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A Sum of All the Parts

Cover of the Book Conscious Robots (Photo credit: Wikipedia)

Consciousness is fascinating and I keep coming back to it. It is personally verifiable in that a person knows that he or she is conscious, but it is difficult if not impossible to tell if a person is conscious from the outside.

When you talk to someone, you and that person exchange words. You say something, and they respond. Their response is to what you say, and it appears to show that the person is a conscious being.



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It’s not as easy as that, however, because it is conceivable that the person is a zombie (in the philosophical sense) and his or her responses are merely programmed reactions based on your words. In other words he or she is not a conscious being.

It seems to me that the best counter to this suggestion is that I am a conscious being and I am no different in all discernible ways from others. It is unreasonable to suggest I am the only conscious being anywhere and that all others are zombies.



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Of course this leaves open the suggestion that some people may be philosophical zombies, but that then raises the question of what the difference is, and how can one detect it. William of Occam would probably wield his razor and conclude that, if one can’t tell, one might as well assume that there are no zombies, as assuming that there are zombies adds a (probably) unnecessary assumption to the simple theory that all humans are conscious beings.

It follows that consciousness is probably an emergent phenomenon related to the complexity and functioning of the brain. It also follows that lower animals, such as dogs, cats and apes are also probably conscious entities, though maybe to a lesser extent that we are.

The only way we can directly study consciousness is by introspection, which is more than a bit dubious as it is consciousness studying itself. We can indirectly study consciousness by studying others who we assume to be conscious, maybe when they have been rendered unconscious by anaesthetics and are “coming round” from them.

In addition, consciousness can be indirectly studied using mind altering drugs or meditation. However we are mainly dependant on verbal reports from those studied this way, and such reports are, naturally, subjective.

Chemical Structure of LSD (Lysergic acid dieth... — Chemical Structure of LSD (Lysergic acid diethylamide) (Photo credit: Wikipedia)

When we introspect, we are looking inwards, consciously studying our own consciousness. There are therefore limits on what we can find out, as the question arises “How much about itself can a system find out?”

A system that studies itself is limited. It can find out some things, but not all. It’s like a subroutine in a bigger program, in that it knows what to do with inputs and it creates appropriate output for those inputs. Its sphere of influence is limited to those processes written into it, and there is no way for it to know anything about the program that calls it.

A subroutine of a larger program uses the lexical, syntactical and logical rules that apply to the program as a whole, though it may have its own rules too. It shares the concept of strings, number, and other objects with the whole program, but it can add its own rules too.

The Universe is like the subroutine in many ways. The subroutine has inputs and outputs and processes the one into the other. In this Universe we are born and we die. In between we spend our lives.



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An aware or conscious subroutine would know that it processes input and creates outputs, but it would have no idea why. We know that we are born, we live and we die. Apart from that we have no idea why.

This sort of implies that while we may use introspection to investigate some aspects of consciousness we will always fall short of understanding it completely. We may be able to approach an understanding asymptotically however – we might get to understand consciousness to the 90% level, so it would not be a total waste of time to study it.

Consciousness seems to be more than a single state, and the states seem to merge and divert without any actions on our part. For instance, when I am driving there is a part of me that is driving the car and a part of me that is route planning, and maybe a part of me that is musing on the shopping that I intend to do.

The part of me that is driving is definitely aware of what is happening around me. I don’t consciously make the decision to slow down when other traffic gets in the way, but the part of me that is driving does so.

Similarly the part that is route finding is also semi-autonomous – I don’t have to have a map constantly in my mind, and don’t consciously make a decision to turn right, but the navigator part of my consciousness handle that by itself.

Those parts of my mind are definitely conscious of the areas in which they are functioning, because if they were not conscious, they would not be able to do their job alone and would frequently need to move to the front of my consciousness disrupting my musing about my shopping.

It’s like part of my consciousness are carved off and allowed to perform their functions autonomously. However if an emergency should arise, then these parts are quickly jolted back into one.

The parts of my mind are definitely conscious as, at a low level, I am aware of them. I’m aware of the fact that I’m following that blue car, and I’m aware that I have to turn left in 200m or so. I’m also aware of my shopping plans, while I’m aware of the music on the radio.

