Paradoxes

In logic, a paradox is a contradictory or implausible conclusion which seems to follow by valid argument from true premises. Aside from Zeno’s paradox, and Maxwell’s demon, proposed by James Clerk Maxwell, Erwin Schrödinger’s cat and Olber’s paradox, all dealt with in other places, the most interesting paradoxes are all logical challenges.

The paradox of the Spanish barber concerns a barber who is asked how business is doing. “Not badly,” says the barber. “I shave everybody in the village who does not shave himself.” The problem: who shaves the barber?

Logicians with a smattering of Greek will
share my delight in this Athenian street sign.

Epimenides of Crete is credited with one of the simplest paradoxes: “All Cretans are liars”, meaning by implication that this statement (made by a Cretan) was necessarily untrue. But if it is untrue, then not all statements made by Cretans are liars, and so on.

There is a more complex form of this paradox, consisting of two sentences. “The next sentence is false” and “The previous sentence is true”. Or "there are two erors in this sentence".

The dilemma of the crocodile: a crocodile seizes a child, but promises to let the child go, if the father guesses correctly whether he will do so or not. If the father offers as his guess the opinion that the crocodile will not return the child, what should an honest crocodile do?

A lawyer is trained by a teacher who says “you must pay me for your tuition after you win your first case”. Several years go by, during which time the teacher gets annoyed because the lawyer has yet to win a case, and sues the lawyer, saying “If I win my case, you must pay me, but if I lose, you have won your first case and must pay me.”

“Not so fast”, says the young lawyer. “If I lose the case, I have yet to win a case and need not pay you. But if I win, then by the court’s judgement, I do not have to pay you.” Does the lawyer have to pay?

Some words describe themselves, so “short” is a short word, but “long” is not a long word, “English” is an English word, but “German” is not a German word, and so on. We call words which describe themselves as autological, while words which do not describe themselves are called heterological.

But what about the word “heterological” — does it describe itself or not? If “heterological” is heterological, then it describes itself, and so it is autological. But if the word is autological, then that means it is a word that does not describe itself … or something.

Paradoxes can be useful ways of extending our knowledge, or at least ways of finding the right questions to ask. Fermi’s conjecture, also known as Fermi’s paradox, was offered by Enrico Fermi. In simple terms, it asks why, if the Galaxy is filled with intelligent and technological civilizations, haven’t they come to us yet?

There are several possible answers to this question (good taste on the ETs’ part, distance, or a recognition that contact with a superior civilisation is damaging to the more primitive one), but as we only have the vaguest idea what the right conditions for life and intelligence in our Galaxy, this paradox probably has no ready answer.

Paradoxes are also useful as a form of the mathematical proof called reductio ad absurdum, an argument which comes to an absurd or contradictory conclusion, hence showing that an initial assumption must be wrong. This shows up well in the so-called grandfather paradox of relativity. Imagine that your grandfather has just built a time machine, which you then use to go back in time, to give your grandfather the plans, so he can later build the time machine.

You reach him at a time where he has yet to meet your grandmother, he refuses to believe you, and in an argument, he steps out into a road, and is run over and killed by a passing car. He has now died before he met your grandmother, so you do not exist, since one of your parents does not exist, and the time machine does not exist, so you cannot be there in any case.

This is one of the arguments physicists use to support their belief in the causality principle. Others say the fact that we have never met time travellers proves that there will never be any, but this is probably one of the most useless of paradox types.

Let's get back to more practical stuff!

Zeno's paradox

Zeno of Elea was a philosopher with a wicked imagination, and he made up a puzzle which can be simply described like this. Suppose you have a hundred-metre race between a man called Achilles and a tortoise. Assume that Achilles runs ten times as fast as the tortoise, and that he gives the tortoise a ten-metre ‘start’.

Zeno said that Achilles can never catch the tortoise for while the man runs the first hundred metres, the tortoise waddles ten metres, and is still ahead. The man runs the extra ten metres, but the tortoise gains an extra metre.

As the man sprints desperately across that metre, the tortoise sneaks a further tenth of a metre, and while Achilles is lunging across that tenth of a metre, the tortoise drifts another centimetre, and so the human can never catch the tortoise. The same argument can be used to show that a thrown spear can never reach its target!

Zeno’s aim was to prove that something we can see happening is impossible, from which it follows that since we can see the impossible happening, our senses must be faulty. In other words, his paradoxes were designed to make people think. Later, Aristotle would argue against Zeno’s ideas, and Zeno’s assumption that space and time were infinitely divisible would make Democritus try to resolve the problem by suggesting that matter was not infinitely divisible, finally coming up with the idea of atoms—and all because Zeno believed the senses could not be trusted —because even though Zeno had proved that Achilles could never catch the tortoise, we know that in real life, he can!

Other paradoxes can be more fun...

That is to say: to be continued

Finding iodine

There are thirty recognized isotopes of iodine, but only one of these, iodine–127, is counted among the stable isotopes, and is found in nature. Radioactive iodine-125 is routinely used in tracing problems with the thyroid gland, and another isotope, iodine-131 has been commonly used to treat overactive thyroid conditions. The “iodine” which is commonly used on small wounds is tincture of iodine, a solution of potassium iodide and iodine in ethanol.

Iodine is element number 53 in the periodic table, atomic weight 126.90. This element was first isolated in 1811 by Bernard Courtois (1777 - 1838). 

Like the Germans in the First World war, the French found themselves restricted by a British naval blockade which stopped them accessing American sources of potash during the Napoleonic wars.  The potassium carbonate was used to make potassium nitrate for French gunpowder, but the seaweed also contained a variety of other chemicals, one of which was an iodide.

In treating seaweed ash with acid to get rid of sulfur compounds, Courtois noticed a purple vapour, which condensed to make crystals of iodine. He later passed this information on to Sir Humphry Davy, who proposed the name “iodine”, from the Greek word for the colour violet, iodes. The credit for suggesting the name is sometimes given to Joseph Gay-Lussac, but this is incorrect.

As mentioned above, iodine is needed in the production of thyroxin, and a deficiency in dietary iodine leads to goitre, so that foods (especially table salt and bread) in many parts of the world now have traces of iodine added, although this is unnecessary in areas where seafood is available.

Part of the hormone ‘picture’ was already there in 1905, because a number of diseases were linked to disorders in particular glands: goiter and cretinism were associated with an enlarged thyroid gland, but this was rightly regarded as a deficiency disease caused by a lack of iodine. Many folk remedies used iodised salts or sea foods rich in iodine, even before we knew iodine existed (the element was detected in 1813). Its role in preventing goitre became more obvious after Eugen Baumann (1849–1896) showed in 1896 that iodine was only concentrated in the thyroid gland.

Curiously, Courtois also discovered that major fascination for undergraduates of a certain kind, nitrogen triiodide, which forms tremendously unstable crystals that will even explode when hot water falls on them.

I have no intention of revealing how I discovered this fact, as I conclude now that I had a lucky escape: Pierre Dulong  lost three fingers and an eye investigating this substance — which may explain why, when he was formulating what is now called “Dulong and Petit’s Law”, he chickened out, and did not investigate tellurium, fraudulently manufacturing the data for that and several other elements.

The reason is probably that when you handle tellurium, it is absorbed, and you get “tellurium breath “. Not to mince words, you stink of stale garlic for months after working with tellurium compounds. Dulong  either feared that, or perhaps he was attached to his remaining fingers and wished to stay that way.

Everything (other than Dulong's fingers, perhaps) is connected.

The art of estimation

Like the previous entry, this comes from my (now) out-of-print volume, The Speed of Nearly Everything.  I may get around to releasing it as an e-book, if enough people think it's a good idea.

*

To a physicist, the notion of an immortal rabbit is quite acceptable. As a boy, my English teacher encouraged me to psychoanalyse Macbeth, even though I objected that we shouldn’t, since Freud hadn’t been invented when Shakespeare was writing. Ever a historically-minded cuss, I argued that it would be more relevant to look at the political situation in London, with a Scot sitting on the throne. Exasperated, he exhorted the class to engage in the willing suspension of disbelief.

And well he might, if he wanted us to accept some of the artifices and conceits of coincidence found in the 19th century novel, but we scientific types were subjected to much hardier fictional nonsense than that.

We routinely solved problems that involve a steel girder of negligible mass, suspended at its centre of gravity by a silken thread, and before we were too far advanced, we heard our first physics joke. It was about the three scientists who were trying to pick the winner of Australia’s premier horse race, the Melbourne Cup, which is held each November.

The mathematician gathers a wealth of data on weather, rainfall, wind, pollen counts and other possible influences, and three years in a row, fails dismally to pick a winner. At the end of those three years, the geneticist has just finished drafting a plan for a breeding program that should, in five generations, produce a winner, but the physicist has got it right, three times in a row.

The others ask him how he did it. He reaches into his pocket and produces an envelope which he turns over. Then he draws a circle on it. “Consider,” he says, “a spherical horse running in a vacuum…”

In fact a spherical cow or spherical horse can be a useful starting point to explore ideas, to get a first approximation that can be extended. Take the yarn about the bumblebee that was shown not to be able to fly: this is usually trotted out as evidence that scientists are thick, but there is a little more to it than that. In 1934, a French entomologist called Antoine Magnan tried to apply an engineer’s equation to bumblebees, and showed that according to that equation, designed for aircraft that did not flap its wings, the bee could not generate enough lift.

A bumblebee, coming in to land (or fall?)

There is a great deal of folklore wrapped around this “event” and who actually was involved, but it appears that the equation was worked out by André Saint-Lagué, and while the incident is often dressed up as “a scientist proving that bumblebees can’t fly”, all that was really shown was that the equation was inadequate to describe the flight of the bumblebee.

Magnan had shown that you can’t apply that particular equation to bumblebees, rather than proving that spherical bumblebees can’t fly, even if real ones, flapping their wings at 130 times a second, move happily along at 3 metres/sec, 11 km/hr or 7 mph. Like Zeno’s paradox (which will be in the next blog entry), Magnan’s calculation merely showed that there was a faulty assumption in there somewhere. The mathematical model was flawed.

When we escaped from the English classroom to the lab, we learned of marvels that could be done with simple apparatus. The muzzle velocity of a bullet could be measured with nothing more than a block of wood, a piece of string, a protractor and a measuring tape.

Our physics teacher, equally as at home with fiction as our English teacher, explained how, in the days of gunpowder and muzzle-loading firearms, slight variations in the ingredients, their amounts and proportions, could make a lot of difference. The most obvious measure was the speed at which a cannon ball or musket ball left the barrel of the gun, or in physics-speak, the muzzle velocity.

The idea was quite simple. You suspend a large block of wood and fire a bullet at it from close range. The bullet lodges in the block, and the energy of the bullet is transferred to the block, which swings like a pendulum. Then one simply has to measure the swing angle and calculate the height the block reaches.

This device even has a name: it is called the ballistic pendulum, and it has been around since the 1742, when it was invented by Benjamin Robins. From the swing, or so we were told, it is a fairly elementary calculation to estimate the energy and hence the velocity of the bullet. Unfortunately, this explanation ignores the 800-pound spherical horse which is rolling around the room.

Some of the energy goes into deforming the bullet and the wood, some is wasted as friction, and to do any calculations, we have to assume that the bullet stops instantaneously (which is as likely as a girder with negligible mass). Of course, if you are trying simply to compare different grades of gunpowder, rather than measuring the muzzle velocities, the losses will be similar in each case, and can be ignored. Whichever powder produces the biggest swing is the best, if everything else is kept constant — and in fairy physics, that always applies.

Robins was born to Quaker parents, but as a mathematician, he tried to make gunnery a science. Along the way, his ballistic pendulum probably showed that Indian saltpetre made the best gunpowder. He died in India in 1751, supervising the construction of forts, and a few years later, the British drove the French out of India, which let them have all that excellent saltpetre for their own use.

Curiously, the pursuit of novel sources for saltpetre during the Napoleonic wars led a French chemist, Bernard Courtois, to discover iodine, but that's another story...the next story, in fact.

That speedy botfly

I revived this excerpt from my out-of-print book The Speed of Nearly Everything when the image on the right turned up in my Facebook feed, coming from Science Humor.

In case you don't look out for details, the hole that the fly was caught in (or poked into) was actually made by a projectile coming from the other side, so the accompanying question about the fly's speed is, at the very least, just a bit misleading, but somebody is going to cite a legend.

When you enquire about fast animals, more often than not, you will read that the fastest animal of all is the deer botfly, which is credited with an amazing 1287 km/hr, though if you convert this to miles per hour, it comes out as a round 800 mph, a figure that smells a little bit like fudged science—and rightly so.

The story begins with a 1927 article by an entomologist called Charles Henry Tyler Townsend, who reported a speed like this in the Journal of the New York Entomological Society. He actually claimed that the fly was clipping along at 400 yards per second, which works out at 818 mph or 1316 km/hr in metric units. As we will see shortly, any preciseness in the conversion is hardly justified.

Townsend reasoned that these flies passed in a blur, and so must have been travelling very fast. On that scientific basis and no other, he credited them with a nice round 400 yards/second.

That story should have been questioned right away, but people wrote it down, passed it on, quoted it uncritically, and never stopped to wonder what would happen if flies were tearing around at supersonic speeds.

As we will see later (next entry in this blog), some people would stop to prove that the bumblebee could not fly, but nobody stopped to consider and demonstrate the impossibility of the botfly claim until 1938, when Irving Langmuir, a Nobel laureate in chemistry, having given it some thought, tested the assumptions.

First, the air pressure on the fly at that speed would be more than half an atmosphere, surely enough to crush it. The energy needed to maintain the flight would be 370 watts, half a horsepower, which would be quite an ask. Aside from anything else, the botfly would use up its own weight in fuel every second, so it would need to be a voracious feeder.

Next, Langmuir had been hit by these flies, and while it hurt, that weight of fly at 1300 km/hr would have left a significant hole, rather like that of a soft bullet, and the fly would have been mashed inside the wound. Instead, the fly bounced off.

Langmuir mocked up a model of the botfly, using solder to make a pellet that was 1 cm long and 0.5 cm wide. He attached this to a string, and whirled it around his head, timing it so he could work out its velocity. He reported that at 13 mph it was a blur, at 26 mph it was barely visible, at 43 mph an observer could not tell which way it was going, and at 64 mph, it was completely invisible.

He concluded that the blur Townsend had seen came from a fly travelling at 25 mph (40 km/hr). His results were published in Science and reported in Time magazine, but legends are tough things, even when they are debunked by Nobel Prize winners. So even today, the same old values keep emerging from the woodwork.

By a curious chance, Langmuir’s name crept into the record books in an entirely different way in 2006 when plasma physicists used a specially designed holographic-strobe camera to capture pictures of matter waves that were travelling at 99.997% of the speed of light.

Known as Langmuir waves, they are generated by intense laser pulses, and may one day lead to “tabletop” versions of high-energy particle accelerators. One step along the way was taking photographs of the waves to see if they behaved the way scientists thought they would. They did, which is more than we can say about the botfly's behaviour.

What is a day?

When does the day start? The ancient Greeks said the day began at sunrise, and ended at the next sunrise. The Babylonians held that the day began and ended at sunset, while up until 1925,

astronomers worked on a “day” which began and ended at noon, and so did the Royal Navy in the days of Captain Cook. Now the astronomers, like us, and the ancient Egyptians, have a day which commences at midnight. Islamic tradition has a day which begins at sunset.

By definition, noon is when the Sun is at its highest point in the sky, so it is always noon somewhere in the world, with a noon zone sweeping along through a degree of longitude every four minutes. Logically, you should be setting your clock forward or back by a minute for each 15-20 kilometres that you go east or west!

This would be far too confusing, and to make life easier, we have split the world into time zones, usually (but not always) 15 degrees across, where everybody keeps the same “official time”. If you are trying to set up a very accurate sundial, you need to make allowance for your position east or west of the true time in your zone.

Some points to ponder: Clockwise is a word used to describe the direction of the shadow of a northern hemisphere sundial. What way does the shadow travel in the southern hemisphere?

If you stop and think about it for a moment, you may be able to deduce where the word “dial” comes from, especially if you know anything about the Latin word dies. If you lack this knowledge, look up “dial” in a good dictionary, and find out where it comes from. From this knowledge, can you say what the most appropriate use of the word “dial” is?

We probably had the idea of the two ways, even before clocks. Widdershins is an old word meaning counter-clockwise. The equally old word which means “clockwise” is deasil.

Somewhere along the way, something happened to humans that made them start using art, that made them start communicating with each other, and generally showing signs of being human, rather than hominid. Could it have been the discovery of time which caused these changes?

It is probably time to look at the things we first used for time-keeping and calendars, the stuff that is out there, beyond our atmosphere, the stuff that even a century ago, people knew was forever unreachable — and rather hard to see in any case.

The barman says, “We don’t serve time travellers in here.”

A time traveller walks into a bar.

Brain size

I will studiously not comment on this next one, much as I would like to:

Bischoff, one of the leading anatomists of Europe, thrived some 70 years ago. He carefully measured brain weights, and after many years’ accumulation of much data he observed that the average weight of a man’s brain was 1350 grams, that of a woman only 1250 grams. This at once, he argued, was infallible proof of the mental superiority of men over women. Throughout his life, he defended this hypothesis with the conviction of a zealot. Being the true scientist, he specified in his will that his own brain be added to his impressive collection. The post-mortem examination elicited the interesting fact that his own brain weighed only 1245 grams.
— Scientific American, March 1992, 8, quoting from an unidentified and untraceable source in Scientific American, March 1942.

SMILE!

Corang River

Australia may be easily divided into two parts: the city and the bush.  Almost all of us live in what is loosely called ‘the city’ — while a smaller number live in ‘the bush’.  Any rural area, any wild or wilderness area, any quiet patch, even in the middle of the city, is ‘bush’.  There are patches of bush, just a kilometre from the Opera House in Sydney, and many larger patches dotted around the landscape.

We have a bush mythology, all about how Australians relate to it, but most of us are wholly city-bred and raised.  Even so, many Australians gain a great benefit from the bush and from bushwalking.  Each weekend, urban Australian may be seen wandering along well-worn tracks through urban bush, while others, rather better equipped, can be found far out in the wilderness of the Watagan ranges, the Blue Mountains, the Budawang Ranges, or some other favoured lonely place.  For my family, the preferred solution is to drive for four hours in the dawn to get to the Budawangs.

The Budawang basement rock was laid down in Devonian times, back 350 million years ago, when fishes ruled the earth, along with their cousins the early amphibians.  We can only guess at the rocks' history after that, but they must have been deeply buried and heated, for they are all now low-grade metamorphics, and that only happens with deep burial and light baking in the depths of the earth.

After baking and squeezing, the rocks were twisted and folded, then slowly revealed again.  The tops of the folds were carved off, leaving a series of tilted quartzite slabs.  These tough ribs of stone were shaped in the days of the early dinosaurs, which must have wandered over their surface, leaving not a trace.  By now, the rocks were 150 million years old, and due for burial once more.

They were covered over in a huge flood of mud and boulders that created massive conglomerates.  Some of the pebbles in this conglomerate are more than a metre across, and the beds stretch for kilometres.  But after the conglomerate had filled in around the old hills of Devonian rock, gentler conditions prevailed.  Sand was laid down over the top, through the Permian, then more Triassic sediments came on top of that.  Finally, the process reversed itself, and the Devonian rocks were brought back to the surface as the covering slowly weathered and eroded away.

The Devonian rocks are survivors.  They took all the world could throw at them before, and now they do so again.  When a river comes up against the uptilted ribs of tough metamorphic rock, the water is forced to fight to find a way through.  Where they can, the rocks dam up the water, creating swamps where the usual harsh dry conditions are replaced by harsh soggy conditions.  Where the river is strong enough, it carves down until it meets the truly immovable rocks: then it backs up to form a lagoon.

A little up the hill from such a place, we find the Permian conglomerate sitting uneasily on the tough Devonian ribs, making what the geologists call an unconformity, where one hand, spread on the cliff face, can span more than a hundred million years of history.  The base of the conglomerate falls apart easily, creating caves and rock shelters where campers can shelter when it rains.

The Budawang Ranges are full of places like this, but for my money, the best place to go in the whole area is the Corang River, at a place where the river backs up against tough tilted Devonian rocks, delayed for a couple of hundred metres.  This makes a pool which is refreshingly chill, even on the hottest day, where mists will rise off the river in the early morning, as the first small birds skim across, looking for insects to eat.

Around the edges of the lagoon, kangaroos thump around, looking for food at night.  It is not a place for those with a heightened imagination.  Wombats charge along their favourite tracks like small tanks, mopokes make dismal sounds in the dark, and possums scream at each other.  This is a good place to bring new chums, people who are marvellous victims when it comes to tall tales.  Here you can regale them with drop-bears, hoop snakes, fanged frogs, bush alligators, wombahs and more.

Out in the wilderness, such tales are believable.  Getting there means a drive of four hours, then a walk for three hours, so we usually camp overnight at the trail head, or drive down in the pre-dawn darkness on deserted roads.  Then we hike in, set up our tents, grab some firewood, and prepare for a couple of days of relaxing, reading and swimming.

Except, that is, when my adult son decides we should go exploring.  This is wild country, but we have been coming here for many years, and we know the land marks, so we elect to do a slightly risky thing, and wander off as a pair.  Strictly, we should take three people, one to stay by any injured person, and one to get help, but this is to be little more than a gentle stroll.

Famous last words!  It has been raining in the last few months, and all the swamps are topped up, trapped behind the quartzite dam walls.  When we decide to plunge into low-lying area for a short-cut, we become well and truly bogged down.  My son, I forgot to mention, is very stubborn: he must have inherited this from his mother, for I am never anything more than unswervingly determined.

Whatever the cause, we decide to push on, rather than going back and around the swamp.  Perhaps it is the range of unusual spiders you can find deep in a swamp: perhaps it is the hope of seeing some of the rarer sundews, small insectivorous plants that live in swamps like these, but we force our way, deeper and deeper, following the trails made by the wombats.  There is no plant alive that can withstand a wombat, which is good when we want a clear track, but wombats are very agile about jumping from tussock to tussock, which is not so good.

Cursing and grunting, we leap, slip and slide our way along.  We have done this sort of thing before, so we are still comparatively dry, but it is all very tiring.  We look around for somewhere to sit, but there is no dry dry place big enough or flat enough to sit on.

I recount the old paradoxical definition of an infinite regress, where a band of soldiers get tired in a swamp.  They form a circle, and each sits on the knees of the one behind, until they are rested.  Arguing the logic of this blunts our discomfort for a while as we slog along.  That and a complex disagreement about the differences between marshes, bogs, fens, swamps, swards and water meadows, which helps to keep our minds active as we go.

All good things come to an end, but so do bad things.  Suddenly, what I still insist is swamp, rises above the water table.  We stop, just at the edge of the mud, looking at the beautiful dry land in front of us.  ‘I've definitely learned one thing,’ he tells me.  Foolishly, I ask him what it was.

‘The fen is muddier than the sward’, he tells me.  Something inside me snaps.  One swift push and he learns the Zen lesson that one person cannot make an infinite regress.  Not in a swamp, anyhow.

Macedonio Melloni, a forgotten genius

This is just a filler to say I aten't dead yet. I'm just fairly busy.

Macedonio Melloni (1798 - 1854) is little-known, which is why this is a brief account. Still, what there is seems quite interesting, and it sets the scene for several other stories, so I will share it with you.

First, some background. William Herschel did many things in astronomy. Among other things, he took the temperature of different parts of the spectrum, and found that the hottest part of the Sun’s spectrum was beyond the visible range. Using a thermometer, he discovered the infrared part of the spectrum.

Now back to Melloni: if you heat a junction between two different metals, you generate a very small current. In order to be able to measure the current, you need to multiply it by linking a number of these junctions together, to make a battery of them, like the pile of cell units that Alessandro Volta made. This is why we call Melloni’s invention a thermopile.

Source: https://muxindia.wordpress.com/2014/11/06/passive-infrared-sensor-pir-sensor/

As well, Melloni developed ways of concentrating the heat from distant sources, and he found out how to use rock salt to make lenses to focus the heat rays, in the same way we use glass lenses to focus light rays. He established that the infrared rays were in every way like light: they could be refracted, reflected, polarised and made to interfere with each other, exactly like light.

Melloni really deserves to be better known, for while Herschel’s discovery of the infrared is a commonplace, few people realise that Melloni’s investigations laid a practical framework within which James Clerk Maxwell could propose the existence of a continuous electromagnetic spectrum.

Everything in science is connected, which is why we can usually spot fraudulent science at a glance. It doesn't connect with all the other bits...