The Optimist’s Paradox

As I was riding up a steep hill this bright morning, taking a load on my bicycle rack for the recycling bins, I reflected that every step (or turn of the pedals) is not only one more step taken but one fewer to be taken, so it’s a double gain. Suppose you have 100 steps to climb, and you climb from step 49 to 50. Before, you had climbed 49 and had 51 to go; after, you have climbed 50 and have only 50 to go; and the difference between the two states, between 49-51 and 50-50, is 2. And likewise for miles to walk, and all other chains of tasks.

So, is this a known, old paradox, dating back perhaps to Zeno?

And, where is the fallacy in it?

Ahem. It might also be the pessimist’s paradox. When you grow one year older…


10 thoughts on “The Optimist’s Paradox”

  1. There was a business consultant firm out of Chicago called George S. May that used a version of this trick as an accounting scam, to convince their clients their results were twice as good as they actually were. Something like, you’ve paid $100 to your vendors, and you’ve reduced your accounts payable by $100, so that’s a $200 improvement!
    As soon as I saw what they were doing, I showed the rep the door. The company is no longer in business.

  2. In current standard English it’s supposed to be “have swum”, but I don’t go for pedantry in these things. We have a family of verbs that go “swim, swam, have swum”, “sink, sank, have sunk”, “drink, drank, have drunk”, but Byron wrote:

    The isles of Greece, the isles of Greece,
    Where burning Sappho loved and sung!

  3. I engage in the exact psycho-trickery that Guy describes when I’m swimming laps. If I intend to swim 40 laps, I sometimes “count” by ticking down the difference between what I’ve already swam compared to what I have left to do, which means that going from the 20th lap completed to the 21st lap completed seems like a difference of two; I then have swam (or is it “swum”) 2 more laps than I have remaining . . .

  4. I love this piece, Guy! Thank you! This was how I endured labour – each time I told myself that every contraction was another one gone, and that meant I was one contraction closer to having my baby in my arms :)

    1. Ruth, this is the best comment of all! If I ever do a new edition of “Think Like a Mother” I’d like to quote you.

  5. Interesting thoughts. Unlike the bike trip a lifespan is not of fixed length. I think this means that every birthday adds something to ones life expectancy – at last, perhaps, a genuine reason for celebrating them!

    Regarding Jack’s half full glass, the trick might be to drink blindfold; then you won’t know when the glass is half full or half empty; yet at the halfway point you will be able to say for sure that you are enjoying the drink, indeed enjoying every sip until the last one, which you wouldn’t have enjoyed so much anyway. It perhaps might be even better not to pour it yourself, as then every sip you take from the glass increases the volume you can expect to be in it.

    1. Walter, I hadn’t found your comment till now because the system seems to have changed: I was getting email notifications of each comment, but for some reason that stopped. So I didn’t “approve” you as a commenter till now. Sorry.

      You and others add dimensions to this little logical game. I’m surprised that you haven’t mentioned chess, at which you are an expert. Whereas the alternation of the left and right pedal doesn’t matter – both work toward the top of the hill – the alternation of white and black moves does: either could lead all the way to the end of the hill! If White’s move opens a clear line for White, it may block a line for Black, so White has gained one advantage and Black lost one and the sum is two.

  6. AND….. yes,, the difference between 49 and 51 is two,,,but the difference between 50 and 50 is zero! So, what is the discrepancy??,,,, and speaking of growing one year older,,, I’ll be 60 in a few weeks, so is that 40 more to go? or 39 plus 1? Happy birthday to me :)

  7. an optimist says the glass is half full; a pessimist say it’s half empty:
    an engineer says the glass is twice as big as it needs to be.
    I know a guy who’s known as 50-50; I guess because his mind is half here and half somewhere else where only he might even know.

  8. I’m not much good at math, but if you’re looking for a rationalization that will help you get through a challenging task, this one seems to work. Conversely, if you want to support a sense of fatalism and doom, it works for that too!

    Here in San Francisco we have weekly curbside pick-up of recyclables. You put all your recyclables in a big blue bin on the curb in the evening, along with a black bin for trash and a green bin for compostables. Then overnight freelance scavengers, some pushing stolen shopping carts and others driving pickup trucks, come and dig through your blue bin pulling out the metal, glass, and plastic that can be sold for scrap. Early in the morning the trash company truck takes what’s left, mostly paper and cardboard. On the one hand it would be much more efficient if the trash company took everything in one trip, but on the other hand I can’t fault people for trying to make a few bucks to survive. It’s an interesting ecology.

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