Deutsch: Servicemenü des Blaupunkt Bremen MP74... — Deutsch: Servicemenü des Blaupunkt Bremen MP74 (Aktivierung: Gerät mit gedrückter “Programm1”- und “Menü”-Taste einschalten). Aktuelle Frequenz: 89,70 MHz (France Musique, Sender Luttange (Metz), PI-Code F203); aktuelle Suchlauffrequenz: 96,80 MHz (bigFM Saarland, Sender Friedrichsthal/Hoferkopf, PI-Code 10B2) (Photo credit: Wikipedia)

While it sounds scary that I’m not totally concentrated on my driving, I believe that this sort of has to happen. If I was totally concentrated on my driving, I would need to stop at every intersection so that I could decide which way to turn.

I would need have my shopping list completely sorted out, to the point of knowing which stores I am going to before even getting the car, and I would have to plan my route precisely. This would not allow for those occasions when passing by something or some shop reminds you that you need something that is not on your shopping list.

This splitting of consciousness allows us to perform efficiently. The only downside is that splitting things too much can result in us becoming distracted. And that is the reason we shouldn’t fiddle with the radio or use cellphones when driving.

The End of (Western) Civilisation

In much of the world capitalism holds sway and there is no denying that, as an economic and social system) it has benefited humanity to a great extent. It has built the great global technological empires that give residents in “western” societies all the consumer goods that we enjoy.

It has provided well for its citizens in general with the standard of living in western societies being the envy of other people in other nations, to the extent that they strive to move to western societies even if their homelands are not embroiled in war and tyranny.

Monument to the fallen in the fight against fa... — Monument to the fallen in the fight against fascism and capitalism in the centre of village Skravena, Bulgaria. (Photo credit: Wikipedia)

But capitalism has its faults, and to some extent it can be likened to a car without brakes. A car without brakes is still driveable, and it is still steerable and can stay on the road until it hits a downhill stretch. Then the driver cannot control the car as it gets faster and faster down the hill and the inevitable will eventually occur.

Capitalism tends to vest power in the businesses and organisations that benefit from it. It tends to concentrate the capital from which it gets its name, of course, in a few individuals and while it benefits most people, there are small but growing number who slide to the bottom of the heap for one reason or another.

The people at the bottom of the heap are not totally without the benefits of the capitalist system. They mostly have televisions for example, which would have been considered a luxury a few decades ago.

However, they often have difficulty with food, accommodation, schooling and medicine. Any jobs that they get will be generally low skilled and low paid. They may even have to work two or more jobs just to get by.

In many western societies houses are becoming more expensive, as measured against family incomes. As houses of their own are out reach for them, they generally rent their accommodation, either from the state or private landlords. Even then they feel the effects of rising house prices in their rent, and often they are forced to rent houses which have serious defects, like damp and mould.

Landlords are of course subject to the capitalism system and are reluctant to spend much money on repairs and so on, as any such expenditure comes out of their pockets. Houses that they let out are often allowed to deteriorate badly.

For those at the bottom of the heap, schooling may be an issue. While the state provides schools, not all schools are the same. A school in an upmarket suburb is almost always better resourced and has better teachers than a school in a poorer area. It’s no surprise, then, when the upmarket schools perform better in terms of qualifications achieved by the pupils.

Access to medical care is often a problem for those at the bottom of the heap. If a trip to the doctor costs $50, as it may well do, then that is a big chunk out of the family budget, and any prescribed medicines will at to the burden. As people get older, they may no longer be able to work and yet this is the time in their lives that they may need medical care more often. In contrast, a people higher up the scale will be more able to pay to have hip operations and so on performed privately.



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People at the top of the scale often look down on those at the bottom as being lazy benefit bludgers who are unwilling to work. This is in most cases untrue. The barriers to rising from poverty to plenty are many, and are in the main insurmountable for many.

The poor are not stupid in the main, though many may not be the brightest of people, and the chances of making it off the bottom rung of the ladder are off putting to many. There are always stories of people making it against the odds, as the saying goes, but in most ways those who succeed in rising up the scale merely reflect the odds.



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999 out of a thousand triers will fail, regardless of drive and ambition. Many will therefore not bother. Even if they did try, they might raise the odds to 2 in a thousand, scarcely any better. It truly is a trap at the very bottom of the heap.

The capitalist system is not able to provide for people in their old age when they are unable to work too. One can put aside a portion of one’s income to help provide for old age but many do not bother, and even they do, the portion that they can put aside depends on their position of the scale of wealth. A poor person, living from hand to mouth, may have no income that he or she can put away for old age.

