Alexander the Great did not Exist AND he had an Infinite Number of Limbs.

NOTE TO READERS - I bet 0.5 KRW that this gets at least one serious comment.

NOTE TO THE INSANE - this article was actually published in a book.

NOTE TO THE INTERESTED - this is what we get up to when the beer runs out.


I Quote:

"Bertrand Russell has defined mathematics as the science in which we never know what we are talking about or whether what we are saying is true. Mathematics has been shown to apply widely in many other scientific fields. Hence, most other scientists do not know what they are talking about or whether what they are saying is true. Thus, providing a rigorous basis for philosophical insights is one of the main functions of mathematical proofs.

To illustrate the various methods of proof we give an example of a logical system.

The perjorative calculus

Lemma 1: All horses are the same colour.

Proof by induction:

It is obvious that one horse is the same colour. Let us assume the proposition P(k) that k horses are the same colour and use this to imply that k+1 horses are the same colour. Given the set of k+1 horses, we remove one horse; then the remaining k horses are the same colour, by hypothesis. We remove another horse and replace the first; the k horses, by hypothesis, are again the same colour. We repeat this until by exhaustion the k+1 sets of k horses have been shown to be the same colour. It follows that since every horse is the same colour as every other horse, P(k) entails P(k+1). But since we have shown P(1) to be true, P is true for all succeeding values of k, that is, all horses are the same colour. Q.E.D.

Theorem 1: Every horse has an infinite number of legs.

Proof by intimidation:

Horses have an even number of legs. Behind they have two legs and in front they have fore legs. This makes six legs, which is certainly an odd number of legs for a horse. But the only number that is both odd and even is infinity. Therefore horses have an infinite number of legs. Now to show that this is general, suppose that somewhere there is a horse with a finite number of legs. But that is a horse of another colour, and by the lemma that does not exist. Q.E.D.

Corollary 1: Everything is the same colour.

Proof:

The proof of Lemma 1 does not depend at all on the nature of the object under consideration. The predicate of the antecedent of the universally-quantified conditional "For all x, if x is a horse, then x is the same colour," namely "is a horse" may be generalized to "is anything" without affecting the validity of the proof; hence, "for all x, if x is anything, x is the same colour." Q.E.D.

Corollary 2: Everything is white.

Proof:

If a sentential formula in x is logically true, then any particular substitution instance of it is a true sentence. In particular then: "for all x, if x is an elephant, then x is the same colour" is true. Now it is manifestly axiomatic that white elephants exist (for proof by blatant assertion consult Mark Twain's The Stolen White Elephant). Therefore all elephants are white. By corollary 1 everything is white. Q.E.D.

Theorem 2: Alexander the Great did not exist and he had an infinite number of limbs.

Proof:

We prove this theorem in two parts. First we note the obvious fact that historians always tell the truth (for historians always take a stand, and therefore they cannot lie). Hence we have the historically true sentence, "If Alexander the Great existed, then he rode a black horse Bucephalus." But we know by corollary 2 everything is white; hence Alexander could not have ridden a black horse. Since the consequent of the conditional is false, in order for the whole statement to be true the antecedent must be false. Hence Alexander the Great did not exist.

We have also the historically true statement that Alexander was warned by an oracle that he would meet death if he crossed a certain river. He had two legs; and "forewarned is four-armed." This gives him six limbs, an even number, which is certainly an odd number of limbs for a man. Now the only number which is even and odd is infinity; hence Alexander had an infinite number of limbs. We have thus proved that Alexander the Great did not exist and that he had an infinite number of limbs. Q.E.D.

It is not to be thought that there are not other types of proofs, which in print shops are recorded on the proof sheets. There is the bullet proof and the proof of the pudding. Finally, there is 200 proof, a most potent spirit among mathematicians and people alike."

Unquote.


Aki's Choralarry:

"All infinitely white horses of the same color belong to either an indefinite number of horses' legs or Alexander the Great."

