A equals B

Weather icon: temperature equal
Weather icon: temperature equal (Photo credit: Wikipedia)

The whole universe is full of inequality. No two galaxies are exactly alike, no two planets are exactly alike, no two grains of sand are exactly alike, no two atoms of silicon are exactly alike. Wait a minute, is that last one correct?

Well, in one sense each atom of silicon is alike. Every silicon atom has 14 protons in its nucleus, and, usually, 14 neutrons. However it could have one or two neutrons extra if it is a stable atom, or even more if it is a radioactive atom. Alternatively it could have less neutrons and again it would be radioactive.

Monocrystalline silicon ingot grown by the Czo...
Monocrystalline silicon ingot grown by the Czochralski process (Photo credit: Wikipedia)

So two silicon atoms with the same number of neutrons in the nucleus are “equal” right? Well, of course a single atom by itself is seldom if ever found in nature, and two isolated similar atoms are very unlikely. But suppose.

An atom of silicon is said to have electron shells with 14 electrons in them. Without going into unnecessary details these electrons can be in a base (lowest) state or in an excited state. With multiple excitation levels and multiple electrons the probability of two isolated atoms of silicon with all electrons in the same excitation state is extremely low.

Atom Structure
Atom Structure (Photo credit: Wikipedia)

In practise of course, you would not find isolated atoms of silicon at all. You would find masses of silicon atoms, perhaps in a random conformation, or maybe in organised rows and columns. One of the tricks of semi-conductors is that the silicon atoms are organised into an array, with an occasional atom of another element interspersed.

Atoms according cubical atom model
Atoms according cubical atom model (Photo credit: Wikipedia)

This has the effect of either providing an extra electron or one fewer in parts of the array. Under certain conditions this allows the silicon atoms and the doping element to pass the extra electron, or the lack of an electron (known as a hole) along the array in an organised manner, a phenomenon known in the macroscopic world as an electric current.

English: Drawing of a 4 He + -ion, with labell...
English: Drawing of a 4 He + -ion, with labelled electron hole. (Photo credit: Wikipedia)

So, while two atoms of silicon may in some theoretical physical and chemical sense be equal, in practice, they will be in different states, in different situations. What can be said about two silicon atoms is that fit an ideal pattern of a silicon atom, in that the nucleus of the atom has 14 protons. Some of the properties and states of the two atoms will be different.

At the very least the two atoms will be in different locations, moving with different velocities and with different amounts of energy. They can never be “equal as such. The best that you could probably say is that two atoms of the same isotope of silicon have the same number of neutrons and protons in their nuclei.

Periodic table with elements colored according...
Periodic table with elements colored according to the half-life of their most stable isotope. Stable elements. Radioactive elements with half-lives of over four million years. Half-lives between 800 and 34,000 years. Half-lives between 1 day and 103 years. Half-lives ranging between a minute and 1 day. Half-lives less than a minute. (Photo credit: Wikipedia)

When we talk about numbers we stray into the field of mathematics, and in maths “equal” has several shades of meaning. When we say that one integer equals another integer we are essentially saying that they are the same thing. So 2 + 1 = 3 is a bit more than a simple equality and in fact that expression can be referred to as an identity.

Algebraic proofs are all about changing the left hand side of an expression or the right hand side of the expression or both and still retaining that identity between the two sides.

Mnemosyne with a mathematical formula.
Mnemosyne with a mathematical formula. (Photo credit: Wikipedia)

In the real world we use mathematics to calculate things, such a velocities, masses, energy levels, in fact anything that can be calculated. Issues arise because we cannot measure real distances and times with absolute accuracy. We measure the length of something and we know that the length that we measure is not the same as the actual length of the object that we are measuring.

Lengths are conceptually not represented by integers but by ‘real numbers’. Real numbers are represented by two strings of digits separated by a period or full stop. Both strings can be infinite in length though the both strings are usually represented as being finite in length.

1 Infinite Loop, Cupertino, California. Home o...
1 Infinite Loop, Cupertino, California. Home of Apple Inc. and one of Silicon Valley’s best known streets. (Photo credit: Wikipedia)

If we measure a distance with a ruler or tape measure, the real distance will usually fall between two marks on the ruler or measure. So we can say that the length is, say, between 1.13 and 1.14 units of measurement. If use a micrometer we might squeeze and extra couple of decimal places, and say that the length is between 1.1324 and 1.1325. With a laser measuring tool we can estimate the length more accurately still.

You can see what is happening, I hope. The more accurately we measure a distance, the more decimal places we need. To measure something with absolute accuracy we would need an infinite number of decimal places. So when we say that the distance from A to B equals 1.345 miles, we are not being exact, but are approximating to the level of accuracy that we need. Hence A is not really equal to B.

Aurora during a geomagnetic storm that was mos...
Aurora during a geomagnetic storm that was most likely caused by a coronal mass ejection from the Sun on 24 May 2010. Taken from the ISS. (Photo credit: Wikipedia)

A particularly interesting case of A not being equal to B is in the mathematical case where one is trying to determine the roots of an equation. There are various method of doing this and there is a class of methods which can be designated as iterative.

One first makes a guess as to the correct value, puts that into the equation which generates a new value which is, if the iterative method chosen is appropriate, closer to the correct value. This process is repeated getting ever closer to the correct answer.

