Monday, August 19, 2013

The Pearl of Great Price--Pascal's Wager Revisited* *

Again, the kingdom of heaven is like a merchant in search of fine pearls who, on finding one pearl of great value, sold all that he had and bought it.” (Matt 13:45,46, RSV).


Blaise Pascal, courtesy "Science Kids" ( )

Among the pile of  Pascal's papers that were to be the “Pensees” was found a proposition that has kept philosophers and theologians occupied for the last 350 years, Pascal's wager:  betting on God is the prudent option. (Notes, below, 1-8)   What new insights can one bring to this, then, after all this time?  I will try to understand the wager from a perspective of contemporary decision analysis, for which the wager was possibly the first instance, and also comment on what happens after one accepts the wager.

First, some background: it is important to keep in mind that although Pascal was a mathematician and physicist of the first order, he did not believe it was possible to show from reason alone that God exists (so much for Anselm and Aquinas!) :

“If there is a God, He is infinitely incomprehensible, since having neither parts nor limits, He has no affinity to us.  We are then incapable of knowing either what He is or if He is.”

On the other hand we can know God by faith:

“But by faith we know His existence;  in glory we shall know His nature.”

The last part of this quote shows the route Pascal wants us to follow: there is an afterlife, and its benefits are infinite.  This being so, the odds for following God are infinite; whatever one might lose in believing, even if there were no God, is finite, whereas that which one can gain from belief,  if there is a God, is infinite:

“But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite.”

Pascal spoke as a counselor of gamblers, for whom (with Fermat) he had developed the first quantitative version of probability analysis. It will be useful, before the wager is recast in a more quantitative format, to give some mundane examples.


In contemporary decision analysis one can proceed in two ways:

1) to examine possible gains and losses for various options, in the absence of known probabilities,  and to choose that option  which would correspond (psychologically or economically) to a preferred strategy:

2) to use known or estimated probabilities for various outcomes and to choose the option with the maximum expected value (see below).     Let's  first assume that  probabilities aren't known,  and see what considerations  might be involved in  choosing an option.   Here is the example:

Investing 10,000 units (dollars or ??) in
1) a savings account at 2% interest;
2) a conservative stock portfolio paying 6% in a good market, and losing 10% in a bad market;
3) a sure thing—an unreported diamond mine in Northern Scotland that your Uncle Angus has told you about—you'll double your investment.

The  table below summarizes the possible outcomes;  the columns represent “state of nature”, that is “good” outcome for a particular option and “bad” outcome ( a – sign means a loss), the rows, the different options.

If you're an optimist, you would of course choose the diamond mine.   If you are a pessimist or  risk-averse, you would choose the option with the least possible loss, the Savings Account (you would follow what is called the mini-max principle in decision analysis(9), choosing the option with minimum possible loss).

Now suppose Uncle Angus was right about the diamond mine—you'd berate yourself for not having invested in it.   This regret is quantified in a decision analysis scenario and used to justify a “mini-max regret” approach (10) for decision making.   For each state of nature (column) you subtract the best outcome to give a negative figure for “regret”.    You then list the worst (that is most negative) regret for each row (option) and choose that option with the least negative worst regret, as shown in the following table:

The option with with the least negative worst regret is the diamond mine, so if you were to follow a mini-max regret approach you would choose that option.  Clearly this is the restatement, in contemporary decision analytic terms, of Pascal's choice for belief, absent a known probability for the existence of God.     Put as a table one would have, symbolically:

There aren't numbers here, but clearly the value for belief in the existence of God (and the afterlife),  X, is much greater than Y (the loss -Y one sustains by belief) or Z (the gain of a possibly hedonistic life that one sustains by unbelief), so the minimum worst regret (least negative) is that for belief in God.

If probabilities for outcomes are known or can be estimated, another approach is to use expected values for each option and choose the option with the maximum expected value.   To get an expected value you multiply each outcome value by the probability for that outcome and sum these products for all the outcomes for a given option.

