Rational versus real

English: Dyadic rational numbers in the interv...
English: Dyadic rational numbers in the interval [0,1] (Photo credit: Wikipedia)
(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An early, outdated representation of an atom, ...
An early, outdated representation of an atom, with nucleus and electrons described as well-localized particles on well-localized orbits.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”?

English: "Ad Infinitum" Oil in Canva...
English: “Ad Infinitum” Oil in Canvas 109 x 152.5 by peruvian painter Ricardo Córdova Farfán (Photo credit: Wikipedia)

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
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.

Wineglass with blue liquid
Wineglass with blue liquid (Photo credit: Wikipedia)

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.

English: A particle motion in an ocean wave. A...
English: A particle motion in an ocean wave. A=At deep water. B=At shallow water. The elliptical movement of a surface particle flattens with increasing depth 1=Progression of wave 2=Crest 3=Trough (Photo credit: Wikipedia)

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.

The Continuum
The Continuum (Photo credit: Wikipedia)

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.

Stonehenge sulis
Stonehenge sulis (Photo credit: Wikipedia)

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.

Particles by fundamental interactions
Particles by fundamental interactions (Photo credit: Wikipedia)

Numbers are fascinating

A little image of aleph_0, smallest infinite c...
A little image of aleph_0, smallest infinite cardinal (Photo credit: Wikipedia)

Numbers fascinate me. What the heck are they? They seem to have an intimate relationship with the “real world”, but are they part of it? If I heave a rock at you, I heave a physical object at you. If I heave two rocks at you, I heave real objects at you. It’s a different physical experience for you, though.

If I heave a third rock at you, again, it’s a different qualitative experience. It’s also a different qualitative experience from having one rock or two rocks thrown at you.

Glyder Fawr
Glyder Fawr (Photo credit: Wikipedia)

Numbers come in three “shapes”. There are cardinal numbers, which answer questions like “How many rocks did I throw at you?” There are ordinal numbers, which answer questions like “Which rock hit you on the shoulder?” Finally there are nominal numbers, which merely label things and answer questions like “What’s you phone number?”

As another example, in the recent 10km walk which I took part in, I came sixth (ordinal) in my age division. That sounds good until I admit that there were only seven (cardinal) entrants in that division. Incidentally, my bib number was 20179 (nominal).

Cardinal numbers include the natural numbers, the integers and the rational numbers and the real numbers (as well as more esoteric numbers). For instance the cardinal real number π is the answer to the question “How many times would the diameter fit around the circumference of a circle?”

It’s a bit more difficult to relate ordinal numbers with real numbers, but the real numbers can definitely be ordered – in other words a real number ‘x’ is either bigger than another real number ‘y’, or vice versa or they are equal. However, there are, loosely speaking, more real numbers than ordinals, so any relationship between ordinal numbers and real numbers must be a relationship between the ordinal numbers and a subset of the real numbers.

English: Example image of rendering of ordinal...
English: Example image of rendering of ordinal indicator º Italiano: Immagine esemplificativa della resa grafica dell’indicatore ordinale º (Photo credit: Wikipedia)

Subsets of the real numbers can have ordinal numbers associated with them in a simple way. If we have a function which generates real numbers from a parameter, and if we feed the function with a series of other numbers, then the series of other numbers is ordered by the way that we feed them to the function, and the resulting set of real numbers is also ordered.

So, we might have a random number generator from which we extract a number and feed it to the function. That becomes the first real number. Then we extract another number from the generator, feed it to the function and that becomes the second real number, and so on.

The random map generator provides a limitless ...
The random map generator provides a limitless supply of colourful terrains of various themes. Open island maps, like this one, allow players to use airstrikes. Cavern maps have an indestructible roof which cannot be passed. (Photo credit: Wikipedia)

What we end up doing is associating a series of integer ordinal numbers with the generated series of real numbers. These ordinal numbers are associated with the ordered set of real numbers that we create, but the real numbers don’t have to be ordered in terms of their size.

Nominal numbers such as my bib number are merely labels. They may be generated in an ordered way, though, as in the case of my bib number. If I had registered a split second earlier or later I would have received a different number. However, once allocated they only serve to show that I have registered, and they also show which event I registered for.

On the occasion that I took part there were two other events scheduled : a 6.5km walk and a half marathon. My bib number indicated to the marshals and officials which event I was taking in and which way to direct me to go.

I’m not a mathematician, but it seems to me that ordinal numbers are more closely aligned to the natural numbers, the positive integers, than to any other set of numbers. You don’t think of someone coming 37 and a half position in a race. Indeed if two people come in at the same time they are conventionally given the same position in the race and the next position is not given.