Traditionally, at least in the last century or so, the state has helped out by providing a guaranteed income for old age in the shape of a pension. However this still has to be paid for and taxes are the way that it is usually done, which means that the richer people will pay for the poorer peoples’ pensions.

It has reached the state in many western societies that welfare, that is schooling, medicine and provision for old age is no longer affordable for society. This is a coming crisis that the current capitalist system cannot avert. It is not exaggeration that it may be the end of civilisation as we know it.



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If the poor can no longer be sustained by the system, then stratification will definitely occur. The “have nots” who outnumber the “haves” will become jealous of them, and that may lead to actual conflict between the classes.

It is not an issue that can be addressed by merely changing the distribution, by effectively taking from the rich and giving to the poor, as this is unfair to those who are considered rich, and will be ineffective anyway, as it provides no incentive for the poor to provide for themselves.



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But if nothing changes, things will get worse and worse until conflict, pestilence, famine and death spread in waves across the Earth. That is, unless we find a better, and fairer societal system.

The original Four Horsemen of Apocalypse. Pane... — The original Four Horsemen of Apocalypse. Panel from X-Factor #24. Art by Walt Simonson. (Photo credit: Wikipedia)

Heaven and Hell



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What if there is an afterlife? Personally I don’t believe in one, but still. What would it be like? The Christian version is that you get judged on your behaviour in this life then go to either heaven or hell. Some versions of Christianity require that you are at the very least baptised, or that you experience being “saved” or that you have taken part in certain rituals before your worldly behaviour is considered.

These lead to awkward questions about, for instance, babies who die before they have a chance to be baptised or whatever. Will they be condemned for ever? I hardly seems fair and various versions of Christianity have ways of getting around this issue. Some suggest that such babies go to a third type of afterlife where they experience neither heaven nor hell – a sort of post life waiting room, which to me seems to be a particularly cruel version of hell!

What is heaven like, though? It must be a pleasant place, as it is a given that it is a reward for a good life, and it has been suggested that it is a place where, the Deity, God, is glorified and worshipped, where all is light, and all the “souls” there experience the joy of being in the presence of the Deity.

Since no one who has died has reported back, the sources for these ideas come from speculation presented as fact, or, as some claim, divine inspiration, an actual message from the Deity. There’s a world of difference between these two options. In other words, it’s either a complete fiction, or absolute fact.

What could we logically deduce, given, for the sake of argument, the existence of an afterlife? It could be that we are reborn in this world, as either a new human or an animal. However, no human can remember any previous existence on this earth, or even on a planet circling a star somewhere across the Universe.



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Of course some people claim to remember previous lives, but strangely, these remembered lives are frequently people who were famous when they were alive, like Anne Boleyn or Shakespeare, and the people who claim to be such reincarnated people often get well documented details of their lives wrong. Of course, history may be wrong, completely wrong, but it is more likely that people who claim to remember previous lives are mistaken or even purposefully deceitful.

If we dismiss the idea of reincarnation, what then? Certainly we can throw away all we know of physics. The only thing that we can be sure of is that things will be different. It may well be that there would be something analogous to our usual physics in play but we have no way of knowing what that would be.

English: Stylized light cone based on the logo for the World Year of Physics 2005. (Photo credit: Wikipedia)

It may be that personality as such would not exist. No meeting with a previously expired spouse in the afterlife would occur, as the personality of the spouse would not exist, and nor would the personality of the recently expired person.

If personalities of the dead did exist in the afterlife, then this could cause issues. What if the recently expired person had remarried after a spouse had previously died. Which of the two spouses would be matched up with the recently expired person? Maybe the physics would be similar to our current physics, but different enough that one person could spend eternity with two or more people?



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I get the impression that there are many more versions or varieties of hell postulated than there are heavens. Visions of hell are often detailed and gruesome. One only needs to look at the right most panel of “The_Garden_of_Earthly_Delights” by Hieronymous Bosch” to see some extreme examples.

Hell is punishment for transgressions in this world. Punishment extends “for all time” which would seem harsh as, surely any sin would be expiated eventually, just as any punishment with the exception of capital punishment in this world is time limited. However this assumes that time in this world is the same as time in the next world, and this may not be so.

