More on Quantum things

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

Schrodinger’s wave equation describes how the quantum state of a quantum system changes with time. Everett’s insight was that the observer of a quantum state was as much part of the system as the observed part of the system. Therefore they were “entangled” in the quantum sense and would be covered by a single quantum state equation.

If the observer and the observed are thus entangled, then so must be an observer who observes the quantum state of the observer and the observed. One can then extend this to the whole universe, which leads to the concept of a wave equation or function which describes the whole Universe.

English: Quantum mechanics travelling wavefunc...
English: Quantum mechanics travelling wavefunctions (Photo credit: Wikipedia)

That there is an equation for the universe is not really surprising and indeed, it is not surprising that it could be a quantum wave equation as the quantum world seems to form the basis of the physical, apparently classically described, world that we see.

I base this idea on the fact that everything that we sees appears to be describable in terms of a deterministic equation. It has been argued that such things as “psi phenomena“, but such claims are yet to be conclusively verified, with many putative examples having been discredited.


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Some people argue for a soul or mind as an example of a non-physical entity, but any such concept leaves a lot of questions to be asked. A non-physical entity cannot, by definition almost, be measured in any way, and there is difficulty in showing how such a non-physical entity can interact with physical ones, and therefore be noticed or detected.

By definition almost, a physical entity, such a body, is only influenced by physical things. If this were not the case we would see physical entities not following the laws of physics. For example, if it is possible to move an object by mind power or telekinesis, one would see the object disobeying fundamental scientific laws, like Newton’s First Law of Motion.

English: Isaac Newton Dansk: Sir Isaac Newton ...
English: Isaac Newton Dansk: Sir Isaac Newton Français : Newton (1642-1727) Bahasa Indonesia: Issac Newton saat berusia 46 tahun pada lukisan karya Godfrey Kneller tahun 1689 Lietuvių: Seras Izaokas Niutonas 1689-aisiais Македонски: Сер Исак Њутн на возраст од 46 години (1689) Nederlands: Newton geboren 4 januari 1643 Türkçe: Sir Isaac Newton. (ö. 20 Mart 1727) (Photo credit: Wikipedia)

The mind is a curious example of a physical entity which is often thought of as being non-physical. After all, a mind does not have a physical location, apart from the skull of the person whose mind it is, and it can’t be weighed as such.

The mind however is a pattern, on the brain, made up of the state of trillions of neurones. It is made up of information, and is much like a computer program which is made up of the state of a few billion physical logic circuits in the guts of the computer.

Vista de la Motherboard
Vista de la Motherboard (Photo credit: Wikipedia)

Open a computer and you won’t see “an image” anywhere. You will see patterns of bits of data in the memory, or on the hard disk, or maybe in transit, being sent to a computer screen. Similarly if you open someone’s skull you will not see an image there either. Just a bunch of neurones in particular states.

The one glaring exception to all the above, is, perhaps, consciousness. It’s hard to describe consciousness in terms of a pattern or patterns of the states of our neurones, but I believe that that is fundamentally what it is.

Schéma d'un neurone , commenté en francais
Schéma d’un neurone , commenté en francais (Photo credit: Wikipedia)

Some people argue that we are conscious beings, (true), and that we consciously make choices (false, in my opinion). When we look closely at any choice that we make, it appears to be that choice is in fact illusory, and that our actions are determined by prior factors.

People seem to realise this, although they don’t acknowledge it. When questioned, there is always some reason that they “choose” in a particular way. Perhaps they don’t have enough cash to choose the luxury option when out shopping, or their desire outweighs their financial state. When pushed people can always think of a reason.

English: A choice of which way to go The choic...
English: A choice of which way to go The choices are a path to Greengore or Intack or the Old Clitheroe Road (Photo credit: Wikipedia)

To be sure, many “reasons” are actually post choice rationalisations, and choices may be based more on emotions than valid rational reasons, but whatever the emotions (such as the desire for an object), the emotions precede decision.

