Pascal’s Wager

Rene DesCartes and Blaise Pascal lived in France about 400 years ago. They were colleagues and they were both mathematicians. Also, they both tried to prove the existence of God.

Rene DesCartes tried to prove it directly, arguing that nothing can exist without God. But his attempt failed; he only got as far as proving that you can’t ask questions if you don’t exist. This is usually summarized as Cogito Ergo Sum, which I discussed in my last post. If you ask me, his attempt was doomed from the start; it’s just too long a chain of ideas and each link in the chain can break very easily.

In order to justify a claim like “You need to believe in Jesus in order to get into Heaven”, you’d have to show all eight of the following: #1 There’s a law which says everything has to come from somewhere, e.g. if you see a shoe, there must have been a shoemaker (Let’s call this the Shoemaker Law). #2 The Shoemaker Law applies to the universe itself. #3 The Shoemaker Law does not apply to whatever created the universe. And whatever created the universe… #4 is able observe what happens as the universe unfolds, #5 cares deeply about the behavior of the creatures which inhabit the universe, and #6 has a plan for rewarding or punishing those creatures based on their behavior. Also, #7 You know what the rewards and punishments are. Finally, #8 You know specifically which behaviors are the ones to be rewarded and which ones are to be punished.

Cogito Ergo Sum doesn’t prove any of the links in that chain, let alone all of them.

In modern times, others have tried to build on DesCartes’s work by making the dubious claim that the Shoemaker Law applies not only to the universe, but to knowledge and logic itself. Their argument is basically “I think, therefore logic is real, therefore God exists”. At best, this allows them to bypass the first three links in the chain I described above. But they conveniently ignore the fact that it doesn’t even address the other five links. You often find such arguments under headings like “Christian Apologetics” or “Presuppositionists”.

Blaise Pascal took a different approach. He fell back on his formula for the Expected Value, which I discussed quite a bit in my three posts about playing the lottery. Here’s the logic which Pascal laid out.

Given the fact that (as we all know) believing in Jesus is what gets you into Heaven, and given the fact that Heaven is an infinite reward, and given the fact that the alternative is Hell, which is an infinite punishment, we can calculate the Expected Value for believing in Jesus. The formula will require some unknown quantities, but as you’ll see in a minute, their precise values don’t change the outcome. First, we need the probability that God exists. Let’s call that “g”. Like all probabilities, this is a number between zero and one. Because you can’t be 100% certain that God does not exist, that means g > 0. It might be 22% or 0.00000004% or it might be 0.000000000000000000001% but whatever it is, it’s not zero. Next, we need to ask the question what does it cost you to believe in a god that doesn’t exist. Let’s call this “c”. Pascal claimed that c was zero, but it still works if c is some other number, as long as c is finite.

EV for believing in Jesus = (g) x (infinite reward) –  (1-g) x (c)

Notice that, if g > 0 and c is finite, this result is always infinity, regardless of the specific values for c and g. Now consider the Expected Value for not believing in Jesus. For this calculation, we need one more number, the value of not believing in Jesus in a world where there is no god. Let’s call this “a”.

EV for not believing in Jesus = (1-g) x (a) – (g) x (infinite punishment)

Notice that, if g > 0 and a is finite, the result is always negative infinity.

Pascal’s conclusion from this is that, no matter how unlikely you think God’s existence might be, whether it’s 50% or 2% or 0.000000000000000001%, it doesn’t matter. When you multiply that probability times the infinite reward of going to Heaven, it’s always a safe bet for you to believe in Jesus.

There are so many flaws with this argument that there’s an entire page on Wikipedia devoted to explaining Pascal’s Wager and its flaws. I’m not going to try to repeat them all. I’ll just point out three which I thought of on my own.

Flaw #1: It uses circular logic.

The whole point of Pascal’s Wager is to try to decide if God exists and what you should do about it. The argument admits the possibility that God might not exist at all. Yet the argument is founded on the assumption that believing in Jesus gets you into Heaven and that Heaven is an infinite reward. If there is no god, then this assumption isn’t true at all. He started his proof for knowledge about God by assuming we have knowledge about God. That’s circular logic.

Flaw #2: It ignores alternatives (such as other religions).

Even if God does exist, that still wouldn’t prove that Heaven is real, or that Heaven is an infinite reward, or that believing in Jesus is what gets you into Heaven. Muslims believe that submission to the will of God is what gets you into Heaven, not belief in Jesus. Some religions believe that God has already decided whether you will get into Heaven or not and nothing you ever do has the power to change that decision. And Pascal conveniently ignored the possibility that there might be more than one god, and perhaps even different heavens. Then there’s one of my very favorite alternatives which I found on youtube: Keight’s Wager (“Keight” is pronounced like “eight”). Put yourself in God’s shoes for a minute. You’ve just created a universe. You’re lonely. You want to invite some people to join you in Heaven. What kind of people would you, God, want to hang out with? It’s easy to imagine that God is really into science. So maybe God would only invite into Heaven people who embrace the scientific method. Now, considering that there’s an amazing lack of evidence proving God’s existence, the only rational conclusion for a scientific-minded person to make is that God does not exist. Therefore, the perfect candidate for who God wants to invite into Heaven is…. an atheist! So, if you want the infinite reward of going to Heaven, your best strategy is to be an atheist. I’m not saying I actually believe Keight’s Wager. I’m just pointing out that Pascal’s Wager rests on unproven assumptions.