Plot of x^3 - 2x + 2, including tangent lines ...
Plot of x^3 – 2x + 2, including tangent lines at x = 0 and x = 1. Illustrates why Newton’s method doesn’t always converge for this function. (Photo credit: Wikipedia)

Of course this process never finishes, so we specify some rule to terminate the process, possibly some number of decimal places, at which to stop. More technically this is called a limit.

To prove convergence, in other words to prove that the process will generate the root if the process is taken to infinity, has proved mathematically difficult. I’m not going to attempt the proof here, but after several attempts from the time of Isaac Newton, this was achieved last century, with the introduction of the concept of limits.

English: A comparison of gradient descent (gre...
English: A comparison of gradient descent (green) and Newton’s method (red) for minimizing a function (with small step sizes). Newton’s method uses curvature information to take a more direct route. Polski: Porównanie metody najszybszego spadku(linia zielona) z metodą Newtona (linia czerwona). Na rysunku widać linie poszukiwań minimum dla zadanej funkcji celu. Metoda Newtona używa informacji o krzywiźnie w celu zoptymalizowania ścieżki poszukiwań. (Photo credit: Wikipedia)

One can then say, roughly, that the end result of an infinite sequence of steps in a process (A) is equal to a required value (B), even though the result no particular step is actually equal to B. You have to creep up on it, as it were.

I’ll briefly mention equality in computer programs and social equality/inequality, if only to say that I might come back to those topics some time.

English: Income inequality in the United State...
English: Income inequality in the United States, 1979-2007 (Photo credit: Wikipedia)

Random musings

sigh-ness#1 (Photo credit: parth joshi)


My musings are pretty random anyway, so here’s some musings on randomness.

Most people have an inkling of what the word ‘random’ means, but if you try and tie it down, it proves to be a concept that is difficult to define. OK, let me start with a dictionary definition from Dictionary.com:

Lacking any definite plan or prearranged order; haphazard

That’s just one of many similar definitions of ‘random’ to be found at Dictionary.com. But hang on a minute – isn’t having no definite plan a plan of sorts. We can imagine Mad King Wotzit from Philopotamia talking with his generals. “Look, we don’t know where the enemy is, and we don’t know many of them there are, and we don’t know if they have muskets, so the plan is to go ahead with no plan and react to circumstances as they arise. Are we all agreed?”

Coup d'oeil #25
Coup d’oeil #25 (Photo credit: ryansarnowski)

I don’t think that definition is strong enough. We often proceed without a plan, but not randomly, and the obstacles in our way may appear haphazard but there will be a reason why every single one exists.

Randomness for a mathematician, a statistician or a philosopher is something deeper. Take, for instance, the tossing of a coin. It may come down head up or tail up and there are no other options (if we declare the case where it lands on its edge as a no throw). So a sequence of throws could go H, T, T, T, H, T…..


Commandant of the Marine Corps James T. Conway...
Commandant of the Marine Corps James T. Conway participates in the coin toss at the New Orleans Saints Military Appreciation Game against the Atlanta Falcons at the Louisiana Superdome. (Photo credit: Wikipedia)

The critical thing is that any toss doesn’t depend on any of the previous tosses, so it has a 50% chance of being heads and 50% chance of being tails. If we have tossed the coin one million times we would ‘expect’ to get 500,000 heads  and 500,000 tails, but, if fact we may get 499,997 heads meaning we tossed a tail 500,003 times. The average number of heads we would get if we did this a number of times would be very close to 500,000, but it might, by chance, be several hundred away.

English: Five flips of a fair coin. Español: C...
English: Five flips of a fair coin. Español: Cinco lanzamientos de una moneda. (Photo credit: Wikipedia)

Suppose we had thrown the fair coin a million times and we came up with 499.000 heads and 501,000 tails, and we continue for another million tosses. Should we expect more heads this time, so that the average comes out right? I believe that it is obvious that if the coin and tosses are fair, then we cannot tell before hand if the gap between heads and tails would close or get wider. The second million, like the first million will result in about 500,000 each heads and tails.

One-tenth penny coins from British West Africa...

One-tenth penny coins from British West Africa, dated 1936 and 1939. (Photo credit: Wikipedia)

Nevertheless gamblers waste their money on the belief that the odds will even up over time. This is therefore known as the Gambler’s Fallacy.


English: Simulation illustrating the Law of La...
English: Simulation illustrating the Law of Large Numbers. Each frame, you flip a coin that is red on one side and blue on the other, and put a dot in the corresponding column. A pie chart notes the proportion of red and blue so far. Notice that the proportion varies a lot at first, but gradually approaches 50%. Animation made in Mathematica–I’m happy to give you the source code if you want to improve the animation or for any other reason. (Photo credit: Wikipedia)

But how do you know if a real coin, as opposed to a theoretical coin is fair. Well, you test it of course. You toss the coin, say 1,000,000 times and see if you achieve 500,000 heads and 500,000 tails. If you get 500,000 heads or near that number, you can say that the coin is ‘probably fair’. What you can’t say, of course, is that the coin is ‘definitely fair’ as the coin could be a dud, but still produce, by chance, the result that a fair coin would.

Shove ha'penny for charity
Shove ha’penny for charity (Photo credit: HowardLake) A coin, at a fair – fair coin?

In addition a real coin is subject to physical laws. Given the starting conditions of the flip, and given the laws of physics, a tossed coin behaves deterministically, resulting in only one possible outcome for the toss. So the toss is not random as people usually use the term. Calculating  what the result might be will likely forever be impossible though.


