5) Proofs and papers

Methods of proof I: generalized logic

In the last hundred years or so, mathematics has undergone a tremendous growth in size and complexity and subtlety. This growth has given rise to a demand for more flexible methods of proving theorems than the laborious, difficult, pedantic, “rigorous” methods previously in favor. This demand has been met by what is now a well-developed branch of mathematics known as Generalized Logic. I won’t develop the theory of Generalized Logic in detail, but I must introduce some necessary terms. In Classical Logic, a Theorem consists of a True Statement for which there exists a Classical Proof. In Generalized Logic, we relax both of these restrictions: a Generalized Theorem consists of a Statement for which there exists a Generalized Proof. I think that the meaning of these terms should be sufficiently clear without the need for elaborate definitions.

The applications of Generalized Proofs will be obvious. Professional authors of textbooks use them freely, especially when proving mathematical results in Physics texts. Teachers and lecturers find that the use of Generalized Proofs enables them to make complex ideas readily accessible to students at an elementary level (without the necessity for the tutor to understand them himself). Research workers in a hurry to claim propriety for a new result, or who lack the time and inclination to be pedantic, find Generalized Proofs useful in writing papers. In this application, Generalized Proofs have the further advantage that the result is not required to be true, thus eliminating a tiresome (and now superfluous) restriction on the growth of mathematics.

I want now to consider some of the proof techniques which Generalized Logic has made available. I will be concerned mostly with the ways in which these methods can be applied in lecture courses — they require only trivial modifications to be used in textbooks and research papers.

The reductio methods are particularly worthy of note. There are, as everyone knows, two reductio methods available: reductio ad nauseam and reductio ad erratum. Both methods begin in the same way: the mathematician denies the result he is trying to prove, and writes down all the consequences of this denial that he can think of. The methods are most effective if these consequences are written down at random, preferably in odd vacant corners of the blackboard.

Although the methods begin in the same way, their aims are completely different. In reductio ad nauseum, the lecturer’s aim is to get everyone in the class asleep and not taking notes. (The latter is a much stronger condition.) The lecturer then has only to clean the blackboard and announce, “Thus we arrive at a contradiction, and the result is established.” There is no need to shout this — it is the signal for which everyone’s subconscious has been waiting. The entire class will awaken, stretch, and decide to get the last part of the proof from someone else. If everyone had stopped taking notes, therefore, there is no “someone else,” and the result is established.

In reductio ad erratum the aim is more subtle. If the working is complicated and pointless enough, an error is bound to occur. The first few such mistakes may well be picked up by an alternative class, but sooner or later one will get through. For a while, this error will lie dormant, buried deep in the working, but eventually it will come to the surface and announce its presence by contradicting something which has gone before. The theorem is then proved.

It should be noted that in reductio ad erratum the lecturer need not be aware of this random error or of the use he has made of it. The best practitioners of this method can produce deep and subtle errors within two or three lines and surface them within minutes, all by an instinctive process of which they are never aware. The subconscious artistry displayed by a really virtuoso master to a connoisseur who knows what to look for can be breathtaking.

There is a whole class of methods which can be applied when a lecturer can get from is premises P to a statement A, and from another statement B to the desired conclusion C, but he cannot bridge the gap from A to B. A number of techniques are available to the aggressive lecturer in this emergency. He can write down A, and without any hesitation put “therefore B.” If the theorem is dull enough, it is unlikely that anyone will question the “therefore.” This is the method of Proof by Omission, and is remarkably easy to get away with — sorry, remarkably easy to apply with success.

Alternatively, there is the Proof by Misdirection, where some statement that looks rather like “A, therefore B” is proved. A good bet is to prove the converse “B, therefore A”: this will always satisfy a first-year class. The Proof by Misdirection has a countably infinite analogue, if the lecturer is not pressed for time, in the method of Proof by Convergent Irrelevancies.

Proof by Definition can sometimes be used: the lecturer defines a set S of whatever entities he is considering for which B is true, and announces that in the future he will be concerned only with members of S. Even an Honors class will probably take this at face value, without enquiring whether the set S might not be empty.

Proof by Assertion is unanswerable. If some vague waffle about why B is true does not satisfy the class, the lecturer simply says, “This point should be intuitively obvious. I’ve explained it as clearly as I can. If you still cannot see it, you will just have to think very carefully about it yourselves, and then you will see how trivial and obvious it is.”