English: Selby Apartments, located on 37th Str...
English: Selby Apartments, located on 37th Street, 37th Avenue, and Marcy Street in Omaha, Nebraska. The view is from 37th Avenue and Marcy, looking northeast. At left is 825 S. 37th Ave; at right is 3710 Marcy. (Photo credit: Wikipedia)

There’s a fundamental difference between natural numbers or the integers, or for that matter the rational numbers and the real numbers. The real numbers are not countable : they can’t be mapped to the natural numbers or the integers. The rational numbers can, so can be considered countable. (Once again, I’m simplifying radically!)

Natural numbers and integers are related to discrete objects and other things. The number of dollars and cents in your bank account is a discrete amount, in spite of the fact that it is used as real number in the bank’s calculations of interest on your balance. If I toss two rocks at you that is a discrete amount.

English: Causal loop diagram (CLD) example: Ba...
English: Causal loop diagram (CLD) example: Bank balance and Earned interest, reinforced loop. Diagram created by contributor, with software TRUE (Temporal Reasoning Universal Elaboration) True-World (Photo credit: Wikipedia)

Even I tip a bucket of water over you, I douse you in a discrete number of water molecules (plus an uncertain number of other molecules, depending on how dirty the water is). However the distance that I have to throw the water is not a discrete number of metres. It’s 1.72142… metres, a real number.

At the level at which we normally measure distances distances don’t appear to be broken down into tiny bits. To cover a distance one first has to cover half the distance. To cover half the distance one must first cover one quarter of the distance. It is evident that this halving process can be continued indefinitely, although the times involved are also halved at each step.

This seems a little odd to me. Numbers are at the basis of things, and while numbers are not all that there is, as some Greek philosophers held, they are important, and, I think, show the shape of the Universe. If the Universe did not have real numbers, for example, then it would be unchanging or perhaps motion would be a discrete process, like movements on a chess board.

If the Universe did not have any integers, the concept of individual objects would not be possible, since if you could point at an object you would have effectively counted “one”. In other words we need the natural numbers so that we can identify objects and distinguish one from one another, and we need the real numbers so that we can ensure that the objects don’t all exists at the same spot and are, in fact separated from one another.

Visualisation of the (countable) field of alge...
Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate degree of the polynomial the number is a root of (red = linear, i.e. the rationals, green = quadratic, blue = cubic, yellow = quartic…). Points becomes smaller as the integer polynomial coefficients become larger. View shows integers 0,1 and 2 at bottom right, +i near top. (Photo credit: Wikipedia)

 

The Search for the Fundamental

Motion of gas molecules Español: Animación mos...
Motion of gas molecules Español: Animación mostrando la agitación térmica de un gas. Cinco partículas han sido coloreadas de rojo para facilitar el seguimiento de sus movimientos. Русский: Хаотическое тепловое движение на плоскости частиц газа таких как атомы и молекулы (Photo credit: Wikipedia)

When does it stop? This screen that I am looking at, the keyboard that I am typing on, the invisible air between my eyes and the screen, even my body, all are composed of atoms, I told and believe. Apart from atoms, all there is is radiation, of various sorts.

The ancient Greek philosophers didn’t know about atoms so proposed various theories, which today seem quaint, but eventually they came around to atomism, and abandoned the other theories. In particular the theory of the four classical elements, earth, fire, water and air was dropped.

The four classical elements, after Aristotle. ...
The four classical elements, after Aristotle. Чотири стихії (за Арістотелем) (Photo credit: Wikipedia)

As I said, the theory now sounds quaint, but, given that the ancient Greek philosophers were not of an experimental frame of mind, the four classical elements could explain much of what could be observed. Everything could have been a mixture of these elements in various proportions.

After all, it appeared to work for colours – all colours that can be displayed on a computer screen can be specified in terms of the amount of the three primary colours of red, green and blue that a single pixel or dot on the screen emits. Why shouldn’t this scheme work for other things than light?

Barycentric RGB
Barycentric RGB (Photo credit: Wikipedia)

However Greek philosophers (and of course, philosophers in other cultures) noticed that, while some things could be broken down into component parts – sugar could be melted and burned, water could be driven off to leave the salts behind, and more importantly alcohol could be evaporated off and collected to make spirits, some things could not be broken down.

Gold, sulphur and phosphorus stubbornly refused to separate into earth, air, water or fire. Of course such stubbornness could be explained by the classical element theory – after all some things are easier to break down than others, but the Greeks eventually dropped the theory in favour of atomism. (This and what follows is highly simplified and condensed).

(Click here for rotating model)
(Click here for rotating model) (Photo credit: Wikipedia)

This is the belief that everything is made up of small indivisible particles which differ from element to element. The lump of gold contains billions of gold atoms, while the sulphur block contains sulphur atoms.

