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When you develop a plan, you have in mind an initial state and a goal, and your task is to develop a sequence of operations that will get you from one to the other. Suppose your task is to avoid a danger or to develop a theorem in geometry.
It seems against intuition that 1011 adequaterly organized neurons should give rise to the consciousness needed to understand and solve the problem, "and yet it happens" (Chalmers)
If we believe both on a parliament-of-the-mind operating system, formed with different neural subnets and, also, that the vote of a particular subnet does not influence its own vote, only the other ones, we could quantify the initial situation with the model of a distracted mind with all the degrees of freedom free, by which we mean that no subnet has entered in a compromise with any other with respect to, for instance, a future dangerous situation. The collection of subnets is -so to say - independent from one another.
If there were only two protominds or subnets (say left vs right hemispheres) in the mind, a crossword panel's initial conditions should appear as follows, previous to any agreement, that is, showing the state of a distracted mind:
Column A Column B
File A Unusable Open to AB (1)
File B Open to AB Unusable
By unusable we mean that one subnet (one hemisphere) does not induce its own vote. In this reduced parliament-of-the-mind, subnet A does not influence the decision or vote of its own subnet. By AB we mean an agreement among subnets A and B. The open elements are explained because no agreement is established among subnets A and B, but an agreement is an open possibility. Just as what happens with crosswords, the tensions among the horizontals and verticals, that guide the agreement process, should be relaxed more or less conveniently and more or less rapidly, according to internal problems and external circumstances.
The degrees of freedom of A are one. At the start, A did not have the necessity to show any agreement at all with B. Now suppose that A and B agreed on how an external danger can be controlled. Just change the word Open by the new word AB, and you have the final situation depicted: the danger is accepted as such. In the meanwhile there are fluctuations and tensions in the agreement- to- be. This means that both subnets have their degrees of freedom (DF) first tensed and then slaved. The phase transition among the distracted mind and the final relaxed mind (passing through a period of excited mind) could be measured by a study of the inventory of the space of degrees of freedom involved. According to Hopfield and our previous work, the computation energy E with the opposite sign, is
At the beginning E = -(2+0)+0 = -2, in the middle E = -(0+0)+2 = 2, and at the end E = -(0+2) + 0 = -2.-E = DF(free+slaved)- DF(tensed)
The phase transition is initialy antientropic and finaly entropic, and in the middle full of bioenergetic fluctuations due to the tensions between the parts of the whole. Let us increase the subnets to a whole of five, which supposes an operating system with five basic sectors (A to E). The agreements between horizontals and verticals (agreements among only two of the sectors) would look as the following crossword matrix with an unusable long diagonal, in which the right column should be considered as a neighbor to the left column and the top file should also be visualized as neighbor to the bottom file:
The letter x means unusable, f means agreement AB, g agreement AC and so on, combining the titles of files and columns. The four lettered words fghi, jklf, mngj, ohkm and ilno, have a logical meaning. Here fghi means that A has relaxed all its restrictions and has agreed with B, C, D and E forming a word that shows the partial compromise. It is partial, because logic shows that the compromise fghi does not involve any other biparty compromise with letters j to o, some or all of them perhaps impossible to satisfy.······A·····B·····C······D·····E A·····x f·····g······h·····i B f x·····j······k·····l C g j·····x······m·····n D h k·····m······x·····o E·····i·····j·····n······o·····x
Suppose now that our minds have three protominds or subnets, extending the first 2x2 matrix to a new 3x3x3 one. We propose now to characterize a similar situation as that pictured with matrix (1). One of the possibilities is that they agree on the recognition of a dangerous situation. Before the danger signal comes, the mind is open and perhaps attentive, but in comparison we could characterize it as distracted at this point. The three dimensional crosswords for A, B and C should look as a box with the following three slices, the second one sandwiched between the other two:
A 3-D matrix showing the possible three-parties agreements.
First (bottom) slice related to A
Column A Column B Column C
File A Unusable Unusable Unusable
File B Unusable Unusable Open to ABC
File C Unusable Open to ABC Unusable
Second (middle) slice related to B
Column A Column B Column C
File A Unusable Unusable Open to ABC
File B Unusable Unusable Unusable
File C Open to ABC Unusable Unusable
Third (top) slice related to C
Column A Column B Column C
File A Unusable Open to ABC Unusable
File B Open to ABC Unusable Unusable
File C Unusable Unusable Unusable
The elements of the matrix should be either unusable because of the repetitions of one of the three letters (most are so forbidden) or open to a triple agreement. The aim to have double agreements (that is AB, AC and BC) does not warrant a triple agreement. The double agreements could be shown with the 2x2 arrangements, like matrix (1), but it is not shown here because it is not a necessary condition to have all the double agreements established before a general triple agreement appears. For instance, the agreement AC is not necessary as an intermediate step if the pair AB agrees directly with C in the parliaments vote. Only two new and almost trivial facts appear when compared with the first analysis of matrix (1). One shows that the open places increase with increasing dimensions, but the increased number involves redundancy: the ABC agreement, once obtained, fills all the open places simultaneously. The other shows that the relation of unusable places to open places increase with N, the number of dimensions. Both trivial rules are monotonic for more dimensions.
That also means that here the degrees of freedom - when open - are initially all free and diminish as such with the serial agreements. The initial situation is the distracted mind (all free), the middle situation shows that most of them are tense: it is the excited mind, and the final one, slaved, is the relaxed mind, so to say, conscious that a danger has to be solved according to the fulll subnets ABC agreement. This peculiar situation takes over and directs the overall behavior, including the possibility of a panic stampede. The phase transition from distracted and free subnets to slaved subnets is paradigmatic for other less dangerous problems to be solved, as the geometric theorem.
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2.feb.1999
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Glosario de Carlos von der Becke.