The hallmark of Proof by Admission of Ignorance is the statement “None of the textbooks makes this point clear. The result is certainly true, but I don’t know why. We shall just have to accept it as it stands.” This otherwise satisfactory method has the potential disadvantage that somebody in the class may know why the result is true (or, worse, know why it is false) and be prepared to say so.

A Proof by Non-Existent Reference will silence all but the most determined troublemaker. “You will find a proof of this given in Copson on page 445,” which is in the middle of the index. An important variant of this technique can be used by lecturers in pairs. Dr. Jones assumes a result which Professor Smith will be proving later in the year — but Professor Smith, finding himself short of time, omits that theorem, since the class has already done it with Dr. Jones…

Proof by Physical Reasoning provides uniqueness theorems for many difficult systems of differential equations, but it has other important applications besides. The cosine function for a triangle, for example, can be obtained by considering the equilibrium of a mechanical system. (Physicists then reverse the procedure, obtaining the conditions for equilibrium of the system from the cosine rule rather than from experiment.)

The ultimate and irrefutable standby, of course, is the self-explanatory technique of Proof by Assignment. In a textbook, this can be recognized by the typical expressions “It can readily be shown that…” or “We leave as a trivial exercise for the reader the proof that…” (The words “readily” and “trivial” are an essential part of the technique.)

An obvious and fruitful ploy when confronted with the difficult problem of showing that B follows from A is the Delayed Lemma. “We assert as a lemma, the proof of which we postpone…” This is by no means idle procrastination: there are two possible denouements. In the first place, the lemma may actually be proved later one, using the original theorem in the argument. This Proof by Circular Cross-Reference has an obvious inductive generalization to chains of three or more theorems, and some very elegant results arise when this chain of interdependent theorems become infinite.

The other possible fate of a Delayed Lemma is the Proof by Infinite Neglect, in which the lecture course terminates before the lemma has been proved. The lemma, and the theorem of which it is a part, will naturally be assumed without comment in future courses.

A very subtle method of proving a theorem is the Proof by Osmosis. Here the theorem is never stated, and no hint of its proof is given, but by then end of the course it is tacitly assumed to be known. The theorem floats about in the air during the entire course and the mechanism by which the class absorbs it is the well-known biological phenomenon of osmosis.

A method of proof which is regrettably little used in undergraduate mathematics is the Proof by Aesthetics: “This result is too beautiful to be false.” Physicists will be aware that Dirac uses this method to establish the validity of several of his theories, the evidence for which is otherwise fairly slender. His remark “It is more important to have beauty in one’s equations than to have them fit experiment” [1] has achieved certain fame.

I want to discuss finally the Proof by Oral Tradition. This method gives rise to the celebrated Folk Theorems, of which Fermat’s Last Theorem is an imperfect example. The classical type exists only as a footnote in a textbook, to the effect that it can be proved (see unpublished lecture notes of the late Professor Green) that… Reference to the late Professor Green’s lecture notes reveals that he had never actually seen a proof, but had been assured of its validity in a personal communication, since destroyed, from the great Sir Ernest White. If one could still track it back from here, one would find that Sir Ernest heard of it over coffee one morning from one of his research students, who had seen a proof of the result, in Swedish, in the first issue of a mathematical magazine which had never produced a second issue and is not available in the libraries. And so on. Not very surprisingly, it is common for the contents of a Folk Theorem to change dramatically as its history is investigated.

I have made no mention of Special Methods such as division by zero, taking the wrong square roots, manipulating divergent series, and so forth. These methods, while very powerful, are adequately described in the standard literature. Nor have I discussed the little-known Fundamental Theorem of All Mathematics, which states that every number is zero (and whose proof will give the interested reader many hours of enjoyment, and excellent practice in the use of the methods outlined above). However, it will have become apparent what riches there are in the study of Generalized Logic, and I appeal to Mathematics Departments to institute formal courses in this discipline. This should be done preferably at undergraduate level, so that those who go teaching with only a Bachelor’s degree should be familiar with the subject. It is certain that in the future nobody will be able to claim a mathematical education without a firm grounding in at least the practical applications of Generalized Logic.

Notes

[1] P. A. M. Dirac, “The evolution of the physicist’s picture of nature,” Scientific American, Amy 1963, p 47.

