Does anyone (else) remember that proof that Julius Caesar had an infinite number of limbs?
Something like,
Julius Caesar was warned to beware the ides of March.
Now, forewarned is forearmed,
But four arms is certainly an odd number of arms for a man.
But four is also even.
And the only number that is both even and odd is infinity.
Therefore, Julius Caesar had an infinite number of arms.
I remember Mom reading us that proof some book decades ago, although I don't think I ever knew what the book was. Anyone remember? I seem to recall that it also proved that Alexander the Great did not exist and he rode a white horse named Bucephalus. And that heaven was hotter than hell (something about the brightness of seven stars).
That was from one of Martin Gardner's math books. It was Alexander the Great who was forewarned not to cross the river or he would die, and since being forewarned is forearmed (four-armed) that means he had six limbs, which is an odd number of limbs for a man, etc. etc., infinite number of limbs.
ReplyDeleteAlexander rode a black horse Bucephalus, but elsewhere in the book he had proved that all horses are white so therefore he did not exist, or something like that.
I think the Heaven being hotter than hell bit was because somewhere in the Bible it describes Heaven having the brightness of seven suns, and that hell has lakes of molten sulfer, from which temperature ranges are calculated, and it turns out sulfur vaporizes at a lower temp than the heat from seven suns.
(I have to break this up to fit into the Comments.)
ReplyDeleteThis is from a wonderful book called A Random Walk in Science: An anthology compiled by R L Weber (c1973, the Institute of Physics).
On the nature of mathematical proofs
By Joel E. Cohen
(Condensed from Opus, May 1961)
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).
Proof. 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 each been shown to be the same colour. It follows then 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.
Theorem 1. Every horse has an infinite number of legs. (Proof by intimidation.)
Proof. 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.
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.’
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 ‘The Stolen White Elephant’). Therefore all elephants are white. By corollary 1 everything is white.
Theorem 2. Alexander the Great did not exist and he had an infinite number of limbs.
ReplyDeleteProof. 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 ‘fore-warned 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.
It is not to be thought that there are not other types of proofs, which in print shops are recorded on 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.
This is also from A Random Walk in Science: An anthology compiled by R L Weber (c1973, the Institute of Physics).
ReplyDeleteHeaven is hotter than Hell
From Applied Optics, II, A14 (1972)
The temperature of Heaven can be rather accurately computed from available data. Our authority is the Bible: Isaiah 30:26 reads, Moreover the light of the Moon shall be as the light of the Sun and the light of the Sun shall be sevenfold, as the light of seven days. Thus Heaven receives from the Moon as much radiation as we do from the Sun and in addition seven times seven (forty-nine) times as much as the Earth does from the Sun, or fifty times in all. The light we receive from the Moon is a ten-thousandth of the light we receive from the Sun, so we can ignore that. With these data we can compute the temperature of Heaven. The radiation falling on Heaven will heat it to the point where the heat lost by radiation is just equal to the heat received by radiation. In other words, Heaven loses fifty times as much heat as the Earth by radiation. Using the Stefan-Boltzmann fourth-power law for radiation
(H/E)^4 = 50
where E is the absolute temperature of the Earth — 300K. This gives H as 798K (525°C).
The exact temperature of Hell cannot be computed but it must be less than 444•6° C, the temperature at which brimstone or sulphur changes from a liquid to a gas. Revelations 21:8: But the fearful, and unbelieving ... shall have their part in the lake which burneth with fire and brimstone. A lake of molten brimstone means that its temperature must be below the boiling point, which is 444•6°C. (Above this point it would be a vapour, not a lake.)
We have, then, temperature of Heaven, 525°C. Temperature of Hell, less than 445°C. Therefore, Heaven is hotter than Hell.
P.S. Joe and Dave, both of you have amazing memories!
ReplyDeleteThose are hilarious, more so now that I've studied number theory. I misremembered which book they were from, probably because I never read either of them.
ReplyDelete