I continue quoting from Alan Lightman's, "A Modern Day Yankee In A
      Connecticut Court and other essays on Science". 

      Conversations with Papa Joe 

      The Fourth Evening

      "The next day I stayed home to prepare some lectures, but my heart
      wasn't in it. I spent the time reading a novel instead, sitting in Papa
      Joe's chair. That night the old gentleman returned as he had promised,
      and wasted no time in getting to the topic of conversation.

      "Now, I'm not afraid of numbers, young man," he began. "A fellow in
      construction for forty years knows numbers." He paused. "But I don't
      understand about equations. And I especially don't understand why you
      put so much stock in them."

      I got out a sheet of paper and wrote down:

                            C = 2 pi r

      "Papa Joe, this says that the circumference of a circle equals its
      radius times two, times pi, a special number close to 3.14."

      "I remember that rule," my great-grandfather said.

      "The real strength of equations is their logic," I said. "You start at
      one point, and an equation tells you what has to come next, according
      to logic. In the example here, you give the radius of any circle, and
      this equation says what its circumference has to be. I think the
      Babylonians or somebody figured the thing out first. They went out and
      measured the radii and the circumferences of a whole bunch of circles,
      of all different sizes, and gradually realized that a precise
      mathematical law held every time. It saved them a lot of trouble when
      they found it. Equations in science are all like this, except usually
      much more involved. They start with some law about nature, and tell you
      what logically follows from the law, step by step. They give rules for
      how things ought to behave."

      "Does every single item in the world have an equation for how it
      behaves?" asked Papa Joe.

      "Most Scientists would say yes--for the physical world, that
      is--although in some cases we haven't yet figured out what the
      equations are."

      "So, if I understand you right, you believe that everything in nature
      follows rules. Whatever the thing is, you'll eventually find an
      equation for it, and it'll stand up and salute."

      "But what's the alternative?" I asked. "To be constantly afraid that at
      any moment houses might float off the earth, or stars might change into
      wheelbarrows, or people might start talking backwards? The world we're
      born into is strange enough as it is. We've got to believe that, at
      bottom, nature is at least rational. Scientists might not discover all
      the rules straight off, but we trust that we'll find them."

      "I can see where that view might bring you some comfort," said the old
      gentleman.

      "What's changed in this century," I continued, "is that we don't have a
      physical feeling for all of the rules we've been finding. The
      Babylonians could draw their circles and measure them with string to
      test their equations. Sir Isaac Newton could compare the prediction of
      his law of gravity with the observed motion of the planets. But many of
      the new rules deal with things we can't touch or see, and some of them
      plain violate common sense."

      "I gave up common sense a few evenings ago," said the old gentleman,
      chuckling, "with the heavens all bursting apart and those things flying
      around the earth looking at invisible light. Your new equations and
      your new gadgets should be very happy together. But I still don't have
      any idea of these new rules you've been talking about."

      "Let me give some examples," I said. "The ones that come to mind are
      from physics."

      "Fine, but please hold to a slow trot, if you don't mind."

      I got up to stoke the fire and pour us some tea. "In the first third of
      the century," I went on, "physicists discovered a new set of rules,
      brimful of equations, called quantum mechanics. Quantum mechanics
      concerns the behavior of atoms, and particles even smaller than atoms.
      One of the rules amounts to this: a subatomic particle can be at
      several places at the same time."

      "Young man, you're galloping."

      "I can't help it. The difficulty is that all our experience with the
      world is based on objects much larger than atoms. Golf balls and
      marbles are things you can pick up with your hands. They have edges.
      They stay where you put them. But as you go to smaller and smaller
      sizes, matter begins behaving differently. When you get down to atoms
      and smaller, your whole idea of a solid object falls apart. A particle
      that size, like an electron, doesn't act like a little sphere with
      sharp edges marking the boundary between itself and the rest of the
      world. An electron acts like a haze, a blur covering all places it
      might be at the moment. If you throw identical marbles with identical
      aim at a wall, they'll all hit the wall in the same spot. But if you do
      the same with electrons, they'll hit it in many different spots. That's
      what the equations of quantum mechanics tell you. And those same
      equations have made very accurate predictions about many other things
      that have been measured and verified. So if you have faith in the
      theory--and physicists these days do--then you have to accept this
      slippery business with electrons. It goes against common sense, but
      there it is."

      The old gentleman had got up from his chair again. "I'm beginning to
      get your meaning," he said. "Would you mind giving me some idea of how
      your physicists go about finding these equations and rules?"

      "It's not much like the Babylonian method of trial and error. For many
      phenomena, we'd never stumble on the right rules that way. There are
      too many choices. Somehow, we've got to sniff out the trial."

      I paused a moment, and Papa Joe took a deep, lingering whiff of the
      aromas drifting his way from the pipe.

      "Simplicity seems the best guide," I continued, "although nobody knows
      why. Scientists these days are constantly searching for the fewest and
      simplest rules possible. Two rules for a thing are better than three. A
      short rule is better than a long one. I know I'm being vague. Let me
      give an analogy. To scientists, nature is a vast game of chess. They
      see the board every now and then with their experiments, study what
      squares the pieces are on, and from this try to figure out the rules of
      play. At first, they might guess that every piece moves one square at a
      time, like a pawn. When this doesn't work, they'll try something
      slightly more elaborate, and so on, but never anything more complicated
      than the facts require. What's astonishing is that this kind of
      approach works remarkably well. It seems that nature loves simplicity.