— (c) Paul V. Dunmore
“The uses of fallacy”
The New Zealand Mathematics Magazine, 7, 15 (1970).

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Methods of proof II: a guide for lecturers

Proof by vigorous handwaving.
Works well in a classroom or seminar setting.

Proof by forward reference.
Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.

Proof by funding.
How could three different government agencies be wrong?

Proof by example.
The author gives only the case n = 2 and suggests that it contains most of the ideas of the general proof.

Proof by omission.
Examples include: “The reader may easily supply the details,” or “The other 253 cases are analogous,” etc.

Proof by deferral.
“We’ll prove this later in the course.”

Proof by picture.
A more convincing form of proof by example. Combines well with proof by omission.

Proof by intimidation.
“Trivial.”

Proof by cumbersome notation.
Best done with access to at least four alphabets and special symbols.

Proof by exhaustion.
An issue or two of a journal devoted to your proof is useful.

Proof by obfuscation.
A long plotless sequence of true and/or meaningless syntactically related statements.

Proof by wishful citation.
The author cites the negation, converse, or generalization of a theorem from the literature to support his claims.

Proof by eminent authority.
“I saw Karp in the elevator and he said it was probably NP-complete.”

Proof by personal communication.
“Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication].”

Proof by reduction to the wrong problem.
“To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem.”

Proof by reference to inaccessible literature.
The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.

Proof by importance.
A large body of useful consequences all follow from the proposition in question.

Proof by accumulated evidence.
Long and diligent search has not revealed a counterexample.

Proof by cosmology.
The negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God.

Proof by mutual reference.
In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown to follow from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.

Proof by metaproof.
A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.

Proof by vehement assertion.
It is useful to have some kind of authority relation to the audience.

Proof by ghost reference.
Nothing even remotely resembling the cited theorem appears in the reference given.

Proof by semantic shift.
Some of the standard but inconvenient definitions are changed for the statement of the result.

Proof by appeal to intuition.
Cloud-shaped drawings frequently help here.

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On the nature of mathematical proofs

Bertrand Russell has defined mathematics as the science in which we never 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.

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.

— (c) Joel Cohen
Opus, May 1961

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Applied perjorative calculus

Theorem. Circular reasoning works.

Proof.

Q.E.D.

Theorem. Zero equals unity, i.e. 0 = 1.

Proof.

Q.E.D.

or, more generally,

Theorem. Any integer equals its successor, i.e. n = n+1 any integer n.

Proof.

Q.E.D.

Theorem. All positive integers an interesting property.

Proof. Assume the contrary. Let A be the lowest non-interesting positive integer. But, hey, that’s a pretty interesting property about A. Hence, we have a contradiction. Q.E.D.

In the same spirit,

Theorem. Every natural number can be unambiguously described in 14 words or less.

Proof. Suppose there is some natural number which cannot be unambiguously described in fourteen words or less. Then there must be a smallest such number. Let’s call it n.

But now n is “the smallest natural number that cannot be unambiguously described in fourteen words or less.” This is a complete and unambiguous description of n in fourteen words, contradicting the fact that n was supposed not to have such a description!

Therefore, all natural numbers can be unambiguously described in fourteen words or less. Q.E.D.

Theorem. All positive integers are equal.

Proof. It suffices to show that for any two positive integers, A and B, A = B. Further, it suffices to show that for all N > 0, if A and B are positive integers which satisfy MAX(A, B) = N then A = B.

We proceed by induction. If N = 1, then A and B, being positive integers, must both be 1. So A = B.

Assume that the theorem is true for some value k. Take A and B with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence (A-1) = (B-1). Consequently, A = B.

Hence, A = B for all positive integers A and B by induction. Q.E.D.

Theorem. The integral of cosine over its first period is zero.

Proof.

Q.E.D.

This was an honest-to-God answer that a calculus student wrote on a final exam! I kid you not — I shared an office with the lucky guy who graded it at the time. As an interesting post script to this story, the student actually demanded he be awarded partial credit, because he got the right answer.

Theorem. A peanut-butter and jelly sandwich is better than life itself.

Proof. A peanut-butter and jelly sandwich is better than nothing.

Nothing is better than life itself.

By transitivity, a peanut-butter and jelly sandwich is better than life itself. Q.E.D.

Theorem. A sheet of paper is a lazy dog.

Proof. A sheet of paper is an ink-lined plane.

An inclined plane is a slope up.