Uni Cricket: Captain PJ and the Coin Toss
Uni Cricket: Captain PJ and the Coin Toss (Photo credit: pj_in_oz)

Do things happen randomly? I don’t believe that real events can be random. If an event is truly random it cannot depend on events that have gone before, because otherwise it would be, in principle, be predictable from the earlier events. The real events that come closest to being unpredictable are decay events and other events at the quantum level, but even there the outcome is fixed, and only the time that the event happens is variable.


English: Simulation of many identical atoms un...
English: Simulation of many identical atoms undergoing radioactive decay, starting with either four atoms (left) or 400 atoms (right). The number at the top indicates how many half-lives have elapsed. Note the law of large numbers: With more atoms, the overall decay is less random. Image made with Mathematica, I am happy to send the source code if you would like to make this image more beautiful, or for any other reason. (Photo credit: Wikipedia)

Computer science requires randomness for various purposes, most notably for generation of keys for ciphers for encryption. However the numbers that are generated are not truly random, but involve some heavy computation with very large integers. Encrypted information requires decryption, which also requires some very heavy computational lifting. Often extra ‘entropy’ is added from mouse movements and key presses.


Thermodynamic system with a small entropy
Thermodynamic system with a small entropy (Photo credit: Wikipedia)

Computer and other physical random numbers can use physical sources such as cosmic rays or the decay of an unstable atom to seed the calculation of a random number. Both the cosmic ray count and the decay of an unstable atom appear to be random locally, but cosmologically both events are the result of the state of the universe and its history to that point in time which is deterministic and deterministic processes are the opposite of random.


Thermodynamic system with a high entropy
Thermodynamic system with a high entropy (Photo credit: Wikipedia)

I feel strongly that the universe is deterministic, and at a classical level this is almost indisputable, but at the quantum level things are not so clear and at our current level of understanding, I believe that it is correct to say that happenings at the quantum level appear to be only statistically predictable. I understand that this is not because of some aspect of quantum mechanics that is currently unknown. There are no ‘hidden variables‘. Some other way around this dilemma may be found, probably involving another way of looking at the problem.



Since the numbers generated by a computational process are not truly random, it is theoretically possible to crack the cipher and decode the message without the key. The numbers involved are so large that this would be extremely difficult and time-consuming using conventional techniques. Quantum computing techniques can theoretically be used to crack current classical encryption schemes.

Mathematical randomness is a totally different thing. Any finite number can be generated by many methods and if the method is known, then the number can’t be called random. This is the basis of a mathematical game where a sequence of numbers is given and the next number is required to solve the puzzle. I don’t like these games because it is possible that two different algorithms may produce the required answer, and an algorithm could be imagined that gives an answer different to the ‘solution’. In other words there is not one unique solution.


A roulette wheel.
A roulette wheel. (Photo credit: Wikipedia)

This makes it extremely hard, if not impossible to decide if a ‘black-box’ algorithm (one where the working are unknown) is producing a random sequence of numbers. Beyond that point, I’m not going to go, as I do not have the knowledge, nor currently the space in this post, to make a stab at a decent discussion. Maybe I’ll come back to the topic.

Toledo 65 algorithm - 8 / 12
Toledo 65 algorithm – 8 / 12 (Photo credit: jm_escalante)


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The Psi thing

Greek psi
Greek psi (Photo credit: Wikipedia)

I read a book recently, a real paper book, which was called “brain wars” and was written by Mario Beauregard, who is a neuroscience professor at the University of Montreal. The book amounts to an attack on materialist philosophy, arguing that the materialist philosophy cannot explain everything, especially the phenomenon of consciousness and “psi” phenomena.

One of the cornerstones of his argument is based around the dualist notion that mind and brain are separate “things”, and indeed one key section from the text, quoted in the blurb on the dust cover as follows:

The brain can be weighed, measured, scanned, dissected, and studied. The mind that we conceive to be generated by the brain, however, remains a mystery. It has no mass, no volume, and no shape and it cannot be measured in space and time. Yet it is as real as neurons, neurotransmitters, and synaptic junctions. It is also very powerful.

A little later he poses the question that the opponents of Decartes posed : “How, they asked, can an immaterial, mental substance act upon the material brain?”

A diagrammatic section of human brain by René ...

Beauregard later quotes Minsky’s statement “The brain is just a computer made out of meat”. For reasons that he goes into in depth later he states that quantum mechanics “has effectively smashed the scientific materialist worldview.” He then complacently concludes that “(m)aterialistic theories, despite their stubborn persistence in the scientific community, cannot solve the mind-brain problem”.

This despite the fact that Quantum Mechanics is completely materialistic and rational!

Marvin Minsky at the KI 2006 artificial intell...
Marvin Minsky at the KI 2006 artificial intelligence conference in Bremen (Photo credit: Wikipedia)

I believe that Minsky’s view is closer to true than the view that there is more to reality than the materialistic view allows. Beauregard is not a computer scientist so he would not know, in detail, how computers work, under the covers. At a basic level running computer is all about signals. These signals flow through the computer like signals flow through the brain’s network of neurons. (Caveat: I’m not a neuroscientist like Beauregard so I may be misrepresenting his field.)

neuron fractal 1
neuron fractal 1 (Photo credit: Anthony Mattox)

At a slightly higher level, a computer runs an operating system. This is program that runs all the time on the computer, running the programs that the user requires, handling the users input by running other little pieces of code, and handling all the bits of equipment (peripherals) that are connected to the computer. Crucially, the operating system can make the peripherals do things, like print the letter “A” on a sheet of paper, or spit out the sheet from the printer. Special purpose computers are the core of the robots that build cars or assemble toasters and pack them  and label them. They can even sort letters, reading ordinary human writing, much of the time accurately.