Saddle Up!

Way back in the years of when, it was speculated that order was needed in math, and geometry was the first victim.

A guy named Euclid decided he would formalise it, so he set out to "axiomise" it....

...and he came up with these, seemingly "self-evident" truths:

Euclid's Five Postulates are:

1. The shortest distance between any two points is a straight line

2. Any straight line can be extended (produced) indefinitely

3. Any point can be surrounded by a circle of any size

4. All right angles are equal to all other right angles

5. Parallel lines never meet

Above you see a translation of them into normal language. The first four don't need much tinkering with, but the fifth was originally stated thus:

"It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles."

Quite a mouthful, and clearly, in that context, NOT "self-evident".

Was there a way around this seeming contradiction? Apparently there still is not.

Quite a while after "Euclid's Fifth", several different people came forward with different interpretations of it. It turned out there were three. Einstein's general Relativity is based on one of the two others.

The three interpretations are really easy to state, but all of which are mutually exclusive:

Either,

a. parallel lines will always be parallel (Euclid),

or,

b. parallel lines will always meet (Riemann et al),

or,

c. parallel lines will always diverge (Riemann et al)

It was by using these "alternate" geometries, that Einstein came up with General Relativity in 1915.

From this acorn of an idea, there sprouted formalised geometry and a little thing explaining in full the anomaly of the precession of the perihelion of the planet Mercury to the tune of 43 seconds of arc per century.

Inifinities.....

...come in all shapes and sizes.

We all know what infinity is, don't we? Or do we?

Sure we do! It's where something goes on forever, never stopping, like the number of numbers. There is no highest number because they go on forever, right? And the decimal portion of Pi goes on forever, does it not? It begins 3.1415926... and never stops and never repeats.

We are so used to this concept of "infinity" that we are almost blase about it.

Allow enlightenment to take place.....

Suppose there is a hotel with an inifinite number of rooms, all occupied by guests.

What does the owner do when a new guest arrives? Why, he puts guest 1 in room 2, guest 2 in room 3, guest 3 in room 4....and so on up to infinity, leaving room 1 empty.

What does the owner do when an infinite number of new guests arrive? This one requires a little more thought.....

Simple, really. He puts guest 1 in room 2, guest 2 in room 4, guest 3 in room 6 and so on up to infinity, leaving an infinite number of "holes" ie vacant odd-numbered rooms. All guests are now occupying rooms. Does that mean the owner somehow now has 2 times an inifinite number of rooms? NO! he still has an infinite number of rooms with a guest in each!

It gets even stranger. Georg Cantor proved that there is more than one "type" of infinity. This guy was involved in set theory, the theory of certain classes of objects, such as the set of cars, the set of brown cars, the set of brown cars with at least 2 wheels, etc.

Sometimes sets can be inside each other, sometimes not. For instance, the set of Green Cars is different from the set of Brown cars, but above the third set "the set of brown cars with at least 2 wheels" is contained inside set 2 "the set of brown cars" AND set 1 "the set of cars".

Cantor took this a little further and argued that sets sometimes can contain themselves. For instance, the set of cars is not a car itself, but the set of all sets IS a set, so it is contained within itself. These are called "self-swallowing" sets. Another example is the set of ALL things that are not brown cars.

So, You have ordinary sets eg "the set of brown cars" which are not members of themselves, and self-swallowing sets eg "the set of all sets".

The question arises: Is the set of all ordinary sets ordinary or self-swallowing?

Inspection shows it is neither. Try it! Assume it is an ordinary set.

So, the set of all ordinary sets belongs to neither group.

The argument put forward above is called the "diagonal" argument for reasons which are not at all clear. Here comes Cantor's same argument, but in the form he proposed.

To begin, suppose we wish to count all numbers. Not just integers such as 1, 2, 3 etc, but ALL numbers, including those with decimals. Well, one of them is pi = 3.1415926.... Another is e = 2.718281828....... and ALL the others in between and around , such 9.80665, and 5.5, 100.99999 and 2.9979. Suppose we list them all in order. For simplicity, let's just consider all those less than 1.