A slow pup is a lazy dog.

By transitivity, a sheet of paper is a lazy dog. Q.E.D.

Theorem. A cat has nine tails.

Proof. No cat has eight tails.

A cat has one tail more than no cat.

Therefore, a cat has nine tails. Q.E.D.

The Salary Theorem. The less you know, the more money you make.

Proof. It is known in universities that knowledge is power, hence

Similarly, it is known in business that time is money, whence

From physics, we have by definition that power is the ratio of work to time, so that

Making the substitutions above, we have

Solving for money, we get:

Thus, as knowledge approaches zero, money approaches infinity, regardless of the amount of work done. Q.E.D.

Theorem. A dollar is equal to one penny.

Proof. We shall use the conventional physics notation and write units behind the numbers. That is, 1 $ means 1 dollar, and 5 c means five cents. Proceeding:

1 $ = 100 c = (10 c)²= (0.1 $)² = 0.01 $ = 1 c.

Q.E.D.

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Analysis of contemporary music using harmonious oscillator wave functions

H. J. Lipkin
Department of Musical Physics
Weizmann Institute of Science

The importance of Harmonious Oscillation in music was well known [1] even before the discovery of the Harmonious Oscillator by Stalminsky [2]. Evidence for shell structure was first pointed out by Haydn [3], who discovered the magic number four and proved that systems containing four musicleons possessed unsual stability [4]. The concept of the magic number was expressed by Mozart, who introduced the ‘Magic Flute’ [5], and a Magic Mountain was later introduced by Thomas Mann [6]. A system of four magic flutes playing upon a magic mountain would be triply magic. Such a system is probably so stable that it does not interact with anything at all, and is therefore unobservable. This explains the fact that doubly and triply magic systems have never been observed.

A fundamental advance in the application of spectroscopic techniques to music is due to Rachmaninoff [7], who showed that all musical works can be expressed in terms of a small number of parameters, A, B, C, D, E, F, and G, along with the introduction of Sharps [8]. Work along lines similar to that of Rachmaninoff has been done by Wigner, Wagner, and Wigner [9] using the Niebelgruppentheorie. Relativistic effects have been calculated by Bach, Feshbach, and Offenbach, using the method of Einstein, Infeld, and Hoffman [10].

There has been no successful attempt thus far to apply the Harmonious Oscillator to modern music. The reason for this failure, namely that most modern music is not harmonious, was noted by Wigner, Wagner, and Wigner [11].

A more unharmonious approach is that of Brueckner [12], who uses plane waves instead of harmonious oscillator functions. Although this method shows great promise, it is applicable strictly speaking only to infinite systems. The works of the Brueckner School are thus suitable only for very large ensembles.

A few very recent works should also be mentioned. There is the Nobel-Prize-winning work of Bloch [13] and Purcell [14] on unclear resonance and conduction. The work of Primakofiev should be noted [15], and of course the very fine waltzes presented by Strauss [16] at the ‘Music for Peace’ Conference in Geneva.

References

[1] G. F. Handel, The Harmonious Blacksmith (london, 1757)

[2] Igar Stalminsky, Musical Spectroscopy with Harmonious Oscillator Wave Functions, Helv. Mus. Acta. 1 (1801) 1

[3] J. Haydn, The alpha-Particle of Music; the String Quartet Op 20 (1801) No 5

[4] A. B. Budapest, C. D. Paganini, and E. F. Hungarian, Magic Systems in Music

[5] W. A. Mozart, A Musical Joke, K234567767 (1799)

[6] T. Mann, Joseph Haydn and His Brothers (Interscience, 1944)

[7] G. Rachmaninoff, Sonority and Seniority in Music (Invited Lecture, International Congress on Musical Structure, rehovoth, 1957)

[8] W. T. Sharp, Tables of Coefficients (Chalk River, 1955)

[9] E. Wigner, R. Wagner, and E. P. Wigner, Der Ring Die Niebelgruppen. I Siegbahn Idyll (Bayrut, 1900)

[10] J. S. Bach, H. Feshbach, and J. Offenbach, Tales of Einstein, Infeld and Hoffman (Princeton, 1944)

[11] E. P. Wigner, R. Wagner, and E. Wigner, Gotterdammerung!! and other unpublished remarks made after hearing ‘Pierrot Lunaire’

[12] A. Brueckner, W. Walton, and Ludwig von Beethe, Effective Mass in C Major

[13] E. Bloch, Schelomo, an Unclear Rhapsody

[14] H. Purcell, Variations on a Theme of Britten (A Young Person’s Guide to the Nucleus)

[15] S. Primakofiev, Peter and the Wolfram-189

[16] J. Strauss, The Beautiful Blue Cerenkov Radiation; Scient’s Life; Wine, Women and Heavy Water; Tales from the Oak Ridge Woods

— (c) Proceedings of the Rehovoth Conference on Nuclear Structure, held at the Weizmann Institute of Science, Rehovoth, September 8-14, 1957.

