Forks & Dice: Bifurcation in Prigogine & Stengers, Order out of Chaos: Man's New Dialogue with Nature

by Corry Shores
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Forks & Dice:

Bifurcation in Prigogine & Stengers,

Order out of Chaos: Man's New Dialogue with Nature

La Nouvelle Alliance: Métamorphose de la science

In the second section of the third chapter in Cinema 2, Deleuze describes a kind of forking where a path of development skews-off wildly at unstable points:

And it is not just the circuits forking between themselves, it is each circuit forking within itself, like a split hair. [...] its repetitions are not accumulations, its manifestation refuse to be aligned, or to reconstitute a destiny, but constantly split up any state of equilibrium and each time impose anew 'meander', a new break in causality, which itself forks from the previous one, in a collection of non-linear relations. [footnote 8: On this notion of forking, cf. Prigogine and Stengers, Order out of Chaos: man's new dialogue with nature, London: Heinemann, 1984, pp. 189-90.] [Deleuze 47bc; 47c; 280a]

Et ce ne sont pas seulement les circuits qui bifurquent entre eux, c' est chaque circuit qui bifurque avec soi-même, comme un cheveu four-chu. [...] ses répétitions ne sont pas des accumulations, ses manifestations ne se laissent pas aligner, ni reconstituer un destin, mais ne cessent de morceler tout état d'équilibre, et d'imposer chaque fois un nouveau « coude », une nouvelle rupture de causalité, qui bifurque elle-même avec la précédente, dans un ensemble de relations non-linéaires [note 7: Sur cette notion de bifurcation, cf. Prigogine et Stengers, La nouvelle alliance, Gallimard, p. 190]. [Deleuze 68d; 69a; 69d]

We will avoid most of what is difficult about bifurcation, even though it is essential for fully grasping the concept. We do so on account of your author's limitations. I probably know less than most readers, who are invited to make corrections and provide better explanations. But also it seems the basic ideas that we need to apply in the context of cinema can be presented in a highly simplified form. So that is what we hope to accomplish in the following.

We will first discuss the notions of linearity and non-linearity, drawing from this page at Mathematics Illuminated.

Consider first x = 2.

And x = 2y

We notice that there is one line that tends the same direction throughout.

The equation is "linear" because its graph (all the "x,y" points on the coordinate plane that satisfy the equation) is a straight line, and also because a small change in the value of x effects a proportional, constant change in y. ("Linear vs. Nonlinear Systems")

But now consider x-squared = 4

And now also x-squared = y

We notice two things about the diagram immediately above. The higher power caused there to be a bifurcation of values. And the lines they graph are not straight or 'linear'.

A nonlinear equation is something that doesn't have just a first power of the independent variable and consequently can't be graphed as a simple straight line. ("Linear vs. Nonlinear Systems")

Mathematics Illuminated then gives the example of pendulum motion to illustrate the difference between linear and non-linear systems.

These so-called nonlinear systems can exhibit some wild behaviors, behaviors that might be considered surprising, behaviors that don't fit so nicely into equations. For example, our simple pendulum behaves very smoothly and predictably as long as it doesn't swing too high.

For larger and larger angles, the range of possible behaviors is more varied than the simple cycling back and forth. For example, if the pendulum has sufficient momentum, it will swing past the horizontal line of the pivot and go all the way around, over the top. If it has a little less momentum than this, it might stall near the vertical position above the pivot, lose the tension of the string, and drop almost straight down under the influence of gravity. Both of these behaviors are examples of nonlinearities. ("Linear vs. Nonlinear Systems," emphasis mine)

(Image obtained gratefully from Mathematics Illuminated)

For the sake of illustration, we might imagine that the pendulum can reach a critical point where there are a number of outcomes, none of which can be predicted. We might then again for the sake of illustration regard there to be a bifurcation point where the development of the system can fork-off into very different directions. Perhaps the slightest infinitesimal fluctuation can cause profoundly different outcomes.

In Order out of Chaos: Man's New Dialogue with Nature (La Nouvelle Alliance: Métamorphose de la science), Prigogine & Stengers describe bifurcation in chemical reactions. Hopefully we can profoundly simplify their explanation without falsifying it too much. The system will be stable if the variables are kept within certain bounds. But when an independent variable is pushed to a critical chaotic point, the dependent variable can veer-off or fork-away into two possible directions of development. They write:

Consider the bifurcation diagram represented in Figure 11.

