(Two notes before you read this: first, I am neither a mathematician nor a scientist, so please correct any errors on that front; and second, many of the ideas and examples for this post came from a fantastic Radio Lab episode called Stochasticity)

“A blade of grass is a commonplace on Earth; it would be a miracle on Mars.  Our descendants on Mars will know the value of a patch of green.  And if a blade of grass is priceless, what is the value of a human being?” – Carl Sagan, Pale Blue Dot: A Vision of the Human Future in Space

The Blade of Grass Paradox (A Modified Version)

Imagine yourself as a blade of grass in a football field.  A hot sun rains exhaustion upon you; the breeze pelts you with pain.  Suddenly you spot a tiny, dark cloud off in the distance—the chances of precipitation are infinitesimal, but how wonderful it would be! Sure enough, a few drops of water start to weave their way toward you: most evaporate, or blow far away, but then a miracle!  A blissful drop falls upon you, just you, and no one else!

From your perspective it seems that your prayers have been answered.  Just think of the chances: there must be tens of thousands of blades of grass in the field, yet somehow that drop chose you.  But let’s expand our perspective to the level of the entire field.  Once we do that, we see that the there was a 100% chance that the drop would fall on a blade grass. So here’s the situation: the blade views this as a miracle while the outside observer is rather unimpressed. For our purposes, we’ll call the observer’s apathy an intuitive understanding of the random workings of probability.  In fact, given enough time, it is a statistical certainty that a drop of water would land upon you, the blade of grass, at some point. Put another way, we have two worldviews, one that is defined by statistical certainty (one of the blades of grass will get wet), and the other defined by what you might call religion (only an external force could’ve steered that drop to me).

Flipping A Coin

Let’s consider another scenario.  You are asked to flip a coin 100 times and record the results.  Each time you flip a coin, the probability of getting heads or tails is 50%, so as you keep flipping things seem pretty ho-hum—a couple of heads in a row, some long sets of heads-then-tails, a couple of tails in a row—until something amazing happens: you get seven consecutive tails.  You do the math and find that the chances of this happening are ½ raised to the 7th power, which is less than 1% (.00781%).  At this vantage point, the outcome seems highly improbable.

But now let’s once again take another look, this time at the entire set of 100 coin tosses.  Run the numbers and you find that if you ask a different question, what is the probability that at some point during those 100 tosses you will get seven consecutive tails, you get a very different answer: about 16.5%.  Going back to those tosses, it’s worth noting one final peculiarity.  Put yourself back in the position of being on that lucky streak of tails.  You’ve tossed six tails in a row, and you are about to do the next toss.  We already know that the chances of getting seven tails in a row is less than 1%, but here’s what’s interesting: the chances of getting tails on that next throw is still 50%.

What’s Going On?

Just as with the blades of grass, how we interpret the event depends upon the question we ask.  Something can be certain (the rain drop will fall on a blade of grass), random (50% chance of getting tails on a particular flip), highly improbable (getting tails seven times in a row) or well-nigh miraculous (of all the blades of grass, the rain drop fell on me.) And here’s where we start to get to my main point: so much of the things we attribute to miracles or gods or good luck are really the results of probability and the extent to which the human brain is bad at processing randomness. 

We all fall victim to things like unrepresentative samples, such as when we judge pit bulls to be vicious animals based on one news report of a pit bull attacking a walker; what we ignore, of course, are the myriad instances where a pit bull does not attack a walker.  We focus on the most emotionally charged outlier at the expense of the more banal average.  This often happens to me at work: I obsess over the one person to whom we lent money that never made a payment, and ignore the 95% of people that pay back our loans.

So What?

If we take the lessons of statistics and apply them to some of the most amazing facets of the universe—that Earth is suited to human life, that human civilization is so advanced, that the human brain has the capacity of self-understanding—we find that those facets, while still amazing, cease to seem miraculous.  In fact, given the age of the universe and the number of planets in that universe, it would be astonishing if at some point and some place in the cosmos life had not come into existence.  Allow enough time to pass and combine it with enough variables—carbon, iron, genes, atmospheric changes—and all of the aforementioned facets become, well, inevitable.

And Where’s The Magic?

Inevitable is not incompatible with wonderful, beautiful, magical or awe-inspiring; neither is ‘probable,’ or ‘it has a 16.5% chance of happening.’ The movements of the stars, the direction of raindrops, the flipping of coins, the countless coincidences that lead us to where we are today can all be explained by randomness and chance.  Yet randomness and chance can only operate—can only exist—within a cosmos consisting of certain laws (the vast majority of which we do not understand), certain elements, certain properties.  Just as it is nearly certain that life will eventually come about somewhere in the universe, it can also be argued that, with a different set of events, a universe in which randomness doesn’t exist could also come about. 

To make life even more delightful, it seems that the more macro our perspective becomes, the less we know about the fundamental operations of the system about which we want to make mathematical claims.  After all, it’s one thing to describe coin flips, but another one entirely to uncover and explain the deepest mysteries of the universe (as a case in point, dark matter—which scientists barely understand—constitutes “84.5% of the total matter in the universe and 26.8% of the total content of the universe.”).  This is not to say that we won’t eventually come to fully understand the laws and workings of nature, but it is important to keep in mind that our lack of understanding of certain ‘cosmic conundrums’ does not negate our complete understandings of other aspects of life.  We may not yet have a unified field theory in physics, but we do understand gravity, natural selection, general and special relativity, quantum mechanics, plate tectonics, and so on.

Conclusions, Arguments and Uncertainties

Thus it seems to me that we are left with the conclusion that religion is probability, and probability is religion.  Wherever we understand the laws surrounding an event we can use statistics to understand the likelihood of that event.  And wherever we do not understand the laws, room must be left for non-statistical explanations.  Here’s a less esoteric example: we know that the human brain ceases to function after death because medical science can irrefutably demonstrate that where there is no electrical function, there is no life as we understand it.  At the same time, because we don’t know what a ‘soul’ is—we can neither prove nor disprove its existence—we can’t say for sure whether a soul survives beyond death. 

My argument is not that math explains everything, or that religion is absurd, but rather that the two, properly viewed, are two sides of the same coin (I couldn’t resist the pun).  No one perspective can be fully correct—for the fundamental lesson of general relativity is that two people moving through space at different speeds experience time differently, and both are ‘right.’ So perhaps no one law can fully explain the totality of what we know as life.  Just as what I weigh on Earth is not what I weigh on Mars, and just as the likelihood of flipping 7 consecutive tails depends on what question I ask about those flips, perhaps there are other moments in time or pockets of the universe where the very laws of physics do not apply.  Of course, there may be a law that can account for both the places where the law applies as well as where it doesn’t—indeed, quantum mechanics and relativity already do so, and the search for a unified field theory is about the quest for that very law.

I think that, at the end of the day, we should be less astounded by occurrences that math reveals to be banal, and more astounded by the fact that math can reveal so much about the world around us, as well as by the fact that there is so much it has yet to explain.

Other Reading

Finding The Higgs Leads to More Puzzles (NY Times)

Radio Lab – Stochasticity