Vladimir A. Lefebvre (1996) The Inexplicable Effectiveness of Metaphysical Reasoning. Psycoloquy: 7(09) Human Choice (7)

IN THE CONSTRUCTION OF MATHEMATICAL MODELS

Reply to Kampis on Human-Choice

School of Social Sciences

University of California, Irvine

Irvine, California

VALEFEBV@UCI.EDU

Kampis (1995) supposes that the problem of free choice can be discussed only in the traditional philosophical context of "mental determinism." He also argues that unfalsifiable assertions cannot be laid at the basis of a theory. In this note I express my view on these issues.

1. This title paraphrases the title of an article by Eugene Wigner (1969), "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." I am turning to Wigner's striking insight to help answer one of Kampis's questions, which may be formulated as follows: Can we hope to obtain new knowledge concerning the real world on the basis of certain propositions whose truth we are unable to establish? Nonetheless I will begin by answering another question, whose sense can be conveyed in the following words: Is it possible to construct a formal model of free choice without deep analysis of the nature of mental determinism?

2. Let us imagine that some professor has given a student the assignment of constructing a mathematical model of free choice. We will assume that the student has a certain level of knowledge both in the field of mathematical modeling and physics and in that of psychology and philosophy. I need the professor in this story only to show that the further progress of the student's thought depends neither on the existence of free choice nor on the student's belief in such existence. The professor's questions have posited a "possible world." In this world the claim that "the subject possesses free choice" is held to be true. The first step that the student must take is to find a theoretical representation of the phenomenon referred to by the words "free choice."

3. I will clarify my idea by an example from mechanics. Each of us has an intuitive notion of what acceleration is. Yet when we proceed to the construction of a world-model in which this phenomenon will manifest itself, we give it a precise theoretical definition: acceleration is the value of the second derivative of the function of time which represents distance. Only after this can we use mathematical analysis to establish the deep relations of the phenomenon of "acceleration" with other mechanical phenomena.

4. We will assume that the student is familiar with the history of science and is aware that free choice is usually opposed to determined choice and that since the appearance of quantum mechanics numerous attempts have been made to reduce the concept of free choice to quantum indeterminacy. At the basis of the quantum description of physical reality is a wave-function. This function posits a distribution of the density of probabilities of possible values of measurements in various points of the space-time continuum. Back in the 1930s John Von Neumann adduced logical arguments to the effect that the quantum mechanical description of the world could not be reduced to a determinist description in which the results of observation would be the unambiguous functions of certain hidden variables. Such an inalienable indeterminism of the physical world was a seductive metaphor for free choice, since the very claim of human freedom implies that the human being can be regarded as the source of causeless changes in the world.

5. On the basis of this reasoning the student is inclined to construct his model using quantum mechanics. He takes down from his shelf a number of popular books from which he learns that the free subject in the process of free choice corresponds to a mixed state characterized by a distribution of probabilities over the set of alternatives, and the act of choice itself is a collapse of the wave function. Yet simple reasoning forces the student to doubt the justification for such an analogy: the "really" free subject is capable not only of choosing one or another alternative, but also of prescribing the probability of his choosing them. If he is in doubt, for instance, the subject can pick up a coin and toss it. The delegation of a particular choice to chance is one of the forms of free choice. Thus the state of the subject at the moment when he is free cannot be correlated with some one particular distribution of probabilities, as required by the metaphor of quantum mechanics. Consequently, the quantum schema within whose framework we must consider probabilities to be determined places too many constraints on the possibilities for making decisions and therefore cannot be used as a basis for the representation of free choice. The student is also compelled to reject any other schema characterizing the free subject by any one particular distribution of probabilities.