If, as sometimes happens, a person has to make a choice between two alternatives, that person can be almost paralysed with indecision. Even then, when a decision is finally made, it can be either a random choice, or maybe the person may say that they made a particular choice because they had decided a different way in another situation, or similar (e.g. they like the colour blue!).

English: Choose your leaders and place your trust
English: Choose your leaders and place your trust (Photo credit: Wikipedia)

If there is no non-physical component to the Universe, as appears very likely, and psi phenomenon do not exist, then everything has a cause. I don’t mean this in the sense that event A causes event B which causes C, but more in the sense that the slope that a marble is on causes it to move in a particular direction.

Causality seems to be a continuum thing, rather than the discrete A causes B case. We can only get an approximation of the discrete case if we exclude all other options. There is a latin term for this : ceteris paribus – all other things being kept the same. “Ceteris paribus” would exclude the case where a wind blowing up or across the slope changes the path of the marble.

English: Picture of marbles from my collection
English: Picture of marbles from my collection (Photo credit: Wikipedia)

For this reason I dislike the Many Worlds Interpretation of Quantum Physics, as it is usually stated. The usual metaphor is a splitting movie film, which results in two distinct tracks in the future. I feel that a better picture would be a marble on a slope with a saddle.

The marble may go left, or it may go right, or it may even follow the line of the saddle. We still require “ceteris paribus” to exclude crosswinds, but there is no split as such. In a quantum model, the marble goes both left and right (and traverses the peak of the saddle with vanishing probability).

Monkey saddle
Monkey saddle (Photo credit: Wikipedia)

The probability that it goes left or right is determined by the wave equation for the system, and has a real physical meaning, which it doesn’t (so far as my knowledge goes) in the splitting metaphor.

I don’t know how my speculations stack up against the realities of quantum mechanics, but I like my interpretation, purely on aesthetic grounds, even if it is far from the mark!


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Many worlds or only one?

English: Position and momentum of a particle p...
English: Position and momentum of a particle presented in the phase space. (Photo credit: Wikipedia)

Scientists often use the concept of a “phase space“, which is basically a representation of all the possible states that a system may be in. For the trajectory of a thrown stone for instance, the phase space would be a four-dimensional space, comprising the three dimensions of space, which define where the stone is, and one of time, which defines when the stone is in a particular position.

The trajectory of the stone is a line in this 4-d space, as the location and time information about the stone is known exactly. However, the stone is not a point and maybe be spinning at the same time that the whole object is flying through the air. This means that the trajectory would actually be a complex four-dimensional worm in phase space.

An animated GIF of a tesseract
An animated GIF of a tesseract (Photo credit: Wikipedia)

What if we were to introduce a probability factor into the experiment? Maybe we would set up the projectile to be triggered by an atomic decay or something similar. We would get a different worm depending on how long the atom takes to decay.

Clearly, if we want to show the all of the possible versions of the worm, the worm now becomes a sort of 4 dimensional sheet. Well, more like a 4-d duvet really, as the stone is not a point object.

Bedding comforter or duvet. Français : Couette...
Bedding comforter or duvet. Français : Couette (literie). Deutsch: Daunendecke, umgangssprachlich Federbett. (Photo credit: Wikipedia)

Within the 4-d duvet, each worm represents a case where the atom has decayed, and each of these cases has a probability associated with it. The probability can be expressed as the probability that the atom has decayed by that time or not, and can run from one to zero.

Actually the probability starts from zero and approaches one but doesn’t quite reach it. In practise in a group of atoms some will decay quickly and others will take longer. If there are a finite number of them, then the chances of any one lasting a long, long time are quite small, and all of the atoms are likely to decay in a moderately short time, a few multiples of the half-life anyway. However there will be a finite but microscopic in the extreme possibility, that an atom will survive for as long as you may consider.


