Parallel worlds or a Continuum?

Spooky Cat
Spooky Cat

A cursory search on the Internet doesn’t tell me one way or another if Erwin Schrodinger owned a cat. Nevertheless he could have owned a cat, so the existence of Schrodinger’s actual cat is unknown to me. David Deutsch might possible argue that Schrodinger’s decision to own a cat or not own a cat resulted in two parallel worlds.

The above is obviously a play on the original scenario outlined by Schrodinger, the famous Schrodinger’s Cat thought experiment. The cat’s state before the box is opened is a strange state, referred to as a superposition, where the cat is both alive and dead. When the box is opened it is argued that this state is somehow resolved with cat being definitely alive or dead.

It's a trap!
It’s a trap!

Suppose that we install a detector in the box with the cat which determines whether or not the cat is dead and notes the time when  it dies. Does this resolve the paradox? After all, if the detector says that the cat died three minutes ago, then we now know exactly when the cat died.

This doesn’t resolve the issue, though, as the detector will also be in a superposition until the box is opened – we don’t know if it has been triggered or not. Of course, some people, including Schrodinger himself, are not happy with this interpretation, and it does seem that, pragmatically, the cat is alive until the device in the box is triggered and is thereafter dead.

Superposition of waves
Superposition of waves

However the equation derived by Schrodinger appears to say that the cat exists in both states, so it appears as if Schodinger’s “ridiculous case” (his words) is in fact the case. Somehow the cat does appear to be in the strage state of superposition.

If we look at the experimenter, he (or she) has no clue before opening the box whether the cat is dead or not. Nothing appears to change for him (or her), but in fact it does. He (or she) is unaware of the state of the cat, so he (or she) is in the superimposed state : He (or she) is unaware whether or not the cat is alive or whether it is dead, which is a superposition state.

Dead Cat Bounce
Dead Cat Bounce

Yet we don’t find this strange. If we remove the scientific gadgets from the box, this doesn’t really change anything – the cat may drop dead from old ages or disease before the box is open. Once again we cannot know the live/dead status of the cat until we open the box.

So, what is special about opening the box? Well, the “when” is very important if we consider the usual case with the scientific gadgets in the box. If we open the box early we are more likely to find the cat alive. If we open it later it is more likely that the cat will be dead. Extinct. Shuffled the mortal coil.

Mortal Coil
Mortal Coil

So it is the probability of atomic decay leading to the cat’s death that is changing. It may be 70% likely that cat is dead, so if we could repeat the experiment 1000s of times 7/10th of the time the cat is dead, and 3/10th of the time the cat is still alive. Yeah, cat!! (There’s also a possibility that the experimenter gets a whiff of cyanide and dies, but let’s ignore that.)

But after the box is opened, the cat is 100% alive or 100% dead. Apparently. How did that happen? Some people claim that something mysterious called “the collapse of the waveform” happened. I don’t think that really explains anything.

 

Collapse of the waveform
Collapse of the waveform

The same thing happens in the real world. If I don’t check my lotto tickets I’m in a superposition state of having won a fortune and not having won a fortune. When I check them I find I haven’t won anything. Again! I must stop buying them. They are a waste of money.

The many worlds hypothesis gets around this by postulating the splitting of the world into two worlds whenever a situation like this arises. After I check my lotto ticket there are two worlds, one where I am a winner and one where I am not. How can I move to the world where I’m a millionaire? It doesn’t seem fair that I stuck here with two worthless bits of paper. does it?

Lotto tickets
Lotto tickets

And what does the probability mean? In the lotto case it is 1 in an astronomical number that I come out a winner and almost 1 that I get nothing. In the cat case it may be 60/40 or 70/30, and in the cat case it changes over time.

If the world splits every time a probabilistic situation arises, then the probabilities don’t actually mean much. What difference does it make if a situation is “more probable” than another situation if both situations come about in the multiverse regardless? It doesn’t seem that it is a meaningful attribute of the branches. What does it mean, in this model that branch A is three times more likely than branch B? Somehow a continuum (probability) reduces to a binary choice (A or B).