6. On the basis of these considerations, I, a student like the one described, do not construct a probabilistic decision-making model of the free subject. Kampis claims in his commentary: "... in Lefebvre's overwhelmingly performance-based model of freedom, the state of having the gift of 'freedom of choice' is characterized by nothing else but a certain behavior in a certain situation of probabilistic decision-making" (Kampis, 1995, par. 6). This is inaccurate. The decision of a free subject, as was explained above, cannot be described in probabilistic terms. He is himself the source of probabilities of choices among alternatives. In particular, he can prescribe to an alternative, a probability equal to one. This means that he is definitively choosing that alternative. The procedure of assigning probabilities, in the framework of the model, is undetermined even in the probabilistic sense, and for this reason the choice of a free subject cannot be described using any probabilistic model whatsoever. For the construction of a formal model of the subject capable of free choice no more detailed analysis of determinism (or indeterminism) is needed. In my article (Lefebvre, 1995) I tried to show that this requires of us only that we translate into the language of mathematical functions the superficial meaning of the sentence "We possess free will and, under some circumstances, free choice." I must emphasize that the construction of such a model itself does not give us any deep understanding of the nature of free choice. What we get is merely a clear theoretical definition of what it means to be free. Nonetheless we arrive by this means at the possibility of developing a general model of choice and of explaining and predicting a number of experimental facts, even of formulating a concrete hypothesis on the character of the connection between mental phenomena and the functioning of neural networks.

7. Moreover, we can now hope to understand what level of reality the phenomenon of free choice might belong to. As we know, quantum mechanics gives us no concrete ideas for an answer to the question: What is the source of the stable probabilities represented by the wave function? If we agree that this is a proper question, then the source must be found behind the scenes of the drama of quantum mechanics. We can suppose that the phenomenon of free choice is another manifestation of this same backstage source.

8. Let us move now to the second question. The claim that the human being possesses free will is valid only in some special possible world, and all consequences of that and other ancillary claims are also valid only in that possible world. Yet we are interested in reality. Within the confines of the psychological laboratory the claims which we obtained through the formal model acquire the status of hypotheses. In accord with Popper's schema we must try to falsify them. If we are successful in that we reject our model. Let us imagine, however, a case where the model's predictions pass all experimental tests. Does the claim that "The human being possesses a capacity for free choice" acquire thereby greater verisimilitude? Like Kampis, I answer this question in the negative, since we cannot exclude the possibility that the same predictions might have been made with another model, bound up with another possible world in which that very claim is held to be false. Therefore I must regard my principle of freedom as metaphysical. It can neither be confirmed nor disproved. This is not a unique situation. Recall, for instance, the second principle of thermodynamics: "A system can produce no work without the transfer of heat from a warmer body to a cooler." This law is formulated with reference to the ensemble of possible states of the system, not one particular state. For that reason it is not susceptible to verification, as in experiments we have to make do with only separate states of the system.

9. The fateful question arises of how we are to obtain knowledge about the world corresponding to our observations using pictures of the world which are in principle unverifiable. Eugene Wigner has called our attention to the fact "... that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it" (Wigner, 1969, p. 124). Not least does such a claim apply to metaphysical thought. It turns out, paradoxically, to be an effective means to the construction of mathematical models, although it has no visible connections with reality (see Adams-Webber, 1995, par. 9). We can only speculate that not only mathematical structures but also metaphysical constructions are bound up with an archetypal stratum of our thinking, correlated in some currently inexplicable fashion with the objective laws of the universe.

Adams-Webber, J.R. (1995) A Pragmatic Constructivist Gambit for Cognitive Scientists. PSYCOLOQUY 6(34) human-choice.2.adams-webber.

Kampis, G. (1995) The Anthropic Principle of What? PSYCOLLOQUY 6(38) human-choice.4.kampis.

Lefebvre, V.A. (1995) The Anthropic Principle in Psychology and Human Choice. PSYCOLLOQUY 6(29) human-choice.1.lefebvre.

Wigner, E.P. (1969) The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In: Saaty, T.L. & Weyl, F.J. (Eds.) The Spirit and the Uses of the Mathematical Sciences. New York: McGraw- Hill Book Company.