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# Volume 05 of W. Ross Ashby's Journal is loading...

 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 velocityIndependence and velocityParameter if not knownProbability 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 mineStatistical mechanics 0973 0974
 Memory potential memoryOrganisation 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 velocityEquilibrium of organisationsIndependence and velocityLevels mechanism ofOrganisation stable 0977 0978
 Dominance chain ofOrganisation chain of organisations 0979 0980
 Organisation number of parametersParameter 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 organisationsOrganisation 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 forDominance test forIndependence test forSubstitution (mathematical) reducibility 0985 0986
 Summary: A (better) restatement of the theorem of 680. Neutral point (of equilibrium - including 'cycle', 'region' etc.) in differential equationSubstitution (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 Break exampleOrganisation break ofSubstitution (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) 0997 0998
 Environment 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 definitionOrganisation breakParameter 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 survivalProbability of survivalSurvival by-product Death achievedIntelligence 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 parameterOrganisation irreversibleParameter 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 variablesNeutral 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 functionDeath achievedNeutral point (of equilibrium - including 'cycle', 'region' etc.) effect of change of parameterParameter and neutral pointSurvival 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 exampleBreak surface interaction of 1025 1026
 Critical surface if not related to essential variablesCritical 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 definitionDominance varyingReversible 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 { dxi/dt = fi(x;y), y'i = ai+bistp{Vi(x;y)} } or { xi = Fi(x0;y;t), y'i = ai+bistp{Vi(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 edgeCritical 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 forDominance chain ofEquilibrium spread ofOrganisation spread ofStep function in differential equations 1047 1048
 1049 1050
 Summary: Differential equations with step-functions are fundamentally unsolvable.Adaptation by breakBreak 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 necessaryBrain necessaryIntelligence brain necessary Break in machineOrganisation 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 organisationOrganisation 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 isomorphismIsomorphism ? 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 breakBrain essentials ofBreak 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 organisationOrganisation 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 breakBreak 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 ReferencesBreak and change of neutral pointEquilibrium range ofNeutral 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 definitionOrganisation change of neutral pointSubstitution (mathematical) dominance in Substitution (mathematical) 'dependance' in 1077 1078
 Matrix of organisationOrganisation 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 approximationStep function eqivalent continuous formSubstitution (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 exampleOrganisation 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 definitionField (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 variableOrganisation 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 variationConscious mind and intrinsic equationsOrganisation change = variationOrganisation intrinsic equationReactions response ? variationSubjective and intrinsic equationsSubstitution (mathematical) intrinsic equation Delay (in substitution) and sub-wholesOrganisation 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 bicycleBicycle 1127 1128
 1129 1130
 Invariant Summary: Some musings on bicycle riding.Environment infinity ofOrganisation [anduls] irritant 1131 1132
 Matrix of organisationOrganisation 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 ofOrganisation 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 startLearning upsets everythingReactions 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 definitionOrganisation definition 1155 1156
 Summary: A refinement of the definition of "organisation". Summary: "Memory" equals change of organisation.Memory as breakOrganisation 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 propertiesCritical 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

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