Spectrum
Spectrum

We could consider that the split is not a split at all, but that reality, the universe, whatever, has another dimension, that of probability. Imagine your worldline, a worm travelling through the dimensions of space and the new one of probability. You open the box and lo! Your worldline continues, and the cat is now dead or alive, but not both.

But which way does it go? That is determined purely by the probabilities, by the throw of the cosmic dice, but once it chooses a path, then there is no other possibility. In the space dimensions you can only be in one place at a time. If you are at A you cannot be at B, and similarly in the probability dimension, if you are at P you cannot be at Q.

Alphabet
Alphabet

However any point P (the cat is still alive!) is merely a point on the probability line. There are an uncountable number of points where the cat is alive and also an uncountable number of points where the cat is deceased. But the ratio between the two parts of the line is the probability of the cat’s survival.

Many Worlds
Many Worlds

Seeing things


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I sometimes suspect that I return to the same topics time and again. Not too often I hope, because that will put people off reading this blog (in case anyone does!) This is possibly a topic which I may have already addressed, but hopefully this post will be interesting anyway.

It seems obvious to me that we all see things differently, and I’m talking about vision here, not “seeing” as a philosophical point of view. Some are short sighted, some long sighted, and others have impaired vision. I see a colour as a shade of blue, while my wife sees it as a shade of green.

Toyota Celica 2.0 GT (ST202) shown in Bright T...
Toyota Celica 2.0 GT (ST202) shown in Bright Turquoise Pearl (colour code 756). (Photo credit: Wikipedia)

One could argue that the difference is merely where the line is drawn, but I think that it is more than that. Apart from the physical differences in the lenses of our our eyes, we may have differences in the physical structure of the rest of our eyes, perhaps in the rods and the cones, and it is highly likely that the physical structures of our brains are different, and our minds (which I think of as the software that runs of the hardware of the brain) are definitely different.

It’s no surprise then that my wife and I disagree on whether a colour is a shade of blue or of green. (Actually we disagree about a lot of things. I believe that it goes with being married for 40+ years!)

Plymouth Valiant 100 of some 40 years ago seen...
Plymouth Valiant 100 of some 40 years ago seen on street in New Orleans (Photo credit: Wikipedia)

In Googling around as I write this post I found an article about the brain’s colour processor. Interestingly it has a section entitled “Color is Personal” which is a part of my theme for this post. This section, however, is not really relevant to my theme as the author then discusses Achromatopsia, where damage to the colour processor causes all sensation of colour to disappear.

It seems that even in our own brains and thinking processes the idea of colour is not fixed. I read another article which describes our own personal perception of colours as “malleable”. The implication of this is that a person might describe a colour as “a shade of green” one day, and “a shade of blue” on another day. Is there no hope of a definitive answer?

Newton's color circle, showing the colors corr...
Newton’s color circle, showing the colors correlated with musical notes and symbols for the planets (Photo credit: Wikipedia)

A physicist could help us out, couldn’t he/she? He/she could measure the frequency of the light and say, definitively, that the colour is blue, or it is green, couldn’t he/she? Well, sort of. This would work for very simple colours, but real world colours are rarely made up of just one colour. The scientist’s scope would likely show a range of frequencies resembling a mountain range. That blue/green colour might have traces or red or violet, and is fairly certain to have more than one peak in the blue/green range.

Albert Einstein showed us that if a scientist was moving at a high speed relative to us, he/she would measure the frequencies in the colour differently from a scientist whose spectroscope was alongside us and not moving or moving at the same speed as us.

General Relativity
General Relativity (Photo credit: Wikipedia)

The ambient light has an effect on the colours that we perceive. A red object in red light doesn’t look red. Other objects of different colours look different in a red light. Similarly, it is difficult to determine the colours of cars and other objects under the yellow/orange sodium lights. According to Wikipedia, the colour of a street light has effects other than simple colour perception – it appears to affect peripheral vision.  New LED technology may be able to remove some of these deficiencies.

There are innumerable effects which affect or perception of colour. The most recently famous illusion is the dress which appears to people to be either black and blue or white and gold, but there are many such illusions. One which I came across a long time ago is the chessboard illusion. In this illusion, two square appear to be different colours, but are in fact the same colour. This illusion is usually shown in monochrome, but the illusion works in colour too, and depends on the shadow of the cylinder to produce the effect.


