Free Will & Determinism

Wherein it is argued that abstracting away from determinism to free will and consciousness is harder than it might seem.

Lets assume for the moment that (be it true or false) that the human brain is intimately tied to quantum-computation mechanisms. Merely introducing such a supposition into the mind-body argument does not inherently or explicitly simplify the mind-body problem.

Why do I say this? Lets take a step back:

Let us assume that free will, as commonly defined, actually exists, and is not an illusion (as is sometimes claimed). Then the above points would seem to imply trouble for mathematics and physics as commonly known and practiced today. How is this? One manifestation of classical (19th century and earlier) physics is its deterministic nature. Clearly this is at odds with the idea of free will. The 20th century quantum mechanics posits an underlying stochastic and indeterministic element. This is equally at odds with the common notion of free will: just because human action is unpredictable does not imply its random.

So if determinism and randomness are not acceptable, what is? Well, lets try to imagine some mathematical framework that might conceivably fit the notion of free will. It would have to be odd: not predictable, so one couldn't use the ordinary vocabulary of algebra and geometry in the ordinary sense: free will seems at odds with statements such as 'if A and B then C', preferring instead 'if A and B then maybe C or D, depending on the choice made'. Nor is it stochastic, so we can't talk in the following fashion: 'if A and B then C 37% of the time'. So how can we talk about free will? What sort of a physical theory, rooted in mathematics, can we construct, that will somehow be shown to support and allow free will? It is not obvious that this is even possible.

Should we take this to be a proof-by-contradiction? 'free will is not tractable by ordinary mathematics, ergo it doesn't exist'? That seems at best absurd.

Lets take a moment to drive this point home. One catch-phrase occasionally bandied about is 'emergent behavior'. For example, consider a collection of round objects of various sizes, with some amount of surface friction. The mathematical equations describing round objects with friction are not terribly complex. Add gravity to the mix, they are still not complex. In particular, one could stare at these equations for a lifetime without realizing that by piling up the round objects, one will get landslides of various sizes. Yet landslides occur, and once one knows that, then one knows how to approach these equations as well, in such a way that landslides will appear in them as well. The landslide is the 'emergent behavior': viz, some aspect that is not obvious from the underlying description of the system. None-the-less, once one knows of the emergent behavior, one can once again pin it down mathematically: landslides occur with certain stochastic distributions. There's an element of randomness and unpredictability, but certainly no element of mystery or magic.

Through similar reasoning, some argue that computers are particularly rich in 'emergent behavior', and that, in particular, it will soon be seen that consciousness and free will will emerge from them. The standard rebuttal to this argument is that computers are utterly deterministic, and that therefore, there is no way in which they could exhibit true free will. The standard counter-argument is that determinism doesn't matter, that by mixing in outside, environmental influences and interactions, the deterministic computer gains a sufficient measure of indeterminism. Unfortunately, these arguments resemble those for a gust of wind: the wind, controlled by partial differential equations, is ultimately deterministic, no matter how difficult it may be to predict, and no matter how much outside influences may affect it. We are not in the habit of ascribing consciousness to gusts of wind; why should another deterministic system, such as a computer, be any different?

Finally, note that adding a component of stochastic behavior does not change the above arguments: thus, saying that we've built a quantum computer out of some anti-ferro-electric spin field of microtuble tublin dimers does not provide an escape hatch. Ultimately, spin field are tractable through standard mathematics, and their behavior can be modeled by computer. Random behavior, random results of quantum measurements do not imply free will. It would seem not to matter if it was 'God playing dice', or the computer's random number generator providing the randomness.

Domain of this Analysis

In the paragraphs above and below, my explicit goal is to tie the metaphysics of free will to possible physical models of how the universe actually works. A few words are in order as to how to bound this discussion. First, to discover the physics behind free will, we must ultimately appeal to concepts that must ultimately be possible to be anchored in the physics that we know today: we must not overtly violate any known laws of quantum mechanics, cosmology and the like. The arguments must ultimately be plausible appeals to possible, but so far unknown, physics. In that sense, I wish to work in the same vein as that explored by Penrose.

