Comparing Apples and Oranges

You know: the very point of saying “it’s like comparing apples and oranges” is that it’s difficult, maybe even impossible, to do so, because —well —because they’re just not the same.  Consider this picture:

 

Forty-nine apples and one orange.   If I put all this fruit in a bag, mix it up and pull one piece out at random, the odds will be 49 to 1 that I’ll pull out an apple.  That is, 49 to 1 against the orange.

Now a question for you: Assuming a random draw, will I be surprised if I pull out an apple?  Answer: no, I won’t.  I fully expect to pull out an apple, due to the odds.  I assume you wouldn’t be surprised either.  I also assume we’d both be surprised if I pulled out the orange, for the same reason.  Am I right?

Now,  I feel as I do without qualification — by which I mean, for example, that if I pick out the orange, my surprise won’t be greater or less depending on whether the orange weighs nine ounces or ten, and I won’t be surprised if I pull an apple from the bag, regardless of the number of leaves on its stem.   The fact is I expect an apple, and as long as I get an apple, I’ll have no cause for surprise.  Right?

But now another question, and this one’s a little harder. What are the odds of my picking out an apple with two leaflets on its stem?  You can scroll back and look at the picture if you want, but try to answer the question without doing so: what are the odds of my picking an apple with two leaflets on its stem?

Ready?

Alright. Hard, wasn’t it?  If you went back to look at the picture, you found there was only one apple with two leaflets on its stem. Knowing that, you determined that the odds against my picking that particular apple were 49:1, the same odds as existed against my picking the orange.  Yet it’s pretty clear, as already determined, I would have been surprised if I’d picked the orange, but I wouldn’t have been surprised if I’d picked the only apple in the bag with two leaflets on its stem.

My real question, then, is why the difference?  And the only answer that makes sense to me comes not from probability theory, but from psychology.  I’m surprised if I draw the orange because, being mindful of the differences between the orange and the apples, I expected an apple. But not being mindful of the uniqueness of the two-leafed apple, I lumped all the apples together and treated them as if they were all the same.  I focused on the fact that the odds against the orange were 49:1, while never forming a similar expectation about the improbability of choosing the two-leafed apple.

Here, then, is my conclusion:  In pulling fruit from the bag, the actual improbability of every single piece of fruit is the same. Yet the perceived improbability of choosing the orange is far greater than the perceived improbability of drawing the two-leafed apple, because… well… because I hadn’t been paying attention to the differences among the apples.

Also, the division of the 50 pieces of fruit into only two categories – apples and oranges – was a subjective choice.  I could have grouped the fruit into large and small, or into three groups based on relative sweetness.  Or according to the number of leaves on the stem, in which case the orange would have been in a group with twenty apples.

Now, in any group of 50 pieces of fruit, no two are going to be exactly alike – the two-leafedness of one will be matched by the graininess of another, the seed count of a third, the sweetness of a fourth, and so on.  But we elect to ignore (or de-emphasize) a whole slew of possible differences, in order to focus on one or two traits.  Only by ignoring (or at least de-emphasizing) other differences do we construct a homogeneous group, treating all 49 of the red fruits the same for purposes of comparison to the orange one — treating them all as “apples” rather than one or two McIntosh, one or two sweet ones, etc.  That’s why I’m not surprised when I pick out that one, unique apple, despite the 49:1 odds against it.

Now consider a related point: that (subjective) decision about what criteria to base comparisons on, while ignoring other criteria, not only explains why we’re surprised if we select the orange, but how we estimate odds in the first place.  In fact, if we consider all their attributes, every piece of fruit is unique. The odds against picking any one are 49:1.  Yet, if we only focus on the uniqueness of the orange, our impression of odds will be vastly different than if we focus on fruit size, or sweetness, or seed count.

It isn’t some sort of unalterable constant of nature that determines how we perceive odds – it’s what we’re mindful of, and our resulting (subjective) expectations.

In an earlier post, Goldilocks and the Case Against Reality, I wrote of the concept that the limited focus which characterizes our brains has been useful to us.  (If I could see every part of the electro-magnetic spectrum, I’d be overwhelmed with information overload, so I’m advantaged by only being able to see visible light.)  My brain is just too small and slow to deal with all the information out there.  Even if I’d happened to notice there was only one two-leafed apple, I could never have taken the time to absorb all the differences among the forty-nine apples.  Compare that, say, to the difficulty of absorbing the different facial features of every person on this tiny, one-among-trillions planet.  I cope with reality by ignoring vast complexities of things I don’t understand, lumping a lot of very special things into groups for the very reason I can’t get my brain to focus on all their differences.

Now, this lesson about comparing apples and oranges teaches me something about God, and I hope you’ll give me a chance to explain.

The astronomer Fred Hoyle is said to have written, “The probability of life originating on Earth is no greater than the chance that a hurricane, sweeping through a scrapyard, would have the luck to assemble a Boeing 747.”  Hoyle apparently used the improbability of life as an argument for the theory of intelligent design. Hoyle’s statement was then quoted in The God Delusion (Houghton Mifflin, 2006), by the atheist Richard Dawkins, who said that the “improbability” of life is readily explained by Darwinian evolution, and declared, “The argument from improbability, properly deployed, comes close to proving that God does not exist.”

Now, whether either of these beliefs makes sense to me, I’ll leave for another day.  My focus is on trying to understand any argument based the “improbability” of life, and its because of what I’ve learned from the fruit.

