Labor supply and demand in a perfect competiti...
Labor supply and demand in a perfect competition labor market (Photo credit: Wikipedia)

A free market is one in which there is no government, monopoly or other authoritative interference in the workings of a market. However there is in practise no such thing, as there are always constraints on a market from one or more of those sources.

For instance, in a small country there may be only two or three organisations which are involved in the whole supply chain, and if they are much the same size there is no drive to compete strongly. If one large competitor decided to drive another large competitor out of the market, it would be expensive and difficult, and would more likely than not trigger monopoly prevention legislative mechanisms.

An example of a cover from a Monopoly video game
An example of a cover from a Monopoly video game (Photo credit: Wikipedia)

On the other hand, a small competitor might be worth an aggressive approach as an attack could be targeted and localised. It would be cheaper and while it might raise a few worries about lack of choice (in an area), it would not trigger any monopoly laws.

An open market goes hand in hand with the laws of supply and demand. Generally these are expressed as graphs showing the intersection of the supply curve (an upwards trending line) with the demand curve (a downwards trending line). Any change in conditions is shown by other lines more or less parallel to the first.

Fig5 Supply and demand curves
Fig5 Supply and demand curves (Photo credit: Wikipedia)

These curves can only be illustrative as they are almost never drawn with quantified axes, and the curves are drawn without the use of any measured data. They are arbitrary. Nevertheless they purport to show the effect of market changes on the equilibrium or balance point where the curves cross.

While the laws of supply and demand may be true in the sense that if either the price or demand changes the other also changes, the graphs are of little practical use, and they are only marginally mathematical, as definite mathematical conclusions cannot be made from them. It is impossible to quantify the effect on demand of raising the price of a can of beans by 10c, for example.

Curried Beans
Curried Beans (Photo credit: Wikipedia)

Nevertheless one could probably use the graphs to suggest that if the price changes in one direction the demand will move in another direction, and these guesses may be used to decide on price changes. It’s definitely a guess, though as the opposite may happen – if you put up a sign saying “Beans now $1.55 per can”, having raised the price by $0.05, you may sell more as you have drawn the customers’ attention to the beans.

The “Free Market”, the “Laws of Supply and Demand”, and the principle of “Laissez Faire” are part of the backbone of Capitalism. Capitalism is a robust economic system which has achieved immense feats and advances. It has harnessed science and sent men to the Moon, given us a computer and communication devices in our pockets. There is no doubt that Capitalism has been hugely successful.

A capitalism's social pyramid
A capitalism’s social pyramid (Photo credit: Wikipedia)

In spite of its amazing successes, there have always been drawbacks to Capitalism. The trend of prices is to rise continually, though at times, they do fall, as demand reduces in recessions and market collapses. These recessions and collapses hurt the poor much more than the rich, as the poor have fewer resources to cope with these setbacks.

Capitalist markets lead to concentration of resources, especially money, in the hands of the rich, and a scarcity of resources in the hands of the poor. It leads to the growth of large market dominating firms, as one firm succeeds while others fail. The successful firm often widens its control of the market by purchasing up and coming smaller firms or older firms who themselves may control a smaller market niche.

Capitalism fosters the growth of the gap between the very rich and the very poor. It is often argued that, in countries where the economic system is Capitalist in nature, the “poor” have much more in the way of consumer items than their parents could have imagined. Most people have a car. Most people have a television. Most have a cellphone.

This is all true, but that is only because these items are both essential and relatively cheap. At the same time, health care is becoming unaffordable for many of the new poor. Schooling is also a huge drain on the poorer families. Many poor people work at multiple jobs to bring up their children and pay for the operations that their parents are coming to need.

As a result, many of the new poor live from day-to-day, with no real opportunity to save for retirement or to lay by a little money to allow for the vicissitudes of life. A small accident that requires time off work and consequently reduction of income becomes a disaster in such a situation.