This differs from the previous diagram in that at the bifurcation point two new stable solutions emerge. Thus a new question: Where will the system go when we reach the bifurcation point? We have here a "choice" between two possibilities; they may represent either of the two nonuniform distributions of chemical X in space, as represented in Figures 12 and 13.

The two structures are mirror images of one another. In Figure 12 the concentration of X is larger at the left; in Figure 13 it is larger at the right. How will the system choose between left and right? There is an irreducible random element; the macroscopic equation cannot predict the path the system will take. Turning to a microscopic description will not help. There is also no distinction between left and right. We are faced with chance events very similar to the fall of dice. (162-163, emphasis mine)

They write a bit later:

If we consider Figure 17 [...] we see that the system already has a wealth of possible stable and unstable behaviors.

The "historical" path along which the system evolves as the control parameter grows is characterized by a succession of stable regions, where deterministic laws dominate, and of instable ones, near the bifurcation points, where the system can "choose" between or among more than one possible future. Both the deterministic character of the kinetic equations whereby the set of possible states and their respective stability can be calculated, and the random fluctuations "choosing" between or among the states around bifurcation points are inextricably connected. This mixture of necessity and chance constitutes the history of the system. (169-170, emphasis mine)

Soon we will discuss the bifurcations of Mankiewicz' movies. When we do so, we will see that we arrive at critical and unstable points in the narrative where a character forks or bifurcates unpredictably. Prigogine & Stengers' bifurcation diagrams will illuminate this concept.

Credits:

Pendulum image and direct quotations regarding linear and non-linear systems obtained gratefully from:

http://www.learner.org/courses/mathilluminated/units/13/textbook/02.php

Linear and non-linear graphs made using the following freeware:

Geogebra

GIMP, and

OpenOffice

The Prigogine & Stengers text citations and images from:

Prigogine, Ilya, and Isabelle Stengers. Order out of Chaos: Man's New Dialogue with Nature. London: Heinemann, 1984.

Also,

Prigogine, Ilya, and Isabelle Stengers. La Nouvelle Alliance: Métamorphose de la science. Paris: Éditions Gallimard, 1979.

Deleuze citations from:

Deleuze, Gilles. Cinema 2: The Time Image. Transl. Hugh Tomlinson and Robert Galeta. London & New York: 1989.

Deleuze, Gilles. Cinéma 2: L'image-temps. Paris: Les éditions de minuit, 1985.

The Laws of Fact and the Facts of Law TF §86 For Dynamism Facts More Real Than Laws: Mechanism Reverses This Attitude. Bergson. Time and Free Will

by Corry Shores
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The Laws of Fact and the Facts of Law

Henri Bergson

Time and Free Will: An Essay on the Immediate Data of Consciousness

Essai sur les données immédiates de la conscience

Ch.III

The Organization of Conscious States; Free Will

De l'organization des états de conscience: la liberté

Part XXIX: Dynamism and Mechanism

Previously Bergson distinguished mechanism and dynamism. We find in our world complex arrangements and rearrangements of materials. Mechanism holds that any given arrangement could not have been otherwise. It was determined to be so on account of previous states and the fixed laws that governed their rearrangement. So even if more complex forms evolve, they would still be predictable from previous states, if we had enough knowledge of these prior conditions. Dynamism starts with the idea that consciousness can produce free actions. The less consciousness is a fact, the more inertia is a factor [so the more determined laws of physics prevail].

§86 For Dynamism Facts More Real Than Laws: Mechanism Reverses This Attitude. The Idea of Spontaneity Simpler Than That of Inertia

Bergson will now show that mechanism and dynamism are based on two very different assumptions regarding the relations between laws and the facts that these laws govern.

[Consider if we saw the spontaneous creative behaviors of an artist at work. We might there regard her consciousness as acting independently of mechanistic laws]. Dynamists think they discover facts that elude the governance of laws. Hence they "set up the fact as the absolute reality, and the law as the more or less symbolical expression of this reality" (141a). Mechanists however find laws operating within those facts. For them, "it is the law which becomes the genuine reality" (141a).

