Number System as Solar System

 

Infinite-Cofinite-Finite Trilemma

           Infinite-Cofinite-Finite Trilemma

 

          A trilemma is a trio of three statements {U,V,W}, any two of which can be true, but not all three.

          Therefore for any trilemma, there are 3 deductive rules:

          From U and V, deduce not-W;

          From V and W, deduce not-U;

          From W and U, deduce not-V.

 

          For instance, from the trilemma:

          Superman can fly;

          Clark Kent can’t fly;

          Superman is Clark Kent.

         

Deduce these three rules:

If Superman can fly;

And Clark Kent can’t fly;

          Deduce: Superman is not Clark Kent.

If Clark Kent can’t fly;

And Superman is Clark Kent;

          Deduce: Superman can’t fly.

If Superman is Clark Kent; 

And Superman can fly;

          Deduce: Clark Kent can fly.

 

          Here is the Some-All-None trilemma:

·       Some A are B;

·       All B are C;

·       No A are C.

 

From this trilemma, you can derive three deductive rules:

If some A are B;

And all B are C;

Deduce: some A are C.

If all B are C;

And no A are C;

Deduce: no A are B.

If no A are C;

And some A are B;

Deduce: some B are not C.

 

          For example, from:

          Some men are heroes;

          All heroes are immortal;

          All men are mortal.

          Deduce:

If some men are heroes;

And all heroes are immortal;

Deduce: some men are immortal.

If all heroes are immortal;

And all men are mortal;

Deduce: no men are heroes.

If all men are mortal;

And some men are heroes;

Deduce: some heroes are mortal.

 

          Here is an infinitary version of the Some-All-None trilemma: the Infinite-Cofinite-Finite trilemma:

·       Infinitely many A are B;

·       All but finitely many B are C;

·       Only finitely many A are C.

 

From this trilemma, you can derive three deductive rules:

 

From: Infinitely many A are B;

And: All but finitely many B are C;

Deduce: Infinitely many A are C.

From: All but finitely many B are C;

And: Only finitely many A are C;

Deduce: Only finitely many A are B.

From: Only finitely many A are C;

And: Infinitely many A are B;

Deduce: Infinitely many B are not C.

 

          For instance, from:

Infinitely many philosophers are Cretans;

All but finitely many Cretans are liars;

Only finitely many philosophers are liars.

 

          Deduce:

From: Infinitely many philosophers are Cretans;

And: All but finitely many Cretans are liars;

Deduce: Infinitely many philosophers are liars.

From: All but finitely many Cretans are liars;

And: Only finitely many philosophers are liars;

Deduce: Only finitely many philosophers are Cretans.

From: Only finitely many philosophers are liars;

And: Infinitely many philosophers are liars;

Deduce: Infinitely many Cretans are not liars.

 

Other examples:

 

Infinitely many days are bliss;

All but finitely much bliss is perfect;

Only finitely many days are perfect.

 

There are infinitely many unicorns;

All but finitely many unicorns are goats;

Only finitely many unicorns are goats.

 

All but finitely many dogs go to Heaven;

Infinitely many dogs are flatterers;

Only finitely many flatterers go to Heaven.

 

All but finitely many cats have nine lives;

Only finitely many vampires have nine lives;

Infinitely many cats are vampires.

 

Infinitely many good deeds are wise;

All but finitely many wise deeds go unpunished;

Only finitely many good deeds go unpunished.

 

All but finitely many Scots are canny;

All but finitely many ghosts are uncanny;

Infinitely many Scots are ghosts.

 

All but finitely many aliens play video games;

Only finitely many angels play video games;

Infinitely many aliens are angels.

 

Only finitely many ducks waltz;

All but finitely many officers waltz;

Infinitely many officers are ducks.

 

Lower and Upper Bounds

Lower and Upper Bounds

 

 

A. Minimum or Less

I freely admit that I take as a rule

That anyone dumber than me is a fool.

But alas! That implies what is quite plain to see;

That my wisdom, at best, is the least that can be.

 

 

B. Upper Bound

 

That anyone wiser than me is a sage

Is a thought I have sadly accepted with age.

There is one consolation I take from this rule;

At worst, I’m the wisest conceivable fool!

