On God and Probability
Blaise Pascal (1624-1662), of triangular fame , was an Enlightenment mathematician and scientist who, like Isaac Newton, took his studies of theology as seriously as he took his studies of the natural world.
Pascal’s most famous contribution to theology (and decision theory) came in one of his posthumous works. In a work titled Thoughts, he described what is now known as “Pascal’s wager” and asserted that it is impossible to prove or disprove the existence of God through human reason. Thus, humans had to “wager” (i.e., make a bet in an uncertain situation) on God’s existence and live out the consequences of their wager.
If you bet that God does not exist and then decide to live hedonistically, then, if God does exist, the cost to you is eternal damnation. But if you bet that God exists and consequently decide to live a sinless life (i.e., a life governed by fidelity to religious teaching), then, if God does not exist, the cost to you is that you “wasted” your life being so honorable. Pascal then concludes that there is good reason to live a sinless life because the probability weighted costs of not living such a life (which are infinitely negative) outweigh the probability weighted costs of doing so [1].
It is a simple enough argument, although the typical Western myopia from the Enlightenment period hobbles it. In human civilization, there have been many more “Gods” than the singular Judeo-Christian one that is Pascal’s focus. To which one should the wager be applied? And which particular interpretation of God’s religious inscriptions should one take as definitive of a sinless life?
Still, the wager is useful in framing how people could consider their actions in the face of seemingly infinite rewards or costs. And extensions to this wage can help us understand how two groups of people could look at the same set of facts and come to different conclusions about sensible subsequent actions.
Climate Context
In 1992, David Orr [2] extended the fundamentals of this wager to climate change. His goal was to provide a framework for understanding how proponents and opponents of climate change policy should evaluate the costs and benefits of mitigation.
I first encountered Orr’s argument in Naomi Oreskes’s Why Trust Science? where she gives it a mathematical framing. Here is a paraphrase of that framing:
Say that the probability of climate change being real is \(p\) and the probability of it not being real is \(1-p\). Say also that humans have the choice between an aggressive climate change policy and no climate change policy. If there is an aggressive policy and climate change is real then, then humanity experiences low climate-change damage \(E\) and some costs \(C\) from their mitigation efforts. If there is an aggressive policy and climate change is not real, then humanity experiences only the mitigation costs \(C\). If there is no policy and climate change is real, then humanity experiences irreversibly high damage \(V\). Finally, if there is no policy and climate change is not real, then humanity experiences neither mitigation costs nor damage.
With this information, we can create a \(2\times2\) matrix of possible costs. Each element is the cost associated with \((X, Y)\) where \(X\) is either an aggressive climate change policy or no climate change policy, and \(Y\)is climate change being real or climate change not being real.
| Climate change is real (\(p \)) | Climate change is not real (\(1-p \)) | |
|---|---|---|
| Aggressive climate policy | Low damage (\(E \)) + Mitigation costs (\(C \)) | Mitigation costs (\(C \)) |
| No climate policy | Irreversibly high damage (\(V \)) | No Damage and No Costs (\(0 \)) |
From this table, we can determine how likely \(p\) climate change needs to be in order for the aggressive climate policy to be beneficial. To find this limiting probability \(p\) we need to compare the expected costs of the aggressive policy with the expected costs of no policy.
The expected value of the costs of an aggressive policy is
and the expected value of the costs of no climate change policy is
In order for the aggressive policy to be worth it, we need \(\langle \text{costs} \rangle_{\text{policy}} < \langle \text{costs} \rangle_{\text{no policy}}\). We want to find the \(p\) that makes this inequality true. Setting the two computed costs equal to each other and then solving for \(p\), we find that this probability must satisfy
$$ \hspace{2cm} p > {C \over V - E}.\hspace{2cm} \qquad (3) $$Or, bounding the results by \(0\) and \(1\),
$$ \hspace{2cm} p_{\text{min}} = \max\left(0, \min\left(1, {C \over V - E}\right)\right) \hspace{2cm} \qquad (4) $$where \(p_\text{min}\) is the minimum probability for climate change that is needed for aggressive climate policy to be beneficial.
Evaluation of evidence
One way to interpret Eq. (4) is as a limit on the probability required for an aggressive policy to be beneficial. Another way is as a proxy for how much evidence is needed in order to enact such a policy.