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We can add another dimension to the phase space, one of probability. This gives us a five dimensional phase space, and the duvet becomes five dimensional. However, an atom decays at a certain time, and there is a single five dimensional worm in the phase space going forward. The space is no longer a phase space though, as a phase space, by definition, describes all possible states of the rock/launcher/atomic trigger, and doesn’t change.

According to the Copenhagen interpretation of quantum physics the state of a quantum system is described by a set of probabilities. When a measurement of the system is made the state becomes certain, and it is said that the waveform described by the probability function has “collapsed”.

Copenhagen
Copenhagen (Photo credit: Wikipedia)

The famous thought experiment of Schrodinger’s Cat is a description of the difficulties of such a case. The cat is enclosed in a box equipped with a mechanism which will release a poison and kill the cat if triggered by the decay of an atom. At some time after the experiment starts the atom may or may not have decayed so the quantum states “decayed” and “not decayed” are superimposed, and therefore so are the states “dead” and “not dead” of the cat.

How do we know if the stone has been fired yet? Well, we go and look to see, and we either see the stone in its launcher or we don’t. Quantum physics says that the stone exists in a superposition of states – launcher and not launched. The question this raises is, if this is so, how does looking at the stone “collapse” the superposition when we look?

Three wavefunction solutions to the Time-Depen...
Three wavefunction solutions to the Time-Dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wavefunction. Right: The probability of finding the particle at a certain position. The top two rows are the lowest two energy eigenstates, and the bottom is the superposition state \psi_N = (\psi_0+\psi_1)/\sqrt{2} , which is not an energy eigenstate. The right column illustrates why energy eigenstates are also called “stationary states”. (Photo credit: Wikipedia)

That quantum superposition is real is indicated by any number of experiments, even though many physicists working in the field (including Schrodinger himself) have expressed discomfort at the idea.

In quantum physics the evolution of everything is defined by the Universal Wave Function. This can be used to predict the future of any quantum physical system (and all physical systems are fundamentally quantum physical systems). Unfortunately for easy understanding, interpretation leads to the superposition problem mentioned above.

Many people have tried to resolve this issue, and the best success has been achieved by the exponents of the Many Worlds Interpretation (MWI), as described by Everett and championed by Bryce DeWitt and David Deutsch. The view of the MWI exponents is that the Universal Wave Function is fundamental and expresses a true picture of all reality. All of it, that is. Not just a physical system and its observer.

Everett’s view, as described in his thesis, is that an observer, as well as the object that he is observing is a subsystem of the system described by the Universal Wave Function. The wave function of these two subsystems does not describe a single state for each of these subsystems, but the states of the two subsystems are superposed, or in Everett’s term, correlated.

en:Many-worlds interpretation
en:Many-worlds interpretation (Photo credit: Wikipedia)

When a particle is observed it may appear to be in state A 70% of the time (correlated with a state A for the observer). Similarly it may appear to be in state B 30% of the time (correlated with a state B for the observer). This led Everett to postulate a ‘split’ of the universe into a state A and a state B.  (The term ‘split’ appears to come from DeWitt’s interpretation of Everett’s work).

The probabilities don’t seem to have a function in this model, and this is odd. The probability that the cat is dead when you open the depends on how long you wait until you open the box. If you wait a long time the cat will more likely be dead than if you opened it earlier.

English: Diagram of Schrodinger's cat theory. ...
English: Diagram of Schrodinger’s cat theory. Roughly based on Image:Schroedingerscat3.jpg (Photo credit: Wikipedia)

This means that the world splits when the cat is put in the box, as from any moment it can be alive or dead, but you do not find out which branch you are in until you open the box sometime later.

I’m ambivalent about the MWI. On the one hand it is a good explanation of what happens when a measurement is made or the cat’s box is opened, and it does away with the need for a waveform collapse, which Everett argued against in his paper. However it is profligate in terms of world creation.