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Antigravity

If you drop a buttered piece of bread, it will fall on the floor butter-side down. If a cat is dropped from a window or other high and towering place, it will land on its feet.

But what if you attach a buttered piece of bread, butter-side up to a cat’s back and toss them both out the window? Will the cat land on its feet? Or will the butter splat on the ground?

Even if you are too lazy to do the experiment yourself you should be able to deduce the obvious result. The laws of butterology demand that the butter must hit the ground, and the equally strict laws of feline aerodynamics demand that the cat can not smash its furry back. If the combined construct were to land, nature would have no way to resolve this paradox. Therefore it simply does not fall.

Thus has man discovered the secret of antigravity! A buttered cat will, when released, quickly move to a height where the forces of cat-twisting and butter repulsion are in equilibrium. This equilibrium point can be modified by scraping off some of the butter, providing lift, or removing some of the cat’s limbs, allowing descent.

Evidence suggests that most of the civilized species of the Universe already use this principle to drive their ships while within a planetary system. The loud humming heard by most sighters of UFOs is, in fact, the purring of several hundred tabbies.

The one obvious danger is, of course, if the cats manage to eat the bread off their backs they will instantly plummet. Of course the cats will land on their feet, but this usually doesn’t do them much good, since right after they make their graceful landing several tons of red-hot starship and extremely pissed-off aliens crash on top of them. No resolution to this problem is yet known.

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Applied Murphy’s Law

Murphy’s Law, in its simplest form, states that “If anything can go wrong, it will.” Or to state it in more exact mathematical form:

where ↵ is the mathematical symbol for “hardly ever.”

To show the all-pervasive nature of Murphy’s work, the author offers a few applications of the law to the electronic engineering industry.

Mathematics

1. Any error that can creep in, will. It will be in the direction that will do the most damage to calculations.

2. All constants are variables.

3. In a complicated calculation, one factor from the numerator will always move into the denominator, or conversely.

General engineering

4. A patent application will be preceeded by one week by a similar application made by an independent worker.

5. The more innocuous a design chage appears, the further its influence will extend.

6. All warranty and guarantee clauses become void on payment of invoice.

7. An important Instruction Manual or Operating Manual will have been discarded by the Receiving Department.

Prototyping and production

8. Any wire cut to length will be too short.

9. Tolerances will accumulate unidirectionally towards maximum difficulty of assembly.

10. Identical units tested under identical conditions will not be identical in the field.

11. If a project requires n components, there will be (n – 1) units in stock.

12. A dropped tool will land where it can do most damage; the most delicate component will be the one to drop. (Also known as the principle of selective gravity.)

13. A device selected at random from a group having 99 percent reliability will be a member of the 1 percent group.

14. A transistor protected by a fast-acting fuse will protect the fuse by blowing first.

15. A purchased component or instrument will meet its specifications long enough, and only long enough, to pass Incoming Inspection.

16. After an access cover has been secured by 16 hold-down screws, it will be discovered that the gasket has been omitted.

(c) “The Contribution of Edsel Murphy to the Understanding of Behaviour in Inanimate Objects,” in EEE: The Magazine of Circuit Design,

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An engineering analysis of Santa Claus

I. There are approximately two billion children (persons under 18) in the world. However, since Santa does not visit children of Muslim, Hindu, Jewish or Buddhist religions, this reduces the workload for Christmas night to 15% of the total, or 378 million (according to the Population Reference Bureau). At an average (census) rate of 3.5 children per house hold, that comes to 108 million homes, presuming that there is at least one good child in each.

II. Santa has about 31 hours of Christmas to work with, thanks to the different time zones and the rotation of the earth, assuming he travels east to west (which seems logical). This works out to 967.7 visits per second. This is to say that for each Christian household with a good child, Santa has around 1/1000th of a second to park the sleigh, hop out, jump down the chimney, fill the stockings, distribute the remaining presents under the tree, eat whatever snacks have been left for him, get back up the chimney, jump into the sleigh and get on to the next house.

