Some notes on a problem of Pascal’s

So, Pascal’s Wager. I am sure you know it, but here’s a formulation anyway:

If you erroneously believe in God, you lose nothing (assuming that death is the absolute end), whereas if you correctly believe in God, you gain everything (eternal bliss). But if you correctly disbelieve in God, you gain nothing (death ends all), whereas if you erroneously disbelieve in God, you lose everything (eternal damnation).

Since I am a graduate student of mathematics, I am not equipped to leave utterances like this alone; and thus, a bit of probability tomfoolery to torture this anyway silly argument follows.

First, some horrendous simplification. Let us consider a system where one can believe in any one of $n$ gods, or disbelieve all of them. Let us further say that either any single one of these gods exists and the others do not; or then no gods exist at all.

Let us suppose for simplicity’s sake that the existence of any particular one of these gods is equally probable, and that probability is denoted by $p_G$, while the possibility of all these gods being fictious is $p_0$. Naturally, $np_G + p_0 = 1$. (We will return to this, later.)

Now, let us suppose (we do a lot of supposing, don’t we?) that the choice to believe any particular one of these $n+1$ choices has consequences wholly determined by that choice; and particularly that we can assign some numerical values to the desirability of the end results of “not believing in a god that exists” ($a_G^-$, trad. “Hell”), “believing in a god that exists” ($a_G^+$, trad. “Heaven”), “not believing in a god when there is none” ($a_0^+$), and “believing in a god when there is none” ($a_0^-$). Here zero denotes the consequences of atheism being true, while G stands for the existence of some god; and please note that we assume that since there are $n$ distinct and different gods, the adverse outcome of one of them being real occurs to all atheists — and to all who believe in any of the $n-1$ other gods!

(I apologize for that exclamation mark. Traditionally, exclamation marks and mathematical symbols co-occur only in crankery of Timecubical dimensions.)

For “simplicity”, we assume all these outcome-values are positive; the indubitable negativity of the various Hell-scenarios and lives wasted in futile worship will be handled with a minus sign.

Having introduced all this notation, we can say that the expected value of atheism, $E(0)$, is the sum of the probabilities of the various outcomes multiplied by their assigned “values”, and can thus be written as:

$\displaystyle E(0) = a_0^+p_0 - n a_G^-p_G$,

while the expected value of believing in one of the gods, or $E(G)$, is:

$\displaystyle E(G) = -a_0^-p_0 - (n-1) a_G^-p_G + a_G^+p_G$.

Now Pascal’s idea was that one should believe in God, because the latter of these values was inestimably greater, or $E(G) > E(0)$; but please note that his simple system contained only atheism and one possible God (or $n=1$); while this system, and the reality it in its crude way emulates, contains several; in this case the indeterminate amount $n$.

If we try to simplify the expression $E(G) > E(0)$, we note that because of our symmetric assumptions (“all gods are the same”), the multiplier $n$ disappears, and after some trivial formula-juggling we are left with this:

$\displaystyle p_G(a_G^+ + a_G^-) > p_0(a_0^+ + a_0^-)$,

or

$\displaystyle p_G > p_0\frac{a_0^+ + a_0^-}{a_G^+ + a_G^-}$,

that is to say that that (and thus any) particular brand of theism is more profitable than atheism if the previous inequality is true. Assuming we are science-literate people momentarily afflicted with the probability-assigning disorder, we can agree that the probability of god’s existence ($p_G$) is not a big number; and being mathematics-literate, we can agree that the rational part of the right-hand side is by necessity something very small, the absolute values of eternal bliss and eternal torment being of necessity more than the rewards of one life well spent, or the losses of one wasted in futile rituals. Since the probability of anything (say of no gods, or $p_0$) is a number between zero and one (mathematically but not realistically including both), the right-hand side is something small as well.

Thus, naively, it seems the problem in insoluble: two very small numbers, and no way to see which is smaller.

To escape this, we make more perilous assumptions, and especially notice that, unlike with Pascal, this “belief in god” is actually “belief in a god”, namely one of the $n$ gods assumed to be choices. Let us assume that the ratio of “earthly outcomes” to “divine outcomes” is some very small number $10^{-k}$, where $k$ is a positive integer. (See first endnote for why this number is not zero.) Let us also assume that for any of the gods the probability of that god’s existence is some very small probability $p_G = 10^{-k}/n$. (See second endnote for matters implicit in this.) Since for the probabilities of gods or none the equality $np_G + p_0 = 1$ holds, we can say that $p_0 = 1 - 10^{-k}$.