Factory Automation with industrial robots for ...
Factory Automation with industrial robots for metal die casting in foundry industry, robotics in metal manufacturing (Photo credit: Wikipedia)

Interestingly people don’t think of robots as mobile computers that can interact with physical objects. The computers in robots run an operating system like your ordinary laptop or desktop, but they are often special versions called “embedded” operating systems.

Open up a computer though, and boot it up. Although you can point to various named parts, like the CPU, or the memory chips, you can’t point to the operating system. It essentially just a pattern impressed on the memory and the various registers and the CPU, and it changes over time. As Beauregard said about the mind, “it has no mass, no volume, and no shape, and it cannot be measured in space and time”. Yet it can influence things, print a letter or paint a car chassis.

June 11, 2007
June 11, 2007 (Photo credit: HeatherKaiser)

It seems that the computer, with its operating system and subsidiary programs, is a good analogy for the brain/mind duality. A big caution here, in that this analogy is just analogy, but it could form the basis of a model of the way that the mind and brain work together. It doesn’t, per se, explain consciousness, but I think that I have, above, provided an explanation of how the supposedly immaterial mind can, through the brain, affect the body, so that we can think above moving a limb, and it happens.

Quantum Physics
Quantum Physics (Photo credit: Jonathan Thorne CC)

Beauregard fastens on “quantum physics” as a possible enabler of psi phenomena, arguing that in quantum physics there is no separation between the mental and the physical. He bases this on what he calls the observer effect : “particles being observed and the observer are linked, and the results of the observation are influenced by the observer’s conscious attempt”.

Hmm. Wikipedia defines the “observer effect” as follows :

In science, the term observer effect refers to changes that the act of observation will make on a phenomenon being observed. This is often the result of instruments that, by necessity, alter the state of what they measure in some manner. A commonplace example is checking the pressure in an automobile tire; this is difficult to do without letting out some of the air, thus changing the pressure. This effect can be observed in many domains of physics.

This is a purely physical effect of measurement – the measuring photon knocks the observed particle slightly off course. Nothing to do with the observer. (A related effect, the Heisenberg principle puts limits on the accuracy with which we can know both the original values of a pair related properties and the subsequent values – roughly speaking).

An optical illusion. Square A is exactly the s...
An optical illusion. Square A is exactly the same shade of grey as square B. See demonstration. (Photo credit: Wikipedia)

I think that Beauregard is actually referring to is an interpretation of quantum mechanics known as the “Copenhagen Interpretation” otherwise known as the “Collapse of the Waveform”. As such he interprets it as saying that the act of observation affects the result of the observation. This is fundamentally not true, because what really happens is that the act of observation merely determines which of probabilities is true. As Wikipedia says :

What collapses in this interpretation is the knowledge of the observer and not an “objective” wavefunction.

In no way does the observer influence the results of the experiment except as a result of the real “observer effect” above, so there is no room there for psi effects.

English: Example of a subject in a Ganzfeld ex...
English: Example of a subject in a Ganzfeld experiment. (Photo credit: Wikipedia)

You may think that I didn’t enjoy the book, but I did! There are unexplained and challenging events described in the book, but I don’t think that it goes anywhere near challenging the materialistic philosophy of science. The only part that I have issue with is when Beauregard challenges what he calls “pseudoskeptics”, those who profess to be skeptics and who are unwilling to look at the evidence for psi phenomenon.

USE IT… (Photo credit: Demetrios Georgalas aka brexians)

In fact these so called pseudoskeptics have probably looked into psi phenomenon at some stage and decided that further consideration is pointless given the diffuse and dubious nature of some evidence and the lack of any information about how this could tie in to or extend in some logical way existing materialistic physics.

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The Hubris of Scientists

Screenshot from the public domain films Maniac...
Screenshot from the public domain films Maniac (1934) showing Horace B. Carpenter as the character “Dr. Meirschultz” (Photo credit: Wikipedia)

Scientists talk about gravity,  mass and probabilities, atoms, Higgs boson, black holes and qasars. Certainly the universe seems to behave as if these concepts represent reality and so scientists are justified in the their assertions and predictions. Nevertheless the assumption that the concepts that scientists use represent reality is debatable.

The scientific method which has been a part of science since 17th century is a set of rules that scientists use to develop and test theories about the scientific view of the world. Basically, the scientist formulates a hypothesis (based on an earlier theory or as a totally new theory) and develops experiments to test the theory. The experiments produce observations which either support or do not support the theory.

English: Flowchart of the steps in the Scienti...
English: Flowchart of the steps in the Scientific Method (Photo credit: Wikipedia)

If the observations agree with the theory they are said to support the theory. If they do not, they are said said to disprove the theory. So far, so black and white. An experiment may be challenged on many grounds. For example the search for the Higgs boson is not done by actually isolating candidate particles and looking at it directly. Instead the expected properties of the Higgs boson, perhaps its mass or energy, the way it interacts with other particles, or other more esoteric properties,  can be used to deduce that, for example, in a particular experiment a peak at a certain point on a graph produced by a scientific instrument could only be the result of the presence in the apparatus for  at least an instant of the required Higgs boson.

