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Location: Great Boreal Deciduous Hardwood Forest, New England, United States

old dude, all hair, swell new teeth

03 November 2005

Bob is a big Nincompoop

You Can't Get There From Here.

[VLEEPTRON EMBARRASING CONFESSION: Our first Proof that There Is An Infinite Number of Primes was defective. We have fixed it. Leave A Comment

* to tell me I'm a Nincompoop

* to ask me what the mistake was, so you can know exactly how I was a Nincompoop. If you didn't catch it, you're a Nincompoop, too.]

[VLEEPTRON TRUE CONFESSION: Vleeptron is indebted to a correspondent, LD, of Princeton, New Jersey USA, for correcting our misspelling 10^10
0 = googol, and 10^googol = googolplex. We are not Perfect. LD has hurt our Self-Esteem.]

Is Vleeptron bothering you with too much math?

Who cares?

Vleeptron knows it hurts. It hurts ten-year-olds, it gives them anxiety and stomach aches. It hurts high-school students. It gives headaches to adults.

It's supposed to hurt. As the football coach and the Army drill sergeant are fond of screaming:

No Pain, No Gain!

I also like this jock bumper sticker:

Pain Is the Feeling
of Weakness Leaving Your Body.

For Math, substitute:

Pain Is The Feeling
of Weakness Leaving Your Brain.

What do you want Vleeptron to do? Stop with the math headaches, and just write lovely long essays about Our Feelings? Will that help your Self-Esteem?

Screw That and Bite Me.

The first time I went to college, I'd already stopped taking that icky painful math stuff, and I studied all Liberal Arts and Humanities courses.

Such beautiful things. Lots of essay questions that began

Describe your feelings about ...

I learned so much about Haiku and My Feelings and Emily Dickinson's Feelings and my other classmates' sensitive Feelings. We shared. We hugged.

There was no such thing as a Wrong Answer in Liberal Arts. All answers that had reasonably good spelling and punctuation were defined as The Right Answer.

If my grammar, spelling and punctuation were excellent (and they always were), I routinely got an A or an A+ . I routinely got final grades of A without going to class or opening the textbook. I was able to sleep a great deal (in and out of class) and concentrate on wild loud Theater Department parties without much interference from Liberal Arts' academic demands.

But after the Army, I went back to college. This time I took only math, electrical engineering, and "hard sciences."

(Hard Sciences are the sciences whose Answers make you use Math. As distinguished from Soft Sciences that make you Describe Your Feelings in Essay Questions.)

Oh boy, what a Cold Shower that was.

Nobody wanted to know about My Feelings anymore.

I was shocked when I took tests, and the tests came back with all this Red Ink. These new horrible professors circled My Answers and wrote:

WRONG WRONG WRONG

(One of these new evil math professors called a student a Nincompoop out loud one day in class. I think that's now a crime, under the No Child Left Behind Act.)

If I wanted a grade above D , I had to wake up at dawn and go to bed at 4 a.m. and read every goddam page of the textbook and do every goddam homework problem. I had to tell my cheerleader friends that I couldn't go to their Skinny Dipping Party. I had to study.

My Math Professors hurt my Self-Esteem.

Back in the Happy Liberal Arts days, if your professor suddenly died, a new professor would take over the class -- and suddenly start teaching us Entirely Different Things about the same subject. The old professor had taught us Day and White. Now the new professor told us it was Night and Black, and would flunk us if we still kept saying Day and White.

Now, in Calculus Class, if the professor suddenly died, the new professor would show up the next day and ask just one question:

"What page did the Dead Guy leave off on?"

Day remained Day. White remained Not Black.

Vleeptron has decided to hurt your Self-Esteem again. Prepare to send an e-mail to The Blog Dean. "This Blog keeps writing about Math! This Blog is hurting my Self-Esteem! This Blog keeps asking Questions I Can't Answer! This Blog doesn't want to know what My Feeling are!"

~ ~ ~

I wish I could remember the name of the stand-up comedian who said

I loved geometry in high school!
And it was so valuable!
Hardly a day goes by
that I don't have to do a Proof!