Assuming that each of these 108 million stops is evenly distributed around the earth (which, of course, we know to be false, but will accept for the purposes of our calculations), we are now talking about 0.78 miles per household; a total trip of 75.5 million miles, not counting bathroom stops or breaks. This means Santa’s sleigh is moving at 650 miles per second — 3,000 times the speed of sound. For purposes of comparison, the fastest man-made vehicle, the Ulysses space probe, moves at a poky 27.4 miles per second, and a conventional reindeer can run (at best) 15 miles per hour.

III. The payload of the sleigh adds another interesting element. Assuming that each child gets nothing more than a medium sized Lego set (two pounds), the sleigh is carrying over 500 thousand tons, not counting Santa himself. On land, a conventional reindeer can pull no more than 300 pounds. Even granting that the “flying” reindeer could pull ten times the normal amount, the job can’t be done with eight or even nine of them Santa would need 360,000 of them. This increases the payload, not counting the weight of the sleigh, another 54,000 tons, or roughly seven times the weight of the Queen Elizabeth (the ship, not the monarch).

IV. 600,000 tons traveling at 650 miles per second creates enormous air resistance — this would heat up the reindeer in the same fashion as a spacecraft re-entering the earth’s atmosphere. The lead pair of reindeer would absorb 14.3 quintillion joules of energy per second each. In short, they would burst into flames almost instantaneously, exposing the reindeer behind them and creating deafening sonic booms in their wake. The entire reindeer team would be vaporized within 4.26 thousandths of a second, or right about the time Santa reached the fifth house on his trip. Not that it matters, however, since Santa, as a result of accelerating from a dead stop to 650 m.p.s.. in .001 seconds, would be subjected to centrifugal forces of 17,500 g’s. A 250 pound Santa (which seems ludicrously slim) would be pinned to the back of the sleigh by 4,315,015 pounds of force, instantly crushing his bones and organs and reducing him to a quivering blob of pink goo.

V. Therefore, if Santa did exist, he’s dead now.

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On the imperturbability of elevator operators: LVII

S. Candlestickmaker
Institute for Studied Advances
Old Cardigan, Wales

Abstract

In this paper the theory of elevator operators is completed to the extent that is needed in the elementary theory of Field’s. It is shown that the matrix of an elevator operator cannot be inverted, no matter how rapid the elevation. An explicit solution is obtained for the case when the occupation number is zero.

1. Introduction

In an earlier paper (Candlestickmaker 1954q; this paper will be referred to hereafter as ‘XXXVIII’) the simultaneous effect of a magnetic field, an electric field, a Marshall field, rotation, revolution, translation, and re-translation on the equanimity of an elevator operator has been considered. However, the discussion in that paper was limited to the case when incivility sets in as a stationary pattern of dejection; the alternative possibility of over-civility was not considered. The latter possibility is known to occur when a Marshall field alone is present; and its occurrence has been experimentally demonstrated by Shopwalker and Salesperson (1955) in complete disagreement with the theoretical predictions (Nostradamus 1555). The possibility of over-civility when no Marshall field is present has also been investigated (Candlestickmaker 1954t); and it has been shown that with substances such as U and I it cannot occur. It is therefore a matter of some importance that the manner of the onset of incivility be determined. This paper is devoted to this problem.

2. The reduction to a twelfth-order characteristic value problem in case operators A, B, and C are looking in the same direction

The notation is more or less the same as in XXXVIII:

Definitions:

γ = first occupant,
B_η = second occupant,
g_g = third occupant,
O = operator,
M(O) = matrix of the operator,
a = acceleration of the elevation of the conglomeration,
Ω_2l = critical Etage number for the onset of incivility,
Ω_2l2 = Ω_2l / π^11/7.

