Where the Theory of Probability Fails

Where the Theory of Probability Fails Author(s): Itamar Pitowsky Source: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Vol. 1982, Volume Two: Symposia and Invited Papers (1982), pp. 616-623 Published by: The University of Chicago Press on behalf of the Philosophy of Science Association Stable URL: http://www.jstor.org/stable/192447 Accessed: 31-10-2016 05:46 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/192447?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Where the Theory of Probability Fails

Itamar Pitowsky

Philosophy Department, The University of Western Ontario

1. Introduction

The purpose of this paper is to present a local solution to the Einstein-Rosen-Podolsky (E.P.R.) paradox by way of a mechanical

analogue (roulette) and then to note some of its consequences for the foundations of mathematics and probability theory. The detailed

mathematical development of the model involves some highly specialized fields in mathematics (set theory, measure theory and group theory). My intention is to avoid these technicalities as much as possible. A complete account that includes proofs and application to physics is in Pitowsky (1982a, 1982b, 1983). Some mathematics is, however, indis-

pensable. I shall denote by x,y,z,w unit vectors in the three dimensional Euclidean Space. If x,y are unit vectors, (x,y) is the (small)

angle between x and y so that 0 ' (x,y) '~ 7r. Let S(2) be the set of all unit vectors. S (2) is the (surface of the) unit sphere in three

dimensional Euclidean space. For a fixed z E S (2) and fixed angle o < e < r let c(z,e) denote the set of all unit vectors that form an

angle e with z. c(z,e) = {w E S (2) | i(z,w) = e}. c(z,e) is a circle on the sphere whose center is on the axis connecting z with -z and

whose radius is sin e (Fig. 1). Let me 8 denote the Lebesgue measure on c(z,e). mze measures the length of subsets of c(z,e), such as arcs, countable unions, or intersections of arcs, etc. We have

ma (c(z,)) = 2Tr sin e and th1s p6 = (2F sin 6) me is a normalized (probability) measure on c(z,e). 2. A Roulette Game

Consider the following gambling device. It is composed of two parts:

PSA 1982, Volume 2, pp. 616-623

Copyright (L 1983 by the Philosophy of Science Association


617

I

'10'

Fig. 1

1) A ball of radius 1 whose surface is divided into two hemispheres, one painted red and the other blue.

2) A pointer which (making an idealized assumption) can pick up an exact mathematical point on the sphere.

To use this device one has to choose at random a direction z E S (2)

and an angle 0 < 0 < 7r. Then one spins the ball around the axes connecting z with -z while the pointer is fixed and oriented towards the points of the circle c(z,e) (Fig. 2).

fC, Fig. 2

The gamble 'red' wins if and only if the sphere stops spinning when the pointer is directed at a red point. There are two kinds of games that one can play with this device:

1) bet on the outcome 'red' or 'blue' before the direction z and

angle e have been chosen at random; 2) bet after z and 6 were fixed.


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I shall refer to these possibilities as 'the first game' and 'the

second game', respectively. There is a vast difference between the two games. In the first game the only relevant pieces of information that one has are the facts that the ball is half red, half blue and

that z,e are going to be chosen at random. In this case, every point on the sphere may be the outcome of the game. The probabilities for 'red' and 'blue' are thus equal O.5.

In the second game, the gambler already knows the values of z and e

and therefore, that the point to be picked by the pointer lies on the circle c(z,6). The probability of 'red' in this case equals the normalized length of the arc of red points on c(z,e). Suppose A denotes

the set of red points (i.e., the red hemisphere) then this probability

equals pe [An c(z,e)I, a number which varies with z,e and can get any value between zero (in the case c(z,e) lies in the blue hemisphere) and one (in the case c (z,e) c A) . 3. A More Complex Roulette Game

A natural generalization of the above game can be realized by a

device in which the set A of red points is not a hemisphere but a

(2) rather more complex subset of S(. In any such device the probability of 'red' in the first game depends on 'global' properties of the set A

(i.e., its Lebesgue measure on the sphere, its 'area') and the probabilities for 'red' in the second game depend on 'local' properties of

A, that is, the (normalized) measure of the set An c(z,e) on c(z,e). I would like to introduce such a roulette game in which A, the set of 'red points', is an extremely abstract subset of S(2). The fact

that such a device is ZogicalZy possible results from the following assertion that turns out to be consistent relative to set theory.

