haiku, humor, math, philosophy, photography, science

They Are Beyond Space & Time

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Taught myself a crash course in digital photo manipulation to respond to

Numbers ~ Pic and a Word Challenge #106

by posting how Plato bounced back from an encounter with intellectual ancestors of Karl Popper.  Hope I did not flunk.

Plato woke up with a nasty hangover after a symposium that had gone badly for him.  Some new sophists who called themselves “natural philosophers” had come to Athens, and the kind of philosophizing they advocated was anything but natural to Plato.

The new sophists spoke about “observations” and “conjectures” and “predictions” rather than abstract reasoning about perfect ideal forms.  Plato could tolerate his student Aristotle’s interest in easy casual observations and simple inferences from them, but the new sophists were different.  They wanted to measure minute details of how the shadows on the walls of Plato’s metaphorical cave flickered.  They would consider anything imaginable as a candidate for “explaining” their observations, even things so fanciful that Homer would never have dared to sing of Odysseus encountering them on his way back to Ithaca.

Instead of trying to establish a conjecture by reasoning to it from first principles, the new sophists wanted to reason from it to a prediction about what they would observe.  Conjectures that led to many diverse predictions matching what was actually observed were to be accepted as true, but only until somebody came up with “better” conjectures that yielded more accurate predictions by more elegant reasoning.  As one of the brasher “natural philosophers” said,

All knowledge is provisional,
never more than the best we have at the moment.

Flummoxed by such craziness, Plato had been hitting the wine harder than usual.  He had passed out just as another “natural philosopher” began replying to the brash one:

Well, that is a little over the top.  For example, …

All that was last night, when stars had carpeted an inky black sky.  Now the sky was light blue, the sun was shining, and Plato’s head was aching.  He winced when he remembered a new sophist’s remark that each star might be something much like the sun but almost inconceivably farther away.  That example of a loony conjecture had prompted a nightmare with Athens (and its circling sun) lost in a humongous whirling vortex of innumerable stars (rather than stationary near the center of the universe, as Athens so obviously was).

The cash bar at the symposium had been pricey, and Plato wondered if he still had enough money to buy some willow bark to ease his headache.  He put his coins on the nearest flat surface and counted them.  Five should be plenty.  Then he noticed that three coins had the side with the face of a leader facing upwards, while two coins had the side with the leader’s mansion facing upwards.  Suddenly, Plato felt much better.  He even felt ready for another encounter with that brash sophist.

Athens_724x505

Plato’s Revenge
|Three plus two was five
|before any mind could know.
|Where do numbers live?

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haiku, humor, math

Two Season-Words; Two Cuts; Several Allusions

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Rules 1 and 2 of
Carpe Diem’s Writing and Enjoying Haiku #3 classical haikurequire a season-word and a cut, which is not the same as requiring exactly one of each.  (Guess who has a math background.)  Dunno how to write a haiku with interchangeable short lines (per Rule 6) that also flows naturally with exactly one cut, but I try to remember that there is a big difference between saying that I cannot do it now and saying that nobody can do it ever.

Hmmm.  Suppose there is exactly one cut, that it is made by punctuation, and that moving the cut is allowed when interchanging the short lines.  This permissive interpretation of Rule 6 did not occur to me until I saw Virginia Popescu’s beautiful haiga, where the haiku still flows naturally with exactly one cut, if we move the dash from after “stone” to after “sun” when interchanging the short lines.  Her response to this episode is also a gentle reminder that my most dangerous assumptions are the ones I do not know I am making.

Maybe I can satisfy Rule 6 with a single stationary cut some time in the future.  Maybe not.  For now, I cut in both places where one line follows another.

This Haiku Is Kosher
 No mosquitoes fly.
 Basho’s frog just meditates.
 The pond stays silent.

zen-frog

Not Quite Kosher
|Zen frog bronze sculpture
|(credit lost, like casting wax).
|Dunno who to thank.

haiku, humor, math, music

Riff on a Faulkner Quote

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The story of my upbeat reinterpretation of a Faulkner quote starts in my kitchen.

The past is never dead.  It’s not even past.

A somber interpretation of this quote comes naturally.

  • The foul stain on America from slavery persists.
  • A mysterious burden is passed down from each generation to the next (as in a post on Na trioblóidí that I found to be simultaneously intriguing, funny, and disturbing).
  • Original Sin.

And so on.

Like many classics, the Faulkner quote can be reinterpreted later, w/o superceding the original intent.  As a quick example of such reinterpretation, consider JS Bach’s Two-Part Invention #11.  It is very quick indeed (about a minute long) and was originally written for solo harpsichord.  Click here to hear it arranged for banjo and marimba, on one track from a Grammy-winning CD, where banjo virtuoso Béla Fleck and friends reinterpret 19 short classical pieces.  We will return to music shortly.