English: Schrödinger's Cat, many worlds interp...
English: Schrödinger’s Cat, many worlds interpretation, with universe branching (Photo credit: Wikipedia)

Another issue is that the split is decidedly binary. The cat is alive in this world and dead in that one. However most other physical processes are, at the macro level anyway, continuous. When a scientist takes a measurement he writes down, for example, 2.5, but this is only inaccurate value as it is impossible to measure something exactly and it may be wrong by up to 0.05 on either side of 2.5 (given the one decimal point value shown).

Consequently, I’d prefer an interpretation where there is no split, but instead a continuum of possibilities as part of a single world. Maybe the single path that we tread through life is an illusion and across the Universe, by virtue of the Universal Wave Function, we experience all possibilities, though to us it feels like we are only experiencing the one.


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I’m going to try something different in this post. I won’t try to stick to a single theme, but will try to create a miscellany of short themes and see how it goes.

Firstly, a friend of mine is a keen photographer who keeps a blog and for five years he has posted one or more photos to his blog every day. It’s a fantastic achievement, and since I have trouble posting once a week, I can only imagine the persistence, application and diligence needed to post once every day.


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I’ve mentioned before that I have started blogs at various times in the past and have been unable to keep them going. This time around, for reasons that I can’t really fathom, I have managed to keep going for coming up to 150 posts now. If Brian succeeds in reaching 5 years of posting he will have posted over 1,870 times. Which is mind-boggling!!

One of the reasons that he has given for dropping his self-imposed regime of daily posting is that he feels that the quality of his posted pictures is possibly suffering from the requirement to post something every day and that the temptation to post a merely adequate (from his point of view) picture just to keep the chain going is strong.

English: Locomotive in KiwiRail livery (not a ...
English: Locomotive in KiwiRail livery (not a particularly good photo, but perhaps adequate until a better is found) (Photo credit: Wikipedia)

I must say that I have not seen any deterioration in his pictures, but I don’t look at them with his eyes. So I will continue to look forward to his posts, even though he will not be posting daily pictures once he reaches the 5 year target.

Of course, this has made me think about this blog and how long I intend to keep it going. I’ve posted nearly 150 times so far which represents a bit under three years. I’d like to reach at least 5 years too, which will be around 250 posts. That’s the current target, so let’s see if I can reach it.

English: 250 West Pratt Street in Baltimore
English: 250 West Pratt Street in Baltimore (Photo credit: Wikipedia)

Next topic. If I was able to ask God a question, I would say “Quantum Physics. What were you thinking!” We like to think that the universe is logical and consistent. If it wasn’t then there would be no guarantee that the sun would not blink out 10 seconds from now. 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0! OK we are fine this time.

I’m told, and I have an inkling about it from my reading and thinking, that Quantum Physics is logical and consistent. However the various popularizations of it appear counter intuitive and paradoxical. How can Schroedinger’s poor cat be both alive and dead? What exactly is a superposition of states? What (if anything) is the “collapse of the wave form”?

Omega-Point-Multiverse
Omega-Point-Multiverse (Photo credit: Wikipedia)

General and Special Relativity were considered mysterious and paradoxical when Einstein first published his papers. At the time someone claimed that only three people in the world truly understood it, but it didn’t take long for it to be taught down to college and undergraduate levels. While strange and challenging, it was soon accepted as true by the majority of people who had come across it, although people still create web sites where they try to prove that Einstein was wrong.

Quantum Physics is also taught at undergraduate levels I believe but it remains (or so I get the impression) as a work in progress. The famous Copenhagen Interpretation was formulated around 90 years ago by Niels Bohr and Werner Heisenburg and there is still no accessible standard interpretation that is accepted by the majority.

English: Experiment suggested by Heisenberg, p...
English: Experiment suggested by Heisenberg, part 1: Wide hole in barrier screen gives only a very general idea of location of photon as it moves toward detection screen. Photons will arrive at a small spot on the detection screen. (Photo credit: Wikipedia)

Niels Bohr once said “Anyone not shocked by quantum mechanics has not yet understood it.” Richard Feynman said “Nobody understands quantum mechanics.” So, God, Quantum Physics. What were you thinking about?