It would look something like

0.1111111111111111....>>>> to infinity
0.2222222222222222....>>>> to infinity
0.3333333333333333....>>>> to infinity....

and suppose our table lists them ALL, one after the other. Obviously there are an infinite number. There is a proof that there is at least one number NOT in our table of the infinitely many "numbers".

It goes like this:

Take the first number, 0.1111111111111.....and change the first column to a "2" creating 0.2111111111111....

Take the second number, 0.222222222222.... and change the SECOND column to a "3" creating 0.2322222222222....

Do the same with the third number. So we will have

0.2111111111111111....>>>> to infinity
0.2322222222222222....>>>> to infinity
0.3343333333333333....>>>> to infinity....

where I have changed the third column in the third number to a "4". Now look at the diagonal. It says 0.234.......to infinity.

It is clear that this number is NOT in our list, as the first column is different from number 1, the second is different for number 2, and so on up to infinity. For any "n", the digit in column n is different from the nth digit of the number "n", so our "made-up" number, derived from adding 1 to the diagonal is absolutely NOT in the list.

But didn't we say that our list was of ALL the numbers all the way to infinity?

The reasoning is that there MUST be another "kind" of inifinity somehow "bigger" than the one we had. In fact, there is not only "one" different kind of infinity, but an infinite many of them, each, in some abstract sense, "bigger" than the last.

This result is true and has had many implications for math, especially for that man Godel again, who said that not everything is proveable within the system. This is a paraphrase of that theorem.

Find the Queen..revisited.

This question first appeared on this blog a couple of years back.

Check out my 10th July 2005 post:

"Five Clues - One for each 10 of 50"

The question was this...

You have three cards, (two aces and a queen), all face down. You are asked to find the queen, and you are asked to pick one card only. Then, the magician looks at them all, and promptly turns face up an ace. What is the probability that your card is the queen?

The answer is a third, since you made the choice when all three were face down.

NOW consider the same scenario...

You have three cards, (two aces and a queen), all face down. You are asked to find the queen, and you are asked to pick one card only. Then, the magician looks at them all, and promptly turns face up, an ace.

Your delicate finger is still on the card you chose before any were revealed, and YOU KNOW the probability is 1/3 that yours is the queen.

So, you have 3 cards, two face down, and an ace face up. You have tiny finger on your face down card, knowing that it is 1/3 that it is the one queen.

Now, lift you finger up, really carefully. You have 3 cards, one face up, an ace, and two face down, the other ace and the queen. You are then asked to make your choice again, so you put your same finger down, ever more carefully, ON THE SAME CARD YOU CHOSE BEFORE.

NOW, what is the probability that your card is the queen?

The astonishing answer is that it is now 1/2.

So....what changed between your lifting your finger off the card, to replacing it back, to cause this change in probability?

Moreover, is the answer different if you had chosen the other card instead?

Comments welcome.

Just what IS the difference between left and right?

Beware: This may damage your health!

The Mathematical equations that physics gives us tell us that Charge (C), Time (T) and Parity (P) ("left-rightness") are equally represented.

That is, you can each reverse all three quite easily and the equations will still look, and behave, the same as before.

The answer to the question "What is the difference between "right" and "left"?" is therefore this : there IS no difference.

Indeed, if an alien tells us he is "blurk-handed" does he mean our left or our right? There is no way of knowing. To most people, this makes sense, even though "left' and "right" "seem" different, there is no descriptive way to distinguish one from the other.

However, I DID say "to most people". Ask anyone who knows any quantum mechanics, and she will say immediately "Yes, there IS a difference". This is because the difference is only visible at quantum level, and it is called CPT violation, discovered by Madam Wu in the 1950's (see History of Science: Parity Violation) . Basically, it is the way an isotope of Cobalt behaves.