The basic equations of the problem on hand are (cf. XXXVIII, eqs. [429] and [587])

(1) ¶α / ¶β = γ ω + n Ѳ j

(2) (5 + π)B_η = a + b + c

(3) x = x

(4) g_g = m v² / 2 = 1

Using also the relation (Pythagoras 520)

(5) 3² + 42 = 5²

we find, after some lengthy calculations,

(6) |M| = 0,

which shows that the matrix of the operator cannot be inverted. The required characteristic values Ω_2l are the solutions of the equation (6). From the magnitude of the numerical work which was already needed for obtaining the solution for the purely rational case (cf. Candlestickmaker and Canna Helpit 1955) we may conclude that a direct solution of the characteristic value problem presented by equation (6) would be downright miraculous. Fortunately, as in XXXVIII, the problem can be solved explicitly in the case when the occupation number is zero. This is admittedly a case which has never occurred within living memory. However, from past experience with problems of this kind one may feel that any solution is better than none.

3. The equations determining the margin at state when the occupying number is zero

For the reasons just given (i.e, because we cannot solve any other problem) we shall restrict ourselves in this paper to a consideration of the cases when the occupation number is zero. In this case Ω_2l satisfies

(7) log Ω_2l = 1,

the solution of which has been obtained numerically; it is approximately

(8) Ω_2l » 2.7,

This result shows that the transition to over-civility occurs between the values 2 and 3 given by Giftcourt (1956), respectively, Bookshelf (1956), a result which should be capable of direct experimental confirmation. The author hopes to deal with this problem next Saturday afternoon.

In conclusion, I wish to record my indebtedness to Miss Canna Helpit, who carried out the laborious numerical work involved in deriving equation (8).

The research reported in the paper has in part been suppressed by the Office of Naval Research under Contract A1-tum-OU812 with the Institute for Studied Advances.

References

Bookshelf, M. F. 1956, J. Gen, Psychol., 237, 476.
Candlestickmaker, S. 1954a, Zool. Jahrb., 237, 476.
_____. 1954b, Parasitology, 237, 476.
_____. 1954c, Zentralbl. Bakt., 237, 476.
_____. 1954d, Trans. N.-E. Cst Inst. Engrs. Shipb., 237, 476.
_____. 1954e, R. C. Circ. mat. Palermo, 237, 476.
_____. 1954f, Adv. Sci., 237, 476.
_____. 1954g, Math. Japonicae, 237, 476.
_____. 1954h, Niol. Bull. Woods Hole, 237, 476.
_____. 1954i, Bull. Earthq. Res. Inst. Tokyo, 237, 476.
_____. 1954j, J. Dairy Sci., 237, 476.
_____. 1954k, Ann. Trop. Med. Parasitol, 237, 476.
_____. 1954l, Trab. Lab. Invest. biol. Univ. Madrid, 237, 476.
_____. 1954m, Cellule, 237, 476.
_____. 1954n, Bot. Gaz., 237, 476.
_____. 1954o, Derm. Zs., 237, 476.
_____. 1954p, J. Pomol., 237, 476.
_____. 1954r, Sci. Progr. twent. cent., 237, 476.
_____. 1954s, Portugaliae Math., 237, 476.
_____. 1954t, Abh. senckenb. naturf. Gesellsch., 237, 476.
Candlestickmaker, S., and Helpit, Canna E. 1955, Compositio Math., 237, 476.
Giftcourt. M. F. 1956, J. Symbolic Logic, 237, 476.
Nostradamus, M. 1955, Centuries (Lyons).
Pythagoras — 520, in: Euclid — 300, Elements, Book I, Prop. 47 (Athens).
Shopwalker, M., and Salesperson, F. 1955, Heredity, 237, 476.

— (c) John Sykes; October 19, 1910

Professor John Sykes’ famous spoof of Professor S. Chandrasekhar so delighted the ‘victim’ that he arranged to have it printed in the format of The Astrophysical Journal. Some librarians bound it in series without noticing.

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A note on piffles

— A. B. Smith

A. C. Jones in his paper “A Note on the Theory of Boffles,” Proceedings of the National Society, 13, first defined a Biffle to be a non-definite Boffle and asked if every Biffle was reducible.

C. D. Brown in “On a paper by A. C. Jones,” Biffle, 24, answered in part this question by defining a Wuffle to be a reducible Biffle and he was then able to show that all Wuffles were reducible.

H. Green, P. Smith, and D. Jones in their review of Brown’s paper, Wuffle Review, 48, suggested the name Woffle for any Wuffle other than the non-trivial Wuffle and conjectured that the total number of Woffles would be at least as great as the number so far known to exist. They asked if this conjecture was the strongest possible.