Pascal did not presume to give a probability for the existence of God and the afterlife.  However he relied on the infinite value of the outcome to give an infinite expected value—any number (however small as long as it's not zero) times infinity is infinity. And as long as the imputed loss is finite, the expected value will be infinite.   This assumption has raised the hackles of philosophers, and counterexamples—such as mixed strategies(2,10) and the “St. Petersburg Paradox”—have been proposed to show how the assumption of an infinite value outcome leads to problems.     In particular, suppose one follows the strategy of choosing the toss of a coin to decide  whether to believe.    The probability will be half that you will  choose to believe, so the expectation value will be infinite,  even though there will still be a probability of one-half  that you have chosen not to believe.    In my opinion these are valid objections, but they ignore the thrust of Pascal's argument, that the gain from belief is so large, that for any non-zero probability of an afterlife, the prudent person will believe.    The statement can be best  put in the forms of odds for the existence of God and an afterlife:

If the odds are greater than the possible loss to gain ratio, then  one should make the wager.    For example, if you believe that the odds for  Great Britain winning the World Cup are 2 to 3  and the bookmakers are giving 1 to 8 odds for Great Britain (win 8, lose 1),  you should bet for, and not against Great Britain.


Who are those who would not accept the wager?   According to Nicholas Rescher (1), the following:

1) the hard-core atheist (if you don't believe in God, you wouldn't believe in the possibility of an afterlife);

2) “the all-out hedonist” (Dr. Faustus?);

3) “the all-trusting disbeliever”, that is, one who believes everyone goes to heaven, that as in St. Teresa's prayer, Jesus will lead all souls to heaven, especially those most in need of his mercy;

4) “the radical skeptic” who disbelieves in all knowledge;

5) theists (e.g Buddhists, Hindus) who believe in God but have a different conception of the afterlife;

6) those who believe in an afterlife but in their evil, like Satan, would rather live in Hell than serve the Lord.

We emphasize again that the argument of Pascal's wager is addressed to the prudential man—the agnostic who believes in the possibility of an afterlife (and God)--and is willing to act so as to gain that reward, even in the midst of doubts.  Is belief then a matter of will?  The agnostic accepts the premise of the wager, but says

“ I am so made that I cannot believe. What, then, would you have me do?”

Pascal responds:

“Endeavor then to convince yourself, not by increase of proofs of God, but by the abatement of your passions.   You would like to attain faith and do not know the way;  you would like to cure yourself of unbelief, and ask the remedy for it...There are people... who are cured of an ill of which you would be cured.   Follow the way by which they began  by acting as if they believed, taking the holy water, having masses said, etc.”

Now can one “fake it until you make it” as Pascal suggests?  Or will the sacraments be ineffective, because the motive of the recipient is mercenary?  Which of the Catechism dicta are appropriate,

(1131)”The sacraments are efficacious signs of grace....They bear fruit in those who receive them WITH THE REQUIRED DISPOSITIONS.” (emphasis added)


(1128) “The sacrament is not wrought by the righteousness of either the celebrant or the recipient, BUT BY THE POWER OF GOD.”  (emphasis added).

The second suggests that if one prays for faith, then the “top-down” approach will work, starting from the head  and eventually through to the heart 15, or, as Pascal suggests:

“ each step you take on this road you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognize that you have wagered for something certain and infinite, for which you have given nothing.”
As a personal note, let me add that Pascal was right and I'll say more about this in a forthcoming blog, "Top Down to Jesus--Bypassing the Road to Damascus".

*This material has appeared, in somewhat different form, as a posting on the Magis Facebook site with a more detailed account of the probability analysis, and as an article in "FAITH" magazine


    1. Nicholas Rescher, Pascal's Wager—a study of practical reasoning in philosophical theology (Notre Dame: University of Notre Dame Press, 1985).
    2. A good summary on the web, with additional references (including other web sites) is given by Alan Hajek:
    3. Daniel Garber, What Happens after Pascal's Wager—living faith and rational belief (Milwaukee: Marquette University Press, 2009).
    4. Itzhtak Gilboa, Theory of Decision under Uncertainty (Cambridge: Cambridge University Press, 2009), pp. 38-40.
    5. Bas C. van Fraassen, Laws and Symmetry (Oxford: Clarendon Paperbacks, 1989), pp.52-53.
    6. James A. Connor, Pascal's Wager—the man who played dice with God (San Francisco: Harper and Collins, 2006).
    7. All quotations are from Blaise Pascal, Pensees W.E. Trotter (Mineola, NY: Dove Philosophical Classics, 2003), #233, pp. 65-69.
    8. Paul Bartha, “Taking stock of infinite value: Pascal's wager and relative utilities.” Synthese 154 (2007), 5-52.
    9. James O. Berger, Statistical Decision Theory and Bayesian Analysis (New York: Springer-Verlag, 1985) p.18
    10. ibid, pp. 377, 387.