 

 

How To Take Over The World

How To Take Over The World

 

I was driving through Chinatown, stuck in a slooow traffic jam. I looked up, and saw a banner slung over the road. It read like this:

(Chinese)(Chinese)(Chinese)(Chinese) 1999 - 2000 (Chinese)(Chinese)(Chinese) 123-456-7890 (Chinese)(Chinese)

I could read none of the text, but I knew the years and I knew the phone number to call. Since then I’ve forgotten the years and the phone number, so the ones I listed above are made up. But I could read those numbers, while I couldn’t read the text.

          And why? Because Hindu-Arabic numbers have taken over the world. They are known in every society on Earth. And that’s good thing. Can you imagine trying to build a car or a computer or a spacecraft or even a flashlight by using Roman numerals? I can’t either; and that is how and why positional arithmetic took over the world.

          Key to the system is a new number, previously ignored: Zero. It means both ‘nothing’ and ‘multiply by ten’. It is the least of the numbers, yet it is vital for making numbers as big as you like.

          Here’s the cream of the jest: we don’t know who invented it! Some nobody from nowhere gives nothingness a name, and poof! We’re a new species! We can do arithmetic! So now we have phones and cars and computers and airplanes! We’ve taken pictures of the far side of the Moon!

          But we don’t know whom to thank for it!

          And that is how to take over the world.

The Numerology Letdown

             The Numerology Letdown

 

          Numerology has long been a disappointment to mathematicians.  The failed romance of numerology goes back to Pythagoras. He proclaimed that number rules the universe; and in a sense we are all Pythagoreans now. (But reformed Pythagoreans; the square root of two and all that.)

You would think that if number rules the universe, then the numbers themselves should be relevant to everyday life; but no. Yes, harmonious musical chords are based on integer ratios; and yes, the sum of the squares of the sides of a right triangle equals  the square of the diagonal;  these Pythagorean insights are valid;  but “five equals marriage” is not. The applications of mathematics deal with the numbers as a class, not as individuals.

Numbers are like Hollywood stars; they have properties and relationships, but no personalities.

 

         The Racist Bone

 

 

            Consider the phrase “I haven’t got a racist bone in my body!” I wonder; what would a racist bone be? After introspection I have decided that the best candidate would be the coccyx; the vestigial tail-bone.

            The coccyx reminds us that our ancient ancestors had tails, and presumably other beastly traits. Among these was probably tribalism, which in us, their descendants, manifests as nationalism, religious fundamentalism, racism and classism.

            So alas, I myself do have a racist bone in my body; or more precisely, a tribalist bone; worse yet, everybody has that same bone; fortunately it’s vestigial.

 

 

            Who First Ate Cheese?

 

 

            Somebody had to be the first to eat cheese. I feel sorry for that someone, but admire that someone’s courage and luck. Surely there had been nothing else to eat, for miles and miles around, and for a long time too. All that was left was this smelly gunk at the bottom of the milk jug. But our heroine ate it, and survived.

 

            Do you think our heroine’s tribe thought cheese to be health food? Not at first, and rightly so; no doubt her tribefolk had all sorts of bad reactions to cheese, starting with lactose intolerance and going on up to obesity and heart disease. But evolution proceeded, and now lactose intolerance is a rarity, and Frenchmen eat cheese yet stay thin.

 

            I wonder about the origin of other healthful foods. Yogurt, for instance. Somebody had to be the first to eat that. Again, it must have been hard times. 

 

            Evolution continues unabated, even within civilization; for now civilization is the environment our genes must adapt to. If milk and cheese are cheap, then lactose intolerance is a genetic defect, and milk becomes health food. If you need readin’, writin’ and ‘rithmetic merely to survive on these mean streets, then so long dyslexia. If flu, measles and the common cold regularly go pandemic in the cities, then your grandchildren, if any, will have kick-ass immune systems.

 

            I predict that in 10,000 years, Cheetos will be a health food.

 

 

Local Optimism

             Local Optimism

Voltaire mocked Leibnitz (in the guise of Dr. Pangloss) for proposing that this is the ‘best of all possible worlds’. But Leibnitz, co-inventor of the calculus, knew the difference between local and global maxima. A global maximum is the largest value that a function reaches, for any input; whereas a local maximum is the largest value that a function reaches, in some neighborhood of the locally-maximizing input.

I therefore propose this modification of Panglossian optimism; Local Optimism, which states that this is the best of all sufficiently similar possible worlds. Any stable world locally optimizes; it’s the best of all nearby possibilities.