The story of my upbeat reinterpretation starts a few years ago.  Tired of having the air in my kitchen be warmer and wetter than elsewhere in the house, I bought a window fan: 2 small quiet fans in 1 housing, meant to be squeezed between sash and sill for blowing air in or out of a window.  I mounted the fan in a doorless doorway, so as to blow air from the dining room into the kitchen.  It does help.  A tall person would need to stoop when passing thru; I do not.

kitchenfan_900x473

To mount the fan, I drilled holes in the fan housing and drove screws thru the housing into wooden supports (cut from scrap lumber) that I attached to the upper corners of the doorway.  I chuckled at the thought that relating horizontal and vertical lengths (along the doorway) to diagonal lengths (of cut lumber) was yet another small consulting gig for Pythagoras.

kitchenfanmount_900x675

Hmmm.  I did not think of Pythagoras as an ancient dead Greek.  I thought of him as an eminent older colleague (long since retired) who is doing quite well for his age and still has consulting gigs.  The past is not past.

Will our civilization endure until I am as old as Pythagoras is now?  (Not w/o some major course corrections.)  Suppose it does.  I doubt that I will have many more consulting gigs.  But Pythagoras will.  Bach’s music will still be cherished and reinterpreted, along with that of other great composers, from Hildegard to Hovhaness and beyond.  Sometimes it is good that the past is not past.

Ad honorem: Hildegard of Bingen, 1098-1179
|Mystic visions or
|migraine headaches? Whatever.
|Her music lives on!

haiku, humor, math, politics

Bhaskara for President!

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Fooey.  He has hardly any name recognition, was not born a US citizen, and has been dead for centuries.  Being more reality-oriented than those who nominated Donald Trump for the job, I cannot seriously promote Bhaskara.  What a pity.

Who’s Bhaskara?  We will get to that question shortly.  First, consider whatever gadget U are using to read this post.  It depends on many things, discovered over many years by many people who (unlike many pols) preferred building up to tearing down.  With many steps omitted (and “depends on” abbreviated to ), a few of those dependencies go like this:

Your Gadget quantum physics coordinate systems Pythagoras’ Theorem

Back in high school, Pythagoras’ Theorem may have seemed like a little fact about right triangles that may have been mildly interesting but did not deserve the effort of slogging thru the book’s tedious proof.  I could read the proof line by line, observe that it was valid, and be glad that I never needed to retrieve it for a test.  Hardly anybody could remember it for more than a few minutes.

Pythagoras’ Theorem turned out to be essential to blogging (and much else), so it would be nice to have a proof that mere mortals could remember, appreciate, and be inspired by.  Enter Bhaskara, 1114-1185.

Bhaskara replaced the usual picture (of 3 squares glued to the sides of 1 triangle) with a picture of 4 copies of the same triangle, arranged to form a big square with a little square inside it:

Pythagoras
(a+b
=
4 · ( ½ · a · b) + c²
The proof is sometimes displayed more tersely, with just the figure.  I prefer to write out a little algebra (while not belaboring why the angles do add up the way the figure suggests).  Tho he did not have modern notation, Bhaskara did have an elegant way to provide more detail for the mathematically fastidious.  He displayed another figure that also puts the 4 copies of the triangle inside a big square with sides a+b.  In the other figure, the area not covered by copies of the triangle amounts to a²+b² because it consists of 2 small squares.  But the not-covered area amounts to c² in the figure displayed above, so we can conclude that

  a²+b²= c²

w/o bothering with algebra and how to compute areas of right triangles.  We just need to bother with drawing both figures.  Wanna try your hand at drawing the other figure?

Googling reveals some variation in what is attributed to Bhaskara. The 1-figure proof I displayed appears in several places (sometimes attributed to Bhaskara and sometimes w/o attribution).  A similar 1-figure proof is commonly attributed to Bhaskara, with a big square of length c.  The 2-figure version that avoids algebra is attributed to Bhaskara in Math in 100 Key Breakthroughs, a nicely illustrated book by Richard Elwes.  Historical accuracy is not crucial at the moment, so I went with the best story w/o worrying about who got it right.

 
OK, I admit that having written a proof of mind-blowing elegance does not really qualify Bhaskara to be POTUS.  Too bad that many people think mind-blowing arrogance can hack it.

Clicking on the “politics” category or tag in this post will display all my uses of acidic humor to cope with the current state of US politics.  But acids are corrosive.  Sometimes, I forego acid and contemplate some of the enduring (so far) glories of modern Western civilization, one of which is that it is not entirely Western.  In particular, we got some elegant math from India and some elegant poetry forms from Japan.

One Way to Stay Sane in 2016

Cherish all that is
true and good and beautiful
(like Bhaskara’s proof).
humor, math

Like a Good Priest

I did not have pen in hand when a bemused radio announcer commented after playing Bach’s 4th Brandenburg recently, so the following quote may not be perfectly exact. It is very very close.

How can anything so complicated and so mathematical be so beautiful?

Imagine a priest who hears one of the great settings of the Mass (or a tour of a Gothic cathedral) followed by

How can anything so complicated and so religious be so beautiful?

That is essentially how I felt. With considerable effort, one could make enough dissonant noise to be as grating as the remark. Scratch a chalkboard with the fingernails of one hand. Bang on the cracks between a few piano keys with the fingers of the other. Step on a cat’s tail and fart loudly. Doing all that would suffice.

Amiens_Rose_Window_640x480

A good priest would redirect any shock or anger at the remark into sorrow and pity for the wayward soul of a heathen who meant no harm. In this one respect anyway, I try to be like a good priest (or a good imam).

Selimiye_Mosque_640x427

Image Sources

Photos were downloaded from Wiki Commons and are used under Creative Commons licenses.