Thus, our result becomes

$\displaystyle 10^{-k}/n > (1 - 10^{-k})10^{-k}$,

or, simplified,

$\displaystyle 1 > n(1 - 10^{-k})$.

What does this mean?

Well, bad news for the theists that wish to use this kind of argumentation, frankly. For theism to be “more profitable” in this crude sense, the above inequality has to hold: and as long as the combined probability of all gods considered ($np_G = 10^{-k}$) is a very small number (and as a probability it is naturally below one), the bracketed part of the right-hand side is very close to one; and in this case, as there are several gods to be considered ($n>1$), theism immediate becomes the less attractive of the two general alternatives. (Though if in Pascal and in a particular movie series “there can be only one”, or $n = 1$, the gamble is profitable in the crude sense described above.)

So there: with these particular assumptions, in this particular model of the problem, Pascal’s Wager is bullshit.

It naturally is so in any context; but in this particular one, it is so by mathematics even allowing that the basic probabilistic nature of the thing is tenable.

* * *

First endnote. The non-infinity of heaven.

Some may quibble that I have missed the entire point of Pascal’s Wager (as I have) with the statement that “Let us assume that the ratio of ‘earthly outcomes’ to ‘divine outcomes’ is some very small number $10^{-k}$, where $k$ is a positive integer”; and insist that since the joys/griefs of Heaven/Hell are infinite in duration and magnitude, the number should not be very small, but actually zero.

I don’t think that’s justified. (Also, it would wreck my argument.)

The number $10^{-k}$ is taken to be the ratio of “how preferable is an atheist’s life, and how horrible a life, your only one, wasted in futile rituals” to “how preferable is Heaven, and how horrible is Hell”. This author thinks there’s something quite horrible in wasting one’s only life, and something very valuable in using it well if that’s all one has; and, being of mathematical bent, the author also wishes to note that the mere fact that Heaven or Hell last forever do not mean their “preference value” (a made-up term) is infinity, no matter how much Pascal opined it to be: after all, one may, just as one solitary example, easily choose an always positive function whose integral from a zero point all the way to infinity is still a limited number, and not infinite.

That is to say,

$\displaystyle \int_1^\infty \frac{1}{x^2}\,dx = 1$,

and because of that Heaven and Hell don’t impress me much. Suck on that, Blaise Pascal.

* * *

Second endnote. On the choice $p_G = 10^{-k}/n$.

If we choose $p_G = 10^{-k}$ instead, we arrive to the inequality

$\displaystyle 0 < n10^{-k}$,

which immediately illustrates the problem of that choice: the simple addition of a god, no matter how curious, makes any theism a better bet, which is absurd! (Whether at this point in this kind of a tomfoolery the word “absurd” has any meaning is left to the reader.)

Indeed, the choice made in the main article ($p_G = 10^{-k}/n$) is better precisely since it (sort of) assumes that the various inferences for the existence of (some) god, and against atheism, are “shared” by the various possible divinities, much in the same way that Ken Ham and Harun Yahya use the same bogus arguments, and more devious theologians give “proofs” for some god, but not for any particular deity. Thus the simple invention of yet another god does not mean that the probability of atheism’s truth is automatically decreased.

If we assume that $p_G = 10^{-k}$ as in this endnote, we only need to follow that with “we assume $10^k$ gods”, and theism is immediately true!)

Finally: the sharp-eyed reader may have noticed the two separate introductions of the number $10^{-k}$ into the logic above. This is purposeful, of course: partly to get at the desired pro-atheistic conclusion (hey, at least I’m honest!), and partly to illustrate that if the ratio of mortal and divine outcomes is somehow “commeasurable” to the probability of some divine reality, Pascal’s argument is in trouble. (And if the theist counters with the “Heaven and Hell are of infinite worth and horror!” canard I don’t see any reason why the atheist could not answer with the equally absolutist “The probability of any god existing is zero, so fuck you too!” retort; things will proceed by their own weight from there.)

Different choices and simplifications will of course lead to different outcomes; but given the fatuity of the argument in the first place, this has been more a mathematical diversion than any serious atheological piece.

One Response to “Some notes on a problem of Pascal’s”

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