If a future event is binary valued (either it happens or it doesn’t), and we are completely ignorant of the probability of occurrence, then we can take the probability of the event to be \(p= 1/2\). As we gather evidence that the event will happen, \(p\) increases. If we gather an infinite amount of evidence that this event will happen, then we find \(p = 1\). On the other side, if we gather an infinite amount of evidence that this event will not happen, then we would find \(p = 0\).
Therefore, the closer \(p_\text{min}\) is to 1, the more evidence is needed to push the probability from complete ignorance to the minimum value needed for climate policy to be worth it. If the \(p_\text{min}=1\), then no finite amount of evidence could convince someone to act, whereas if \(p_\text{min}=0\) then even a large amount of evidence against the event could still encourage one to act.
The issue then in applying Pascal’s wager to Climate policy is that different groups of people have different values for \(E\), \(C\), and \(V\) and thus disagree on the amount of evidence needed to view climate policy as valuable. We can coarsely divide the two groups of people into “Environment Focused” people and “Economics Focused” people.
- Environment Focused: Many people believe that reduced fossil fuel use, cleaner air, and less animal extinction are all positives that far outweigh the reduction in economic production or the loss of jobs that would result from climate change policy. In such a framing, the mitigation costs \((C)\) of climate policy are actually negative, meaning that regardless of the momentary costs, there is a net benefit to enacting climate policy \(V \gg E + C\).
In this case, Eq.(4) would give us \(p_{\text{min}}=0\), which suggests that even if climate change was not happening, there would still be value in implementing climate change policy.
- Economics Focused: on the other side, there are people who believe that any aggressive climate policy would be economically disastrous (i.e., yielding a high and positive \(C\)) and that there is likely little that humans can do to affect global climate. Thus, they estimate that the damage (if any) the earth would experience would be essentially the same whether we implement climate change policy or not (i.e., \( V \approx E\)).
In this second case, Eq.(4) gives us \(p_{\text{min}} \approx 1\), meaning that these skeptical folks would need much more evidence to decide to implement climate change policy. This is perhaps why any disagreement in climate science is reason enough for some people to be hesitant in implementing any policy changes. If you need a probability of \(99\%\) in order to act, and only \(97\%\) of experts believe climate change is happening, you would say we don’t have enough information.
Post-Modernism and Evidence
This model makes clear that the debate about climate change has less to do with scientific evidence and more to do with a priori values that lead people to interpret scientific evidence in very specific ways.
If you care deeply about ecology and you are already partial to the arguments that humans can change (and are changing) the environment in deeply negative ways, then not acting on climate change seems stupid.
But if you care less about an abstract ecological disaster, whose most pernicious effects you likely won’t experience in your lifetime, then economic arguments are more important to you, and it would seem wasteful to act when evidence is “insufficient.”
This framing applies to things well beyond climate change and can help people understand why multiple groups of people could look at the same set of information and interpret it in wildly different ways. The fact that there are differing interpretations itself is perhaps the most important piece. Post-modernism, constantly lambasted by political sectors as responsible for the moral and intellectual deterioration of 21st-century academia, is valuable here for it makes clear that even if there is a so-called objective reality, such a reality must be filtered through human lenses, which are inevitably riddled with at worst bias and at best very specific values.
Failing to acknowledge this background of values does not make them go away. It just makes one’s ideological opponents harder to understand.
Footnotes
[1] This claim of minimal cost only works on an individual level. Communities or nations of people do not simply practice their faith by living self-contained, honorable lives. Instead, there is often a proselytizing fury (usually with militaristic intent) that goes with a nation believing in a God, and certainly a religious crusade would be cast as actively harmful (or more harmful than it already is) in the case where God does not exist. But I digress.
[2] Orr, David W. "Pascal's wager and conomics in a hotter time." Ecological Economics 6.1 (1992): 1-6.
[3] Another complicating aspect is that costs are felt immediately, but benefits are deferred both in time and to other people. In the case of an aggressive climate policy, a current generation takes on the costs \(C\) of the policy, but the world-saving benefits are largely experienced by a later generation, primarily in the form of avoiding the irreversibly high damage \(V\) that would result from inaction. This makes a single individual's sense of the costs even more abstract: In the moment, there really is no point in comparing \(E\) and \(V\). Cynically, they are both zero because they will be experienced by someone else.