But we're going to do one today. Ouch. Ouch. Ouch. Get out your bottle of Generic Aspirin.

~ ~ ~

Remember Ramanujan's Taxicab, when he told the cabbie to drive him to Infinity, and Step On It?

Well, as the farmer in Maine told me when I was lost and asked him for directions:

You Can't Get There
From Here.

We Mere Mortals can drive to Big and Large and even Vast. I can give you directions to

3,458,002,671,887,991,380

When you get there, send me back a postcard, please.

googol is the name George Gamow's 10-year-old nephew gave to

1000000000000000000000000000000000
0000000000000000000000000000000000
000000000000000000000000000000000

but if you want to carry it in your pocket, you can just call it 10^100 .

I can tell a cab driver how to get you to googol.

Then Gamow asked his nephew what he'd call

10^
1000000000000000000000000000000000
0000000000000000000000000000000000
000000000000000000000000000000000

(ten to the googol power, or 10^googol )

and Little Nephew called that a googolplex.

Carl Sagan said:

"A piece of paper with a googolplex
written out on it could not be stuffed
into the known universe."

But a googolplex is still a Finite Number. Eventually it ends. So You Can Get There From Here.

But you can't ever reach Infinity.

Like Ramanujan, you can think about Infinity, you can imagine Infinity, you can dream about Infinity.

You can even Know a few Facts about Infinity, just as I know a few Facts about Ulan Bator, although I've never been there.

I will get to Ulan Bator someday. (Someone who went there brought me back some local cigarettes and currency!) If I win the lottery and have $20,000,000 to buy a ticket from the Russians, I might even get to spend some time on the International Space Station.

But I'll never get to Infinity.

But here's a Souvenir from Infinity, a Postcard from a Place no human being can ever actually reach.

Here's how to stand on a chair and gaze upon Infinity, and bring back a Fact about Infinity. Just like Ramanujan.

~ ~ ~

A Prime Number is any positive whole number greater than One which can only be evenly divided by Itself or by One.

Here are the first few Primes:

2 3 5 7 11 13 17 19
23 29 31 37 41 43 47 53 ...

The whole numbers which aren't here are called Composites -- they CAN be evenly divided by some whole number not 1 and not itself.

Notice that the only Even Prime is 2.
5 is the only Prime that ends in the digit 5.
All other Primes are Odd, and end in either 1, 3, 7 or 9.

Are there more Primes?

Yes. Lots. The Greek mathematician and astronomer Eratosthenes (286 BC - 194 BC) invented a tool called The Sieve which supercomputers still love to use to find Primes quickly and efficiently.

(Very Large Primes are actually Very Valuable and worth a Lot of Money. Whodathunkit?)

Notice that as the numbers get larger, the Primes get sparser, thinner; the Primes get rarer.

So ... maybe they'll get So Rare that they stop entirely.

Do the Primes ever end?
Is there one Biggest Prime Number,
and after that, no more?

Euclid (325 BC - 265 BC) wondered about that.

~ ~ ~

I believe I can prove
that the Primes
never end.

For any Prime Number, however large, I believe I can prove that there'll always be a larger Prime Number.

In other words, I'll try to prove that

There's an Infinite Number of Primes.

To start, let me write all the Primes, lowest to higher,
this way:

P(1) = 2
P(2) = 3
P(3) = 5
P(4) = 7
P(5) = 11
P(6)
P(7)
P(8)

etc.

Now just for the sake of this proof, I'll say that the Primes Do End -- There Is a Largest Prime. I say there's no Prime greater than this Largest Prime.

I'll call this Largest Prime P(n) .

We don't know what n is. We don't know how large P(n) is.

All we know is that it's a (very large) Prime.

Now I'm going to make a special number based on P(n).

First, I'll multiply together
all the Primes less than and including P(n) .