Capitalism stratifies society and the bottom strata, often those with a lack of education or intelligence, lags behind those who are in higher strata. Those at the highest levels tend to outstrip those at lowest levels until their wealth, to those in lower strata, appears as meaningless numbers. What the difference between $100 and $1000 to those at the bottom? It’s a huge amount. What about the difference between $10 billion and $100 billion? It’s irrelevant.

English: Memorial to a wealthy benefactor
English: Memorial to a wealthy benefactor (Photo credit: Wikipedia)

Capitalist market forces tend to favour those who already have over those who don’t and the barriers that prevent those in the lower strata from moving up are immense. Those few who make are the lucky ones. Yes, luck plays almost as big a part in entrepreneurial success as luck does in winning the lotto.

Capitalism is the best economic system that we have ever had, without a doubt. It is however not without its flaws. Socialism is not a good economic system, but purports to deal with the issues of poverty by redistribution of wealth. (Maybe I’ll do a piece on socialism’s flaws at some time).

Karl Marx (1818-1883)
Karl Marx (1818-1883) (Photo credit: Wikipedia)

Capitalism however does not deal with poverty or the poor. Some effects do trickle down and today’s poor appear rich in comparison with the corresponding strata in the past, but the fundamental poverty still exists.

It would be nice to think that there is some other system, waiting for someone to discover it. The odds are probably good, as no system lasts forever. What it would look like I’ve no idea. We would need to get a much better scientific view of the so-called social sciences to really solve this fundamental problem.

Iconic image for social science.
Iconic image for social science. (Photo credit: Wikipedia)


Simple Arithmetic

Addition, division, subtraction and multiplica...
Addition, division, subtraction and multiplication symbols (Photo credit: Wikipedia)

There periodically appears on the Internet an arithmetic type of puzzle. Typically it will be a string of small natural numbers and a few arithmetic operations, such as “3 + 7 x 2 – 4” and the task is to work out the result.

The trick here is that people tend to perform such a series of calculations strictly from left to right, so the sequence goes:

3 + 7 = 10, 10 x 2 = 20, 20 – 4 = 16. Bingo!

Most mathematicians, and people who remember maths from school would disagree however. They would calculate as follows:

7 x 2 = 14, 3 + 14 = 17, 17 – 4 = 13. QED!

Why the difference? Well, mathematicians have a rule that states how such calculations are to be performed. Briefly the calculations is performed from left to right, but if a multiplication or division is found between two numbers, that calculation is performed before any additions or subtractions. In fact the rule is more complex than that, and a mnemonic often used to remember it is “BODMAS” or “BEDMAS” (which I’m not going to explain in detail here. See the link above).

This rule is only a convention and so is not followed everywhere, so there are various “correct” answers to the problem. Also, there are still ambiguities if the conventions are applied which could cause confusion. However, most people with some mathematical training would claim that 13 is the correct answer.

Interestingly, computer programming languages, which are much stricter about such things, codify the precedence of operations in a calculation exactly, so that there can be no ambiguity. It is the programmers task to understand the precedence rules that apply for a particular language.

example of Python language
example of Python language (Photo credit: Wikipedia)

In most cases the rules are very, very similar, but it is the documentation of the language which describes the rules of precedence, and wise programmers study the section on operator precedence very closely.

There are ways of specifying an arithmetic problem uniquely, and one of those (which is sometimes of interest to programmers) is “Reverse Polish Notation“. Using this my original puzzle becomes “3 7 2 x + 4 -” which looks odd until you understand what is going on here.

Illustration of postfix notation
Illustration of postfix notation (Photo credit: Wikipedia)

Imagine that you are traversing the above list from left to right. First you find the number “3”. This is not something you have to do, like “+”, “-“, “x” or “/”, so you just start a pile and put it on the bottom. The same goes for “7” and “2”, so the pile now has “3” on the bottom and “2” at the top and “7” in the middle.

Next we come across “x”. This tells us to do something, so we pull the last two things off the pile and multiply them (7 x 2 = 14) and stick the result, “14” back on the stack which now contains “3” and “14”. The next thing we find is “+” so we pluck the last two things off the pile “3” and “14” and add them, putting the result “17” back on the (empty) pile.