One possible way the Higgs boson might be prod...
One possible way the Higgs boson might be produced at the Large Hadron Collider. Similar images at: http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/Conferences/2003/aspen-03_dam.ppt (Photo credit: Wikipedia)

In a similar way, we don’t detect an electric current directly. Instead we rely on electromagnetic theory which predicts that moving electrons should produce a magnetic field and that magnetic field would interact with a static magnetic field of a permanent magnet perhaps to produce a force on the permanent magnet hence moving a needle. Behold! We detect an electric current. Actually what we see is the movement of a needle and we infer the electric current from that observation.

Sometimes the chain of inference is short, as in the electric current experiment, while in others it is very much longer. I expect that the detection of the Higgs boson falls into the latter category, but I could (easily) be wrong. It is apparent that the more links that there are in the chain of inference, the higher the likelihood that one of the links might be debatable.

How to deduce various data with the observatio...
How to deduce various data with the observation results (Photo credit: Wikipedia)

So, faced with an experiment that supposedly tests a theory, the result does not absolutely prove or disprove the theory. If the experiment appears to show agreement with the theory, an opponent of the theory may cast doubt on the experimental method or in the theories that the theory being tested relies on. He or she would claim that the result doesn’t show what it purports to show. In addition he or she might point out that one experiment does not prove the theory as the next experiment could show the opposite. One experimental failure is enough to disprove the theory.

My cooking companions this evening- Zak dispro...
My cooking companions this evening- Zak disproved the “watched pot” theory. (Photo credit: who_da_fly)

Or is it enough to disprove it? Not really because the proponent of the theory  could claim that some currently unknown effect or other is preventing the experiment from producing the correct observations. So debate follows, more experiments follows, and in the end, a consensus is achieved. History will record that theory A was generally accepted until so-and-so’s experiment replaced it with theory B. Or that theory A was extended by theory B and confirmed by so-and-so’s experiment. Or similar. Much more black and white!

Scientists explain experimental results in terms of theories. For instance when sodium is introduced into a flame (perhaps in the form of sodium chloride – salt) and the light from the flame is passed through a prism then a bright yellow line is seen. Scientists explain this as the result of the transition of an excited electron from an elevated orbit to a lower one. This explanation depends on several, maybe many, other explanations, such as an explanation of what ‘excited’ means and what ‘electron’ means and what ‘orbit’ means. In many cases these explanations are based on mathematics, and an explanation is based on concepts each of which requires explanation.

sodium flame test
sodium flame test (Photo credit: Wikipedia)

So therein lies the hubris of scientists. Their attempts at explanation of observable facts is a bottomless pit of explanation on explanation. There is no ultimate explanation. The universe is and does what it is and does.

So, am I saying that science is pointless? No, I am merely saying that we need to be careful and not treat our explanations as anything other than very clever descriptions of those bits of the universe that we are have seen.

Contents of the universe according to WPAP 5-y...
Contents of the universe according to WPAP 5-year results (Photo credit: Wikipedia)

I like the analogy of the sheet. Suppose you have an object hidden behind a sheet. You are allowed to make pin pricks in the sheet, one at a time. The universe is the object behind the sheet and each pin prick is an observation. As you make more and more pin pricks in the sheet you see more and more of the object behind the sheet. You may discover that a line of pin pricks is showing red. You form a theory that behind the line joining the existing pin pricks, between the existing pin pricks and, with less certainty, beyond the end pin pricks in the line, everything is red. To check this theory you make a pin prick between two existing pin pricks and find that the new pin prick shows red. The theory is supported by this new observation.

Scientists have been creating these pin pricks for centuries and now have a pretty good idea of the shape of the universe (and a pretty holey sheet!). Nevertheless there are parts of the object behind the sheet, the universe, that they haven’t yet uncovered, and maybe never will.

An example of simulated data modelled for the ...
An example of simulated data modelled for the CMS particle detector on the Large Hadron Collider (LHC) at CERN. Here, following a collision of two protons, a is produced which decays into two jets of hadrons and two electrons. The lines represent the possible paths of particles produced by the proton-proton collision in the detector while the energy these particles deposit is shown in blue. (Photo credit: Wikipedia)

As an example of the type of thing that I mean, consider the so-called dark matter. Scientists appear to have pretty much discovered what constitutes matter but they can’t account for some aspects of certain large scale phenomenon observed in the universe and have hypothesised a new type of matter called ‘dark matter’, which doesn’t appear to interact with normal matter except gravitationally. It’s like suddenly finding some pin pricks showing blue in a line that is otherwise red. Something unexpected that needs explanation.

I accused scientists of ‘hubris’ above. That’s not entirely fair as hubris implies arrogance and while scientists confidently create explanations for phenomena that they study, I believe that most would concede that their explanations could (with very low probabilities, I would guess) prove to be erroneous.

''I think that it's important for scientists t...
”I think that it’s important for scientists to explain their work, particularly in cosmology. This now answers many questions once asked of religion.” – Stephen Hawking (Photo credit: QuotesEverlasting)

The number of the universe.