Eternity may pass in a different way in the next world – maybe how eternity feels to the sufferer may depend on the seriousness of the transgressions. That may be fairer than for a minor transgressor to suffer through an endless time, judged by the standards of this world.

In accounts of heaven and hell both have inhabitants who have never (so far as I understand it) ever been mortal or human. Angels inhabit heaven and devils and demons live in hell. In both realms the job of the inhabitants appears to be to provide direction for the souls or entities from this world. Angels presumably assist in providing the dead souls with their appropriate rewards and devils and demons punish the transgressors.



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Over the centuries people have imagined or vouchsafed a vision of heaven and hell. It seems that both heaven and hell have a hierarchy of inhabitants. The hierarchies are power hierarchies with the Deity at the top of the one, and the anti-Deity at the top of the other one. This is either a reflection of the prejudices of the person who is imagining these realms, or an interesting structural characteristic or heaven and hell.

Naturally hell is nasty and heaven is nice. I don’t know if it is some cultural bias as a result of my background and upbringing, but it seems to me that the stories and legends of hell are much more detailed than the stories and legends of heaven. My thought is that people are more interested in the gruesomeness of hell than the niceness of heaven.

Certainly the Wikipedia articles on heaven and hell support this theory, as the section on the Christian hell is much longer than the corresponding article on the Christian heaven. The Christian heaven merely has “many mansions“, while the Christian hell is more complex. In fiction especially as exemplified by Milton’s Paradise Lost, and in myth as in Egyptian beliefs hell can be envisaged as complex hierarchies.

But basically, we have no way of knowing for sure if heaven and hell are real and separate realms from our standard world. There can be no physical contact between such realms and our own, since physics describes only the behaviour of things within a single realm and it has no purview outside of that.

We can’t know for sure, so while we may think that they exist and believe in heaven and hell, we should look for our ethics only in this world. Any other course is illogical.

Photography, Ancient and Modern

19th century studio camera, with bellows for f... — 19th century studio camera, with bellows for focusing (Photo credit: Wikipedia)

When I looked up the date when photography was invented I was surprised that it was first tried in 1800, if you allow the word to mean the capturing by some means or other an image created by some means or other!

While optics were known and understood well before this time, no one apparently thought of using glass to create images of views, people or anything else. If they did, there appears to be no record. Also the technology didn’t exist to record any image so created, so it would have been pointless to do so anyway, though artists may have been able to benefit from an image projected onto a canvas as a guide.

The geometry of a pinhole camera as seen from ... — The geometry of a pinhole camera as seen from the X2 axis (Photo credit: Wikipedia)

So before “the camera” we had “the camera obscura”. A camera obscura is basically a darkened room with an image created by a pinhole camera projected onto a white screen. They are fascinating to visit and I highly recommend visiting one.

When the means for creating an image (a lens) came together with a means of capturing the image photography was born. At first the techniques were hit and miss with wet plates coated in chemicals to capture the image and simple lens to create the image, and a long exposure time.

But techniques and technologies quickly improved and exposure times came down. In the early days, when sitting for a photograph, the long exposure times meant that braces had to be used to prevent the sitter from inadvertently moving and spoiling the photograph.

The image of a photographer in those days was of a man hiding under a dark cloth doing mysterious things with his camera, maybe firing off a tray of flash powder to record the image on a glass plate, then dashing off to his “dark room” to process and fix his image onto the glass plate with dangerous chemicals. The end result was an image with light and dark reversed, a so-called “negative”.

New dark room, Boston Camera Club, Bromfield St. (Photo credit: Wikipedia)

Some of these early cameras were works of art, with shining brass trimming, great leather bellows and polished wood bodies and plate holders, all mounted on a substantial tripod. Great brass screws could move the lens closer to or further from the body which held the plate and often could move the lens up and down or from side to side to compensate for perspective distortions.

Several things eventually brought photography in reach of the man in the street. Firstly, it became possible to record the images onto a strip of plastic, which meant that the camera only had to be loaded once in a while. At the same time, it became possible for you to hand your film to the local chemist or apothecary and have the films developed and positive images printed on cards.

The rise of mass production allowed cameras to be produced very cheaply, and the Box Brownie arrived, costing two British pounds. Anyone could be a photographer. To be sure corners had to be cut, so the Brownie was a simple box with small and very simple lens, and the shutter was simply a plate that usually blocked light from entering.