Next up is “4” which we put on the pile, and finally, we have “-“, so we pull the two last elements from the pile (“17” and “4”) and subtract the second from the first, giving “13” (Yay!) and that is the answer which we put back on the stack. The stack now contains nothing but the answer.

This looks confusing, but that may be because we are used to the conventional left to right way of doing things. It is actually easier for a computer to understand the RPN version of the puzzle and there are no ambiguities in it at all. Technically, it’s a lot simpler to parse than the conventional version.

Image for use in basic articles dealing with p...
Image for use in basic articles dealing with parse trees, nodes, branches, X-Bar theory, linguistic theory. (Photo credit: Wikipedia)

Parsing is what happens when you type a command into a computer, or you type something complex, such as a credit card number into the checkout section of a web site. The computer running the web site takes your input and breaks it up if necessary and checks it against rules that the programmer has set up.

So, if you type 15 numbers or 17 numbers into the field for the credit card number, or you type a letter into the field by mistake, the computer will inform you that something is wrong. Infuriatingly, it may be not be specific about what the trouble is!

Español: Un Guru meditation en una amiga
Español: Un Guru meditation en una amiga (Photo credit: Wikipedia)

Anyway, back to the arithmetic. It grates with me when people make simple arithmetical errors and then excuse themselves with the phrase “I never was much good at maths at school”! That may well be true, but to blame their problems with arithmetic of the whole diverse field of mathematics.

It’s like saying “I can’t add up a few numbers in my head or on paper because I missed the class on elliptic functions“! It’s way over the top. For some reason people (especially those who can’t get their head around algebra) equate the whole of mathematics with the bit that they do, which is the stuff about numbers, which is arithmetic.

English: Weierstrass p, Stylised letter p for ...
English: Weierstrass p, Stylised letter p for Weierstrass’s elliptic functions from Computer Modern font (obtained by TeX command \wp) Deutsch: Weierstrass p, stilisierter Buchstabe p für die elliptische Funktion von Weierstrass in der Computer-Modern-Schrift (generiert durch das TeX-Kommando \wp (Photo credit: Wikipedia)

As we evolved, we started counting things. It’s important to know if someone has got more beans than you or that you have enough beans to give everyone one of them. We invented names for numbers and names for the things (operations) we did on them.

We did this without much thought about what numbers actually are. We as a species have only relatively thought deeply about numbers fairly recently, and we only discovered such things as real numbers and geometry in the last couple of thousand years so it is not surprising that the average brain has yet to expand to cope with the more advanced mathematical concepts.

Graphic showing the relation between the arith...
Graphic showing the relation between the arithmetic mean and the geometric mean of two real numbers. (Photo credit: Wikipedia)

This could be why so many people these days equate fairly simple arithmetic with mathematics as a whole – our brains are only now coming to grips with the concept that there is more to maths than simply manipulating numbers with a very few simply operations.

It may be that the average human brain never will get to grips with more advanced maths. After all, people can survive and thrive in the modern world with on a rudimentary grasp of mathematics, the arithmetic part.

English: the arithmetic sequence a_n=n Deutsch...
English: the arithmetic sequence a_n=n Deutsch: die arithmetische Folge a_n=n (Photo credit: Wikipedia)

Some human brains however do proceed further and much of modern society is the result of mathematics in its wider sense applied to the things that we see around us. For instance,  we could not have sent men to the moon without advanced mathematics, and technology relies heavily on mathematics to produce all sorts of things. It’s a good things that some brains can tell the difference between the field of arithmetic and mathematics as a whole.

English: A .gif animation of the vibration cor...
English: A .gif animation of the vibration corresponding to the third smallest eigenvalue of the electric pylon truss problem from EML 4500 HW 6. (Photo credit: Wikipedia)

Here be Dragons

Dragon Green
Dragon Green (Photo credit: Wikipedia)

Dragons, big scaly fire breathing reptiles. So many of our folk tales and even many modern tales include dragons as an important component, usually as a hostile force. Of course in many tales the dragon is merely a device to give the hero some seemingly impossible difficulty to overcome.