Secondly, to better expose the workings, one might well need to expand the discussion into broader domains of metaphysics and ontology. We need to be aware that the question of free will is tied to the broader questions of being, of existance, such as Heidegger's 'Da-sein'. Chairs did not exist before man, yet the concept of 'chair' seems to be timeless; or is it? If indeed, the Universe and Everything In It is a part of Physics, and is subject to physical laws, then 'Being-ness' is something that is ultimately intertwined with Time. We are able to reason and contemplate because we are able to remember the past, and to think about it. 'Being-ness', likewise, seems to be a memory of the past that we are able to call on in the present, as we flow through time and reason about existance. Free will is our appearent ability to shape the future; but free will can also be viewed as our ability to pose questions, and then maybe answer them: it presupposes memory and existance. Free will seems to operate, at least in part, in the Platonic realm of concept and being; thus, if we fail to sketch out the physics for free will, it may be because we've failed to sketch out the physics underlaying platonic reality first.

Escaping the Trap

OK, the above presented some standard arguments and counter arguments. Lets now look for chinks in the armor.

Types of Quasi-Indeterministic Turing Systems

Lets review the different ways in which Turing machines can be made to appear to be quasi-indeterministic. In other words, the behavior of certain computer programs, although ultimately deterministic, may be hard to predict, because 'small' changes in input can lead to 'large' changes in output.
Modeling Chaotic Dynamics
One way in which computers can be made to resemble indeterministic systems is when they model chaotic dynamics. Such modeling tends to make the outcome very sensitive to initial conditions. Note that the 'initial conditions' are a quasi-continuous input (i.e. 'floating point numbers'), and the output is quasi-continuous as well (i.e. are also 'floating point numbers').

Discretely-valued NP-Complete Problems
NP-Complete problems such as the the Traveling Salesman or Linear Programming have discretely different outcomes that cannot be continuously transformed into one another. That is, two different solutions to the traveling salesman problem are inherently discrete. The distances might be quite close to one-another, but the distances are still differ by a finite value, and there is no continuous-valued solution set. Algorithms used to solve the traveling salesman problem, such as simulated annealing, or neural networks, do tend to be sensitive to continuous-valued intermediate states and inputs: e.g. in simulated annealing, the outcome is sensitive to the cooling schedule in unpredictable ways. This class of problems are interesting because they resemble the 'either-or' type decisions that we associate with free will.

Random Input
Programs that respond to ongoing external input (random, continuous or discrete). In one way, one is tempted to label these programs as 'open systems' in the sense that they are not closed off from the environment. In another way, they resemble 'closed systems' which do not read all of their input all at once, but rather a little bit at a time, reacting to each new stimulus. That is, responding to input 'in real time' is not fundamentally different than recording the input, and then analyzing the recording at a later time. This identity ceases to be the case when the computer then performs actions to change the external environment: the changed environment may not be predictable, and thus, the end result is no longer deterministic in the computational sense. See 'Open System' below. Note, however, if the environment is governed by classical deterministic differential equations, then ultimately, the behavior of the combined system is ultimately predictable (however difficult it may be to do so in practice).

Open System
Hameroff gives an interesting example of a discretely valued outcome subject to continuous-valued random disturbances: A purely-deterministic, robot sailor hoping to dock at one of three ports. While still far at sea, the smallest of wind gusts may make the sailor choose a different port; once closer in, only the largest of gusts that might blow the sailor from the mouth of one port to the mouth of another will change the outcome.

By having a computer interact with a strongly-(chaotically-)mixing external world, outcomes become very hard to predict. However, this does not imply that the combined system is non-deterministic. After all, the wind currents can be modeled mathematically to arbitrary precision, and results can be predicted within the error bounds imposed by chaotic divergence.

Phase Transitions/Critical Exponents
Consider the vibrations of a long, weighted pole. In real life, as the pole is lengthened, it eventually breaks. Mathematically, we can determine the length at which the pole becomes unstable (it is the length at which minor perturbations grow without bound). This problem seems to be algorithmically non-tractable: beyond the critical length, a minor perturbation in the millionth decimal place will grow to a large bend in finite time.