From about the start of the scientific revolution, people started to work out the rules of chemistry, and the ‘why’ of chemical reactions. Why did carbon in coal burn away and leave an ash? We know that the carbon in the coal burns using the oxygen in the air and creates oxides of carbon which are gasses and not easily detectable, but the experiments which led to this knowledge were preformed in the era of the scientific revolution.

So, matter is composed of atoms. That seemed to be the end of the story, as the vast majority of chemical experiments could be explained in terms of atoms, but exactly why atom A reacts in fixed proportions with atom B, but won’t have a bar of atom C. These relationships were noted but not really explained.

By the middle of the 19th century scientists began to detect problems with the “atoms as billiard balls” model. Electrons were discovered and soon related to chemistry, answering the above question. The new model, “atoms as small planetary-like systems”, had a small positively charged, and solid nucleus surrounded by a swarm of negatively charged electrons, with the electrons taking a major role in determining the chemistry of the atom.

It was discovered that many elements behaved as if each atoms of the element weighed the same, but some elements broke this rule. The gas Chlorine for example has an atomic weight of 35.45. In other words each atom weighed about 35 and half times as much as a Hydrogen atom.

It was eventually discovered that not all Chlorine atoms weighed the same. Most had an atomic weight of 35 but some (about half) had a weight of 36. To cut a long story short it was discovered that the supposedly solid nucleus was composed of a collection of other particles called protons and neutrons.

English: Liquid Chlorine in flask for analysis.
English: Liquid Chlorine in flask for analysis. (Photo credit: Wikipedia)

While the number of protons and electrons determine the chemistry of an atom almost completely, the number of neutrons contribute mass to the atom and barely affect the chemistry.

While electrons appear to be truly fundamental particles and cannot be broken down further, the protons and neutrons are composed of particles called quarks. For reasons mentioned in the Wikipedia article quarks cannot be found in isolation, but are only found in other particles.

English: The quark structure of the proton. Th...
English: The quark structure of the proton. There are two up quarks in it and one down quark. The strong force is mediated by gluons (wavey). The strong force has three types of charges, the so-called red, green and the blue. Note that the choice of green for the down quark is arbitrary; the “color charge” is thought of as circulating among the three quarks. (Photo credit: Wikipedia)

In addition to protons and neutrons, quarks make up other sub-atomic particles such as mesons. Scientists have discovered or postulated bosons which are particles that bind quarks and other fundamental particles together. From then on, things get complicated!

I haven’t mentioned the photon, which is bosonic, or the neutrino which is a fermion. All fundamental particles fit into one of these two families, and all sub-atomic interactions are the result of the rather incestuous exchange of these particles in their various groups and a strict set of rules. So far so good.

English: Enrico Fermi
English: Enrico Fermi (Photo credit: Wikipedia)

However, there are still questions to be answered. Are these particles truly fundamental or do they have components, which may or may not be particles in the classical sense? What are the sizes of these particles, if such a concept is appropriate at this level? Have we found them all? What about dark matter?

Scientists have abandoned the first question. They don’t generally refer to particles as fundamental. They have seen a long list of fundamental particles turn out to be not so fundamental after all.

Sizes of the particles may not make sense at the particle level, but the various theories may indicate sizes for some of them. There are difficulties over the size of the electron for instance. If it were a point object rather than having something that equates to size, then that causes difficulties with some theories.

As for the third and fourth questions, it appears that scientists may have found all the particles that explain ordinary matter, but naturally cautious, they don’t rule out other forms of matter such as the so called “dark matter” and “dark energy“. Dark matter and dark energy apparently interact with gravity and (from the Wikipedia article) and the Weak Nuclear Interaction.

pie chart of dark matter and normal energy rat...
pie chart of dark matter and normal energy ratio taken from en.wikipedia (Photo credit: Wikipedia)

My original question was “When does it stop?” By this I meant, which particles are truly fundamental and which have components that determine their properties? This question remains open, but if you have followed through my exposition, you will probably see that this is a question without an easy answer.

 

Mainframes were different

IBM System 360/65 Operator's Panel The IBM Sys...
IBM System 360/65 Operator’s Panel The IBM System/360 Model 65 first shipped in 1966 This photo was taken by Mike Ross of corestore.org . He has given permission via email “Feel free to make use of … my pictures under the GNU license. All I ask is that they are credited & linked…”. –agr 19:07, 17 Aug 2004 (UTC) (Photo credit: Wikipedia)

Mainframes were … different. Today we have devices that have orders of magnitude more processing power than the old mainframes. The old mainframes were big machines that performed business tasks for large organisations and that cost millions of dollars. The phone in your pocket shows you your email, let’s you Facebook or tweet your friends where ever you might be and what ever you might be doing. You can even use it to phone people!