On pressing the shutter lever a spring was tightened until the plate flicked over, giving the lens a brief look at the outside world. The film strip was held on one spool and transferred to another. A window in the back of the camera showed the backing strip of the film and you would the film on until the next number showed in the window.

The user aimed the camera by using a crude “viewfinder”. This was simply a window in the top of the camera which allowed the photographer to look down on a tiny mirror which reflected the view in front of the camera. It was around a centimetre wide and pointed only more or less in the same direction. To switch from landscape mode to portrait mode you turned the camera over and looked into a similarly minute viewfinder!

Of course things rapidly moved on from there. People loved the Box Brownie and soon handheld cameras of all sorts appeared. Some had two linked lens arranged piggy back style as in the Rollieflex, and some had a single lens, like the Hasselblad. Those two were high end machines, and featured switchable components and high quality lenses and other accessories. They tended to be favoured by professional photographers.

Most other cameras were built on a different design, though. Most featured a eye level viewfinder, and cheaper ones usually didn’t have interchangeable accessories. Most moved away from the spool to spool system to the 35mm cassette. Really cheap cameras eventually had a drop in cassette system.

Two other big changes were the introduction of colour film and the single lens reflex system. Many photographers used to monochrome film were appalled by the advent of colour film and swore never to change to it. Most amateurs adopted it with enthusiasm of course. Eventually everyone (almost) used colour.

The single lens reflex system allowed the photographer to see exactly what he was shooting as the viewfinder looked through the same lens through which the image was captured. Generally the viewfinder was eye level but the Hassleblad was an exception retaining the waist level view point.

The more expensive cameras had knobs, dials and buttons all over them, but the cheaper varieties had only a few, and some did not have any controls. All used film cassettes and most had flash devices for low light level conditions.

Then along came digital. Film disappeared, to be replaced by flash storage. Most cameras lost their viewfinders, which were replaced by small screens covering the whole of the back of the camera. Many settings could be set using the screen and a few buttons, and the cameras sizes shrunk. Some these days are credit card size.

But now, it seems that the so called “compact digital cameras” have briefly had their day. Every smartphone has an embedded camera, and people are not buying the compacts. Some smartphones now come with Leica lens technology.

The result? Billions (if I’m not mistaken) of absolutely atrocious photographs spamming the Internet. From cute cats to drunken revellers, everything is now floating around out these. But I’m optimistic. Real photographs and real photographers are still out there. Somewhere.



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The Space Between the Stars

Space is big. The Voyager spacecraft (Voyager I and II) were launched in 1977 and are, forty years later, only just entering interstellar space. Though the exact point at which space becomes “interstellar” is debatable.

The Voyagers will take 40,000 years or so to reach one of the stars in the “local” group. That’s about one fifth of the time that humans have existed as a separate species. Or 400 times as long as the length of time that a human is able to live. If a generation is around 20 years long, that is about 2,000 generations. It is a long, long time, and we may well be extinct as a species by then, for one reason or another.

The “local group” of stars is an arbitrary group of stars which are (relatively) close to the Sun. I’m unsure whether they really constitute a group of bodies bound by gravity or whether they are close to the sun by chance. Of course if any of the stars in the local group are bound by gravity, then the stars would form a binary or multiple star system.

Of course, our star and all the others in the local group are part of our galaxy, the Milky Way. Specifically we are part of one of the arms of the Milky Way, which is a spiral galaxy. All stars in the Milky Way are bound by gravity, with the possible exception of stars which are merely passing through the Milky Way at this time.

Just like stars, galaxies seem to form groups, which then form super-groups and so on. All these structures are several orders of magnitude larger than the prior smaller ones, are more complex and contain more matter.

The majority of space however is just space. The gaps between the bits of matter, stars, systems, galaxies, groups and so on contain almost nothing, or a seething sea of virtual particles depending on how you look at it.

The “almost nothing” consists of a very small number of particles (usually hydrogen atoms or nuclei, protons) in a cubic metre. For comparison the best vacuum that can be created on Earth may contain several million atoms in that volume. This of the same order of magnitude as molecular clouds as observed by astronomers. Molecular clouds are among the densest clouds observed in space.

The stars, planets, asteroids and similar bodies comprise only a very small part of the Universe and the average density of the Universe is much the same as the density of empty space. In other words, the Voyagers are heading into areas where the conditions are more typical of the Universe than those around our star.