Re-iterated Monte-Carlo
Lets throw dice. If an algorithm likes the result, we are done. If it doesn't, then roll again. ...
Damn this stuff gets muddled fast. There is a distinct difference between the determinism embodied in an algorithm (and the inherently discrete-step fashion in which we talk about this determinism), and the determinism of continuous-valued differential/integral equations. In fact, the determinism embodied in differential equations is much harder to talk about, since there are cases where in some sense, even an 'infinite number of decimal places of precision' seem to not be enough.

The Continuum Limit

The determinism embodied in differential equations is much harder to talk about, since there are cases where in some sense, even an 'infinite number of decimal places of precision' seem to not be enough. Under what circumstances can we argue that something 'smaller than infinitesimal' can have a finite effect?

This is at the limit of my knowledge of the subject, but here goes:

Devil's Staircase
The devil's staircase is a construction the graphs the Farey-addition of real numbers. Alternately, it can be expressed as a construction with continued fractions (Conway). It has the curious property that all derivatives are zero at all rational numbers, yet, none-the-less, the function is strictly monotonically increasing, viz. f(x) < f(y) whenever x < y.

This is but one of a whole class of functions that are well defined on rational values, but bizarre or undefined at irrational values. (See an artistic exploration in my Art Gallery).

Insofar as there may be any actual, physical systems that embody these types of mathematically pathological behaviors, they cast particularly long and troublesome shadows onto the metaphysical discussion of free will. They introduce an awful lot of hand-waving and wiggle-room into all arguments.


In "The Gestalt of Determinism" we learn that determinism is inherently undecidable.

Gestalt: can we make an algebra out of statements about undecidability? Another classic example: 'even God cannot change the value of 2+2, or the value of Pi'. This appears to be a concrete statement about the powers of God, a discussion of whom is in most other respects impossible.

The Nature of Free Will

Lets take a step back, and look at the nature of free will a bit more carefully.

At best, free will seems to be constructive: as in one of Conway's games, we are given a menu of choices, with each choice made resulting in a new game to play. But that misses the point: we want to talk about free will, and not the choices that free will is presented with. We want to talk about the how and why of decision making process, rather than the outcome of it. How can we begin to imagine what sort of (microscopic) physics could talk about it objectively without also removing its very course of action?

With that in mind, lets review what we do seem to know about consciousness and its place in the physical universe.

umm, with that as the introduction, we now come to the main point, which is .... 'I have discovered the most marvelous proof, for which the margin of this web page is too small...'. Ha ha. Later duude.

Links, Bibliography


Formal Systems
Formal mathematical systems consist of (a finite) set of syntactical rules. The expression of these rules define what is say-able in such systems. Since there is a strict correspondence between what is say-able and what is computable (via Church), one is limited in formal systems to what one can compute. Yet we also know that there are things that are not computable, ergo, these are unreachable to formal systems.

Indeed, Penrose (Emp. new clothes) works at length to establish this point in layman's terms. The upshot is that formal systems employ axioms as their 'inputs'. These axioms are understood by humans as somehow being 'obviously true' statements. These axioms are not provable within the formal system. Mathematics sits on shaky foundations in that we, as humans, do not know that any given set of axioms does not lead to a set of internal contradictions within a formal system. We have to believe that something like Peano's axioms are not self-contradictory; we cannot formally prove it. (See Clear and Certain Notions, a conversation exploring the issue of intuitionist viewpoints (e.g. 'platonic mysticism') vs. the limits of formal systems).

On a related note, we have additional metaphysical questions about the way in which humans use language: Do we in fact use language to communicate ideas that are not expressible as formal systems? Certainly, any act of communication that we engage in as human necessarily results in a finite-length string, and so, naively, we might try to ask whether such a string might be 'computable'. But this seems irrelevant, and beside the point, since it fails to address what the mind does in understanding the communication.

March, August 2000
Linas Vepstas