      "Take the case of the electron," I went on. "The precise equations for
      electrons were worked out by Professor Dirac fifty years ago. Now Dirac
      was a theorist, a pure pencil-and-paper man. I suspect he'd never been
      under the hood of a car in his life. But he had great faith in this
      idea of simplicity. So for the electron, he figured out the simplest
      and prettiest rules possible, consistent with the other rules he
      already knew. And his rules have held up for fifty years, tested by
      countless experiments. A more complicated theory would have been wrong.
      Out of his theory, by the way, came an unexpected prediction of a new
      kind of particle never before seen, a close cousin of the electron,
      called a positron. Professor Dirac wasn't looking for positrons; they
      just marched out of his equations for electrons. A few years after his
      prediction, real positrons began turning up in the Lab."

      "Remarkable," said Papa Joe.

      "There are quite a few stories like that one. With every success,
      scientists have gotten more sure of themselves. In recent years,
      physicists have staked their reputations and millions of dollars
      hunting subatomic particles predicted by their theories."

      My great-grandfather whistled softly. "I'd hate to be ruined by a
      positron that wouldn't come out of the brush," he said. "You know, I
      reckon it would be easier for me to follow you if I knew more math."

      "Well, I'm cutting some corners, it's true," I said. "But you've been
      keeping up better than I would on something this new."

      "What I admire most in these scientists," said Papa Joe, "is how
      they're willing to trust their equations against common sense. I don't
      believe I could follow the plans for a house that seemed upside down.
      That takes faith."

      "I agree. You'd want to be darn sure of your architect. And you
      wouldn't move in right away." We sat for a time without talking,
      listening to the faint bark of a dog down the street.

      "Tell me about some other theories that seem contrary," said the old
      gentleman.

      "You remember the black holes from last night?"

      "Yes, They were my favorites."

      "Black holes were predicted by Professor Einstein's new theory of
      gravity. According to the theory, if you went to live near a black hole
      and then came back to earth, you'd be much younger than if you'd stayed
      here. The gravity of the thing slows down time in its vicinity."

      "Confound it," shouted the old gentleman. "I'll go along with your
      fuzzy atoms and particles, whatever they're good for, But time is time.
      A year is a year, isn't it? I must have misheard you."

      "You didn't mishear me, Papa Joe, although I agree that the idea seems
      preposterous. You see, Professor Einstein's theories propose that the
      flow of time is not fixed, as it seems. Time depends on motion and on
      gravity. The effect is tiny unless you're moving at extremely high
      speeds or being pulled by a very high gravity, and that's why you don't
      notice it. But sensitive instruments and clocks have verified the
      effect. It's taken me years to get used to the idea."

      "Now that I think of it, I remember a big commotion over one of
      Einstein's predictions being proved."

      I nodded. "You're probably remembering the famous experiment during the
      solar eclipse of 1919. One of Einstein's theories predicted that light
      should be attracted by the sun, the way a planet is. The effect is very
      small, because light travels much faster than planets, but it's there
      and it's measurable. To test the prediction, you examine some stars
      just past the edge of the sun. According to Einstein, the starlight
      should be deflected by the sun on its way to the earth, and the images
      of the stars should be slightly distorted. Some astronomers did the
      measurement at the first convenient eclipse, when stars could be seen
      near the sun, and confirmed the effect. These days, most scientists
      believe just about every prediction of Einstein's theories. even the
      ones not yet proved."

      "I wonder whether Professor Einstein was bothered by this odd business
      with time slowing down," said Papa Joe.

      "I don't think so," I replied. "From what I can tell, Einstein believed
      that the new ideas were logical and natural, given certain facts. He
      had this wonderful way of starting from scratch, without taking
      anything for granted. And he never expected to experience all the
      mysteries of nature with his body. To him, it was pleasure enough to
      get a mental glimpse now and then, and imagine the rest."

      As I got up to stretch, the church clock in town chimed eleven. The old
      gentleman was back at his spot near the window, looking out at the
      night. I joined him there. Sirius, the brightest star in the sky, was
      in easy view, as well as half a dozen constellations--celestial
      pictures of hunters and serpents and lions and dogs, ancient visions of
      men and women looking for order.

      "You know," said Papa Joe, "I believe your faith is contagious. These
      last few nights I've felt so tiny I could fit inside an atom, and so
      big I could step from one star to the next." He paused, staring out the
      window. "I proposed to your great-grandmother on a night like this."
      Papa Joe turned and took a long look around the room. "You take care of
      that pipe."

      I stood for a moment beside my great-grandfather, shoulder to shoulder,
      and then he melted away.
      
      __________________________________________________________________


      Conversations with Papa Joe appeared serially in "Science 84-86", which
      was my first introduction to the writings of Alan Lightman. It was a
      great joy to discover this story as one of many essays in the
      collection titled, "A Modern Day Yankee In A Connecticut Court". I
      chose to reproduce "Conversations with Papa Joe" over four months, in 
      its entirety, to treat you to a fine story--but also to a story that
      teaches some very fundamental ideas about astronomy and how we do
      astronomy and science. Papa Joe was certainly a likable character--a
      curious man with a good head on his shoulders.