Voyager 1 is currently within the heliosheath ... — Voyager 1 is currently within the heliosheath and approaching interstellar space. (Photo credit: Wikipedia)

I mentioned virtual particles earlier. While virtual particles show up as short-lived particles that briefly come into existence in some particle interactions, the virtual particles that I refer to come into existence in a vacuum as pairs and almost immediately mutually annihilate. Though they do not interact with other matter, they do have an effect which can be measured.

Mathematicians have a different concept of space. In mathematics space is a (usually) three dimensional construct that serves merely to separate and give structure to such things as points, lines, planes, volumes and shapes. Point A is distinguished from point B by the distance between them and also the orientation of a line joining them.



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In a simple case every point has (usually) three coordinates which define its position relative to some fixed point or origin and fixed coordinate system. The coordinate system can be any system that locates the point.

For instance, you can describe the point A’s position as “Face along a given axis, rise up until you are level with point A. The distance moved up is one coordinate. Rotate left through an angle until a line parallel to the plane you rose up from passes from you through the point A. The angle you turned through is the second coordinate. The third coordinate is the distance along the line from you to the point.

I’ve described a cylindrical coordinate system, but the coordinate system may be any system that gives three unique (in that system) coordinates for point A. A common system is the Cartesian system of three mutual perpendicular axes. Another is the spherical system, defined by two angles and a distance.

Of course, such systems can be generalised to more dimensions or fewer, depending on the needs of the mathematician. Most people can understand simple two dimensional graphs which are usually drawn using a two dimensional Cartesian coordinate system.

Of course scientists use mathematical models for various purposes. For instance the scientist may wish to know the probability of one hydrogen atom in the interstellar space meeting another such molecule. Since we have only about one such atom in every cubic metre, the probability is going to be small, but, of course, we can assume as lone a time period as we wish.

Gravity has to be figured in, to be sure, but a long time will be required for such atoms to collide. If the atoms are by chance moving slowly relative to each other, they may stick together and form the basis of a particle of matter. Such a clump might attract other atoms and before long (well actually after literally an astronomical length of time) a star will form.

It must have happened otherwise we would not be here. We are the result of matter aggregating and then exploding. All the atoms in our bodies that are not hydrogen were made in the centres of stars. The stars have to have exploded to allow these atoms to end up in our bodies.

A long time has passed since the birth of the Universe. In that time matter has crept together hydrogen atom by hydrogen atom until great collections of atoms have compressed in the centre to the point where nuclear reactions have occurred. Hydrogen fused to helium, then to heavier elements all the way up to Uranium.

Ball-and-stick model of the haem a molecule as... — Ball-and-stick model of the haem a molecule as found in the crystal structure of bovine heart cytochrome c oxidase. Histidine residues coordinating the iron atom are coloured pink to distinguish them from haem a. Colour code: Carbon, C: grey-black Hydrogen, H: white Nitrogen, N: blue Oxygen, O: red Iron, Fe: blue-grey Structure by X-ray crystallography from PDB 1OCR, Science (1998) 280, 1723-1729. Image generated in Accelrys DS Visualizer. (Photo credit: Wikipedia)

At which point the stars have exploded throwing all the elements out into the Universe. These elements then crept together again to produce new stars like our sun and gaseous and rocky planets orbiting them. Prior to this there were no rocky planets and no life. We live in the Universe Mark II.

Capitalism

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 (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 (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.

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.

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.



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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.



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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.

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) (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.

Simple Arithmetic

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).



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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.

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 (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.



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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 (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.

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.

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.

Here be Dragons

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).

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.



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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 (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 (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.

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.

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 http://www.33rdinfantrydivision.org/archi... — From http://www.33rdinfantrydivision.org/archivesphotos/may5_flamethrower.jpg source information from 33rdinfantrydivision.org : 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.

Let’s be Rational – Realer Numbers

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…

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).

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.

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.

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.

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.

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].

Rational versus real

(My last post was very late because I had taken part in a 10km walk on the Sunday and spent the week recovering.)

There’s a fundamental dichotomy at the heart of our Universe which I believe throws some light on why we see it the way we do. It’s the dichotomy between the discrete and the continuous.

A rock is single distinct thing, but if you look closely, it appears to be made of a smooth continuous material. We know of course that it is not really continuous but is constructed of a mesh of atoms each of which is so small that we cannot distinguish them individually, and which are connected to each other with strong chemical and physical bods.