Flaw #3: The exact same logic leads to conclusions which are obviously wrong.

Suppose I show up at your door selling a bottle of water which came from the Fountain of Youth. Given the fact that (as we all know) drinking water from the Fountain of Youth bestows upon you the gift of immortality, and immortality is an infinite reward, let’s calculate the Expected Value for purchasing this bottle. We need the probability that I’m telling the truth about the water. Let’s call it “t”. You can’t be 100% sure that I’m lying, so t > 0. I didn’t specified the asking price for the bottle of water; let’s call it “p”.

EV = (t) x (infinite reward) – (1-t) x (p)

As long as p is a finite number and t is not zero, this formula always comes out to infinity. Therefore, you should definitely buy the bottle of water from me, no matter how much money I ask from you, and no matter how slim the chance is that I might be telling the truth. If the reward is infinite, then your only logical course of action is to hand over all your money.

Clearly, this is wrong-headed. Only a fool would hand over all their money to a stranger selling bottles of “magic” water. To suggest that logic demands that this must be the best course of action is just ridiculous.

.  .  .  .  .

Remember in my last post when I said that the Expected Value formula doesn’t work very well when you use very large numbers? Well, here’s a case where Pascal tried to apply the formula to INFINITE numbers, and it failed miserably. Frankly, he should have known better. But he was desperate. He knew deep down that he was 99.9% convinced that God doesn’t exist, but he badly wanted to keep clinging to some tiny scrap of hope. He couldn’t face the idea if giving up his belief. So he slapped together this appalling collection of bad logic and said he would keep on believing in Jesus anyway.

I can sympathize with Pascal’s situation. I struggled for years before I could finally give up my belief. After holding on to it for such a long time, it was very difficult to let go. I was a believer from childhood up until my early thirties.

I think that if I had started questioning my beliefs in my fifties or sixties, it would have been even harder to let go. I’m not sure if I would have been able to do it.

In conclusion… if you’re a believer who wants to try to bring me back into the light and you think to yourself “Hey! I know what to say to an Atheist. I’ll say what if you’re wrong? That’ll get him”… don’t even bother. I have spent way more time asking myself that very question than you ever will.


You can’t know anything with 100% certainty.

When people say “It’s impossible to know with 100% certainty that there is or isn’t a god.”, I respond that it’s impossible to know with 100% certainty ANYTHING. If “100% certainty” is your benchmark, then nobody knows anything about anything and we can all just give up on ever trying to find any knowledge at all. Obviously, in the real world, we have to make a judgment call and say “In this situation, 99% certainty is good enough for me to make a decision.” or maybe it’s 95%, or 99.999%, depending on the situation. It seems to me an awful lot of time gets wasted quibbling over whether someone who is 99.7% sure the aren’t any gods should be called an atheist or an agnostic.

Even scientific facts (like “water freezes at 32 degrees Fahrenheit”) are subject to change when more evidence comes in. For example, as recent as 20 years ago, it was considered a “fact” that male pattern baldness was caused by a sex-linked gene on the X chromosome. Now we know that the original study which made that determination was faulty. The “fact” that you inherit baldness exclusively from your mother’s side of the family turns out to be simply not true.

Even the example of water freezing at 32 degrees F isn’t 100% true. The truth is more complicated than that, depending not just on temperature but also pressure. The so-called triple point of water happens at .01 degrees C and 611.73 Pa (roughly .006 atmospheres). Around 2,000 atmospheres, water can remain liquid all the way down to zero Fahrenheit. Read the wikipedia article about “ice”.

Heck, even in an ordinary real-life setting, if you put a bowl of water outside and the meteorologist on the radio tells you that it’s 31 degrees outside, can you be 100% sure that the water will freeze? Of course not. The weather report could be mistaken. The water could have trace amounts of salt in it, which changes the freezing point. The bowl might be in direct sunlight, preventing it from freezing.

And in the bigger picture, the only reason that you think you know that water freezes at 32 degrees F is that you remember having been told this fact by other people. But you can’t be 100% certain that your memory is accurate. People forget things all the time and make mistakes. Maybe the correct number is 23 and not 32 but you have some combination of Alzheimer’s disease and Dyslexia. Sure, the chances of that being true are very very slim but it’s not zero.