Arguably we fritter away most of the enormous amount of processing power and storage that our hand held devices provide. We watch videos on them, yes, even porn, and play endless silly games on them. And cats. Cats have taken over the Internet and thereby our connected devices.

A six-week old kitten.
A six-week old kitten. (Photo credit: Wikipedia)

The fundamental concept of the mainframe is one single powerful(!) computer with all devices, such as card readers, terminals (screen and keyboard, no mouse), printers, tape units and other devices, more or less directly and permanently connected to the computer.

Interestingly, when you typed something into a terminal using the keyboard, it wasn’t sent immediately to the mainframe but was recorded in a buffer in the terminal. When one of a number of keys was hit the whole buffer was sent to the mainframe. These special keys were “Return”, which is similar to the “Enter” key, a “PF Key”, which is similar to a function key, and a few others.

Return
Return (Photo credit: Wikipedia)

This meant that you could type and edit stuff at your terminal and it would only be sent when you were finished. That is different from the model used by PCs and other modern devices, where every single key press that occurs is sent to the target computer, including typos and correction to typos.

Of course, when you press a key on your PC keyboard, the computer that is the target is the one that the keyboard is connected to, and what you type in goes into a buffer, but the principle still applies. The effect is more obvious if a lot of people are connected to a multi-user computer and are using it heavily, when the response to hitting a key takes a second or so to echo back to the screen.

Multi-user computers are not common these days as they are not trivial to set up, and computers and networks have become so fast that it is generally easier to change the model and access applications over the network rather than use the direct connect model that was used in the early days.

A lot of the features of modern computing devices originated in mainframes. Mainframes originally ran one job or task at a time, but soon they became powerful enough to run many jobs at the same time. Mainframe operating systems were soon written to take advantage of this ability, but to achieve the ability to run multiple jobs, the operating systems had to be able to “park” a running job while another job got a slice of the processor.

Punched card in the 80-column-format according...
Punched card in the 80-column-format according to the IBM standard. The card was used at the beginning of the 1970s at the University of Stuttgart (Germany) for the input of Fortran programms in the IBM mainframe. The plain text of the coded line is on the top left. (Photo credit: Wikipedia)

To do that the operating system had to save the state of the process, especially the memory usage. This was cleverly achieved by virtualising memory usage – the job or task would think that it was accessing this bit of memory but the memory manager would make it use that bit of memory instead. The job or task didn’t know.

For instance the job or task might try to read memory location #ff00b0d0 (don’t worry about what this means) and the memory manager would serve up #ffccbod0 instead. Then a moment of so later another job or task might try to read that memory location. It would expect to find its own data there not the first task’s data, and the memory manager would this time serve up, say, #ffbbb0d0.

SVG Graph Illustrating Amdahl's Law
SVG Graph Illustrating Amdahl’s Law (Photo credit: Wikipedia)

The key point is that the two tasks or jobs access the same address, but the address is not real, it is what is known as a virtual address, and the memory manager directs the request to different real addresses. This allows all sorts of cunning wheezes – a job or task can address more memory than the machine has installed and memory locations that have not been used in a while can be copied out to disk storage allowing the tasks in the machine to collectively use more memory than physically exists!

Exactly the same thing happens in your phone, your tablet, or your PC. Many tasks are running at the same time, using memory and processors as if these resources were dedicated to all the tasks. (Actually not all tasks are running at the same time – only as many tasks as there are processors in the processor chip can be running at the same time, but the processors are switched between task so fast it appears as if they are. The same is also true of mainframes).

Diagram showing the memory hierarchy of a mode...
Diagram showing the memory hierarchy of a modern computer architecture (Photo credit: Wikipedia)

Of course, there’s a downside to the mainframe model and that is that if the mainframe goes down, everyone is affected. In the early days of the PC era, every PC was independent and if it went down (which they often did) only one or two people, those who actually used the computer were directly affected. So if the Payroll computer crashed it didn’t affect Human Resources.

Soon though the ability to connect all the computers over a network became possible and computing once again became centralised. Things have changed, but the corporate server or servers now fulfil the role that once belonged to the mainframe.

All the advantages of centralisation have been realised again. Technical facets of the operation of computers have been removed from those whose job was not primarily computing, much to the relief of most them I’d suspect. Backups and technical updates are performed by those whose expertise is in those fields, rather than by reluctant amateurs in the field.

However, the downside is that a centralised computing facility is never as flexible as the end users would like it to be and, somewhat ironically, that as an outage of the old mainframe used to affect many people, so will an outage of a server or servers in the current milieu.

Ganglia report showing editing outage, when Wi...
Ganglia report showing editing outage, when Wikipedia server srv156 stopped responding (Photo credit: Wikipedia)