Sometimes dragons are mere beasts, but in some tales they are intelligent, if malevolent, beasts. Smaug, in “The Hobbit” by J R R Tolkien is of the latter kind. He sits on a pile of treasure and is furious when Bilbo Baggins steals a golden cup. He later accuses Bilbo of trying to steal from him (which is true).

Smaug as he appears in the animated film.
Smaug as he appears in the animated film. (Photo credit: Wikipedia)

The patron saint of England is Saint George, who was an early Christian martyr. Saint George is noted for slaying a dragon to save a princess. The princess was intended as a sacrifice to the dragon who was causing sickness in the inhabitants of the local town.

In legends, once a dragon has been killed, it’s body, blood and teeth could be used for various purposes. Sometimes the blood was beneficial to humans, conferring invincibility or other virtue, or it could be poisonous. The teeth could be sowed to raise armies, sometimes of skeletons.

Most fictional and mythical dragons are scaly reptiles, but one of the odd ones out is the furry creature called the “Luck Dragon” in the film “The NeverEnding Story”. This dragon had a head resembling that of a dog, front limbs and a tapering furry body which merged into a tail.

Most fictional dragons are noble creatures, but the “Swamp Dragons” created by Terry Pratchett in his discworld series of books which are altogether baser than the “Noble Dragons“. Swamp dragons are small creatures, are almost always ill (because of their diet) and are prone to explode if very ill or excited.

English: The Nine Dragon Wall in the Beihai Pa...
English: The Nine Dragon Wall in the Beihai Park, a large imperial garden in central Beijing (Photo credit: Wikipedia)

On one occasion one exploded after being enraged at the sight of itself in a mirror, imaging that it was in the presence of a rival. It does appear though that the fraught gastric processes may have a reason – a swamp dragon is described as flying on its stubby wings by emitting gasses created by its digestive processes.

Some dragons can apparently be tamed. In Anne McCaffrey’s Pern of books series, a partnership has developed between the flying dragons and humans to deal with the threat of “thread” which comes from a companion planet and is inimical to all life forms on Pern.

All the Weyrs of Pern
All the Weyrs of Pern (Photo credit: Wikipedia)

These dragons, which similar the standard dragon type from mythology, are large enough to be ridden by humans, and breath fire to kill the thread on being fed a particular type of rock. The dragons are genetically modified from the much smaller fire lizards and communicate with their riders by telepathy.

One unique ability of these dragons is to teleport from place to place carrying their riders with them. It also becomes apparent that they can also time travel while teleporting, Unsurprisingly, Terry Pratchett created a cameo parody or homage to the Pern books and their dragons in the first book of his discworld series, “The Colour of Magic”.

The Discworld as it appears in the SkyOne adap...
The Discworld as it appears in the SkyOne adaptation of The Colour of Magic. (Photo credit: Wikipedia)

Where there are dragons, the untamed variety, The more I think about them, the more I remember cases of dragons in literature and in films. A fairly recent example is the film “How to Train Your Dragon“. The dragons are at first treated as hostile, but the aspiring dragon killer, Hiccup, finds an injured dragon, it transpires that the dragons are friendly creatures and only attack humans because the humans are attempting to exterminate them.

The modern dragon is built along the same physical plan, whatever the media they are described in. Dragons are reptiles, usually lay eggs, mostly have four legs or limbs and a pair of wings. Mostly they breathe fire, and where this is touched on, it is usually implied that the fire is generated internally by ingesting and digesting rocks.

Saint George and the Dragon at Casa Amatller
Saint George and the Dragon at Casa Amatller (Photo credit: Wikipedia)

However, early myths about dragon describe dragons as more akin to large serpents, even to the extent of having no limbs. Indeed, in early texts, the word used for dragons also means serpent.

Interestingly, although England’s patron saint is a dragon killer, the red dragon has come to symbolise Wales. “Y Ddraig Goch” is a red dragon and can be found on the Welsh national flag. He attains ascendency over an invading white dragon who symbolises the Saxons, after a long battle and an interval when both dragons were imprisoned in a hill in Snowdonia.

