Against Physical Tendencies

Graduate Session, 14:30, 12 July 2008
Martin Cooke (Glasgow)
Aristotelian Society/Mind Association Joint Sessions, Aberdeen

Five plausible properties of tendencies (§1) and some realistic physics (§2) leads to two contradictory results (§4, §5), indicating the implausibility of quantum-mechanical probabilities being tendencies (§6).

 

§1

Popper’s propensities interpret indeterministically the quantum-mechanical probabilities associated with radioactive decays etc. Under the single-case (or tendency) interpretation, probabilities would exist were there just one event of the relevant kind. Tendencies would have at least the following properties:

T1) Given some independent tendencies, if each might be T (for suitable T) then they might all be T.

T2) Finite multiplicativity of independent probabilities.

T3) Outcomes, however unlikely, may be conditioned upon.

E.g. fair coin-tosses arising from indeterminism at the atomic level: Many outcomes are possible, but most of them are not fair coin-tosses. To say that, were there two of them, the chance of two heads would be 25% is to condition upon unlikely outcomes; but it would clearly not matter how unlikely such fair coin-tosses were, just that they were independent etc.

T4) Finite additivity of mutually exclusive probabilities.

T5) Given some collectively exhaustive and equally likely possibilities, if, on each one, T is certain (for suitable T) then T is certain.

 

§2

Countably many (aleph-null) collectively exhaustive and mutually exclusive possibilities cannot be equally likely (§4, §5), but a subset of the possible results of aleph-null fair coin-tosses does behave like that (§3), and random events akin to coin-tosses are likely to occur in such numbers as aleph-null. There may be aleph-null particles in this universe (or even in parallel universes), if space is infinite, each existing at the start of a half-life, a duration for which the probability of decay is ½.

And vacuous spatio-temporal regions are associated with some probability of virtual particles appearing (or so the Casimir effect indicates). The probability (of some such fluctuation) is higher for larger regions and lower for smaller ones, so (at least if space-time is a continuum) there would be spatio-temporal regions within which the probability of (such) virtual particles appearing is ½, which I call ‘Regions.’

So, if this universe is everlasting (as the increasing rate of expansion of the universe indicates) then endless sequences of Regions will exist in this universe (as they would were space infinite). And even if this finite universe collapses, there may then be another Big Bang and so on, either endlessly (whence aleph-null particles) or else until an everlasting universe (whence aleph-null Regions); and of course, aleph-null Regions might exist in parallel universes.

I call any such process, occurring in aleph-null instances (without entanglement) in the universe(s), a ‘Toss,’ and its two equally likely outcomes ‘Heads’ and ‘Tails.’ Tosses can be causally independent of each other, so aleph-null Tosses can be probabilistically independent of each other: Were space infinite, each Toss could lie outside the light-cones of all the others. And were there aleph-null parallel universes, we could have one Toss per universe.

As for a temporal sequence of Regions (and similarly for particles over half-lives), while the outcomes of earlier Regions might have a slight effect upon later probabilities, that could be compensated for by having the location, size or shape of the later Regions depend upon earlier outcomes (and note that they depend upon earlier non-Toss events anyway), as follows. Cf. two fair coin*-tosses, the first being the fair toss of some fair coin, but the second coin* being a red fair coin if the first fair coin*-toss yields a head, a blue fair coin if it yields a tail. While the second coin* is of indeterminate colour to begin with, while it is not causally independent of the first, it is probabilistically independent of it. So, consider a branching tree of possible Regions: The outcome of the first (or trunk) Region (i.e. Head or Tail) existing, the second Region* is whichever possible 2^nd (or branch) Region is indeed Region. So, a sequence of Regions* (from some relatively determinate tree) is a probabilistically independent sequence of Tosses (non-Toss events just add complexity).

 

§3

Some of the possible outcomes of some simple endless sequence of Tosses (e.g. of Regions*) begin with a finite number of Heads, followed by an endless sequence of Tails, and such outcomes will be called ‘Integers.’ The Integer n has n Heads followed endlessly by Tails. Since Tosses are independent, so each Integer is possible (via T1, with T being the Toss being a Tail).

To say that some possibilities are equally likely is to say of any two of them, that neither should be expected more than the other; and Integers are equally likely: Were they not equally likely, some Integer p, would be more likely than some other, say q. Those two simple endless sequences differ in q – p Tosses, if q > p (and in only p – q of them otherwise), so p being more likely than q has to be due to the difference in a finite number of Tosses, so (via T2, since Heads and Tails are equally likely) p cannot be more likely than q. Note that Integers may not have numerical probabilities; but if they do then, since they are equally likely, those probabilities must (via T2) be less than any positive real number—e.g. they might all be 0 or some nonzero infinitesimal.