2 x 3 x 5 x 7 x ...
... x P(n-2) x P(n-1) x P(n)

Now I'll Add One to that number,
and call the result

Q = [ 2 x 3 x 5 x 7 x ...
... x P(n-2) x P(n-1) x P(n) ] + 1

Obviously Q is larger than P(n) .

Now there are only two possibilities:

Either Q is Prime,
or Q is Composite.

=======================================

If Q is Prime,
we've found a Prime
which is larger than P(n) .

So my original contention,
that P(n) was the Largest Prime,
was False.

=======================================

The only other possibility: Q is Composite.
So it can be evenly divided by some Prime.

Can Q be evenly divided by any Prime
which is less than or equal to P(n) ?

We'll divide Q by every Prime
from 2 to P(n) .

Can Q be evenly divided by 2?

2 x 3 x 5 x 7 x ... x P(n-1) x P(n)

CAN be evenly divided by 2;
2 is a Factor,
because we built that product that way.

But to make Q

Q = [ 2 x 3 x 5 x 7 x ...
... x P(n-1) x P(n) ] + 1

we added 1 .
So Q divided by 2 would leave Remainder = 1 .
Q can't be divided evenly by 2.

Can Q be evenly divided by 3?

Again, we added 1 to make Q,
so again, Remainder = 1 .

Can Q be evenly divided by 5?

Again, we added 1 to make Q,
so again, Remainder = 1 .

And so on,
until we try to divide Q by P(n-1) .
And again, because we added 1 to make Q,
Remainder = 1 .

Finally we try to divide Q by P(n) itself.
Again, because we added 1 to make Q,
Remainder = 1 .

No matter what Prime P(n) is,
we've tried to divide its Q
by every possible Factor from 2 to P(n) .

We always get Remainder = 1 .

So Q can't be evenly divided
by P(n) or by any Prime less than P(n) .

So Q may be a Composite,
but the Prime that divides into it evenly
must be bigger than P(n) .

So if Q is Composite, again,
there's a Prime bigger than P(n) .

So again, my original contention
that P(n) is the Largest Prime
is False.

So no matter how large P(n) is,
we can always find a Prime
which is larger than P(n) .

So the Prime Numbers never end.

There Are Infinitely Many Primes.

Which is what I set out to prove. :)

~ ~ ~

Okay! We're just mere Human Beings, so we can never REACH Infinity.

But Euclid showed us how to PROVE something about Infinity!

That didn't hurt too much, did it?

And your Self-Esteem ... it got BIGGER, didn't it?

That pain oooh ouch is just Weakness Leaving Your Brain!

2 Comments:

Anonymous j said...

since i'm a total Nincompoop and don't give a damn about other people's feelings nor have i any kind of sel-esteem or even find Mathematics and all the other "hard sciences" (!!) fascinating subjects, i must ask

what's the point on proving Infinity?...

(long time no seen, Bob! Playwrite Guy is back from the dead)

22:01  
Blogger Bob Merkin said...

hey hey hey j!!!

how was the dead? *I* would certainly write a play about that!

Well, to start small ... what's the point of travelling to exotic places like Ulan Bator? Well ... people who go there (the physicist Feynman wanted to go there all his life, never quite made it) find it fascinating. It's in the human spirit (some humans anyway) to want to go to faraway places. (cf. Vasco da Gama, etc.)

Infinity is a unique place -- so exotic that metaphysicians don't even call it a "place," but they do call it a "locus." Wherever the heck it is, it seems to exist

* independent of the Time and Space humans are stuck in and familiar with;

* independent of human thought and discovery -- it's just always "there," whether humans are here or not.

Certain personalities find what they can discover about Infinity to be shockingly beautiful and mysterious.

Plato says human beings can't touch pure mathematical objects ("Platonic Objects") with their hands or their feet, they can't see them with their eyes -- but we CAN have contact with Platonic objects with our minds.

It's a kind of human, spiritual adventure unlike any others.

I'll stop now. And welcome back! Did you see the big New York Times article about Bairro Alto? Not as good as your travelogue, but very interesting!

What's up with Theater?

03:07  

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