Flags and things. There is a big debate in this country at the moment about whether or not to change the flag. People often refer to the Canadian flag as a case where a new flag was adopted after public discussion. It’s not a good case study though as the actual adoption of the winning flag involved a farcical mistake : “Through a six-week period of study with political manoeuvring, the committee took a vote on the two finalists: the Pearson Pennant (Beddoe’s design) and the current design. Believing the Liberal members would vote for the Prime Minister’s preference, the Conservatives voted for the single leaf design. The Liberals, though, all voted for the same, giving a unanimous, 14 to 0 vote for [it]”. (From Wikipedia).

Flag of Spanish Vexilology Society (Asociación...
Flag of Spanish Vexilology Society (Asociación Española de Vexilología) (Photo credit: Wikipedia)

The process for our new flag was decided on early. Firstly all submissions would be reviewed, and a “top 40” would be selected by a panel composed of, basically, a bunch of celebrities and other. No disrespect to them, but they were not flag experts, and if they were chosen to sort of represent the man/woman in the street that is what they achieved.

Of the top 40, four were “approved” by the ruling National cabinet. It’s not too clear how this was done, but only conspiracy theorists would contend that three out of the four contained the fern emblem that the Prime Minister favoured and that he somehow influenced the selection.

Stjórnarráðið in Reykjavik, the seat of the ex...
Stjórnarráðið in Reykjavik, the seat of the executive branch of Iceland’s government (Photo credit: Wikipedia)

Now we have the top four we are supposed to vote in a referendum to pick one to go up against the current flag in a second referendum. Interestingly a group has been formed to promote a fifth design (originally in the top 40) over the top four. We will see where if anywhere that this movement goes. I’d guess it will eventually fail.

One hundred words to go. Interest is high on the attempt of Jarryd Hayne, a former Australian rugby league player who has secured a spot in San Francisco 49ers NFL team. (That’s what is called “American Football” everywhere except the USA). Good luck to him, I say. I expect to see a surge of popularity for the sport in this part of the world.

RLWC - Fiji v Ireland, Gold Coast 2008. Fijian...
RLWC – Fiji v Ireland, Gold Coast 2008. Fijian fullback Jarryd Hayne. (Photo credit: Wikipedia)

He seemed to bring something new to the American game, but time will tell if opposition coaches find ways to nullify his effect or whether other players will adopt some of his style, which to my naive eye seems to be a more fluid running game. Or maybe he really is an exceptional player. Time will tell.

Well, I quite enjoyed jumping around through various topics but I don’t think that I will do it every time. I’ll just take it as it comes.

A show jumping course
A show jumping course (Photo credit: Wikipedia)

 

Philosophy and Science


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Philosophy can be described, not altogether accurately, as the things that science can’t address. With the modern urge to compartmentalise things, we designate some problems as philosophy and science, and conveniently ignore the fuzzy boundary between the two disciplines.

The ancient Greek philosophers didn’t appear to distinguish much between philosophy and science as such, and the term “Natural Philosophy” described the whole field before the advent of science. The Scientific Revolution of Newton, Leibniz and the rest had the effect of splitting natural philosophy into science and philosophy.

Statue of Isaac Newton at the Oxford Universit...
Statue of Isaac Newton at the Oxford University Museum of Natural History. Note apple. (Photo credit: Wikipedia)

Science is (theoretically at least) build on observations. You can’t seriously believe a theory that contradicts the facts, although there is a get-out clause. You can believe such a theory if you have an explanation as to why it doesn’t fit the facts, which amounts to having an extended theory that includes a bit that contains the explanation for the discrepancy.

Philosophy however, is intended to go beyond the facts. Way beyond the facts. Philosophy asks question for example about the scientific method and why it works, and why it works so well. It asks why things are the way they are and other so called “deep” questions.


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One of the questions that Greek philosopher/scientists considered was what everything is made of. Some of them thought that it was made up four elements and some people still do. Democritus had a theory that everything was made up of small indivisible particles, and this atomic theory is a very good explanation of the way things work at a chemical level.