Theorem. There exists a subset A c S(2) with the foZZowing properties:

1) w E A iff -w ' A, that is, S (2) \A = -A = {-w I w E A}. 2) For aZZ z E S (2) and 0 < e < i: cos2 (e/2) z E A pze

[A

n

c

(z,e)]

zsin (8/2) z ' A

=

2

(1)

First, a note about the proof. The theorem can be proved with the aid of the axiom of choice and some higher axiom of set theory such as the continuum hypothesis or (the weaker) Martin's axiom. (In fact, what we need on top of the axiom of choice is the following axiom that relates measure and cardinality: Every subset of a Lebesgue space

whose cardinaZity is strictZy Zess than the continuum is Lebesgue measurabZe and has measure zero. This axiom might deserve the name 'the measure theoretic continuum hypothesis'.) Moreover, the axiom of choice is indispensable in the proof (see below) and there are good


619

indications that the theorem is in fact independent of Z.F.C. (which means that some axiom on top of the axiom of choice is indispensable in its proof).

All these observations make the very mathematical existence of A dubious (at least for some philosophers). In any case, it is at least

logically consistent (relative to set theory, that is) to pretend that we have a device in which the set of 'red points' is the set A of the theorem.

What is the probability of the outcome 'red' in the first game

(that is, before z, e where chosen at random)? The intuitive answer

is 1/2 because w is a 'red point' iff its opposite, -w, is a 'blue point' and thus A is in fact 'half the sphere'. This intuition, how-

ever cannot be phrased in terms of the Lebesgue measure of A on the sphere, simply because A turns out to be nonmeasurable (which means that the concept of area cannot be extended to be defined on sets such

as A). Solovey (1970) has constructed a model of set theory in which the axiom of choice fails to be true and in which every subset of a

Lebesgue space is Lebesgue measurable. Hence the axiom of choice is indispensable in the proof of my theorem.

The calculation of probabilities for the second game is simpler. When z,e have already been fixed the gambler has to take the following steps:

a) Check whether z itself is 'red' or 'blue'.

b) If z is 'red' then the probability of the outcome 'red' is

cos2(8/2) and if z is 'blue' the probability of the outcome 'red' is sin2(8/2). 3. The E.P.R. Experiment as a Second Game Instead of talking about 'red' and 'blue' we can talk about spin

values. Suppose that every electron at each given moment has a

definite spin in every direction w E S(2) which is either 'up' (that is, +1/2 h where h is Planck's constant divided by 2T) or 'down' (i.e., -1/2 h). The spin in the w direction is 'up' if and only if the spin in the opposite direction -w is 'down'. Suppose, moreover, that the set of directions for which the spin is 'up' is the set A in

the theorem. If this is the case we can imitate the roulette game by a physical system.

The first game corresponds to betting on the electron spin ('up' or 'down') in a randomly chosen direction. The outcome is 'up' with frequency 1/2 in accordance with our intuition (not in accordance with measure theory, however).

The physical analogue of the second game is as follows:

a) Take an electron and fix a direction z and an angle e.


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b) Measure the electron spin in the z direction making sure not to

disturb the spin values in other directions. c) Bet on the outcome ('up' or 'down') of a subsequent measurement of

the spin in a direction chosen at random among those directions

that form angle 0 with z. The only obstacle to an actual realization of such a game lies in

the second step because it is a highly confirmed hypothesis of quantum theory that a spin measurement in the z direction may cause a flip in the spin values in other directions, and there seems to be no way to control this phenomena. The E.P.R. experiment is designed to circum-

vent this difficulty. It is known that in some specific cases two interacting electrons emerge from the interaction in the so called, singlet state. This means, operationally, that the two electrons have opposite spins in every direction in which it is measured. Also, the

singlet state seems to be stable in the sense that the one-one opposite relation among the spin values is not spontaneously destroyed. Bearing

this in mind we can construct a physical analogue of our roulette: a) Take an electron pair in the singlet state; fix a direction z and angle e.

b) Measure the spin of the first electron in the -z direction making sure that at the time of measurement the second electron is already far away.

c) Bet on the outcome ('up' or 'down') of a subsequent spin measurement in the +x direction performed on the second electron, where x

is chosen at random among the directions that form angle e with z. Since the first measurement on the first electron is performed when the electrons are considerably separated it is not likely to influence

the spin values of the second electron. This is the principle of locality which I assume to be true.