Bergson will now explain why mechanism assigns a higher reality to law, while dynamism assigns higher reality to fact. It is because they define the term simplicity in different ways.

For mechanists, a principle is simple when its effects can be foreseen and even calculated. [Inertia is a principle that can be explained by laws of physics. So] under this definition of simple, inertia is seen as a simpler principle than freedom. [Also, homogeneous conditions allow for clearer analysis and prediction, hence] the homogeneous would be simpler than the heterogeneous. [A mathematical abstraction of a real physical situation will give a consistent outcome, while the actual experiment of this situation will produce a margin of error. Thus for mechanists,] the abstract is simpler than the concrete.

Dynamists however would note that we all have an immediate knowledge of our own free spontaneity. Then, we acquire the notion of inertia secondarily by defining it in terms of what we already know, our own freedom. So inertia is a derived concept, and freedom is the immediately intuitable notion from which inertia is derived. (142a) So let's see how dynamists would look at the primitive notions in mechanism (inertia, homogeneity, abstraction). The dynamist sees these ideas as really being derived from a combination of several richer notions. When combined, these notions in a sense neutralize each other "just as darkness may be produced by the interference of two lights" (141d)

But under both views, we define the inertia of matter by saying that because of inertia, matter "cannot move or stop of its own accord, that every body perseveres in the state of rest or motion so long as it is not acted upon by any force" (142a-b). We see that [because the motion of a body changes only by means of something acting upon it,] activity is inherent to the idea of inertia. [Now, if for example we like the dynamists saw concrete facts as what is simple and primary, rather than abstract laws, then we might say that nothing can predetermine the changes of those states. So we would view human action as being free and spontaneous. But if instead we prioritized abstract laws, we would say that resulting states were the necessary consequence of the laws governing the prior states. Hence under this mechanistic view we would regard human action as determined. Thus Bergson writes that] "It is therefore natural that, a priori, we should reach two opposite conceptions of human activity, according to the way in which we understand the relation between the concrete and the abstract, the simple and the complex, facts and laws" (142b).

[Directory of other entries in the Time and Free Will series.]

Images of the pages summarized above, from the English translation [click to enlarge]:

Images of the pages summarized above, from the original French [click to enlarge]:

Bergson, Henri. Time and Free Will: An Essay on the Immediate Data of Consciousness. Transl. F.L. Pogson. New York: Dover Publications, Inc., 2001. Available online at:http://www.archive.org/details/timeandfreewill00pogsgoog

Bergson, Henri. Essai sur les données immédiates de la conscience. Originally published, Paris: Les Presses universitaires de France, 1888. Available online at:http://www.archive.org/details/essaisurlesdonn00berguoft

Motions & Machines. TF §85 Mechanism, Dynamism, and Free Will. Bergson. Time and Free Will

by Corry Shores
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Motions & Machines

Henri Bergson

Time and Free Will: An Essay on the Immediate Data of Consciousness

Essai sur les données immédiates de la conscience

Ch.III

The Organization of Conscious States; Free Will

De l'organization des états de conscience: la liberté

Part XXIX: Dynamism and Mechanism

Previously Bergson discussed our social tendencies. We normally translate our confused flux of inner states into distinct and fixed forms. We betoken them with a symbol or word, and then we place them into a homogeneous ideal space. Bergson concluded by suggesting this produces problems when we analyze causality, freedom and personality.

§85 Mechanism, Dynamism, and Free Will

Bergson will now discuss the rivalry between mechanism and dynamism. Dynamism begins with "the idea of voluntary activity, given by consciousness" (140b). [A lack of action would be no voluntary activity, so] dynamism represents inertia "by gradually emptying this idea" (140b). [It seems that matter would be inert until acted upon, hence] "it thus has no difficulty in conceiving free force on the one hand and matter governed by laws on the other" (140b).

