The Mathematical Basis for the Inapplicability of Probability to Invalid Functions
There are frequent statements in the literature that facts can be assigned a probability. These usually come in the form of asserting that the fact is improbable (usually “highly improbable” or “surprising”). Some writers making this claim should know better. Others are best advised to stay away from mathematics altogether. The intuitive (and mostly correct) response is that facts don’t have probabilities. They just are. It is equally incorrect, however, to try to assign probabilities to speculative possibilities. By speculative possibility I mean a “probability function” every one of whose arguments (whose integrand if the probability function can be integrated) is a function f(x) = x/∞. The most rigorous way to show this is via the mathematics of probability. A random variable or probability function is a rule or function that assigns a numerical value to an outcome of an experiment. A probability distribution is a list of all possible values of a random variable. The sum of probabilities in a probability distribution or sample space must = 1. So the probability of all possible valid outcomes is 1 and the probability of excluded or invalid outcomes is 0. For example, the probability that a coin toss will result in heads or tails is 1; the probability that the coin will not land on the ground is 0. A valid probability is typically a rational number, r, such that 0 ≤ r ≤ 1. The probability that a single coin toss will result in heads is ½. A probability of 0 is impossibility and a probability of 1 is inevitability or fact. Probability density functions or PDF’s of continuous random variables are accumulation functions whose values are probabilities (sums of the probabilities of continuous random variables over a given interval). Constants are functions that assign a constant value to every argument. If “n” stands for a constant, then the probability function or random variable of n assigns the value 1 to n and the value 0 to everything else. In this way a constant can be regarded as the outcome – the only possible outcome – of an experiment. The probability distribution P(f(x) = n) would contain only one member whose value is greater than 0. Facts (or at least factual numerical measurements) are constants. I shall group the probability functions of constants and speculative possibilities together and call them “invalid probability functions.”
The graph of the constant function f(x) = n is a horizontal straight line parallel to the x-axis and intersecting the y-axis at n. E.g.:
The graph of the PDF of a constant is a horizontal line parallel to the x-axis that intersects the y-axis at 1. It satisfies the requirements for a PDF only at intervals that are single points. It does so, however, for every single point.
A speculative possibility is a probability function P whose integrand is f(x) = x/∞. That is, the random variable for any value of x yields x/∞. Thus it yields 0 for every value of x (assuming x/∞ = 0; if x/∞ is undefined, then P(x/∞) is invalid because it does not have a defined integrand). Its graph is a horizontal line that coincides with the x-axis. It does not fulfill the requirements of a PDF since there is no sample space S for f(x) such that P(S) = 1. Since there is nothing to accumulate, the graph of a speculative possibility would look the same as the graph of its random variable:
Assuming the integrand of P = F(x) is f(x), then f(x) = F' (x) = 0. The speculative possibility yields the same result as its random variable, viz. the constant 0 for any value of x in the random variable. It is not a valid probability function since there is no interval at all whose value = 1. Speculative possibilities, however, are not the same as the probability function associated with the constant function f(x) = 0. The latter yields the probability 1 for any value of x just like any other constant; the former yields the probability 0 at x = 0 just as it does for any other value of x. In terms of probability calculations the constant 0 should be looked upon as the value of the integrand of a probability function whereas x/∞ in the case we are considering is the value of a probability function (the probability that some outcome x will obtain).
If you follow a speculative possibility of possible universal constants construed as a probability function to its logical conclusion, the universe doesn’t exist! There is a point to grouping constant functions and speculative possibilities as invalid functions for the purposes of discussing the “probability” of a fact because those who try to assign a probability (or an improbability) to a fact usually put themselves in a hypothetical situation where, before the fact was actualized (for example, before the appearance of the universe), alternatives appear to have been available. Of course this imaginary scenario begs the question as to whether the facts were indeed “chosen” among alternatives. Moreover, it cannot be evaluated using the mathematics of probability because by hypothesis there are an infinite number of possible outcomes for the value of the universal constant each with exactly the same probability, namely x/∞. Since the value of every possible argument of a speculative possibility is 0, it is impossible to construct a sample space such that the sum of the probabilities of the members of the sample space = 1. The mathematical impossibility, in my opinion, undermines the validity of loose talk about improbabilities in theological and cosmological speculation.
The error, I believe, on the part of those who try to assign probability values to speculative possibilities is that they misconstrue continuous probability functions or PDF’s, those functions that assign a theoretically infinite number of random variables to a theoretically infinite number of events. The point is that a well-formed PDF still assigns a real n, such that 0 ≤ n ≤ 1 for every one of the potentially infinite number of random variables within the sample space. And it still requires an interval for definite integration (its sample space) whose accumulated probability value = 1. A speculative possibility does not satisfy this requirement since it assigns 0 to every random variable in its range.
Speculative possibilities are also not the same as improper functions because, for one thing, the sample space represented by a real number interval on the graph of the speculative possibility need not equal or approach either ∞ or -∞. In case it did so, a speculative possibility would remain invalid because it would still assign the value 0 to every one of its arguments. Even an infinitely long sample space on the graph of a speculative possibility could not be constructed such that the sum of the probabilities of its arguments = 1.
There remains the case where the observed or hypothetical constant is supposed to be a real number interval with more than one member such as would be the case where the actual constant could be any number within a margin of measurement error. Obviously the fact that a measurement may have a margin of error does not mean that the constant does not in fact have a single value. And there is no evidence that the notion of a range of actual values for a universal constant would make any sense in the context of physical theory. Moreover in much of mathematics the expression “∞/∞” (which would express the probability of a continuous number interval being an actual value out of a continuous set of alternatives) is undefined. The proposals for defining “∞/∞” do not help. For if ∞/∞ were to be defined as = 1, the result would be that all number segments would be actual values of, say, a single universal constant. Given any rational number, for example, that rational number would be the correct (and observable?) value of the ratio between gravitation and the strong nuclear force, an obvious absurdity. The proposal that “∞/∞” is meaningful in case the numerator is greater than the denominator of the expression depends on the assumption that the numerator of the expression expresses an uncountable infinity and the denominator expresses a countable infinity. I suspect that any meaningful discussion of the universal constants would limit their values to the rational numbers in which case the property of density of a given segment would not imply non-countable infinity. But, even if the irrational numbers could be viable values for the universal constants, there is no reason that the infinity of the denominator may not include irrational numbers also and consequently be uncountable. It would probably have to for, if irrationals are actual values of the numerator, then irrationals would have to be possible values as well. In fact for any ∞/∞ whose numerator denotes a set of all the real numbers within the margin of error of a measurement and the denominator denotes a set that is complementary to the numerator set, then the cardinality of both the numerator and the denominator equals the cardinality of the continuum or .
Mill (probably because of his reservations about the empirical applicability of the mathematics of probability) knew this on a more informal level: “…in the absence of any ground from experience for estimating (a thing’s – WD) degree of probability, it would be idle to attempt to assign any.” Logic p. 376