The above procedure resembles a singZe instance of the E.P.R. experiment (i.e., a measurement on one pair of electrons) though there is one inconsequential difference. In the E.P.R. experiment the first and second measurements are in fact performed simultaneously (so that no

one has time to bet). It is easy to construct a version of the E.P.R. experiment in which this is not the case; the results will not be different. When we repeat the experiment in the E.P.R. set-up we do it with fixed z and x and a random sample of electron pairs. Adding one more

assumption, this turns out to be equivalent to a case in which we repeat the experiment with the same pair of electrons, with z fixed and with x varying randomly on c(z,e); i.e., the E.P.R. experiment is equivalent to the roulette. I shall not prove but only mention which extra assumption is needed. The property of the set A given by the formula (1) is rotationally invariant, that is, if we transform the space by an orthogonal trans-


621

formation a the set A' = aA = {aw w E Al also satisfies formula (1) when we interchange A with A?. (This does not happen when A is a hemisphere.) The extra assumption needed here is that for each electron the set of directions for which the spin is 'up' is of the form aA where A is fixed (the same for all electrons) and a some orthogonal transformation. It is known that the relative frequency of 'spin up in the x direction given spin up in the z direction? as measured in the

E.P.R. experiment is cos 2(/2), i.e., equals the expectations of our roulette.

4. What Happened to Bell's Inequality?

Nothing, it is still valid. The difference between my approach and other approaches lies in its interpretation. I claim that my model explains why physicaZZy observed (as opposed to counterfactual) frequencies violate Bell's inequality.

In my model the conditional statement: "If z is 'up' and i(z,x)

= e then the probability that x is 'up' is cos2 (6/2)" is true (given the existence of the set A). Yet the conditional expectation cos 2(/2) violates Bell's inequality. There is no logical contradiction here and part of the reason is that A is nonmeasurable. It turns out that if we want to recover these expectations on a hidden variable space with a probability measure P, then for all x the subset of hidden variables which correspond with the state 'spin up in the x direction' is P-nonmeasurable.

It follows that theory to include in the framework could be shown to

what we need here is an extension of probability some species of nonmeasurable 'events'. It is only of such an extension that the relative frequencies converge to the expected value ('most probably'). Such an extension of probability is developed in Pitowsky (1983).

5. Some Philosophical Remarks

According to my model the violation of Bell's inequality by observed

frequencies results from the nature of the distribution of spin values over directions and not from a violation of locality. Thus it seems that a consistent 'realistic' and local 'explanation' of the E.P.R. paradox is after all possible.

Realistic in what sense? The very existence of the set A is dubious and depends on some higher axioms of set theory. If these axioms are mere creations of the mind, that is, bear no physical significance, one can hardly maintain that I have got a 'real explanation'. Consistent, maybe, but hardly realistic.

But is it the case that set theory is not and could not be significant in physics? Maybe set theoretical properties of families of

space-time points are important. If this is the case then the existence or nonexistence of abstract creatures and the truthfulness or


622

falsity of certain axioms becomes the subject of an empirical study in very much the same way that the geometry of space time bears empirical significance. If this is the case the violation of Bell's inequality may be taken as a piece of evidence that some nonmeasurable distribu-

tions exist in reality. I shall leave it to the reader to decide which of these alternatives is more reasonable. Final comment about probability. The E.P.R. experiment may be taken as a parody on the Dutch Book argument. Suppose that a gambler does not

know the nature of the distribution of spin values in all directions but knows only that w is 'up' iff -w is 'down'. Suppose, moreover,

that he is to bet in the second game. Being a rational guy and having knowledge of probability theory he arrives at Bell's inequality. This does not tell him how to bet but rather how not to, but, as it turns

out, he is to lose a lot of money. The irony is that the E.P.R. exper-

iment is not merely a thought experiment. We can actually construct a device which makes the statement 'A person is rational iff he is not' true.

I believe that all this is an example of the fact that probability theory cannot be taken as synthetic apriori. There exists no univer-

sal concept of probability that could be justified once and for all. Each scientific theory applies its own concept together with its own idiosyncratic definitions of 'randomness', 'independence' and the like. The theory of probability is no more justified than the theory in which

it is embedded. "Our statements about the external world," said Quine "face the tribunal of sense experience not individually but only as a

corporate body" (Quine 1951, p. 41).


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References Pitowsky, I. (1982a). "Resolution of the Einstein-Podolsky-Rosen and

Bell Paradoxes." Physical Review Letters 48: 1299-1302. ----. (1982b). "Answer to Comments." Physical Review Letters

49: 1216.

-. (1983). "Deterministic Model of Spin and Statistics." Physical Review D 27: 2316-2326. Quine, W.V.0. (1951). "Two Dogmas of Empiricism." The Philosophical

Review 60.: 20-4 3. (As reprinted in From a Logical Point of View. New York: Harper & Row, 1961. Pages 20-46.) Solovey, R.M. (1970). "A Model of Set Theory in Which Every Set of the Reals is Lebesgue Measurable." Annals of Mathematics 92:

1-56.