T. Brown, “A collection of 250 papers on Woffle Theory dedicated to R. S. Green on his 23rd Birthday” defined a Piffle to be an infinite multi-variable sub-polynormal Woffle which does not satisfy the lower regular Q-property. He stated, but was unable to prove, that there were at least a finite number of Piffles.

T. Smith, L. Jones, R. Brown, and A. Green in their collected works “A short introduction to the classical theory of the Piffle,” Piffle Press, 6 gns., showed that all bi-universal Piffles were strictly descending and conjectured that to prove a stronger result would be harder.

It is this conjecture which motivated the present paper.

— (c) A. K. Austin
“Modern Research in Mathematics”
Math. Gaz. 51 (May 1967) 150.

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On the supposed evilness of girls

— T. Komplexify, Ph.D.
Weizguyy Institute of Smart Axes

Abstract

In this article, we discuss the classic proof that girls are evil. The author will briefly discuss the origins of the problem and review the classic proof. The author then indicates a mathematical flaw in the argument, invalidating the statement. The article concludes with a revised and corrected statement of the result.

I. Introduction

I recently received an email discussing the differences between men and women from various mathematical and engineering points-of-view. Most of it was extremely funny, and sooner or later all shall certainly appear within the mathematico-humorist community, properly researched, and appended with standard references in the literature.

However, one portion of the email included a mathematical “proof” of the fact that girls are evil. This proof is doubtless familiar to many readers, having circulated a few times in mathematicians’ inboxes. However, for those readers unfamiliar with this well-known proof, we present it now.

II. Statement and classical proof

Theorem. Girls are evil.

Proof. It is axiomic in all cultures that girls require both time and money, and any man with either a deficiency in available “quality time” or “disposable income” knows that this a joint-proportion, whence

Similarly, it is has been proved that “time is money” [1], whence

Substitution yields

We also know that “money is the root of all evil” [2], whence

Substituting again yields

Squaring on the right-hand side of the equation yields

establishing the result. Q.E.D.

III. Identifying and resolving the flaw

The above “proof,” so-called, is widely known to mathematicians, leading to the widespread belief that girls are evil.

It will therefore come as a surprise to find that the proof above is flawed, and indeed, the result is incorrect. There is a subtle flaw in the above argument that seems to have escaped most diligent readers for quite some time. In the interest of correcting this mis-truth, which has improperly vilified girls as being evil, we present now the correct statement and its proof.

Theorem (Corrected). Girls are absolute evil.

Proof. Arguing as above allows us to conclude

However, let us more intently examine the consequences of money being the root of all evil. A moment’s thought shows that it is incorrect to conclude that

To see this, recall that evil is a inherently negative concept [3]. We cannot take square roots of negative quantities in the real world, lest we are will to assume that money is imaginary. (Graduate students in particular may choose to investigate this concept further [4].) Thus, we are therefore forced to conclude that

Substituting again yields

Squaring on the right-hand side of the equation yields

establishing that girls are absolutely evil. Q.E.D.

IV. Conclusion

We sincerely hope this clears things up.

V. Notes

1. I. Walker, “Time is money, professor proves,” CNN.com (2002) May 29
2. The Bible, King James Version (1611), I Timothy, Chapter 6, Verse 10
3. cf. Q. Smith, “An Atheological Argument from Evil Natural Laws,” (1991) Section 2.
4. This idea is explored somewhat in K. Marx, Das Kapital (1861).

The research reported in the paper has in part been suppressed by the National Silence Foundation.

— from komplexify.com

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The Thermodynamics of Heaven

The temperature of Heaven can be rather accurately computed. Our authority is in 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, viz. 300 K. This gives H as 798 K (525^o C).

The exact temperature of Hell cannot be computed, but it must be less than 444.6^o C, the temperature at which brimstone or sulfur 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^o C. (Above this temperature it would be a vapor, not a lake.)

We have, then, temperature of Heaven, 525^o C. Temperature of Hell, less than 445^o C. Therefore, Heaven is hotter than Hell.

(c) Applied Optics, vol. 11, A14, 1972.

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The Thermodynamics of Hell

A thermodynamics professor had written a take home exam for his graduate
students. It had one question:

Is Hell exothermic (gives off heat) or is it endothermic (absorbs heat)? Support your answer with proof.

Many of the students wrote proofs of their beliefs using Boyle’s Law (gas cools off when it expands and heats up when it is compressed). One student however gave the following answer: