The Banach Tarski Theorem

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There’s a mathematical theorem (the Banach Tarski theorem) which states that

Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

This is, to say the least, counter intuitive! It suggests that you can dissect a beach ball, put the parts back together and get two beach balls for the price of one.

This brings up the question of what mathematics really is, and how it is related to what we loosely call reality? Scientists use mathematics to describe the world, and indeed some aspects of reality, such as relativity or quantum mechanics, can only be accurately described in mathematics.

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So we know that there is a relationship of some sort between mathematics and reality as our maths is the best tool that we have found to talk about scientific things in an accurate way. Just how close this relationship is has been discussed by philosophers and scientists for millennia. The Greek philosophers, Aristotle, Plato, Socrates and others, reputedly thought that “all phenomena in the universe can be reduced to whole numbers and their ratios“.

The Banach Tarski theorem seems to go against all sense. It seems to be an example of getting something for nothing, and appears to contravene the restrictions of the first law of thermodynamics. The volume (and hence the amount of matter) appears to have doubled, and hence the amount of energy contain as matter in the balls appears to have doubled. It does not appear that the matter in the resulting balls is more attenuated than that in the original ball.

The Banach–Tarski paradox: A ball can be decom...
The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. (Photo credit: Wikipedia)

Since the result appears to be counter intuitive, the question is raised as to whether or not it is merely a mathematical curiosity or whether it has any basis in reality, It asks something fundamental about the relationship between maths and reality.

It’s not the first time that such questions have been asked. When the existence of the irrational numbers was demonstrated, Greek mathematicians were horrified, and the discoverer of the proof (Hippasus) was either killed or exiled, depending on the source quoted. This was because the early mathematicians believed that everything could be reduced to integers and rational numbers, and their world did not have room for irrational numbers in it. In their minds numbers directly related to reality and reality was rational mathematically and in actuality.

English: Dedekind cut defining √2. Created usi...
English: Dedekind cut defining √2. Created using Inkscape. (Photo credit: Wikipedia)

These days we are used to irrational numbers and we see where they fit into the scheme of things. We know that there are many more irrational numbers than rational numbers and that the ‘real’ numbers (the rational and irrational numbers together) can be described by points on a line.

Interestingly we don’t, when do an experiment, use real numbers, because to specify a real number we would have write down an infinite sequence of digits. Instead we approximate the values we read from our meters and gauges with an appropriate rational number. We measure 1.2A for example, where the value 1.2 which equals 12/10 stands in for the real number that corresponds to the actual current flowing.

English: A vintage ampere meter. Français : Un...
English: A vintage ampere meter. Français : Un Ampèremètre à l’ancienne. (Photo credit: Wikipedia)

We then plug this value into our equations, and out pops an answer. Or we plot the values on a graph read off the approximate answer. The equations may have constants which we can only express as rational numbers (that is, we approximate them) so our experimental physics can only ever be approximate.

It’s a wonder that we can get useful results at all, what with the approximation of experimental results, the approximated constants in our equations and the approximated results we get. If we plot our results the graph line will have a certain thickness, of a pencil line or a set of pixels. The best we can do is estimate error bounds on our experimental results, and the constants in our equations, and hence the error bounds in our results. We will probably statistically estimate the confidence that the results show what we believe they show through this miasma of approximations.

Image of simulated dead pixels. Made with Macr...
Image of simulated dead pixels. Made with Macromedia Fireworks. (Photo credit: Wikipedia)

It’s surprising in some ways what we know about the world. We may measure the diameter of a circle somewhat inaccurately, we multiply it by an approximation to the irrational number pi, and we know that the answer we get will be close to the measured circumference of the circle.

It seems that our world resembles the theoretical world only approximately. The theoretical world has perfect circles, with well-defined diameters and circumference, exactly related by an irrational number. The real world has shapes that are more or less circular, with more or less accurately measured diameters and circumferences, related more or less accurately by an rational number approximating the irrational number, pi.

Pi Animation Example
Pi Animation Example (Photo credit: Wikipedia)

We seem to be very much like the residents of Plato’s Cave and we can only see a shadow of reality, and indeed we can only measure the shadows on the walls of the cave. In spite of this, we apparently can reason pretty well what the real world is like.

Our mathematical ruminations seem to be reflected in reality, even if at the time they seem bizarre. The number pi has been known for so long that it no longer seems strange to us. Real numbers have also been known for millennia and don’t appear to us to be strange, though people don’t seem to realise that when they measure a real number they can only state it as a rational number, like 1.234.

English: The School of Athens (detail). Fresco...
English: The School of Athens (detail). Fresco, Stanza della Segnatura, Palazzi Pontifici, Vatican. (Photo credit: Wikipedia)

For the Greeks, the irrational numbers which actually comprise almost all of the real numbers, were bizarre. For us, they don’t seem strange. It may be that in some way, as yet unknown, the Banach Tarski theorem will not seem strange, and may seem obvious.

It may be that we will use it, but approximately, much as we use the real numbers in our calculations and theories, but only approximately. I doubt that we will be duplicating beach balls, or dissecting a pea and reconstituting it the same size as the sun, but I’m pretty sure that we will be using it for something.

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I see maths as descriptive. It describes the ideal world, it describes the shape of it. I don’t think that the world IS mathematics in the Pythagorean sense, but numbers are an aspect of the real world, and as such can’t help but describe the real world exactly, while we can only measure it approximately. But that’s a very circular description.

