I'd better explain first how this talk came to be given. Some time ago I gave a talk to the Quintics society on ''Paradoxes and Unplayable games,'' and while preparing that I came across a book by Augustus de Morgan, called ''A Budget of Paradoxes.'' In de Morgan's sense, a paradox is a viewpoint not commonly held: this doesn't always imply that it is incorrect, though often it may be spectacularly eccentric. De Morgan's book gives an extremely detailed catalogue of such paradoxes, some of which I shall mention, along with more recent examples of amateur mathematics.
Augustus de Morgan was born in 1806 and could therefore boast that he was aged n in the year n2, i.e. he was 43 in the year 1849. (Anyone born in 1980 will be able to make a similar claim.) He was educated at Trinity and then became a professor at University College, London, dying in 1871. Nowadays he is best remembered for de Morgan's laws in set theory:
where the ' denotes complement.
De Morgan's book shows that he had a wide range of other interests: for example he collects together various anagrams of his own name such, as ''Great Gun! Do us a sum!'' and ''Gus! Gus! A mature don!'' In collaboration with Whewell, Master of Trinity, he attempted ''Cabbala alphabetica'' -- what we would nowadays call pangrams -- namely using all 26 letters of the alphabet to form phrases. They tended to cheat by using i in place of j and u in place of v, however. One of their best is their advice to a young man: ''Get nymph; quiz sad brow; fix luck,'' which means ''Get married; be cheerful; watch your business.''
So let's look at some of these paradoxes from a modern point of view. In 1798 one Charles Palmer published a work entitled: ''A treatise on the sublime science of heliography, satisfactorily demonstrating our great orb of light, the sun, to be absolutely no other than a body of ice! Overturning all the received systems of the universe hitherto extant; proving the celebrated and indefatigable Sir Isaac Newton, in his theory of the solar system, to be as far distant from the truth, as any of the heathen authors of Greece or Rome.''
Mr Palmer's idea was that the Sun was a huge lens made of ice, focussing the radiance of God onto the earth. There seem to be certain difficulties in this theory, e.g. why doesn't the ice melt? but maybe some Science Fiction author could come up with a rationalization of it.
There is, or was, a real problem here: what is the true source of the sun's energy? Kelvin, a very distinguished physicist, later calculated that burning coal or other conventional fuels wouldn't produce enough heat and, living in the age before nuclear power was discovered, came to another erroneous conclusion: the energy came from the gravitational collapse of the sun. This theory has other unwanted consequences, e.g. the sun won't last for as long as we expected, but it was a very good try.
Knowing that the sun is made of ice, we might wish to turn our attention to comets, and read a book written in 1856, entitled: ''Comets considered as volcanoes, and the cause of their velocity and other phenomena thereby described.'' As de Morgan rather briefly puts it: ''The title explains the book better than the book explains the title.'' Well, I suppose this might make sense as Science Fiction a la Arthur C. Clarke (and indeed ''2061 -- Odyssey three'' is set largely on Halley's comet).
A more mathematical theory of astronomy was published by one D.T. Glazion (price one penny and probably overpriced), under the title ''Important discovery in astronomy, communicated to the Astronomer Royal, December 21st, 1846. That the Sun revolve round the Planets in 25748 2/5 years, in consequence of the combined attraction of the planets and their satellites, and that the Earth revolve round the Moon in 18 years and 228 days.'' I find it difficult to see how he gets these figures. If you were to neglect the mass of the earth (!) and calculate the period of orbit of a point orbiting the moon at the same radius, you would find that, because the moon's mass is 1/81 of the earth's, the orbital period of the earth round the moon should be 9 months. Maybe someone else can produce a calculation giving 18-odd years.
To be fair to Mr Glazion, even Einstein (and indeed Newton) got one or two things wrong. For example, Einstein did not realise that a static universe was impossible (this was pointed out by Friedmann in 1922), and tried several fudges to get round this problem. See Stephen Hawking's best-selling book ''A brief history of time'' for further details.