English: Measurement unit

Anything that can be measured can be encoded in a single number. Take for instance the trajectory of a stone thrown into the air. Its position in relation to the point of launch and the time it has taken to reach that point can be encoded into a set of numbers, three for the spacial dimensions and one for the time dimension. This can be done for all the points that it passes through. These individual numbers can then be encoded into a single number that uniquely identifies the trajectory of the stone.

Or, a physicist can describe the motion of the thrown stone by using generic equations and plug in the starting position and starting velocity of the stone, which can then be encoded, probably in a simpler fashion than the above point by point encoding.

Throwing Stones

If we can imagine a set of equations that describe all the possible physical processes (the “laws of nature”?) and we can imagine that we can measure the positions of all the particles (including photons,’dark matter’ and any more esoteric things that might be out these), then we could encode all this in a huge number which we could call the ‘number of the universe’. Such a number would be literally astronomical and I do mean ‘literally’ here.

The most concise expression of the state of the universe over all time is probably the universe itself and the laws that govern it. Each individual particle has its own attribute, like charge, mass, position and so on as well as things like spin, charm and color. Some of these change over time and some are fundamental to the particle itself – if they change so does the nature of the particle. The rest of the universe consists of other particles which have a lesser or greater effect on the particle, all of which sum together to describe the forces which affect the particle.

English: Position and momentum of a particle p...

There are a couple of things which might derail the concept of the number of the universe. Firstly there is Heisenberg’s Uncertainty Principle and secondly there is the apparent probabilistic nature of some physical processes.

What follows is my take on these two issues. It may make a physicist laugh, or maybe grimace, but, hey, I’m trying to make sense of the universe to the best on my abilities.

uncertainty principle

People may have heard of the Uncertainty Principle, which states that there are pairs of physical properties which cannot both be accurately known at the same time. You may be able to know the position of a particle accurately, but you would not then be able to tell its momentum, for example.

It is usually explained in terms of how one measures the position of something, which boils down to hitting it with something else, such as a photon or other particle. The trouble here is that if you hit the particle with something else, you change its momentum. This is, at best, only a metaphor, as the uncertainty principle is more fundamental to quantum physics than this.

Staccato aerophagia waveform. Its characterise...

Wikipedia talks about waveforms and Fourier analysis and an aspect of waves that I’ve noticed myself over the years. If you send a sound wave to a frequency analyser you will see a number of peaks at various frequencies but you cannot tell how the amplitude of the wave changes with time. However, if you display the signal on an oscilloscope you can get a picture of the shape of the wave, that is the amplitude at any point in time, but not the frequencies of the wave and its side bands. Err. I know what I mean, but I don’t know if I can communicate what I mean!

The picture above shows a spectrum analysis of a waveform. I don’t have the oscilloscope version of the above, but below is a time-based view of a waveform.

English: sinusoidal waveform

In any case, the uncertainty doesn’t imply any indeterminacy. A particle doesn’t know its position and momentum, and these values are the result of its properties and the state of the rest of the universe and the history of both. This means that the uncertainty principle doesn’t introduce any possible indeterminacy into the number of the universe.

On the second point, some physical processes are probabilistic, such as the decay of a radioactive atom. I don’t believe that this has any effect on the number of the universe. The number incorporates the probabilistic nature of the decay, including all the possibilities.

There is an interpretation of quantum physics called the “Many Worlds Interpretation“, where each possible outcome of a probabilistic process splits off into a separate world, resulting in an infinity of separate worlds. I don’t believe that this tree of probabilistic worlds is a useful view of the situation.

English: Schrödinger's Cat, many worlds interp...

No, I think that there is a probabilistic dimension, just like time or space. All the things that can happen, ‘happen’ in some sense. The probability of you throwing 100 tails in a row with a fair coin is very small, but it is possible. As I see it the main objection to this view is the fact that we only see one view of the universe and we don’t appear to experience any other possible views of the universe, but this is exactly the same with the dimensions of space and time. We only experience one view of space at a time as we can’t be in two places at the same time. While we could be in the same place at two times they are two distinct views of the universe.

In any case the number of the universe encompasses all probabilities so if you still adhere to the single probability model of the universe, our universe and all possible universes are encoded by it. The question then becomes how you can extract the smaller number that encoded the single universe that we experience. I believe that that is not a question that needs to be answered.

The question that does remain open is – why is that number the number of our universe? Why not some other number?

English: Level II Multiverse: every disk is a ...


The Story of Nothing, in Arizona
The Story of Nothing, in Arizona (Photo credit: cobalt123)

Nothing is an interesting concept with many different aspects. Maths, science, philosophy and many other fields of endeavour have their own overlapping concepts of nothing, zero, null or just the absence of anything.

Some computer languages have a concept of ‘null’. This is not the same as the concept of ‘zero’. To use the usual analogy of pigeonholes, numbers and other things in computers are conceptually stored like objects stored in pigeonholes. Each pigeonhole must have a location, sort of like ‘third row down, fourth hole in the row’. A pigeonhole could be empty or it could contain a number or a string of characters or more complicated objects that the computer recognizes. It could optionally have a label so that it can be found quickly.

Pigeon Holes
Pigeon Holes (Photo credit: Graela)

A computer moves things around and in the process it manipulates them. Given this analogy, what is ‘nothing’ to a computer?  It could mean several things. It could mean the number zero, stored in a pigeonhole or it could refer to an ’empty string’ stored in a pigeonhole. (An ’empty string’ is like the object ‘where’ when the individual letters ‘w’, ‘h’, ‘r’, and the two ‘e’s have been removed. It is represented by two ). It can be a more complicated object that hasn’t been completely set up. Alternatively it could refer to an empty pigeonhole. It could even refer to a label which has not yet been allocated to a pigeonhole. Pity the poor programmer who has to keep all these ‘nothings’ separate in his or her mind (and a few others that I’ve not mentioned!).