Friday, August 2, 2013

The Theology of Water--Is Design Intelligent?

"The water that I shall give him will become in him a fountain of living water, welling up into eternal life. This is a new kind of water, a living, leaping water, welling up for those who are worthy. But why did Christ call the grace of the Spirit water? Because all things are dependent on water; plants and animals have their origin in water. Water comes down from heaven as rain, and although it is always the same in itself, it produces many different effects, one in the palm tree, another in the vine, and so on throughout the whole of creation. It does not come down, now as one thing, now as another, but while remaining essentially the same, it adapts itself to the needs of every creature that receives it."  
Quoted in the "Office of Readings" (Monday, Week 7 of Easter),  from a catechetical instruction by St. Cyril of Jerusalem.

Atomic structure of ice; O's represent oxygen atoms;
H's represent hydrogen atoms;  blue lines represent
chemical bonds;  red lines, hydrogen bonds.
The title of this post, "The Theology of Water", is taken from a short story by Hilbert Schenck in a collection of science-fiction stories with a religious theme, "Perpetual Light", which I read several months ago.   
In this story, after fruitless searches in the rest of the solar system, some middle-aged astronaut scientists explore Titan,  the largest moon of Saturn, to find life.     Titan is unique amongst solar system satellites in having an atmosphere, albeit a very cold one.
The scientists don't find life in any form, but they do find a strange type of water:  freezing and melting points much lower than "earth" water, but still with the unusual feature of solid water (ice) lighter than liquid at the freezing point, and with other differences in the thermodynamic properties.  The different properties are in fact those that would be suitable for life on this cold world, if life existed.   In testing the Titan water, the scientists turn it into earth-type water and realize that they are the life for which water is intended.

I dispute the essential scientific point of this story, that water at comparable temperatures and pressures would be different on Titan than on earth.    The properties of ice-- its relatively high melting point (compared to what one might expect doing a Periodic table comparison), it being lighter than liquid water--and the unusual thermodynamic properties of water can be  traced ultimately to fundamental bonding properties, specifically to the properties of the hydrogen bond  (see the illustration above), which in turn can be explained (in principle) by fundamental physics--quantum mechanics and electrostatics. 
Nevertheless, in telling the story, Schenck makes this important point: the properties of water  are tightly linked to the properties of the planet earth in order to provide an environment suitable for life (that is to say, carbon-based life as we know it).    Here are those properties (and I quote from the story--all temperatures are in degrees Centigrade--0 degrees Centigrade is the normal freezing point of water):
1)  liquid water has a maximum density at 4 degrees.   If it didn't (if the maximum density was at the freezing temperature), the cold water would sink to the bottom of the ocean and earth's  average surface temperature would be more than 20 degrees lower;
2) if the vapor pressure or the unusually high heat of vaporization of water is changed, either too much or not enough cloud would exist, which, in either case, would be a meteorological disaster;
3) if the density of ice is greater than that of liquid water at the freezing point (for most substances the density of the solid is greater than that of the melt), the ice would sink to the bottom of the oceans and the oceans would be perpetually frozen at the bottom, leading to massive winds at the surface;
4) if the high specific heat of liquid water is reduced, the temperature stabilizing effect of the ocean is lowered, and more storms and lower average temperature results;
5) the properties of water are optimized for the tilt of the earth's axis (23.5 degrees from the vertical)--if it were 0 degrees tilt, the temperature stabilizing effect would be too large, with complete cloud cover and ice-caps down to 40 degrees latitude
6) in the story, the properties of water are set for a mean earth temperature that is optimum for metabolism at 98.6 degrees Fahrenheit (and guess to what temperature that corresponds?)

Our biochemistry crucially involves the chemistry of water and hydrogen bonding.   The structure and reactions of proteins, enzymes, and DNA is critically dependent on hydrogen bonding, internally and to other biochemical molecules.    This blog isn't an appropriate context for even an abbreviated biochemistry lesson, but here are some web sites  about biochemistry and about the role of hydrogen bonding in DNA and proteins that will give some simple ideas to start.   