We might as well include one flat-earther in this discussion, the best I know being one S. Goulden, who published a work entitled ''Zetetic astronomy: Earth not a globe'' in 1857. His theory was that the earth is flat, surrounded by ice, and he apparently gained great praise from provincial newspapers for his ingenuity. The Leicester Advertiser thought that ''the statements would seem very seriously to invalidate some of the most important conclusions of modern astronomy,'' while the Norfolk Herald was clear that ''there must be a great error on one side or the other.''
Goulden seemed to have prepared himself for all possible objections: he could explain how the earth was circumnavigated, why a ship's hull disappears (when outward bound) before the mast-head, why a pendulum vibrates with less velocity at the Equator than the Pole, the cause of day and night, winter and summer, and so on. Apparently the distance of the Sun from London is 4,028 miles, which may explain some anomalies. He describes some of his work as a Challenge to Mathematicians, explains that the Moon is self-luminous, NOT a reflector, and finally that the Earth is the only material world and that it will ultimately be destroyed by fire.
A non-mathematical aside. It is not just amateur mathematicians who come up with unusual ideas. It would probably be tactless of me to mention names, but there are several professional mathematicians with eccentric views -- for example, one prominent member of the Pure Maths Department has recently been telling people of his theory that the Duke of Bedford was murdered by the Inland Revenue... I know also of at least one angle trisector who actually has a Ph.D. in Mathematics!
Isaac Frost (1846) decided that Newton had got it wrong: ''You will observe that Newton's system shows the earth traverses round the sun on an inclined plane; the consequence is, there are four powers required to make his system complete:
1st. The power of attraction. 2ndly. The power of repulsion. 3rdly. The power of ascending the inclined plane. 4thly. The power of descending the inclined plane.
You will thus easily see the four powers required, and Newton has only accounted for two; the work is therefore only half done.''
Newton was a popular victim of the paradoxers: the Count de Predaval in 1842 produced the ''Provisional Prospectus of the Double Acting Rotary Engine Company.'' This company wished to provide energy by means of a perpetual-motion machine: a drum with one vertical half in mercury, the other in a vacuum: the drum working round for ever to find an easy position. The prospectus promises ''Steam to be superseded: steam and electrical convulsions of nature never intended by Providence for the use of man.''
Designing perpetual motion machines is quite easy, but for some reason nobody ever gets round to building a working model. Let us leave the world of Physics with a result due to one William Pope which may explain why: ''The friction of the air is the cause of magnetism.''
There seem to be very few good independent theories of Probability, although the ''Almanach Romain sur la Loterie Royale de France'' collected all the drawings of the French lottery (two or three, each month) from 1758 to 1830. You were supposed to make your fortune by observing that any number that has not appeared for a long time must come up soon (a popular misinterpretation of the various Laws of Large Numbers).
Complex Numbers have always fascinated amateur mathematicians. This being an Adams Society talk it is appropriate to refer to Douglas Adams who in one of his ''Hitch-Hiker's Guide to the Galaxy'' books has Marvin, the super-intelligent but paranoid robot, revealing that he's just found out a new meaning to the square root of -1 (though we don't learn what it is). One F.H. Laing decided that the square roots of -a2 were +a and -a, because if you multiply +a by -a you get -a2. As de Morgan puts it: ''The author seems quite unaware of what has been done in the last fifty years.'' In fact mathematicians are quite happy with their notions of the square root of minus one, although the general public may believe otherwise.
Not all the ideas in de Morgan's book were as foolish as that last one. One J.W. Nystrom decided that all arithmetic, weights, measures and coins should adopt the ''tonal'' system, base sixteen. Mathematicians (who count on the 5 digits of each hand) have conventionally used 10 as the base: but computer scientists (who use the 4 fingers of each hand as well as their toes), have used 16 for some time, calling the sixteen digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The tonal system was to use Noll for Naught, and then count 'An, De, Ti, Go, Su, By, Ra, Me, Ni, Ko, Hu, Vy, La, Po, Fy, Ton; and then Ton-an, Ton-de etc.' So one rather large number comes out as ''Detam-memill-lasan-suton-hubong-ramill-Posanfy.''