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One brain is very like any other brain. When a scientist shows someone a colour on a card, the same areas of the brain show activity in all individuals, if we exclude some cases where brain function is abnormal for some reason. We can’t delve very much deeper into this issue as we don’t know what this activity signifies, beyond the bare fact that the person was shown a card with a colour on it. We certainly can’t tell if they see it as a shade of blue or a shade of green, and we can’t tell what their subjective experience is when the brain activity occurs.

In some individuals a number or letter may invoke a sensation of colour. Such people might have the sensation of seeing something green when they think of or read the number 6. I don’t know if this imprinted behaviour because the person was presented with a green symbol when first learning their numbers or whether or not it was merely a chance association that arose at a different time, or indeed if it was because of some neurological happening or trauma that has allowed the association to happen.

English: A graph or how the brain interprets color
English: A graph or how the brain interprets color (Photo credit: Wikipedia)

Anyhow, when we see something, there are many stages to the process that  starts with light leaving the object, reaching our eyes, being refracted by the lens of the eye to form an image on the retina at the back of the eye, being sensed by the rods and cone cells in the retina, and sending signals to the brain, which then processes the data.

The amazing thing here is that the image sent to the brain is pretty messy. The eye is not a perfect sphere, the retina is curved in three dimensions and the resolution is pretty rubbish. The retina has at least one major gap in it, rods and cones are not evenly distributed across the retina. Our perception however, is smooth and break free. We have our image processing hardware and software in the brain to that for that.

Retinoblastoma retina scan before and after ch...
Retinoblastoma retina scan before and after chemotherapy (Photo credit: Wikipedia)

It means we can watch a soccer match, and we can see the black and white panels or the ball rotating as it spins across the television screen, when the unprocessed image that reaches our eyes may be quite blurred. Seeing is believing!


Embed from Getty Images

Embed from Getty Images

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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Time waits for no man

time travel
time travel (Photo credit: flyzipper)

Time is an odd thing. We say it passes, but it sometimes feels more like we are travelling through. As the old joke goes, we all travel through time – at a rate of one second per second.

While that might bring a smile, it does raise a question about time travel, because if one travels through time, one presumably travels through it at some rate or other, say ten years per minute. The problem with that it is that we are measuring a rate, which is a change of some variable with respect to another, but in this case we are measuring the rate of change of time with respect to time as well.

English: Acceleration as derivative of velocit...
English: Acceleration as derivative of velocity along trajectory (Photo credit: Wikipedia)

The time intervals over which the traveller is passing are measured in the usual way by a clock, but how does the traveller measure his time by? He could carry a clock with him, which he could then use to estimate his progress along the standard time scale. In other words the time traveller would somehow have to carry his own time scale with him which is different to the usual time scale.

Illustration of a light cone, based on Image:L...
Illustration of a light cone, based on Image:Light cone.png (Photo credit: Wikipedia)

Of course, Einstein’s Special Relativity shows that, in a way, we do carry our own time frames around with us, and if we are in motion relative to some other frame then time passes differently the two frames.  Of course, to simply travel in time, we would not want to travel in space, so we can’t use Special Relativity to allow us to use a different time frame, so far as I can see.

Also, we can only travel forward in time by using this loophole. No matter how fast we move relative to someone else, we both move forward in time, so we can’t use Special Relativity to go back in time and kill dear old Grandad.

The Grandfather Paradox
The Grandfather Paradox (Photo credit: Kaptain Kobold)

Einstein’s General Relativity considers that space-time (the conjunction of space and time) possesses curvature, and some theories use this to allow backwards time travel. However these solutions produce “closed time-like curves” which is not so much time travel as a time loop, perhaps like the loop in “Groundhog Day” where Bill Murray’s character repeatedly awakes to the same day.

groundhog day!
groundhog day! (Photo credit: NapaneeGal)

It appears that we need to look further for some way to travel in time. If we can’t use current physics, we will need to consider something more “science fiction” than modern physics. Of course science fiction time travellers don’t seem to explain their travels in more than a cursory way, because, after all, the mechanism is only secondary to the story line.

Time Machine Clockwork
Time Machine Clockwork (Photo credit: Pierre J.)