Mechanism takes the opposite course. [It does not regard forces as being free. Rather, all events are determined necessarily from prior ones. This can also mean that more and more complex forms could perhaps evolve. However, it also means that with enough knowledge of the previous states, we could predict these more complex arrangements.] Bergson writes:

It assumes that the materials which it synthesizes are governed by necessary laws, and although it reaches richer and richer combinations, which are more and more difficult to foresee, and to all appearance more and more contingent, yet it never gets out of the narrow circle of necessity within which it at first shut itself up. (140c)

Les matériaux dont il opère la synthèse, il les suppose régis par des lois nécessaires, et bien qu'il aboutisse à des combinaisons de plus en plus riches, de plus en plus malaisées à prévoir, de plus en plus contingentes en apparence, il ne sort pas du cercle étroit de la nécessité, où il s'était enfermé d'abord. (107c)

[Directory of other entries in the Time and Free Will series.]

Images of the pages summarized above, from the English translation [click to enlarge]:

Images of the pages summarized above, from the original French [click to enlarge]:

Bergson, Henri. Time and Free Will: An Essay on the Immediate Data of Consciousness. Transl. F.L. Pogson. New York: Dover Publications, Inc., 2001. Available online at: http://www.archive.org/details/timeandfreewill00pogsgoog

Bergson, Henri. Essai sur les données immédiates de la conscience. Originally published, Paris: Les Presses universitaires de France, 1888. Available online at: http://www.archive.org/details/essaisurlesdonn00berguoft

[Labels continuation for: The Wheel is Come Full Pixel: Analog vs. Digital; Bostrom and Sandberg's Brain Emulation, Examined and Critiqued. Section 4]

by Corry Shores
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This entry continues the labels list for the posting titled:

The Wheel is Come Full Pixel: Analog vs. Digital; Bostrom and Sandberg's Brain Emulation, Examined and Critiqued. Section 4

Awake! Arise! or be Forever Disorganized; Bostrom and Sandberg's Brain Emulation, Examined and Critiqued. Section 3

by Corry Shores
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[The following is tentative material for my presentation at the Society for Philosophy & Technology Conference this summer.]

[Other entries in this series.]

Corry Shores

Do Posthumanists Dream of Pixilated Sheep?

Bostrom and Sandberg's Brain Emulation,

Examined and Critiqued

Section 3:

Awake! Arise! or be Forever Disorganized

Bostrom’s & Sandberg’s Roadmap presupposes a physicalist standpoint. So everything has a physical basis. Minds emerge from the brain’s pattern of physical dynamics. If you replicate this pattern-dynamic in some other physical medium, the same phenomena should likewise emerge. They write that “sufficient apparent success with [Whole Brain Emulation] would provide persuasive evidence for [this theory that consciousness may be realized in multiple distinct physical forms, or what’s called] multiple realizability.” (Bostrom & Sandberg 14)

Our mind’s emergence requires a dynamic process. Paul Humphreys calls it diachronic pattern emergence. (Humphreys 438)

According to emergentist theories, all reality is made-up of a single kind of stuff. But its parts aggregate and assemble into dynamic organizational patterns. The higher levels exhibit properties not found in the lower ones. Yet, the higher level would not exist were it not for its constituent lower level. (Clayton, 2-3)

Todd Feinberg suggests water, for example. The H2O molecule does not itself bear the properties of liquidity, wetness, and transparency. However, an aggregate does. (Feinberg, 125) Emergent features go beyond what we may expect from the lower level. Hence the higher levels are greater than the sum of their parts.

In our brains, no one single neuron is conscious. Yet our minds emerge from the complex dynamic pattern of all our neurons communicating and computing in parallel. Roger Sperry offers compelling evidence. There are "split brain" patients whose right and left brain hemispheres are disconnected from one another. Nonetheless, they maintained unified consciousness. But there is no good account for this on the basis of neurological activity. (Clayton 20)

William Hasker follows Sperry. He says that mental properties “manifest themselves when the appropriate material constituents are placed in special, highly complex relationships.” (Hasker, 189-190) He offers the analogy of magnetic fields, which he says are distinct from the magnets producing them. For, they occupy a much broader space. The magnetic field is generated because its “material constituents are arranged in a certain way – namely, when a sufficient number of the iron molecules are aligned so that their ‘micro-fields’ reinforce each other and produce a detectable overall field.” Once generated, the field exerts its own causality, which affects not only the objects around it, but even the very magnet itself. Hence Hasker’s analogy: just as the alignment of iron molecules produces a field, so too the particular organization of the brain’s neurons generates its field of ‘consciousness.’ (190) This emergent consciousness-field permeates and haloes our brain-matter, occupying its space and traveling along with it. (192)