In addition the sixteen months of the year were to be called:
Anuary, Debrian, Timander, Gostus, Suvenary, Bylian, Ratamber, Mesudius, Nictoary, Kolumbian, Husamber, Vyctorius, Lamboary, Polian, Fylander, Tonborius,
which is at least fairly poetic. How you divide 365 days into 16 months does not seem to be specified. Still, given that when I was young we used bases 12 and 20 for our money and we still use bases 16, 14, 8 and 20 for our weights and 12, 12, 3, 22, 10 and 8 for measures, the above proposal might have been quite reasonable.
And so finally to p, for which a lot of different values have been proposed, most since the days when Archimedes decided that it was between 3 10/71 and 3 1/7. Most people have heard that the state of Illinois tried to legislate that p was equal to 4 -- though I'm not sure if they intended to prosecute people for drawing circles whose circumferences were not four times the diameter (or selling rolls of wallpaper with similar properties?) More seriously, this is the first year in which Cambridge undergraduates will not be taught in any course that p is transcendental (i.e. not the solution of an algebraic equation a(n)xn + a(n-1)xn-1 + ... + a(1)x + a(0) = 0, with a(0), a(1), ..., a(n) all integers). I therefore expect to see a new generation of mathematically highly educated circle squarers!
There are many other values of p in the literature, including the square roots of 10 and g. One Royston G. Roadley-Battin recently turned up at the Pure Maths department with a 100-page document proving that p is equal to the square root of two plus 7/4 (which makes 3.164), and much earlier, in 1749, one M. de Fauré maintained that it was 256/81. He applied to John Bernouilli to attest his solution (consisting of a mighty collection of equations): Bernouilli replied in guarded (not to say obscure) terms that ''Following the suppositions of this Memoir, it is so evident that t must equal 34, y=1, and z=1, that this doesn't need either proof or authority to be recognised by all the world.'' Or, as de Morgan puts it, ''It would seem it is easier to square a circle than to get round a mathematician.''
When dealing with circle-squarers, many mathematicians adopt a ''one letter'' rule: it is generally agreed that the author is entitled to one letter, either explaining why p is known not to be the value proposed, or pointing out the mistake in the supposed argument. Any more than that, and the correspondence may never end. Editors of Journals are usually terser still, and write back along the lines of ''Thank you for your contribution: it has been filed.''
De Morgan's main antagonist was definitely one James Smith, who was very prolific of pamphlets proving that p is 25/8, and whose tone became less and less conciliatory. Three of his publications are as follows:
''The problem of squaring the circle solved, or the circumference and area of the circle discovered'' (1859),
''The British Association in Jeopardy, and Dr Whewell, the Master of Trinity, in the stocks without hope of escape'' (1865), and
''The British Association in Jeopardy, and Professor de Morgan in the Pillory without hope of escape'' (1866).
They don't write them like that any more, although a few years ago I did see a paper with the title ''Some errors of M.M. Rao.'' It was to do with Probability Theory and observed that Rao had produced several famous results in Probability Theory, but it should be pointed out that several of them were false as stated. It caused a mild controversy, with people writing in to defend Rao against his attacker (who may well have been right in this case). Personally I'd be quite happy to see certain eminent professors in the pillory without hope of escape...
Mr James Smith proved that p must be 25/8 by a powerful logical technique. Recall that for Reductio Ad Absurdum one proves a proposition A by assuming not-A, adding in B, known to be true, and deducing a contradiction. Mr Smith's technique might be dubbed Reductio De Absurdo -- it proves proposition A by assuming A, adding in B, known to be false, and deducing a contradiction. So he begins by postulating a circle with diameter 1 and circumference 25/8, after which he never looks back.
The following letter from Whewell to Smith is worth reproducing as a fine example of Victorian courtesy:
The Lodge, Cambridge, September 14th, 1862
Sir - I have received your explanation of your proposition that the circumference of the circle is to its diameter as 25 to 8. I am afraid I shall disappoint you by saying that I see no force in your proof: and I should hope that you will see that there is no force in it if your consider this:- in the whole course of the proof, though the word circle occurs, there is no property of the circle employed. You may do this: you may put the word hexagon or dodecagon or any other word describing a polygon in the place of Circle in your proof, and the proof would be just as good as before. Does this not satisfy you that you cannot have proved a property of that special figure -- a circle?
