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)
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
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
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: 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
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.
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
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.
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
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_{i}stp{V_{i}(x;y)} } or { x_{i} = F_{i}(x^{0};y;t), y'_{i} = a_{i}+b_{i}stp{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: 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
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.
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: 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
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
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
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
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!
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
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
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 2^{n} 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
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)
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.