Beyond faulty memory, there’s also the possibility that you are not who you think you are at all and everything you think you remember about your past is actually an elaborate hallucination. You could be lying in a hospital bed, in a coma, on some distant planet, dreaming that you’re an Earthling, and all the so-called facts you think you learned on Earth are just figments of your imagination. Sure this idea seems far-fetched, but you can never be 100% certain that it isn’t true.

I’m not saying that facts don’t exist, or that nothing is true. I’m just saying that, as a human being, our knowledge of the facts is never 100% certain.

The only fact I can think of that might come close to being 100% certain is Rene DesCarte’s Cogito Ergo Sum, “I think therefore I am”. But even that statement is very limited. It only applies to the person who is doing the thinking. And it doesn’t really explain what it means to exist. If I’m part of a simulation, living inside a computer, is it fair to say that I “am”? Cogito Ergo Sum doesn’t even prove that your brain has any physical substance, let alone the body which you believe contains your brain. It also doesn’t explain what I am. It just says that I am. And I’m still not entirely convinced that it’s 100% certain. Maybe there’s a flaw in the logic that we haven’t discovered yet.

But most of the time, in day-to-day life, it’s pointless to worry about this stuff. All you need is to be convinced that it’s probably okay and the risks are small. Could a speeding car kill you? That’s a sizable risk, so it’s prudent to take precautions like staying on the sidewalk and waiting for the signal at the crosswalk and looking both ways before crossing the street. But it would be overreacting to never leave your house just because you can’t be 100% sure that a car won’t drive up onto the sidewalk and kill you. There are no guarantees in life. Just accept the fact that, sooner or later, everybody dies, and make the best judgment calls you can in each situation. If you spend your life terrified of death, you miss your chance to enjoy the life you have.

Playing the Lottery, part 3

Now that we’ve discussed Pascal’s calculations of “expected value” and the various shenanigans the Lottery uses to make the prize look bigger than it really is, let’s talk about the end results of being a so-called winner.

First, ask yourself what money is good for. Seriously. There are things you can get if you have money which you can’t get if you don’t have money. For example, if you have $2 you can get a hamburger, which can mean a lot if you’re very hungry. If you have just $10 you can get a good meal at a restaurant. If you have $50 you can spend the night in a hotel room instead of sleeping on the street. If you have $1,000 you can rent an apartment for an entire month. With $10,000 you could buy a nice used car or an older motorhome. With $100,000 you could buy a small house or a really nice motorhome. With $1,000,000 you could buy a really nice house plus have enough money left over to buy food for yourself for several years. Each of these examples demonstrates the utility of money. With it, you can get something you need or want, without it you can’t. This might affect your happiness level (or might not) but it certainly can affect your health and your safety.

But how much utility can you get from $2,000,000 compared to $1,000,000? You can buy a house that’s twice as big. But will that make you twice as happy? Will it make you twice as safe? Will it make you twice as healthy? Certainly not. The difference between struggling to find food and shelter vs. having a nice house full of food is about $4,000 per month. Beyond that, having more and more money only adds a tiny amount to the list of things you can do and how healthy/safe/happy you’ll be.

Of course, if you got $10,000,000 you could give most of it away to other people. Then both you and 9 of your friends could each have a house full of food. But my point remains that $10 million in the hands of one person is not 10x better than $1 million.

Consider the following game. I’ll put 30 six-sided dice into a shoebox and shake it up. You buy a ticket from me for $1,000 and then we open the box. If all 30 of the dice have landed on 6, I pay you $1,000,000,000,000,000,000,000,000,000. That’s one billion billion billion dollars, also known as an octillion. It’s many many times all the money on Earth right now. If you had that much money, it would literally be impossible for you to spend it all because there simply isn’t enough stuff on Earth for you to buy. Let’s calculate the expected value for this game. The chance of rolling 30 sixes is 4.52337×10^-24, which is .00000000000000000000000452337 .

EV = (4.52337×10^-24)x($1 octillion) – (1-4.52337×10^-24)x($1,000) = +$3,523.37

As you may remember, any positive expected value at all means that the game is tilted in favor of the player and it’s a “good” bet. In this case, you are risking $1,000 and expect to make a profit of $3,523.37 which is a fabulous return on your investment. Would you do it? Would you actually pay me $1,000 for a ticket to play this game?

I submit that, even if you believed that I’d be able to follow through on the promise of paying out if you win the bet, it still would be foolish for you to spend $1,000 on a ticket. The amount of happiness you’d get from winning simply isn’t worth what you give up by having to pay me $1,000.

This demonstrates a fundamental flaw in the Expected Value formula. It assumes that getting 100x as much money has 100x as much utility or will make you 100x as happy, and that’s simply not true. The larger the numbers involved, the less useful the formula becomes.

However, there’s a positive result from buying a lottery ticket which has nothing to do with winning. Just buying a ticket gives you a chance to dream about changing your life. If the amount of happiness you get from dreaming about becoming a millionaire makes you happier than keeping the cost of that ticket, then it might be money well spent.