Or you may do this: calculate the side of a polygon of 24 sides inscribed in a circle. I think you are a Mathematician enough to do this. (An optimistic assumption (JRP)) You will find that if the radius of the circle be one, the side of the polygon is .264 etc. Now the arc which this side subtends is according to your proposition 3.125/12=.2604, and therefore the chord is greater than its arc, which you will allow is impossible.
I shall be glad if these arguments satisfy you, and
I am, Sir, your obedient Servant,
W. WHEWELL
Clearly this did not satisfy Mr Smith, as 3 years later he was writing pamphlets putting Whewell in the pillory!
De Morgan had more to say on the subject of paradoxers:
''If I had before me a fly and an elephant, having never seen more than one such magnitude of either kind; and if the fly were to endeavour to persuade me that he was larger than the elephant, I might possibly be placed in a difficulty. The apparently little creature might use such arguments about the effect of distance, and might appeal to such laws of sight and hearing as I, if unlearned in those things, might be wholly unable to reject. But if there were a thousand flies, all buzzing, to appearance, about the great creature; and, to a fly, declaring, each one for himself, that he was bigger than the quadruped; and all giving different and frequently contradictory reasons; and each one despising and opposing the reasons of the others -- I should feel quite at my ease. I should certainly say, My little friends, the case of each of you is destroyed by the rest. I intend to show flies in the swarm, with a few larger animals, for reasons to be given.''
More recently, Underwood Dudley has produced a book, named ''A Budget of Trisections,'' which clearly owes something to de Morgan. There weren't many trisections in the original book, and Prof. Dudley has made a fine collection. My favourite is from someone with initials E.S.D. (1975), who wrote to the German publishing firm Springer-Verlag, marking it ''Attention: Mr Verlag'' (the English equivalent might be to write to Penguin Books Limited, care of Mr Limited, or possibly Mr Penguin). This person also asserts that the circle is a 12,000 sided figure -- which changes the nature of Mathematics entirely!
A few days ago a letter reached the Pure Mathematics notice-board from one Zhang Lei in China. He has disproved the Four Colour Theorem (that every map on the plane can be coloured in at most four colours without colouring neighbouring regions the same), and for a mere $1,200,000 will send you the solution, with the proviso that you have three months in which to find a mistake. A better bargain was offered by the American who had proved Fermat's Last Theorem (that you can't solve xn + yn = zn in positive integers if n > 2) and was prepared to sell you the proof for only $1,000 ($5,000 if you wanted to publish it under your own name!)
The doyen of modern paradoxers is probably George N. Kayatta R.M. (the R.M. stands for ''Renaissance Man,'' a title he has adopted in his bid to get Nobel prizes in the Arts and Sciences simultaneously). His fundamental equation unites Einstein and Pythagoras in a wholly original manner:
He explains that p can have many different values, by pointing out that ''The velocity of p is not constant,'' and so relativistic effects allow expansion and contraction of p. It may be worth mentioning that if you take the speed of light (186,000 miles per second), normalise it to 1.86 and square it (obtaining c2) you obtain about 3.459, which is p + 1/p.
Kayatta has also translated the Bible into verse bearing a style all his own -- it took him 11 years. The beginning of Genesis reads:
In the Darkness of the God There was nothingness and Odd There was beingness and Void There was seeingness annoyed And therefore from the outer eye Flew the future of the Why
and he maintains this level of poetry throughout.
Kayatta's mathematics has been described thus: ''Theories that reveal devastating insights into the disconstancy of the speed of light, the sub-zero of zero, the new p, and the inaccuracy of computer analysis of megadistance and velocity. This may change the entire facade of mathematical reality.''
So what are the lessons to be drawn from this survey of paradoxers? Firstly, for obvious reasons, paradoxing is likely to be more successful if you try non-Mathematical subjects, such as claiming that Queen Elizabeth wrote Titus Andronicus, or whatever. Your views are less likely to be refuted. Secondly, being interested in Mathematics may imply a lack of sanity, but there is regrettably a lot of evidence for the converse implication as well.
Last updated on December 9th 2004 by Jonathan Partington