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1941
Volume 05
0970+03
0970+04
0970+05
0970+06
Dominance and velocity Independence and velocity Parameter if not known Probability of parameter 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.
Summary: If, from a given system, we remove knowledge of a variable, we must introduce probability to replace it. (But see next paragraph)
Environment in parts Summary: The idea is suggested that the old memories, as organisations, may be present implicitly rather than explicitly.
0975
0976
Dominance and velocity Equilibrium of organisations Independence and velocity Levels mechanism of Organisation stable 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.
Equilibrium of organisations Organisation exploring organisation Oddments [11]: Stability of organisation 0982, examples 701, 0606. 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.
0981
0982
Field (of substitution) exploring 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.
Neutral point (of equilibrium - including 'cycle', 'region' etc.) control of 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.
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).
Step function defined 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.
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.
Neutral point (of equilibrium - including 'cycle', 'region' etc.) effect of change of parameter Organisation irreversible Parameter and neutral point 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.
Parameter changes of state of equilibrium 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)
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.
Break definition Dominance varying Reversible process breaks 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".
Step function in differential equations 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)
1041
1042
Break surface no free edge Critical surface has no free edge 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.
Summary: Substitutions may, perhaps, define an infinite continuous group.
Simplicity meaning of Summary: "Simplicity", "wholeness", etc are perhaps clarified by the discussion above.
1045
1046
Break equations for Dominance chain of Equilibrium spread of Organisation spread of Step function in differential equations 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.
1047
1048
1049
1050
Adaptation by break Break and adaptation Summary: Differential equations with step-functions are fundamentally unsolvable.
1051
1052
Adaptation brain necessary Brain necessary Intelligence brain necessary 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)
Break in machine Organisation in machinery, examples Society [12]: Organisation has two complexities: number of variables and number of parameters, 984. In man-made machines, 1054.
1053
1054
Summary: Examples are given in ordinary machinery of "change of organisation" and "break". Both are rare.
Dominance definition Summary: Our definition of "dominance" of 960 is correct. See 1077 for a fuller survey.
1055
1056
Break to change organisation Organisation self change = break Summary: The idea of a system, like the brain, altering its own organisation necessarily implies the presence of step-functions and
breaks.
1057
1058
Differential equation and linear partial differential equations 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.
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: It is shown conclusively that "isomorphism" does not necessarily imply "group".
Summary: Some examples are given showing how a statement may be quite true about the whole and yet quite untrue of all the parts.
Organisation two meanings united Summary: Although a general system has no tendency to survival by adaptive behaviour, yet a "brain" has. Details are given. (see 1068)
1063
1064
Organisation definition Summary: A definition of 'organisation' is given which covers both dynamic, machine, organisations, and static, pattern ones.
Dominance definition Organisation change of neutral point Substitution (mathematical) dominance in 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".
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.
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.
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
Organisation no "right" 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.
Equilibrium change of 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.
1099
1100
Break surface layers of, protect variable, also dependant
Variable central, protection of 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.
Organisation joining two organisations 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.
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.
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!
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.
Summary: Preliminary discussion of a machine falling, temporarily, into parts.
1133
1134
Adaptation growing 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.
Affect of my problem Summary: A statement of my present emotional position.
Break number of Organisation number of 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.
1139
1140
Break surface causes fresh start Learning upsets everything Reactions new upsets old Summary: n breaks provide 2n organisations. To give 10 different organisations every second throughout a man's life we need only 35 breaks!
Reflex, conditioned and break-theory 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.
1141
1142
Environment as complex number Summary: Each single environment is a (hyper) complex number.
1143
1144
1145
1146
Equilibrium "normal" Equilibrium essential definition 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.
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)
Machine definition Organisation definition 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)
1155
1156
Memory as break Organisation and memory Summary: A refinement of the definition of "organisation". Summary: "Memory" equals change of organisation.
Adaptation is equilibrium Summary: "Adapted" behaviour equals the behaviour of any system around a point of normal equilibrium. (1148)
1157
1158
Equilibrium Courant's definition Summary: All my theory explains the "trial and error" method in terms of non-living matter. All that, but nothing more.
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.