Zero (Photo credit: chrisinplymouth)

In mathematics we have the concept of zero, but this is a fairly newly introduced concept. Some number systems, such the Roman Numeral system do not have a zero, and it was a big conceptual jump to add zero to the mathematical number systems. After all, what do you hold when you have two oranges and you give them away? Nothing! You can’t see zero oranges in your hands, unless you are a modern mathematician of course.

So mathematically ‘nothing’ is zero then? It could be, though ‘nothing’ could be integer zero, ‘0’, rational zero, ‘0/any number’, real number zero, ‘0.0’, complex zero, ‘0 + 0i’, or many many other versions of zero. Maths also has a concept of a set, which is just a collection of objects, which can be pretty much anything. An analogy often used is to liken a set to a bag which contains any sort of object. Statisticians are fond of sets which comprise a set of balls which can be of more than one colour but are usually otherwise identical. If all the balls are removed from the bag, what do you have? A bag with nothing in it! It is usually referred to as an ’empty set’. Note the similarity with the ’empty string’ mentioned above. There’s nothing coincidental there.

Illustration of Function (mathematics).
Illustration of Function (mathematics). (Photo credit: Wikipedia)

There are other sorts of ‘nothing’ in mathematics. A mathematical ‘function’ is a way of relating ‘variables’. The details don’t matter, just the fact that functions have ‘zeros’. They may have one or more zeros or they may have none. Having no zeroes could be considered a sort of ‘nothing’, in a way, though the functions in question are no less proper functions than any other. I’m sure that there are other more esoteric ‘nothings’ in maths.

In physics things should be clearer, right? In physics a vacuum is created is all matter is removed, leaving … nothing. Except that it appears to be impossible to actually remove everything from a container leaving nothing. Even the best pumps will leave a considerable numbers of atoms floating around inside the container. Other methods of emptying the container may reduce slightly the number of atoms in it, but we can’t even reach the very low densities found in the gas clouds visible to astronomers. Even in the depths of space between the galaxies we still find the occasional atom, usually of hydrogen.

Vacuum Pump
Vacuum Pump (Photo credit: Sascha Grant)

Maybe we should look between the atoms for nothing? Most people have an image of an atom as a sort of miniature solar system with the nucleus standing in for the sun and the electrons standing in for the planets. Unfortunately the analogy breaks down if you look closely. Electrons are only found in certain orbits around an atom and even that is an over-simplification. Their location depends on a probability function and in some views this means that the electron is sort of smeared out in space and doesn’t have a strict location and you can’t say specifically that it is ‘there’ at a particular location, only that it has a particular possibility of being there.

One consequence of this is that you can’t say that is isn’t at a particular location, so it is impossible to declare that there is nothing at a particular point in space at any one time. If you consider all the particles in the universe, they all have a probability of being there, so you might be surprised not to find a particle there at a particular moment in time.

Vacuum polarization
Vacuum polarization (Photo credit: Wikipedia)

In addition to this, I have read article which describe ’empty space’ as a seething mass of pseudo particles or virtual particles. These come in pairs of particle and anti-particle which are continually coming into existence, mutually annihilating each other out of existence again. Viewed in this way it is difficult to describe ’empty space’ as containing nothing, so we still haven’t found ‘nothing’. Although physics has the concept it is hard to find a physical instance of it.

The Big Bang era of the universe, presented as...
The Big Bang era of the universe, presented as a manifold in two dimensions (1-space and time); the shape is right (approximately), but it’s not to scale. (Photo credit: Wikipedia)

Cosmologists talk about the “Big Bang” when everything came into existence. Before the Big Bang, they say, there was nothing. Nothing! But what does this mean. I like to think of it by analogy. If you take a piece of paper and draw a circle on it, you can consider this circle to contain all space and time and everything that exists in space and time. If you draw a line horizontally through it you can label the big inside the circle as ‘time’. Note that the line should not extend beyond the circle.

The point where the line reaches the left hand side of the circle is the Big Bang. The point where the line reaches the right hand side of the circle is the point where everything collapses on itself and space and time cease to exist.

Some cosmologists think that there will not be a collapse, so the curve is not a circle but a curve open to the right. This doesn’t affect my argument – everything and every time is included inside the curve.

English: Shows slices of expansion of universe...
English: Shows slices of expansion of universe without an initial singularity (Photo credit: Wikipedia)

If you now draw a line vertically, not extending beyond the curve, and label it ‘space’. If you move the line to the left, the graphical distance between the top point and the bottom shrinks. Moving the line to the left moves it back along the time axis and represents an earlier state of everything. When the line just touches the curve the point of intersection of the two lines represents the Big Bang.

What about the points outside of the curve? This is where the analogy breaks down. Since we have included all space and time inside the curve the points outside the curve do not represent real points in space and time at all. In short, they do not exist. We could loosely say that nothing exists outside the curve of space and time, but that is not true. ‘Nothing’ is a concept based on space and time, being the opposite of ‘something’ or the potentiality for ‘something’ and as such needs a space-time framework to mean anything. If there is no space and time, there can be no ‘something’ and therefore ‘nothing’ is meaningless. Beginners in science and astronomy might ask what is beyond the boundary of the universe, but the question doesn’t mean anything. The universe contains everything.