English: Welsh Dragon
English: Welsh Dragon (Photo credit: Wikipedia)

Dragons are associated power. Having scales and claws, and being able to breath fire, are attributes that give them strong defensive and offensive capabilities. Their size gives them strength and they are a very great challenge to any heroes who take them on. Often they can only be defeated by trickery or luck, such as when Smaug was killed because he had a small unprotected area on his belly which allowed the hero to shoot fatally in that one spot.

Dragons are associated with magic, with wizards, witches, princes and princesses and supernatural items and events of all sorts. “Dungeons and Dragons” melds all these factors into a table top and role playing game which was popular in the 1970s and 1980s.

Dungeons & Dragons game in IV Getxo Comic Con.
Dungeons & Dragons game in IV Getxo Comic Con. (Photo credit: Wikipedia)

However, dragons as such do not appear to be a large factor in the game, which revolves more around the characters who may be clerics, fighters or magic users. Dungeons and Dragons does have monsters and while some may be dragons, there are many other types of monsters, which may or may not be controlled by other players taking part in the game.

Finally, to bring this ramble through the topics dragons to a close, I will mention one other dragon that I recall from films, and that is the one which appeared in film and book “Doctor No” by Ian Fleming.

From source information from : S/Sgt Bill Seklscki fires a flame thrower at a Japanese position near Manacag, Luzon, P.I. Jan 25 1945. Photo: National Archives. Webmaster note: 33rd did not land on Luzon until Feb 1945. Date of picture could be a mistake. (Photo credit: Wikipedia)

In the book and film James Bond is sent to Crab Key to investigate Doctor No. Rumours abound about the “dragon” which roams the island, deterring anyone from visiting. In the end the fire breathing dragon turns out to be a vehicle fitted with a flamethrower. This goes to show that while fictional and mythical dragons may be common, real dragons are scarcer than hen’s teeth.

Dr. No as seen in the James Bond Jr. animated ...
Dr. No as seen in the James Bond Jr. animated series. (Photo credit: Wikipedia)




Let’s be Rational – Realer Numbers

Symbol often used to denote the set of integers
Symbol often used to denote the set of integers (Photo credit: Wikipedia)

Leopold Kronecker said “God made the integers, all else is the work of man”. (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”). However man was supposedly made by God, so the distinction is logically irrelevant.

I don’t know whether or not he was serious about the integers, but there is something about them that seems to be fundamental, while rational numbers (fractions) and real numbers (measurement numbers) seem to be derivative.

English: Note: The irrational and rational num...
English: Note: The irrational and rational numbers make the set of real numbers. (Photo credit: Wikipedia)

That may be due to the way that we are taught maths in school. First we are taught to count, then we are taught to subtract, then we are taught to multiply. All this uses integers only, and in most of it we use only the positive integers, the natural numbers.

Then we are taught division, and so we break out of the world of integers and into the much wider world of the rational numbers. We have our attention drawn to one of the important aspects  of rational numbers, and that is our ability to express them as decimal fractional numbers, so 3/4 becomes 0.75, and 11/9 becomes 1.2222…

Parts of a micrometer caliper, labeled in Engl...
Parts of a micrometer caliper, labeled in English. Someone can replace this with a prettier version anytime. (Photo credit: Wikipedia)

The jump from there to the real numbers is obvious, but I don’t recall this jump being emphasised. It barely (from my memories of decades ago) was hardly mentioned. We were introduced to such numbers as the square root of 2 or pi and ever the exponential number e, but I don’t recall any particular mention that these were irrational numbers and with the rational numbers comprised the real numbers.

Why do I not remember being taught about the real numbers? Maybe it was taught but I don’t remember. Maybe it isn’t taught because most people would not get it. There are large numbers of rational accountants, but not many real mathematicians. (Pun intended).

Square root of two as the hypotenuse of a righ...
Square root of two as the hypotenuse of a right isosceles triangle of side 1. SVG redraw of original work. (Photo credit: Wikipedia)

In any case I don’t believe that it was taught as a big thing, and a big thing it is, mathematically and philosophically. It the divide between the discrete, the things which can be counted, and the continuous, things which can’t be counted but are measured.