Two different possible mechanisms spring to mind.

Firstly, one way is to assume a sort of parallel world. A time traveller can enter this parallel world from any point in time and re-enter the standard universe at a different point, earlier or later in time. The traveller travels in time by analogously travelling in space in a world which has its own space-time with one of its space dimensions parallel to the conventional world’s time dimension.

parallel worlds
parallel worlds (Photo credit: aloshbennett)

Secondly, the author can conjecture a viciously curved space-time so that the characters can, at certain locations, move from one part of space-time to another part of time which is either earlier or later in the time dimension. Typically the character will “step sideways” or something to jump between times, either with the help of a machine or maybe not.

HELP ME HELP MYSELF!
HELP ME HELP MYSELF! (Photo credit: eyewashdesign: A. Golden)

One such tale of the second sort is “By His Bootstraps” by Robert A Heinlein, is which the main character passes through a portal to a distant future, only to entangle himself with later (and, relatively, earlier) versions of himself. He encounters a mysterious character who identifies himself as “Diktor”.   I’ll leave it there, as I don’t want to spoil the story for those who haven’t read it yet.

By His Bootstraps
By His Bootstraps (Photo credit: Wikipedia)

An example of the first sort is “The Corridors of Time” by Poul Anderson, where the main character is recruited into a war raging up and down the “corridors of time”.

Most stories, however, don’t specify in more than a cursory way the physics that is supposedly employed by the time traveller.

Train travel
Train travel (Photo credit: Wikipedia)

The effects of time travel are what is explored by these stories. If the traveller goes back in time he or she must interact with the world at the earlier time from when he started. There are time stories in which the traveller changes things so the state of the future time from which he can is changed.

Future World 2012 (Explored)
Future World 2012 (Explored) (Photo credit: Scottwdw)

This is the premise behind the Terminator series of films, where the Cyborg assassin (played by Arnold Schwarzenegger) is sent back in time to kill the mother of John  Connor, the leader of the rebellion against the killer machines. Obviously, in the future from which the Terminator comes, John Connor is born so the Terminator is trying to change the future from which he comes.

The Terminator
The Terminator (Photo credit: Dunechaser)

The obvious paradox here is that the Terminator risks changing the future into one in which he was never created. In which case he would not be able to come back in time to kill Sarah Connor.

The other sort of time travel story treats time as if it were immutable. Any events that happen are eventually shown to have logically been the consequence of the time travel in the first place. For instance, the mysterious stranger on the street turns out to be the time traveller, keeping an eye on the earlier version of himself. All is explained so that the stream of events is logical both from the time sequential point of view and also from the point of view of the traveller. The aforementioned “By His Bootstraps” is a story of this sort.

The Mysterious Stranger
The Mysterious Stranger (Photo credit: Wikipedia)

Is it possible that events in the past could have changed? We would not know it! As far as we would be concerned the past event would have have always happened, since there would be a temporal progression from the past event to our current time. Of course the language is tricky here, as it does not handle such matters as changes to the past.

Einstein's Theory Fights Off Challengers (NASA...
Einstein’s Theory Fights Off Challengers (NASA, Chandra, 04/14/10) (Photo credit: NASA’s Marshall Space Flight Center)

To use the physicists’ favourite analogy of a rubber sheet, if time is considered to be along one dimension of the sheet, and space along the other, then an event which changes the past is like someone pulling a point on the sheet to one side, which affects all the points from both the future and the past of the point which has been moved. But the sheet itself remains intact.

Rubber sheet undergone drying in smoke. Three ...
Rubber sheet undergone drying in smoke. Three versions are there in the picture. 1. Fresh sheet. 2. Little dried in sun light. 3. Dried in smoke. (Photo credit: Wikipedia)

However, I think that this scenario is unlikely. There doesn’t seem (at the present time) any physical mechanism by which time travel could be achieved, and even though it appears that time travel is logically possible (under the sort of scenario as in “By His Bootstraps”) in a deterministic universe, the simplest conclusion is that time travel is most likely unachievable in this universe.

A "jumpgate" of the X Universe, part...
A “jumpgate” of the X Universe, part of a space-travel network. (Photo credit: Wikipedia)

 

 

 

 

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 ...