          Local optimism suggests that there may be many stable worlds, some better than ours. There may also be inherently unstable worlds, that are the worst of all sufficiently similar possible worlds.

          Any continuous path from one stable world to another must begin by getting worse.

          Any continuous path from one stable world to a better one must, in between, pass through the worst world on the path.

          A path from one stable world to a better one that never gets worse must be discontinuous.

          Analogs of local optimism are confirmed - and fundamental - in biology and physics; Darwinian evolution for biology and the Law of Least Action for physics. Local Optimism has sufficient scientific support to appeal to the likes of Leibnitz; yet also sufficient satiric undertones to appeal to the likes of Voltaire.

          For why does the universe minimize action? Is it lazy?

Cadaeic

        Cadaeic

 

          A “cadaeic” is a mnemonic for the number pi. “Cadaeic” itself is a cadaeic, if you use the letter-code A=1, B=2, and so on; then ‘cadaeic’ spells out 3-1-4-1-5-9-3; whereas pi’s expansion is 3.1415926…; so cadaeic is good to seven places if you round to the nearest digit.

          Most cadaeics use word-length coding, for instance:

          “How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”

          Counting out word lengths, that’s 3-1-4-1-5-9-2-6-5-3-5-8-9-7-9; pi to fifteen digits. Here are others, some submitted by my Trigonometry students:

 

          How I, even I, adore campfires in August, while the bears scavenge.

 

          “Can I have a merry Christmas at Santa’s house?” she asked.

 

          Pie, I want a piece, blueberry or cherry peach pie helps maniacal cipherers conjure fantasies for an apt mnemonic that brings to faulty mind all the thougts for an apropos, memorable image.

 

          “Hah!”  I roar. “I never calculate pi, having known the shady, half-cute, contrived antique mnemonics!”

 

          Now I need a super difficult pi puzzle, afore our human capacity retaining numbers dissolves and we get mnemonic work.

          Can I have a small container of coffee? These are tough concepts, demanding nightly examining for me.

 

          Now I, with a small packhorse

          Go toward those low hills

          Constant beckoning

          Driving endlessly

          For in the distance

          Days beyond my memory

          Lies the one sleeping eye.

         

          Now I will a rhyme construct

          By chosen words the young instruct

Cunningly devised endeavor

Con it and remember ever

Widths of circles here you see

Sketched out in strange obscurity.

 

Sir, I bear a rhyme excelling
in mystic truth and magic spelling;
Numerical sprites elucidate
and to the circular form relate;
If Nature gain, who can complain
yet my critics fulminate. Finis.

 

            Here are some e mnemonics:

 

 

            To compute a quantity to multiply, a treasure to increase, just apply exponents.

 

It enables a numskull to memorize a quantity of numerals.

To destroy a building we detonate a quantity of hydrogen bombs.

I’m forming a mnemonic to memorize a function in analysis.

In Florida, a kangaroo is escaping a teenager by bounding over gates.

We wrecked a building on Thursday, a hospital on Saturday, with eight engineers.

He repeats; I shouldn’t be tippling, I shouldn’t be toppling here!

He laughed, I screamed. We embraced a paradise of squeezes. ‘Twas love’s grandiose remarkable gift, silly to die madly for people. Faithfully be juvenile, embrace with passion. I was crazy to forget, stupid to fall, weakening quickly, holding love’s embrace.

 

Shot by the Arrow

Shot by the Arrow

 

I doubt that perfect government is possible. Consider Kenneth Arrow’s “Impossibility Theorem”. It is a mathematical proof that no social choice function can always have all four of these virtues:

1) it is fair; it divides power equally,

2) it is decisive; it answers all question set to it,

3) it is linear; it has no illogical preference loops A>B>C>A,

4) it is responsive; it never defies a consensus.

 

Or in other words: any society, however constructed, must at times be unfair, or indecisive, or illogical, or perverse. Cruelty, weakness, folly, and perversity are political evils; so Arrow’s Theorem proves the inevitability of political evil.

The proof is mathematical; it has nothing to do with sin, or sex, or money, or power, or human nature, or what two kids stole from a fruit tree long ago. The theorem is cold, not hot; its diagnosis is impersonal. You can no more practice perfect government than you can draw a round square. It would apply even to a society of saints and angels.

In a way this is comforting: political evil isn’t necessarily anyone’s fault. But it’s also disturbing: political evil is necessarily everyone’s problem.