The way the divide is usually presented is that the rational numbers (the fractions and the integers) plus the irrational numbers make up the real numbers. Another way to put it, as in the Wikipedia article on real numbers, is that “real numbers can be thought of as points on an infinitely long line called the number line or real line”.

Collatz map fractal in a neighbourhood of the ...
Collatz map fractal in a neighbourhood of the real line (Photo credit: Wikipedia)

Another way to think of it is to consider numbers as labels. When we count we label discrete things with the integers, which also do for the rational numbers. However, to label the points on a line, which is continuous, we need something more, hence the real numbers.

Real numbers contain the transcendental numbers, such as pi and e. These numbers are not algebraic numbers, which are solutions of algebraic equations, so are defined by exclusion from the real numbers. Within the transcendental numbers pi and e and a quite large numbers of other numbers have been shown to be transcendental by construction or argument. I sometimes wonder if there are real numbers which are transcendental, but not algebraic or constructible.

A rather sexy image of Pi from the german wiki...
A rather sexy image of Pi from the german wikipedia. (Photo credit: Wikipedia)

The sort of thing that I am talking about is mentioned in the article on definable real numbers. It seems that the answer is probably, yes, there are real numbers that  are not constructible or computable.

Of course, we could list all the constructible real numbers, mapped to the real numbers between 0 and 1. Then we could construct a number which has a different first digit to the first number, a different second digit to the second number and so on, in a similar manner to Cantor’s diagonal proof,  and we would end up with a number that is constructed from the constructible real numbers but which is different to all of them.

English: Georg Cantor
English: Georg Cantor (Photo credit: Wikipedia)

I’m not sure that the argument holds water but there seems to be a paradox here – the number is not the same as any constructible number, but we just constructed it! This reminds of the “proof” that there are no boring numbers.

So, are numbers, real or rational, just labels that we apply to things and things that we, or mankind as Kronecker says, have invented? Are all the proofs of theorems just inventions of our minds? Well, they are that, but they are much more. They are descriptions of the world as we see it.

Apollonius' theorem
Apollonius’ theorem (Photo credit: Wikipedia)

Whether or not we invented them, numbers are very good descriptions of the things that we see. The integers describe things which are identifiably separate from other things. Of course, some things are not always obviously separate from other things, but once we have decided that they are separate things we can count them. Is that a separate peak on the mountain, or is it merely a spur, for example.

Other things can be measured. Weights, distances, times, even the intensity of earthquakes can be measured. For that we of course use rational numbers, while conceding that the measurement is an approximation to a real number.

Tape ruler
Tape ruler (Photo credit: Wikipedia)

A theorem represents something that we have found out about numbers. That there is no biggest prime number, for example. Or that the ratio of the circumference to the diameter is pi, and is the same for all circles.

We certainly didn’t invent these facts – no one decided that there should be no limit to the primes, or that the ratio of the circumference to the diameter of a circle is pi. We discovered these facts. We also discovered the Mandlebrot Set and fractals, the billionth digit of pi, the bifurcation diagram, and many other mathematical esoteric facts.

Mandlebrot Fractal made with Paint.NET
Mandlebrot Fractal made with Paint.NET (Photo credit: Wikipedia)

It’s like when we say that the sky is blue. To a scientist, the colour of sunlight refracted and filtered by the atmosphere, peaks at the blue wavelength. The scientist uses maths to describe and define the blueness of the sky, and the description doesn’t make the sky any the less blue.

The mathematician uses his tools to analyse the shape of the world. He tries to extract as much of the physical from his description, but when he uses pi it doesn’t make the world any the less round as a result. Mathematics is a description of the world and how it works at the most fundamental level.

English: Adobe photoshop artwork illustrating ...
English: Adobe photoshop artwork illustrating a complex number in mathematics. (Photo credit: Wikipedia)

[I’m aware that I have posted stuff on much the same topic as last time. I will endeavour to address something different next week].