Democritus and his fellow philosopher/scientists had, it is true, some evidence to go and to be fair so did those who preferred the four elements theory, but the idea was more philosophical in nature rather than scientific, I feel. While it was evident that while many substances could be broken down into their components by chemical method, some could not.

Antoine Lavoisier developed the theory of comb...
Antoine Lavoisier developed the theory of combustion as a chemical reaction with oxygen (Photo credit: Wikipedia)

So Democritus would have looked at a lump of sulphur, for example, and considered it to be made up of many atoms of sulphur. The competing theory of the four elements however can’t easily explain the irreducible nature of sulphur.

My point here is that while these theories explained some of the properties of matter, the early philosopher/scientists were not too interested in experimentation, so these theories remained philosophical theories. It was not until the Scientific Revolution arrived that these theories were actually tested, albeit indirectly and the science of chemistry took off.

Model for the Three Superior Planets and Venus...
Model for the Three Superior Planets and Venus from Georg von Peuerbach, Theoricae novae planetarum. Image enhanced for legibility. The abbreviations in the center of the diagram read: C[entrum] æquantis (Center of the equant) C[entrum] deferentis (Center of the deferent) C[entrum] mundi (Center of the world) (Photo credit: Wikipedia)
Before that, chemical knowledge was very run by recipes and instructions. Once scientists realised the implications of atomic theory, they could predict chemical reactions and even weigh atoms, or at least assign masses to atoms, and atomic theory moved from philosophy to science.

That’s not such a big change as you might think. Philosophy says “I’ve got some vague ideas about atoms”. Science says “Based on observations, your theory seems good and I can express your vague ideas more concretely in these equations. Things behave as if real atoms exist and that they behave that way”. Science cannot say that things really are that way, or that atoms really exist as such.

English: Adenine_chemical_structure + atoms nu...
English: Adenine_chemical_structure + atoms numbers (Photo credit: Wikipedia)

Indeed, when scientists took a closer look at these atom things they found some issues. For instance the (relative) masses of the atoms are mostly pretty close to integers. Hydrogen’s mass is about 1, Helium’s is about 4, and Lithium’s is about 7. So far so tidy. But Chlorine’s mass is measured as not being far from 35.5.

This can be resolved if atoms contain constituent particles which cannot be added or removed by chemical reactions. A Chlorine atom behaves as if it were made up of 17 positive particles and 18 or 19 uncharged particles of more or less the same mass. If you measure the average mass of a bunch of Chlorine atoms, it will come out at 35.5 (ish). Problem solved.

English: Chlorine gas
English: Chlorine gas (Photo credit: Wikipedia)

Except that it has not been solved. Democritus’s atoms (it means “indivisibles”) are made up of something else. The philosophical problem is still there. If atoms are not indivisible, what are their component particles made of? The current answer seems to be that they are made of little twists of energy and probability. I wouldn’t put money on that being the absolute last word on it though. Some people think that they are made up of vibrating strings.

All through history philosophy has been raising issues without any regard for whether or not the issues can be solved, or even put to the test. Science has been taking issues at the edges of philosophy and bringing some light to them. Philosophy has been taking issues at the edge of science and conjecturing on them. Often such conjectures are taken back by science and moulded into theory again. Very often the philosophers who conjecture are the scientists who theorise, as in famous scientists like Einstein, Schroedinger and Hawking.

:The Black Hole, Los Alamos
:The Black Hole, Los Alamos (Photo credit: Wikipedia)

The end result is that the realm of philosophy is reduced somewhat in some places and the realm of science is expanded to cover those areas. But the expansion of science suggests new areas for philosophy. To explain some of the features of quantum mechanics some people suggest that there are many “worlds” or universes rather than just the one familiar to us.