An early, outdated representation of an atom, ... — An early, outdated representation of an atom, with nucleus and electrons described as well-localized particles on well-localized orbits. (Photo credit: Wikipedia)

If we restrict ourselves to the usual chemical and physical processes we can determine to a large extent determine what the atoms are which comprise the rock, and we can make a fair stab at how they are connected and in what proportions.

We can explain its colour and its weight, strength, and maybe its magnetic properties, even its value to us. (“It’s just a rock!” or “It’s a gold nugget!”) We have a grab bag of atoms and their properties, which come together to form the rock.

English: Gold :: Locality: Alaska, USA (Locali... — English: Gold :: Locality: Alaska, USA (Locality at mindat.org) :: A hefty 63.8-gram gold nugget, shaped like a pancake. Very beautiful and classic locality nugget. 4.5 x 3 x 0.6 cm Deutsch: Gold :: Fundort: Alaska, Vereinigte Staaten (Fundort bei mindat.org) (Photo credit: Wikipedia)

The first view of atoms was that they were indivisible chunks with various geometric shapes. This view quickly gave way to a picture of atoms as being small balls, like very tiny billiard balls. Then the idea of the billiard balls was replaced by the concept of the atom as a very tiny solid nucleus surrounded by a cloud of even tinier electrons.

Of course the nucleus turned out not to be solid, but to be composed of neutrons and protons, and even they have been shown to be made up of smaller particles. Is this the end of the story? Are these smaller particles fundamental, or are they made up of even smaller particles and so on, “ad infinitum”?

It appears that in Quantum Physics that we have at least reached a plateau, if not the bottom of this series of even smaller things. As we descend from the classical rock, through the smaller but still classical atoms, to the very, very small “fundamental” particles, things start to get blurry.

The electron, probably the hardest particle that we know of, in the sense that it is not known to be made up of smaller particles, behaves some of the time as if it was a wave, and sometimes appear more particle like. The double slit experiment shows this facet of its properties.

Diagram of the double-slit experiment (Photo credit: Wikipedia)

The electron is not unique in this respect, and in fact the original experiments were performed with photons, and scientists have performed the experiment even with small molecules, showing that everything has some wave aspects, though the effect can be very small, and is for all normal purposes unnoticeable.

A wave as we normally see it is an apparently continuous thing. As we watch waves rolling in to the beach we don’t generally consider it to consist of a bunch of atoms moving up and down in a loosely connected way that we call “liquid”. We see a wave as distributed over a breadth of ocean and changing in a fairly regular way over time.

At the quantum level particles are similarly seen to be distributed over space and not located at a particular point. An electron has wave like properties and it has particle like properties. Interestingly the sea wave also has particle like properties which can be calculated. Both the sea wave and the electron behave like bundles of energy.

You can’t really say that a wave is at this point or that point. A water may be at both, albeit with different values of height. If the wave is measured at a number of locations, then by extension it has a height in between locations. This is true even if there is no molecule of water at that point. The height is in fact the likely height of a molecule if it were to be found at that location.

By analogy, and by the double slit experiment, it appears that the smallest of particles that we know about have wave properties and these wave properties smear out the location of the particle. It appears that fundamental particles are not particularly localised.

It appears from the above that at the quantum level we move from the discrete view of particles as being individual little “atoms” to a view where the particle is a continuous wave. It points to physics being fundamentally continuous and not discrete.

There’s a mathematical argument that argues against this however. Some things seem to be countable. We have two feet and four limbs. We have a certain discrete number of electrons around the nucleus of an atom. We also have a certain number of quarks making up a hadron particle.

Other things don’t appear to be countable, such as the positions a thrown stone can traverse. Such things are measured in terms of real numbers, though any value assigned to the stone at a particular instance in time is only an approximation and is in fact a rational number only.

At first sight it would appear that all we need to do is measure more accurately, but all that does is move the measurement (a rational number) closer to the actual value (a real number). The rational gets closer and closer to the real, but never reaches it. We can keep increasing the accuracy of our measurement, but that just gives us a better approximation.

It can be seen that the set of rational numbers (or the natural numbers, equivalently) maps to an infinite subset of the real numbers. It is usually stated that the set of real numbers contains the rational numbers. I feel that they should be kept apart though as they refer to different domains of numbers – rational numbers are in the domain of the discrete, while the real numbers are in the domain of the continuous.