Biologists interested in alien life have considered biochemistries other than carbon-based/H2O.  (See the Wikipedia article on hypothetical types of biochemistry .)   Of these, one based on ammonia, NH3, seems most likely.    However the hydrogen bonds between ammonia molecules are only half as strong as those between water molecules.   Also, the temperature range for liquid ammonia is much lower than that for water, -78 to -33 degrees, so chemical reactions would proceed much more slowly, possibly too slowly for life-sustaining reactions.

So, the chemistry of hydrogen-bonding is one of those "finely-tuned" realities of nature that enable human life to exist.  We recall the Anthropic Principle, used to explain the fine-tuning of physical constants and cosmological facts (among which are the age of the universe and the unlikely existence of a large moon for our planet) that enables the existence of intelligent, carbon-based life.   I have not invoked the improbability of such fine-tuning, because probability, as a quantitative measure, is not properly applied to a single entity, and there is but one universe--we can know no other despite the speculations of metaphysical cosmologists.

How then do we justify the unlikelihood of such fine tuning, cosmological, physical and chemical?    And when I use the term unlikelihood, I'm not referring to the improbability of picking one white ball out of a bag of a zillion black balls.    Rather, I'm saying that we can think of all sorts of other universes, with different physical constants and laws, for which our type of life would not be possible.    Indeed, it is hard to imagine how any of the operative laws/constants might be nudged just a little bit and still allow for our kind of life.

Such fine tuning for hydrogen-bonding physics and chemistry should not, I believe, be tossed as another ingredient into the Intelligent Design" (ID) stew.     As I understand  ID, its principal tenet is opposing the Darwinian model for evolution (common descent).   Proponents of ID argue that gradual changes in form or biochemistry that might enhance survival (the cornerstone of the Darwinian survival-of-the-fittest program) are not sufficient to achieve the drastic differences in morphology and the "irreducible complexity" of various biochemical schemes.   

To my mind this is a "God of the gaps" type argument--to attribute that which we don't understand to specific divine intervention.   Moreover, a God who frames fundamental physics so that variety and complexity grows "naturally" from a unified beginning is much more to be admired and worshiped than a God who assembles, Leggo-like,  all the objects of a Young Earth (including evidence for a 4.5 billion year old earth and a 14 billion year old universe).    Paul Davies puts it very well:

“...the hypothesis of an intelligent designer applied to the laws of nature is far superior than the designer ...who violates the laws of nature from time to time by working miracles in evolutionary history. Design-by-laws is incomparably more intelligent than design-by-miracles.[emphasis added]”  (The Cosmic Jackpot: Why our universe is just right for life." p.200)
"Design-by-laws" (in Davies' felicitous phrase) is just how the anthropic principle can be interpreted.    Since a  full discussion of the anthropic principle would require a much lengthier blog, I'll defer that.   But I would like to end with one further comment.    This is a blog entitled "Reflections of a Catholic Scientist".    And, as a Catholic scientist, my God is much more than a creator, a demiurge who designed the universe engine and pressed the starter button.    My God is a Trinity,  a personal God, who intervenes from time to time in history, who sustains the laws of physics that make the universe-engine chug along, and who came to us in the person of His son, verified by historical revelation.    

About Me

My photo

Retired, cranky, old physicist.   Convert to Catholicism in 1995.   Trying to show that there is no contradiction between what science tells us about the world and our Catholic faith.   Intermittent blogs and adult education classes to achieve this end (see   and

Extraordinary Minister of Communion volunteer to federal prison and hospital; lector, EOMC.
Sometime player of bass clarinet, alto clarinet, clarinet, bass, tenor bowed psaltery for parish instrumental group and local folk group.

And, finally, my motivation:
“It is also necessary—may God grant it!—that in providing others with books to read I myself should make progress, and that in trying to answer their questions I myself should find what I am seeking.
Therefore at the command of God our Lord and with his help, I have undertaken not so much to discourse with authority on matters known to me as to know them better by discoursing devoutly of them.”
St. Augustine of Hippo, The Trinity I,8.