This is really in the realm of philosophy as it is, as yet, unsupported by any evidence (that I know of, anyway). There are philosophers/scientists on both sides of the argument so the issue is nowhere near settled and the “many worlds interpretation” of quantum mechanics is only one of many interpretations. The problem is that quantum mechanics is not intuitively understandable.

Diagram of one interpretation of the Nine Worl...
Diagram of one interpretation of the Nine Worlds of Norse Mythology. (Photo credit: Wikipedia)

The “many worlds interpretation” at least so far the Wikipedia article goes, views reality as a many branched tree. This seems unlikely as probabilities are rarely as binary as a branched tree. Probability is a continuum, like space or time, and it is likely that any event is represented on a dimension of space, time, and probability.

I don’t know if such a possibility makes sense in terms of the equations, so that means that I am practising philosophy and not science! Nevertheless, I like the idea.

Displacement of a continuum body, from a refer...
Displacement of a continuum body, from a reference configuration to the current configuration. Continuum mechanics. (Photo credit: Wikipedia)

 

Why Pi?

Based on Image:P math.png
Based on Image:P math.png (Photo credit: Wikipedia)

If you measure the ratio of the circumference to the diameter of any circular object you get the number Pi (π). Everyone who has done any maths or physics at all knows this. Some people who have gone on to do more maths knows that Pi is an irrational number, which is, looked at one way, merely the category into which Pi falls.

There are other irrational numbers, for example the square root of the number 2, which are almost as well known as Pi, and others, such as the number e or Euler’s number, which are less well known.

Illustration of the Exponential function
Illustration of the Exponential function (Photo credit: Wikipedia)

Anyone who has travelled further along the mathematical road will be aware that there is more to Pi than mere circles and that there are many fascinating things about this number to keep amateur and professional mathematicians interested for a long time.

Pi has been known for millennia, and this has given rise to many rules of thumb and approximation for the use of the number in all sorts of calculations. For instance, I once read that the ratio of the height to base length of the pyramids is pretty much a ratio of Pi. The reason why this is so leads to many theories and a great deal of discussion, some of which are thoughtful and measured and others very much more dubious.

Menkaure's Pyramid
Menkaure’s Pyramid (Photo credit: Wikipedia)

Ancient and not so ancient civilisations have produced mathematicians who have directly or indirectly interacted with the number Pi. One example of this is the attempts over the centuries to “square the circle“. Briefly squaring the circle means creating a square with the same area as the circle by using the usual geometric construction methods and tools – compass and straight edge.

This has been proved to be impossible, as the above reference mentions. The attempts to “trisect the angle” and “double the cube” also failed and for very similar reasons. It has been proved that all three constructions are impossible.

English: Drawing of an square inscribed in a c...
English: Drawing of an square inscribed in a circle showing animated strightedge and compass Italiano: Disegno di un quadrato inscritto in una circonferenza, con animazione di riga e compasso (Photo credit: Wikipedia)

Well, actually they are not possible in a finite number of steps, but it is “possible” in a sense for these objectives to be achieved in an infinite number of steps. This is a pointer to irrational numbers being involved. Operations which involve rational numbers finish in a finite time or a finite number of steps. (OK, I’m not entirely sure about this one – any corrections will be welcomed).

OK, so that tells us something about Pi and irrational numbers, but my title says “Why Pi?”, and my question is not about the character of Pi as an irrational number, but as the basic number of circular geometry. If you google the phrase “Why Pi?”, you will get about a quarter of a million hits.

Animation of the act of unrolling a circle's c...
Animation of the act of unrolling a circle’s circumference, illustrating the ratio π. (Photo credit: Wikipedia)

Most of these (I’ve only looked at a few!) seem to be discussions of the mathematics of Pi, not the philosophy of Pi, which I think that the question implies. So I searched for articles on the Philosophy of Pi.

Hmm, not much there on the actual philosophy of Pi, but heaps on the philosophy of the film “Life of Pi“. What I’m interested in is not the fact that Pi is irrational or that somewhere in its length is encoded my birthday and the US Declaration of Independence (not to mention copies of the US Declaration of Independence with various spelling and grammatical mistakes).