English: Illustrates the relationship of a cir...
English: Illustrates the relationship of a circle’s diameter to its circumference. (Photo credit: Wikipedia)






English: Holiday in village
English: Holiday in village (Photo credit: Wikipedia)

I should imagine that going on holiday, for many people would be a relatively new thing. While those with money might decide to shift operations from home to another location, which might or might not be near a beach, those who work from them would mostly have no respite from day to day toil, since their employers would still require looking after as usual.

As ordinary people became wealthy, and the old social structures faded away for the most part, it would have become more usual for ordinary people to go away, just as their employers used to.

Rangiputa, Karikari Peninsula, Northland, New ...
Rangiputa, Karikari Peninsula, Northland, New Zealand. Rangiputa is a beach and bach (holiday home) community on the west side of the peninsula (Photo credit: Wikipedia)

The word “holiday” itself is a  contraction of “holy day”, and on holy days there were celebrations and less formal work. The word has come to mean a day on which one does not have to work. Most countries these days would have statutory holidays on which which people would not have to work. There may be other restrictions, such as legislation that shops should remain closed.

It’s understandable that some countries require shop closures, as this means that shop staff get the holiday too, but many countries these days allow shops to stay open if they wish and some of the best retail days are on statutory holidays. Usually shops that stay open are required to compensate staff who are required to work.

English: Brixham - Harbourside Shops These sho...
English: Brixham – Harbourside Shops These shops mainly cater to the holiday trade who visit the harbour. (Photo credit: Wikipedia)

Holidays are disruptions to normal schedules. When one goes away, one is in a different environment and one has to make do. Even something as simple as making a cup of tea may be complicated by the need to find a spoon, a cup, and a teabag, not to mention the need to figure out the operation of a different jug!

These things are not an enormous issue, and in fact draw attention to the fact that one is on holiday. All schedules are voided and one can do whatever one wants. Often this may amount to doing nothing.

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A “holiday industry” has evolved, which provides accommodation, and resources for those temporarily away from home. It also provides entertainments or “attractions” if the holiday maker doesn’t just want to lay on the beach. The holiday maker may do all sorts of things that he or she doesn’t usually do, from the exciting (bungy jumping or similar) to the restful (a gentle walk around gardens or maybe a castle visit or may a zoo).

These facilities are all staffed by helpful people who arrange things so that the holiday maker can enjoy his or her self without worries. These people are of course employed by the facilities, but many of them enjoy their work very much anyway. It’s a sort of bonus for helping people.

English: Ultra Dynamics Dowty Turbocraft water...
English: Ultra Dynamics Dowty Turbocraft waterjet boat (Photo credit: Wikipedia)

Holiday makers must also be fed, and this has become a huge industry too. In any seaside towns so-called fast food outlets can be found in abundance, along with more up market restaurants and cafés, for more leisurely eating. For many people one of the advantages of being on holiday is that one doesn’t have to cook, and one can choose to eat things that one doesn’t normally eat.

Holidays can be expensive. Since we are close to the Pacific Islands, like Tonga, Samoa and Fiji, many people fly out to the islands on their summer holidays. This means flight and accommodation has to be booked and paid for.

English: Great Frigate Birds (Fregata minor) o...
English: Great Frigate Birds (Fregata minor) on Johnston Atoll, Pacific Island Wildlife Refuges (Photo credit: Wikipedia)

When the holiday makers arrive at their destinations, they have to pay for food and entertainment. Other expenses may be for sun screen cream, snacks, tours, tips, and the odd item of clothing which may have been accidentally left at home.

Holiday entertainment may comprise guided tours, or visiting monuments or zoos. Amusement parks are often an attraction as are aquariums. All this can cost a lot, but unless you are content to veg out on the beach, you’ll have to pay for it. Even vegging out on the beach comes at a cost, from sun protection through to drink to offset the dehydration caused by the sun.

English: Roller coaster, M&Ds Theme Park, Stra...
English: Roller coaster, M&Ds Theme Park, Strathclyde Country Park The larger and older of the two roller coasters, at the very southern end of the park. (Photo credit: Wikipedia)

So, why do we throw over the usual daily regime, and drag our family on an often uncomfortable road, sea, or plane trip, to a location where we know little of the environment, which will cost us money, to spend the days traipsing from “attraction” to “attraction” spending more money and feeding on often costly food of unknown quality or provenance?

Part of the answer is that the daily regime becomes boring and descends into drudgery. Removing ourselves from the daily regime allows us to escape that drudgery for a while. As far as the cost goes, well, one is prepared to spend a certain amount of money to escape the drudgery for a while.

Money for All
Money for All (Photo credit: Wikipedia)

Removing ourselves from the usual means that we can try the unusual. We may try Mexican food, or Vietnamese food. Or even Scottish cuisine if we choose. The world is our oyster.

We can try sports and pastimes that we have never tried before. Bungee jumping. Skiing, water or snow. We can visit a “Theme Park”, ride a roller coaster, or other ride. We can scare ourselves and excite ourselves.

Skiing (Photo credit: Wikipedia)

We can experience different cultures, different scenery, but at the end of the day we know that we will be returning to our mundane lives. We have at the back of our minds the cosy ordinariness of our usual lives, as a sort of safety harness.

We know our comfortable house will be there for us to return to, and while we may enjoy the beds in our hotel, motel, holiday home or tent, we look forward to the return to our own beds. We look forward to drinking the brands of coffee and tea that we prefer and fill the fridge with the foods that we prefer to cook.

English: Hotel room in the Waldorf Hilton, Ald...
English: Hotel room in the Waldorf Hilton, Aldwych, London. (Photo credit: Wikipedia)

Few people would want to live in hotels and sleep in strange beds as a way of life, but there are some people who do so. While we enjoy being on holiday, as a break from our usual lives, we would probably not want to live that way for an extended period. Those who do are unusual people.

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