Wonderment of Pondering

“When walking, you see things that you miss in a motor car or on the train. You give your mind space to ponder”.

Tom Hodgkinson

Tom Hodgkinson is the founder of the magazine, The Idler, who also wrote the book, “How to be Idle”.  Obviously I’m not here to endorse the idea to anyone still at work, earning a living.  I’m here to tickle the mind of those fortunate to have the free time to indulge on such activities that do not necessarily require going outdoors, or break a sweat on some physical activity, or anything requiring manual labor. But I’ll have you know that the mind, in its own right, can be made to engage in activities that could rival physical labor, consuming calories in the process. The brain may only weigh 5% of our total body weight but it consumes 25% of the energy the body produces. The mind can travel to places far longer than a 26.2 mile run, or through an impervious wall; or across oceans; even through time. And it will cost nothing.  A cup of steaming coffee helps but not necessary.

“Many things that don't really mean so much of anything are wonderful.”

― C. JoyBell C.

One could be doing wood working or some other worthy hobby and in between steps of activities one will always find the mind wandering into something else.  That’s when I usually take a break.  In those breaks when the hands and eyes are free to uncoordinate (hobbies and sports always require hand/eye coordination) the mind whirs into activity to fill the gap. It may have been idling at some moments when the activity becomes routine but it is winding up to uncoil when given the chance. The idle mind would go to ponder at any moment.  I’ve collected a few of those wandering wonders:

  

1. First, let's do the numbers:

When I was a little kid I thought a thousand was such a huge number. Now retired, I woke up one morning and our government owes 20 trillion dollars.  I have not even begun to wonder what it feels like to have or owe $1 billion. 

Now, to deal with big numbers we make short cuts to say or write them. K is now a thousand, so we had Y2K at the turn of the century – Year Two Thousand. Then to deal with the past, naturalists would shorten “one hundred thousand years ago” to 100 KYA. Paleontologists and cosmologists then came up with 67 MYA, or 4 BYA.  You guessed it: 67 million years ago and 4 billion years ago, thus explaining when the dinosaurs began to perish after an asteroid impact, and the moment our solar system was created, respectively.

In 1920 when mathematician Edward Kasner was contemplating the largest whole number he can think of, his nephew, Milton Sirotta suggested the word googol, to stand for a number followed by lots of zeros. Expressed as an exponent, it would be 1 X 10¹⁰⁰, which means 10 multiplied by 10 a hundred times (10X10X10...). If we have each and every person (man, woman and child) count down one million grains of sand from the total of grains in a googol each (to save time and effort) there are not enough people in the entire world to account for all of it in a googol (not even close as a fraction of one per cent).  Not content with the googol and wanting to come up with yet another big number, Kasner coined a googolplex – 1 followed by a googol of zeros: 1 X 10^googol. Paraphrasing Carl Sagan, assuming we have the capability to write down in zeros the number of particles in a googol, it will be physically impossible because there is not enough space in the entire "known" universe to hold all of it if they were written as an average hand-written zero, about 1/8 inch in diameter formed into a pellet. How crazy is that? And, get this, a googol or googolplex does not have much of a mathematical significance or utility.  But as large as a googolplex is, there is one even larger, represented only by a single character.

As a side note, I suspect Google intentionally made its name a homophone of googol as a subtle suggestion that the search engine company covers a googol of stuff.

 2.  Let’s one-up the Googolplex.

Three hundred years ago, both Isaac Newton and Gottfried Leibniz independently discovered calculus.  In this branch of mathematics we are confronted with this symbol ∞. As a word it is written as infinity. It is essential to integral calculus, appears in mathematical operations but is not considered a number, yet it is used as a numerical limit.  We can say 1 + ∞ = ∞.  1 X 10¹⁰⁰ + ∞ = ∞; and get this, ∞ - 10^googol still equals∞ (that is ∞ minus a googolplex!!); in other words, no number can ever be larger than ∞ and neither does it devalue to anything less than∞. However, you cannot subtract∞ from ∞. It is indeterminate! It’s not only in mathematics that ∞ plays a role although it does not follow any of the mathematical rules. But  calculus without it makes calculations impossible to do. 

Beyond the math infinity is also physically real.  Aside from symbolizing the extent of the universe it actually manifests itself in other mind boggling phenomenon.

When a very massive star collapses after it had exhausted all of its hydrogen, it will implode from its own gravitation that, at some point, its gravitational field becomes infinitely strong as to pull all of its mass into an infinitely small point. It becomes a black hole. Einstein’s theory predicted it although he was actually skeptical that anything as large as a star could be reduced to a single point – smaller than an atom. And now the phenomenon of black holes is as common as there are galaxies. The gravitational pull of a black hole is so strong it will pull surrounding stars until they too become part of it and the infinite gravity prevents anything to come back out once inside; not even light can escape, hence a black hole.

3. Let’s get a sample of Infinity

There are quite a number of examples from day to day activities.  Some people claim to have infinite patience while waiting in line at the Department of Motor Vehicles or the Driver’s License office.  Some pet owners think the same way about training a cat to do anything.  It takes infinite intelligence to decipher the tax code or assemble anything without a single screw or nut left over .  A husband prays for infinite wisdom when he asks his visibly upset wife what’s wrong and not understanding what it means when she replies, “Nothing”.  I know to stop these analogies when I get close to the limit so on to the serious stuff.

We know a square is a polygon with four sides. If we keep adding sides, using straight lines, soon the square becomes circular.  We keep doing it and sure enough we have a circle.  A circle is essentially a polygon with an infinite number of sides. Proof? See photos below.  I learned to cut a perfect circle using my table saw (that cuts only in straight lines !!). I would begin with a square board with a center hole for it to pivot on a jig. I would cut additional sides, first making it an octagon and by repeatedly cutting more and more sides, what began as a square board is turned into a perfect circle in just a few cuts - technically making it a polygon with an infinite number of sides.  

Someone, a long time ago, while I was at school, told me that infinity is arrived at by dividing a number, any number, by zero.  For a while I believed it but now I know better.

Say, divide 1 by .01, thus 1/.01 = 100; 1/.0000000001= 1 billion. If we keep adding zeros after the decimal point in the denominator but always keeping 1, or any number for that matter, at the end, the answer keeps getting bigger and bigger.  Try 1/.00000000000000000000000000000000000001 = ?  So, if we do the math with 1/.00000000000000000 … 1, with zeros so unimaginably never ending, we get a glimpse of what infinity is in numerical terms.  However, we will only get there for as long as the denominator gets to be infinitely small, approaching but not quite zero.  Almost like a circular argument, isn’t it?  And it brings us back to the circle.

Well, unlike infinity, and human that I am, incapable of an infinite musing, I must end it here.

For now.  There could be a sequel because there is an endless material to ponder.

 (I can tell the interest level by the readership feedback I get.  I see that, outside of the U.S., I get more readership in France, which leads me to wonder... and ponder... hmm).