Now, we can consider two simple endless sequences of Tosses (e.g. via the odd and even elements of one), and both may yield Integers—it is unlikely that they will, but possibilities (and probabilities where they exist) conditional upon them both being Integers can be considered (via T3). E.g., in some possible world Mr E, observing a simple endless sequence of Tosses in a finite time (as a bifurcated supertask, see Earman, J. and J. D. Norton (1993) ‘Forever Is A Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes’, Philosophy of Science 60, 22-42),  and seeing that he has an Integer, thinks, “If Miss Tree’s simple endless sequence is also an Integer, is it likely to be a bigger one?” It seems to him that it is, because only finitely many of the possible Integers are smaller than his (aleph-null are bigger). But if they were both Integers, Miss Tree, similarly observing her supertask, would have reached the opposite conclusion, similarly. So that is an impossible world.

 

§4

But maybe that was too quick (e.g. subjective probabilities bring their own problems—although they ought to conform to objective probabilities where they exist—and bifurcated supertasks are physically unrealistic, although no so that they have not been used in metaphysical arguments) so consider the subset of the set of all the possibilities for two simple endless sequences of Tosses that is the set of pairs of Integers, which I call ‘P’. Probabilities within P are those conditional upon the outcomes both being Integers.

Can much be said about such conditional probabilities (within P)? Let two Integer variables (corresponding to any two simple endless sequences of Tosses) be n and m. Within P, either

B) either m is bigger than n, or else m = n is odd, or else

S) either m is smaller than n, or else m = n is even.

Since n and m are obtained in exactly the same kind of way, it will generally be the case that whatever can be said of n can also be said of m (with appropriate alterations).

Simply by the symmetry of the situation, it is plausible that neither B nor S is more likely than the other. And since either B occurs, or else S does (exclusively and exhaustively), so it is plausible that the conditional probability (within P) of S is 50% (via T3).

 

§5

What else can be said about that probability? Since n and m are independent of each other (and since T3), it is plausible that we may consider n first, and then m. Suppose that the conditional probabilities (within P) of the various values of m were all 0.

Whatever the value of n, the conditional probability of S (within P, and given that value of n) would also equal 0 (via T4, because given any n only finitely many values of m give rise to S) so it would not be a number bigger than 1%.

And were they all some nonzero infinitesimal then (via T4) the conditional probability of S (within P and given any value of n) would be at most infinitesimal (note that if n = 1 then B), so again, it would not be a number bigger than 1%.

There may be no such numerical probabilities of the m, but similarly the tendency for S (within P and given some value of n) may also lack a measure, and if so then there would be, associated with that tendency, no number bigger than 1%. So, suppose there is such a conditional probability of S (but again, none of the m). Whatever the value of n, either

G_n) m > 100·n – 99, or else

L_n) m < 100·n – 99.

Now, were it the case (counterfactually) that some combinations of n and m were excluded, in such a way that, for n > 1, G_n could not occur, and m was not equal to n, then for n > 1 the conditional probability of S would be 1%, as there would then be (n – 1) positive integers smaller than n, and 99·(n – 1) bigger. And it is surely plausible that the actual possibility of those combinations would not make that value bigger than 1%, since for given n either all those possibilities give rise to B or else, if m = n is even, all but 1 of them do.

So, since the conditional probability (within P) of n taking some value is 1, and since the conditional probability (within P and given that value of n) of m then being such that S is plausibly (from §5) not bigger than 1% (even if it exists), so it is plausible (via T5, with T being that the conditional probability of S is not a number bigger than 1%, if T might be such an assertion) that the conditional probability of S (within P) is not a number bigger than 1%.

 

§6

But it is plausible (from §4) that the conditional probability of S (within P) is 50%. Consequently it is plausible that one of my assumptions was false.

Might one of those results (in bold) simply imply the falsity of the other? But were the result of §4 false, the result of taking n first (a low probability of S) would still be contradicted by the result of taking m first (a low probability of B).

The result of §5 might be false, but this is precisely what is in question, and T5 has a general air of plausibility about it beyond this context (e.g. Simpson’s paradox would not arise because the partition has to be into equally likely parts). Consider the corner of an infinite square of pairs of positive integers, its diagonal being (1, 1), (2, 2), (3, 3) and so forth. A column might represent Mr E knowing his Integer and seeing that Miss Tree’s Integer would, were there one, be quite unlikely to be smaller (that column’s entries being equally likely), and yet random pairs could conceivably be equally spread out, above and below the diagonal, yielding an average of 50%. Still, that only reproduces the contradiction, rather than resolving it. There is also what Mr E can see, paradoxically; and what Mr E and Miss Tree could get is there to be got even if they aren’t there.

And the physics presupposed by the aleph-null Tosses was quite plausible, insofar as at least one of the possibilities mentioned in §2 is almost certain to be the case. (And it is surely implausible that space must be finite because the tendency interpretation is true, etc.) And an interpretation of probability that did not include properties equivalent to T1-5 would hardly be a theory of tendencies (although the meaning of ‘propensity’ is a little imprecise). So it is empirically implausible that quantum-mechanical probabilities are tendencies. After all, many physicists do prefer to work with more frequentistic interpretations, some even rejecting indeterministic interpretations of quantum mechanics.

 

§7

Still, were tendencies the most realistic interpretation of physical probability (as they are, according to my 2006 ”What are my Chances?”) and were the actuality of the simply infinite relatively unsupported empirically (as it is), then I would have shown the implausibility of that actuality (via supertasks and such).