I agree that the odds are against a hurricane assembling a 747, and against life’s existence exactly as it is today.  But is my surprised reaction to such improbabilities any different than my surprise at the random drawing of an orange, but not at the two-leafed apple?  Imagine, for a moment, that some other configuration of scrap parts had been left in the hurricane’s wake – one that appeared entirely “random” to me.  Upon careful inspection, I find that a piece of thread from the pilot’s seat lies in the precise middle of what was once the scrap heap.  A broken altimeter lies 2 meters NNE of there.  The knob of the co-pilot’s throttle abuts a palm frond 14.83 inches from that.  The three hinges of the luggage compartment door have formed an acute triangle, which (against all odds) points precisely north; the latch from the first class lavatory door is perched atop the life jacket from Seat 27-B….

I trust you get the picture.  Complex?  Yes.  Unique?  Yes.  So I ask, what are the odds the triangle of hinges would point exactly north?  The odds against that alone seem high, and if we consider the odds against every other location and angle, once all the pieces of scrap have been located, what are the odds that every single one of them would have ended up in precisely the configuration they did?

In retrospect, was it just the assembly of the 747 that was wildly against the odds?  It seems to me that every unique configuration of parts is improbable, and astronomically so.  Among a nearly infinite set of possible outcomes, any specific arrangement ought to surprise me, no?  Yet I’m only surprised at the assembly of the 747.  What I expect to see in the aftermath of the hurricane is a helter-skelter mess, and I’m only surprised when I don’t.

But on what do I base my expectation of seeing “a helter-skelter mess?” Indeed, what IS a “helter-skelter mess”?  Doesn’t that term really mean “all those unique and unlikely arrangements I lump together because, like the apples, I’m unmindful of the differences between them, unmindful of the reasons for those differences, ignorant of how and why they came to be as they are?”

Suppose, instead, that with the help of a new Super-Brain, I could not only understand all the relevant principles of physics, all the relevant data – the location, size, shape and weight of every piece of scrap in the heap before the storm — and suppose further that when the storm came, I understood the force and direction of every molecule in the air, etc.  With all that data, wouldn’t I be able to predict exactly where the pieces of scrap would end up?  In that case, would any configuration seem improbable to me?  I suggest the answer is no.  There’d be one configuration I’d see as certain, and the others would all be patently impossible.

Compare it to a deck of cards.  We can speak of the odds against dealing a certain hand because the arrangement of cards in the shuffled deck is unknown to us.  Once the cards have been dealt, I can tell you with certainty what the odds were that they’d be dealt as they were: it was a certainty, given the order they had previously taken in the deck.  And if I’d known the precise arrangement of the cards in the deck before it was dealt, I could say, with certainty, how they would be dealt.  Perfect hindsight and foreknowledge are alike in that neither admit of probabilities; in each case — in a state of complete understanding — there are only certainties and impossibilities. The shuffling of a deck of cards doesn’t mean that any deal of particular cards is possible, it means that we, the subjective observers, are now ignorant of the arrangement that has resulted. The very concepts of luck, probability and improbability are constructs of our limited brains.  Assessments of probability have developed as helpful ways for human beings to cope, because we live in a world of unknowns.

Now, let’s return to the scrap heap, one more time.  But this time, we don’t have an all-knowing Super-Brain.  This time, we’re just a couple of ants, crawling across the site after the hurricane has left.  On the off-chance that the hurricane has left a fully assembled 747, would we be mindful of how incredibly unlikely that outcome had been?  I suspect not. A 747 has no usefulness or meaning for an ant, so we probably wouldn’t notice the structure involved, the causes and purposes of each part being where it is. From our perspective as ants, that assembled 747 might as well be a helter-skelter mess — an array of meaningless unknowns.

Now, after traversing the 747, something else catches our little ant eyes. Immediately, we scramble up the side of the anthill, plunge into the entrance, race down the pathway to the Queen’s deep chamber, and announced with excitement that something truly amazing has happened.

“It’s surely against astronomical odds,” I say. “I wouldn’t believe it myself, had I not seen it with my own two eyes!”

“What is it?” the Queen’s courtiers demand to know.

“A great glass jar of sweet jelly has appeared,” you say, “just outside the entrance to our anthill!  That jelly could have landed anywhere in the jungle.  What are the odds it would land just outside the entrance to our hill?  A thousand to one?  A million to one?  There must be a reason…”

Well, there probably is some reason, it seems to me.  But the difference in approaches taken by people and ants to the perceived “improbabilities” here reminds me of comparing apples to oranges.  It’s not just that apples are different from oranges.  Whether “God” made us or not, we’re all unique, in many, many ways.  Some of us — I’ll call them the oranges — attribute perceived improbability to “plain ol’ luck.” Others, like one-leafed apples, attribute it to intelligent design.  Others, like leafless apples, say that improbability nearly proves the non-existence of God.  I say, what we perceive as improbable depends on whether we’re ants or people.  Our surprise varies widely, depending on the criteria we’re (subjectively) mindful of.  But as unique as we are, we’re all alike in one respect: we all have limited brains, and that’s why we need concepts like probability —to cope with our profound lack of understanding.

So, call me a two-leafed apple if you like, but when I encounter the improbable — the fact that the grains of sand on a beach happen to be arranged exactly as they are, and the snowflakes in a blizzard move exactly as they do — I try to remember that what I experience as “randomness” is just a name I give to what I can’t get my mind around.  “Improbability” tells me nothing about God, one way or the other, except that, if God does exist, she gave me a brain that’s incapable of fully understanding the uniqueness of things, or why any of it exists.

And I’m okay with that.

— Joe

 

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