It always emits electrons in the same direction, no matter which way you point it. If you turned it upside-down, it should give off the elctrons in the opposite direction, but it doesn't.

So all the alien has to do is say, "my blurk (or sklurk) hand is in the direction of emission of the cobalt electrons", and we will know which of OUR hands he is referring to.

The upshot is that nature is right-handed (or left-handed, we cannot tell which!).

Now that we have proved that left/right are unequal, to cancel it out we should prove that black/white are equal, but we should be doubly careful of those "Ped Xing"s.

Beware of the Phone!

In England, where I am based now (Oxford University, actually), there is a major Government policy towards people who DO NOT smoke cigarettes, cigars, pipes, etc in public placees.

It is now law for places like pubs and restaurants to have designated areas for those people who do not indulge.

The reason, which has only come to light because of research, is that cigarettes give off smoke, as does the smoker, which is pssively inhaled by those who do not smoke if they share the same space on a bus, etc.

It is a phenomenon called "passive smoking" and has been proven to cause death just like normal smoking does, but taking longer.

The late, great Roy Castle, who came from Liverpool and who was a great comedian, entertainer and dancer (he was once in the Guinness Book of World Records for recording a song during which the performer played the most solo musical intruments), died from passive smoking, having performed, in his youth, in many, smokey Jazz bars and clubs. His speciality was the trumpet.

However, he was diagnosed with lung cancer owing to smoking, even though he had never touched a cigarette in his life. A research center in Liverpool for this condition was built in his memory.

It is now a fact of life in some places and rules have been introduced to countermand the practice of "passive smoking".

Now we come to the strange effects that perhaps mobile phones have on us. Researchers claim that overuse of the mobile phone, or cellphone, will harm the brain, owing to the proximity between the head and the microwaves entering the units. Of course, effects will not become apprarent for another 25 years or so, just like smoking kills in that time. However, despite this, it hasn't stopped so-called "eco-aware" companies from selling us hands-free headphones/earplugs so that we can talk without the handset being close to our heads.

Far-sighted, or just plain politics gone mad?

Either way, this writer is willing to hazard a bet that there will eventually be areas designated especially for those who do not use cellphones, not because of the noise all these talkers produce, but because of the "passive radiation" they would be giving the rest of us.

What may they call it - - - - "passive personal microwaving"?

On Firmly Inedecipherable Prepositions of Mathematics and Related Systems

For those who like logic paradoxes, this one is a beaut, Bruce. It is called

THE PARADOX OF THE UNEXPECTED HANGING.

It concerns a condemned prisoner and his Judge. The man, already sentenced to death by hanging, is told two things by the Judge.

1) “You will be hanged at noon on one day of next week, Monday through Friday.”
2) “The execution will be a surprise to you: You won’t know the day of the hanging until the executioner knocks on your cell door at 1155 on the day of the hanging.”

In his cell, the prisoner reflects thus:

“I can’t be hanged on Friday, since by Thursday night I will know that I will be hanged Friday, so it won’t be a surprise. “

He smiles.

“So if the hanging is not on Friday, it cannot be on Thursday either, since by Wednesday night I will also know that my execution will be on Thursday. “

His smile broadens.

It broadens further when he realises he can extend this logic right up to Monday, coming to the fabulous conclusion that he cannot possibly be hanged at all, because the Judge apparently has made an erroneous statement.

He is really smiling now.

However, he actually has to go beyond surprise into astonishment when, on Tuesday at 1155, the executioner turns up and he is hanged at noon, expiring with the most comical look of surprise on his face you have ever seen.

So what was wrong with the prisoner’s argument? Did the Judge make an erroneous statement? The sentence was carried out to the letter, all in accordance with the Judge’s two statements, yet the prisoner’s logic seems ironclad.

Much has been written about this paradox, and I have my own take on it.

However, surprisingly enough, I will await the comments of those in the know before divulgence.