If there were other universes, with their own space and time, they would have to be right alongside our universe (that is an analogy of course – language fails us in this situation) as there is nothing to be between the two universes. If you were able to travel from one universe to the other, a concept which I don’t believe stands up to examination, you probably wouldn’t notice the difference. Maybe nothing is a sort of inability to be. But that language implies an intent, which implies a lot of other things and maybe leads to pantheism and I don’t wish to go there.

Absolutely Nothing is Allowed Here
Absolutely Nothing is Allowed Here (Photo credit: Vicki & Chuck Rogers)

Well, I’ve used over 1300 words to talk about ‘nothing’, so I will stop here. What comes after the end of this post? Why, nothing, of course!

Why do things make sense?

Make it make sense
Make it make sense (Photo credit: edmittance)

Things pretty much make sense. If they don’t we feel that there is a reason that they don’t. We laughingly make up goblins and poltergeist to explain how the keys came to be in the location in which they are finally found, but we, mostly, have an underlying belief that there are good, physical reasons why they ended up there.

Things appear to get a little murkier at the level of the quantum, the incredibly small, but even there, I believe that scientists are looking for an explanation of the behaviour of things, no matter how bizarre. One of the concepts that appears to have to be abandoned is that of every day causality, although scientists appear to be replacing that concept with a more probabilistic version of  the concept of causality. But I’m not going to go there, as quantum physics has to be spelled out in mathematics or explained inaccurately using analogies. I note that there is still discussion about what quantum physics means.

English: Schrödinger equation of quantum mecha...
English: Schrödinger equation of quantum mechanics (1927). (Photo credit: Wikipedia)

We strive for meaning when we consider why things happen. When a stone is dropped it accelerates towards the earth. This is observation. We also observe the way in which it accelerates and Sir Isaac Newton, who would have known from his mathematics the equation which governed this acceleration, had the genius to realise that the mutual attraction of the earth and the stone followed an inverse square law and, even more importantly, that this applied to any two objects which have mass in the entire universe.

English: Mural, Balfour Avenue, Belfast Mural ...
English: Mural, Balfour Avenue, Belfast Mural on a gable wall on Balfour Avenue in Belfast (see also 978903). The mural “How can quantum gravity help explain the origin of the universe?” was created by artist Liam Gillick and is part of a series of contemporary art projects designed to alert people to the ‘10 remaining unanswered questions in science’ at public sites across Belfast. (Photo credit: Wikipedia)

So, that’s done. We know why stones fall and why the earth unmeasurably and unnoticeably jumps to meet it. It is all explained, or is it? Why should any two massy objects experience this attraction? Let’s call it ‘gravity’, shall we? How can we explain gravity?

Well, we could say that it is a consequence of the object having mass, or in other words, it is an intrinsic property of massy objects, which if you think about it, explains nothing, or we can talk about curvature of space, which is interesting, but again explains nothing.

Curved Spaces
Curved Spaces (Photo credit: Digitalnative)

Can you see where I am going with this? Every concept that we consider is either ‘just the way things are’ or requires explanation. Every explanation that we can think up either has to be taken as axiomatic or has to be explained further. Nevertheless most people act as if they believe that there is a logical explanation for things and  that things ultimately make sense.

It is possible that there is no logical explanation of things, and that the apparent relationships between things is an illusion. I once read a science fiction story where someone invented a time machine. Everywhere the machine stopped there was chaos, because there were no laws of nature and our little sliver of time was a mere statistical fluke. When they tried to return to the present they could not find it. This little story demonstrates that although we appear to live in a universe that is logical and there appears to be a structure to it, this may just be an illusion.

English: Illustration of the difference betwee...
English: Illustration of the difference between high statistical significance and statistical meaningfulness of time trends. See Wikipedia article “Statistical meaningfulness test” for more info (Photo credit: Wikipedia)

If we do live in a logical universe we not be able to access and understand the basis and structure of it. We may see things “through a glass darkly”. We may be like the inhabitants of Plato’s Cave. Everything we experience we experience through our senses, so our experience of the world is already second-hand and for many purposes we use tools and instruments to view the world around us. Also, our sense impressions are filtered, modified and processed by our brains in the process of experiencing something. We can take prescribed or non-prescribed drugs which alter our view of the world. So how can we know anything about the universe.

Alternatively there may be order to the universe. There may be ‘laws of nature’ and we may be slowly discovering them. I like the analogy of the blanket – a blanket is held between us and the universe but we are able to poke holes in it. Each hole reveals a metaphoric pixel of information about what lies behind the blanket. Over the years, decades, centuries and millennia we have poked an astronomical number of holes in the blanket, so we have a good idea of the shape of what lies behind it.

Cámara estenopéica / Pinhole camera
Cámara estenopéica / Pinhole camera (Photo credit: RubioBuitrago)

So why do things make sense? Is it because there is a structure to the universe that we are either discovering or fooling ourselves into believing that we are discovering, or is there no structure whatsoever and any beliefs that there are illusions. Maybe there’s another possibility. Maybe the universe does have the structure but it is an ‘ad hoc’ structure with no inherent logic to it all!

Highly Illogical
Highly Illogical (Photo credit: Wikipedia)