Suppose whole brain emulation continually falls short. This could support Todd Feinberg’s argument that the mind does not emerge from the brain. He agrees with Searle that

the naïve idea here is that consciousness gets squirted out by the behavior of the neurons in the brain, but once it has been squirted out, then it has a life of its own (Searle, 1992) (qt. in Feinberg 126)

Feinberg does in fact think consciousness results from the interaction of many complex layers of neural organization. However, no level emerges, because none are more independent than any other. Our vision illustrates. We see a wide variety of stuff. But we can recognize singularities like our grandmother. Much visual information must be processed through many layers of neuron-circuits until finally arriving at the “grandmother cell.” Yet all layers must work together at once to achieve this recognition. The brain is a vast network of circuits far too interconnected to discern higher and lower levels of organization. (Feinberg 130-131)

But perhaps Feinberg, so to speak, looks too much among the iron atoms and so he never notices the surrounding magnetic field. Nonetheless, his objection may still be problematic for whole brain emulation. Bostrom & Sandberg write:

An important hypothesis for WBE is that in order to emulate the brain we do not need to understand the whole system, but rather we just need a database containing all necessary low-level information about the brain and knowledge of the local update rules that change brain states from moment to moment. (Bostrom & Sandberg 8)

But if Feinberg’s holistic theory is correct, we cannot only emulate the lower levels and expect the rest to spontaneously emerge. For, we need already to understand the higher-levels in order to program the lower ones. Thompson et al. write:

The brain is thus a highly cooperative system: the dense interconnections among its components entail that eventually everything going on will be a function of what all the components are doing. (Thompson, Varela, & Rosch 94a-b)

Thus the behavior of the whole system resembles a cocktail party conversation much more than a chain of command. (96a)

Consciousness results from neural activity. But it might do so in a way that is not perfectly suited to emergentist theories. Hence whole brain emulation might provide evidence indicating whether and how our minds relate to our brains.

[Next entry in this series.]

Clayton, Philip. "Conceptual Foundations of Emergence Theory." in The Re-Emergence of Emergence: The Emergentist Hypothesis from Science to Religion. Ed. Philip Clayton and Paul Davies. Oxford: Oxford University Press, 2006. More information and partial preview available at: http://books.google.be/books?id=KJF1ydg3HJQC&hl=en

Feinberg, Todd E. "Why the Mind is Not a Radically Emergent Feature of the Brain." in The Emergence of Consciousness. Ed. Anthony Freeman, Thorverton, UK: Imprint Academic, 2001. More information and partial preview available at: http://books.google.com/books?id=YBnLgsAOe6AC&printsec=toc&dq=Why+the+mind+is+not+a+radically+emergent+feature+of+the+brain&lr=&source=gbs_summary_s&cad=0#PPA136,M1

Hasker, William. The Emergent Self. London: Cornell University Press, 1999. More information and limited preview available at: http://books.google.be/books?id=dCW023Hc1q4C&hl=en

Humphreys, Paul. "Synchronic and Diachronic Emergence." Minds and Machines. Vol.18, Number 4, December, 2008, pp.431-442. More information and online text available at: http://www.springerlink.com/content/d442431150343t17/?p=f1cef51a00d346b582d2c3ad1386c814π=1

Sandberg, A. & Bostrom, N. (2008): Whole Brain Emulation: A Roadmap, Technical Report #2008‐3, Future of Humanity Institute, Oxford University. Available online at:http://www.fhi.ox.ac.uk/Reports/2008-3.pdf

Searle, J. R. The Rediscovery of the Mind. Cambridge: MIT Press, Bradford Books, 1992. (Cited in Feinberg)

Varela, Francisco J, Evan Thompson, & Eleanor Rosch. The Embodied Mind: Cognitive Science and Human Experience. Cambridge, Massachusetts: The MIT Press, 1991. More information and limited preview available at: http://books.google.be/books?id=QY4RoH2z5DoC&hl=en