Jonah Bossewitch	PHI 327

April 6, 1996	Chad Mohler

Midterm Exam

I.	The nature of the "Determinism vs. Indeterminism" issue is twofold: For centuries people have attempted to define these concepts and examine what they actually imply. They have also tried to determine which of these terms best describes our universe (not necessarily in that order). In recent years, the utility of this distinction has been challenged, as people are often more concerned with answering the question "How does the Universe behave?" rather than categorizing that behavior in "metaphysical" terms. The boundary between deterministic and indeterministic has been made fuzzy by the various senses of determinism that have been formulated, forcing us to qualify any assertions we make involving these terms.

Early formulations of determinism often invoked the concept of causality, which, as Earman points out (Course Packet #1, p. 5), explains "a vague concept - determinism- in terms of a truly obscure one." More sophisticated conceptions of determinism were advanced after the development of modern physics, and later these were rigorously formulated mathematically. Essentially, they define determinism in terms of the transitions between states, and make no explicit mention of predictability. A system is deterministic iff state X always evolves into state Y, and state Y is unique. A system is bi-deterministic iff state Y could only have evolved from state X, and state X is unique.

Laplace romantically imagined a God-like being, wielding the most powerful equation in the world, who could predict all the future states of the universe given the present state (in fact, this was how he defined determinism). The problems with the approach lie in how much power we are willing to grant this being. If his computational power is limited to a Turing machine, he will run into some problems that might be theoretically impossible to solve (since the initial states would be impossible to specify), or would require more time to compute than the lifespan of the Universe. Furthermore, unless this creature can manipulate real numbers with arbitrary precision, chaos will quickly overtake his computations, causing them to diverge radically from their real values.

Settling whether a given natural system is deterministic or not is meaningless without specifying which sense of determinism is intended, and from this perspective is somewhat arbitrary. However, certain boundary cases might clearly indicate that a system behaves one way another. For instance, if it could be proven that certain computations could only be carried out in a short amount of time nondeterminsitically, and some creatures exhibit the ability to solve such problems, then they must be employing nondeterministic algorithms. Thus, in principle, there might exist certain systems which empirically suggest in/deterministic behavior (if we bolt the skeptics outside the door).

II.	A common cause is an event C, that always precedes two correlated events A and B. The common cause principle states that given any two correlated events (A,B) there must exist a third event C which is their common cause. Under this principle, failure to find C indicates an incomplete scientific theory that is simply ignorant of the true common cause. Formally, this can be expressed as follows:

If (1) P(AB) > P(A)P(B), then there is an event C such that

(2) P(AB|C) = P(A|C)P(B|C) & (3) P(AB| C) = P(A| C)P(B| C) & (4) P(A|C) > P(A| C) & (5) P(B|C) > P(B| C).

The common cause principle combined with perfect correlation implies determinism, or to be more precise, is one way of defining determinism. From an epistemological standpoint, we can never be sure that we have identified the "real" common cause. At best, all the theorizing in the world can only produce a common cause model that accurately describes and predicts the behavior of the system.

Here at Princeton, there is a longstanding tradition that it is bad luck for undergraduates to walk out the main gate in front of Nassau Hall until graduation day. A Senior Philosophy major became interested in this superstition, and found that there is a statistically significant positive correlation between students who walk out the main gate prematurely, and students who fail out of school. At first, she was tempted to view this correlation as evidence for a direct causal relationship between walking through the gate and failing out of school - the curse lived!. However, remembering the common cause principle, she hypothesized that there might be a third factor that was correlated with both of the other events. In fact, she determined that students who violate popular beliefs tend to have rebellious personalities, and that most people who fail out of school also have rebellious personalities. Students with a rebellious personality were more likely to fail out of school, and these same students were also more likely to walk through the main gate. Thus, she modeled the correlation by positing a third hidden variable, the students personality, which was the ultimate cause of both A and B. Of course, this explanation might be the best one, but it might not be correct. Perhaps the eagles sitting atop the gate have more influence on our lives than any of us dream.

III.	Most classical physicists professed to adhere two principles: "(1) Value Definiteness - every observable (quantity or determinable property) has at all times one of its possible values. (2) Veracity of Measurement - the outcome of a (genuine, good) measurement of an observable on a given system reveals the value which that observable had at the initiation of the measurement." (lecture notes, 2/29) Before the discoveries in QM, it was possible to have faith in these principles since they were consistent with the data, even though the skeptic could always reject them. Both principles make claims about unobserved events, and by definition, cannot be verified empirically (although more sophisticated theories of confirmation might support them).

Even before the age of QM, there were some problematic cases in classical physics that challenged these two principles. The color of iodine is violet in its solid state, brown in its liquid state, and invisible in its vapor state. If the principle of value definiteness is true, then shouldn't we be able to assign a color value to the iodine vapor? One way to refute this challenge is to argue that color is not an observable. For centuries philosophers and physicists have grappled with a class of qualities which seem to be response dependant, and are not intrinsic to the object. A quality which is observer dependant might not qualify as a uniquely determinable property that the principle of value definiteness applies to.

There exists a parallel to the iodine case, which challenges the veracity of measurement principle from within the classical domain. If a thermometer is put in a hot cup of coffee, its mercury column rises, but the final reading is not the initial temperature of the coffee (because the coffee cooled a little when it heated the thermometer). Does this process still satisfy the veracity of measurement principle? In this case, since the observable being measured, temperature, can be independently defined and measured (by measuring pressure, volume, conductance, or other know properties which are correlated with temperature), our theory can accurately predict the error that the thermometer introduces to the measurement. This precalculated error can be factored into the measurement (as long as the value definiteness principle holds) to allow us to call this process a measurement, despite the fact that the measurement changes the quantity of the observable being measured. Including our outside knowledge, we are able to reveal the initial value of the observable, and this is what is crucial for the veracity of measurement principle.

IV.	"We call state x an eigenstate of observable A, and k an eigenvalue of A, the corresponding eigenvalue for state x exactly if the following holds: PxA(k) = 1... The probability of outcome k, if observable A is measured on a system in state x, equals 1." (lecture notes 3/12, 3/7). This is equivalent to the vector notation O|B> = @|B>, where |B> is the eigenvector of the O operator (or observable), @ is the eigenvalue associated with |B>, and @|B> is an eigenstate of O (Albert's notation, p. 29). Eigen- , meaning "proper" or "pure" can be thought of as the lucky vector which only has its length, not its direction, transformed by the operator. Physically this means that when an observable is measured while the system is in an eigenstate, the result of the measurement will certainly be the eigenvalue corresponding to that eigenstate.

(a) If there exists more than one eigenstate corresponding to different eigenvalues for a given observable, then by the definition of an eigenvalue, Px1A(k1) = 1, Px2A(k2) = 1, Px3A(k3) = 1...

But, state z is orthogonal to state x iff PzA(k) = 1 and PxA(k) = 0 (lecture notes 3/7). Since k1 will certainly be observed when A is in state x1, the probability of observing k2, k3, ... when A is in x1 is zero (Px1A(kn) = 0, for n > 1), and this is true for all the cyclic permutations of x. Hence, eigenstates corresponding to different eigenvalues are orthogonal.

(b) When the photon is in state |A(r) = 0> it is certain not to pass the A(r) filter.

(c) When an observable has more than one eigenvalue for a given eigenstate, any mixture (or superposition) of these pure states will also be a k-eigenstate of the observable. So, a mixture of degenerate k-eigenstates can be an eigenstate of an observable.