Pi constant
Pi constant (Photo credit: Wikipedia)

What I’m interested in is why this particular irrational number is the ratio between the circumference and the diameter. Why 3.1415….? Why not 3.1416….?

Part the answer may lie in a relation called “Euler’s Identity“.

e^{i \pi} + 1 = 0

This relates two irrational numbers, ‘e’ and ‘π’ in an elegantly simple equation. As in the XKCD link, any mathematician who comes across this equation can’t help but be gob-smacked by it.

The mathematical symbols and operation in this equation make it the most concise expression of mathematics that we know of. It is considered an example of mathematical beauty.


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The interesting thing about Pi is that it was an experimental value in the first place. Ancient geometers were not interested much in theory, but they measured round things. They lived purely in the physical world and their maths was utilitarian. They were measuring the world.

However they discovered something that has deep mathematical significance, or to put it another way is intimately involved in some beautiful deep mathematics.

English: Bubble-Universe's-graphic-visualby pa...
English: Bubble-Universe’s-graphic-visualby paul b. toman (Photo credit: Wikipedia)

This argues for a deep and fundamental relationship between mathematics and physics. Mathematics describes physics and the physical universe has a certain shape, for want of a better word. If Pi had a different value, that would imply that the universe had a different shape.

In our universe one could consider that Euler’s Relation describes the shape of the universe at least in part. Possibly a major part of the shape of the universe is encoded in it. It doesn’t seem however to encode the quantum universe at least directly.

English: Acrylic paint on canvas. Theme quantu...
English: Acrylic paint on canvas. Theme quantum physics. Français : Peinture acrylique sur toile. Thématique physique quantique. (Photo credit: Wikipedia)

I haven’t been trained in Quantum Physics so I can only go on the little that I know about the subject and I don’t know if there is any similar relationship that determines the “shape” of Quantum Physics as Euler’s Relation does for at least some aspects of Newtonian physics.

Maybe the closest relationship that I can think of is the Heisenberg Uncertainty Principle. Roughly speaking, (sorry physicists!) it states that for certain pairs of physical variables there is a physical limit to the accuracy with which they can be known. More specifically the product of the standard deviations of the two variables is greater than Plank’s constant divided by two.

English: A GIF animation about the summary of ...
English: A GIF animation about the summary of quantum mechanics. Schrödinger equation, the potential of a “particle in a box”, uncertainty principle and double slit experiment. (Photo credit: Wikipedia)

In other words, if we accurately know the position of something, we only have a vague notion of its momentum. If we accurately know its velocity we only have a vague idea of its position. This “vagueness” is quantified by the Uncertainty Principle. It shows exactly how fuzzy Quantum Physics.

The mathematical discipline of statistics underlay the Uncertainty Principle. In a sense the Principle defines Quantum Physics as a statistically based discipline and the “shape” of statistics determines or describes the science. At least, that is my guess and suggestion.


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To return to my original question, “why Pi?”. For that matter, “why statistics?”. My answer is a guess and a suggestion as above. The answer is that it is because that is the shape of the universe. The Universe has statistical elements and shape elements and possibly other elements and the maths describe the shapes and the shapes determine the maths.

This is rather circular I know, but one can conceive of Universes where the maths is different and so is the physics and of course the physics matches the maths and vice versa. We can only guess what a universe would be like where Pi is a different irrational number (or even, bizarrely a rational number) and where the fuzziness of the universe at small scales is less or more or physically related values are related in more complicated ways.


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The reason for “Why Pi” then comes down the anthropological answer, “Because we measure it that way”. Our Universe just happens to have that shape. If it had another shape we would either measure it differently, or we wouldn’t exist.


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Random musings

sigh-ness#1
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.

 

TESORO DE CORAL, NOSTOC
TESORO DE CORAL, NOSTOC (Photo credit: PROYECTO AGUA** /** WATER PROJECT)

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...
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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