Summary: "Break-surfaces" are examined and some properties noted.
Break surface layers of, protect variable Organisation break
1095	1096


Summary: A statement is given of the theorem that a multilayer of break surfaces "encourages" the representative point to stay in that region.
1097	1098

Summary: It might be suggested that with a million neurons the chance of getting them all properly adjusted is negligibly small. The answer is that there is usually no such thing as the right solution. We count as suitable any organisation whatsoever so long as it gets the equilibrium where we want it. Organisation no "right"
Summary: After studying the fixed points in a dynamic world (i.e. neutral points) I presume the next step would be to take a lot of neutral points and set them moving. Equilibrium change of
1099	1100

Break surface layers of, protect variable, also dependant
Summary: A layer of break surfaces keeps within bounds not only the variables concerned, but any other variable which is a direct function of them. Variable central, protection of
1101	1102

Variable deputising

1103	1104


Summary: A variable may add further break-surfaces for its further protection by deputising, i.e. by controlling another variable so that the latter breaks if the first goes too far. And this leads to the important observation that it does not matter where or why a break occurs as long as it occurs. From my point of view, all that is wanted is some change of organisation and it doesn't matter how or why it is done. Any change is as good as any other change. Organisation joining two organisations
1105	1106

Parameter for joining two machines
Organisation splitting into parts
1107	1108

Summary: We discover how to join and unjoin two machines. Also we notice that if a machine is at a neutral point it is possible, under restricted conditions, to separate and rejoin without disturbing the state of equilibrium.

1109	1110

Summary: A red letter day. A problem in the application to the brain is solved. Environment several

1111	1112


Summary: If a machine with variables x has break-variables h with V-surfaces which surround an x region, and if we join this to any machine y, then the presence of the h's and the V's will tend to keep the x's within the V-region. And when the machine has settled to equilibrium, disconnecting the machine y and putting on another one, z (or changing parameters R) merely starts the x-machine changing its organisation again until it has found a new equilibrium, with the x's still inside the V-region. O.K., O.K!
1113	1114

Equilibrium list of examples

1115	1116


Summary: A list of examples of equilibrium in biology.
1117	1118

Equilibrium to two environments
Reflex, conditioned and break-theory
1119	1120


Summary: If two environments keep occurring, a system will break till it finds an organisation making it stable to both. Reactions two basic types of
1121	1122

Summary: "Reaction" is divided into "response" and "variation". Summary: The intrinsic form of a substitution might prove interesting. Break as variation Conscious mind and intrinsic equations Organisation change = variation Organisation intrinsic equation Reactions response ? variation Subjective and intrinsic equations Substitution (mathematical) intrinsic equation
Delay (in substitution) and sub-wholes Organisation degree of
1123	1124



1125	1126

Summary: It is concluded that if a whole is to be (almost) separated into two parts, the variables concerned at the "join" must be (almost) constant. Delay is not an important factor. Summary: After all these years I conclude that "vectors" are not what I want.
Balance on bicycle Bicycle
1127	1128



1129	1130

Invariant
Summary: Some musings on bicycle riding. Environment infinity of Organisation [anduls] irritant
1131	1132

Matrix of organisation Organisation splitting into parts
Summary: Preliminary discussion of a machine falling, temporarily, into parts.
1133	1134

Summary: We want to get adaptation on a scale, so that we can show that systems, under certain conditions, will move from lesser to greater adaptation. Adaptation growing
Summary: A statement of my present emotional position. Affect of my problem
1135	1136

Equilibrium partial
Adaptation partial
1137	1138


Summary: If an organisation stops at a field which is only partly stable this does not really matter; for if the danger of breaking is large, it will soon break and try new fields, while if the danger is small then there is little to worry about. Break number of Organisation number of
1139	1140

Summary: n breaks provide 2ⁿ organisations. To give 10 different organisations every second throughout a man's life we need only 35 breaks! Break surface causes fresh start Learning upsets everything Reactions new upsets old
Summary: Does the acquisition of a new reaction upset all the older one's as demanded by my theory? The answer seems to be "yes" but it may in some cases be of zero extent. Reflex, conditioned and break-theory
1141	1142

Summary: Each single environment is a (hyper) complex number. Environment as complex number

1143	1144



1145	1146


Summary: The definition of "equilibrium" is taken up from 1092, and made much more precise. It is concluded that it belongs to a path A special type of common occurrence is defined and given the name of "normal" equilibrium. Equilibrium "normal" Equilibrium essential definition
1147	1148

Organisation one or two? Organisation splitting into parts

1149	1150



1151	1152

Summary: (1) Changing coordinates in two machines is apt to make one of them. (2) Changing to normal coordinates splits a machine into independent parts. (Cf. 3868)

1153	1154

Summary: A review of Jennings' book. Organisation of organisations
Summary: The "constants" i.e. variables whose changes make observed behaviour may themselves be activities composed of other variables. And these "constants" whose changes make.... This needs specifying from the organisational point of view. (See 1193) Machine definition Organisation definition
1155	1156

Summary: A refinement of the definition of "organisation". Summary: "Memory" equals change of organisation. Memory as break Organisation and memory
Summary: "Adapted" behaviour equals the behaviour of any system around a point of normal equilibrium. (1148) Adaptation is equilibrium
1157	1158

Summary: All my theory explains the "trial and error" method in terms of non-living matter. All that, but nothing more. Equilibrium Courant's definition
Summary: Courant's definition of equilibrium. On closer reading, as R and ρ may be small to any degree, it appears that Courant's definition does not allow finite cycles like that of 1144.
1159	1160



1161	1162



1162+01	1162+02



1163	1164



1165	1166



1167	1168



1168+01	1168+02



1169	1170



1171	1172


Summary: The sheets give the mathematical theory up to about Oct '42; but, of cource, not at all completely.
1173	1174

Break surface further properties Critical surface if not related to essential variables
Summary: A clarification of the concept of a "break-surface". Summary: The conditioned reflex is not clear yet. Reflex, conditioned and break-theory
1175	1176



1176+01	1176+02

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