Catastrophe Theory

By Salix alba – Own work, CC BY-SA 3.0,

I’ve already covered the topic of fractals and Chaos Theory, but the arrival and popularity of these two obscures a slightly earlier and rather similar mathematical topic which has a number of things in common with them, although it’s a lot “smoother”. This was Catastrophe Theory.

On 28th July 1975, BBC-2 broadcast a ‘Horizon’ documentary entitled ‘Happy Catastrophe’ which got a larger response from its viewership than any other ‘Horizon’ episode. It clearly captured the public’s imagination, attracting more correspondence than any other ‘Horizon’ up until that point, and in fact stuck in my own mind more than most other programmes at the time. Looking back at it, I found a number of other episodes in the mid-’70s quite memorable, such as the one on epilepsy and another on Erich von Däniken, which I mention here, but certainly this is one of them, and in fact epileptic seizures themselves could be modelled using catastrophe theory (CT) itself. To an extent, I want to blog about CT today, but I’m also interested in why it was so popular, and why it seems to be largely ignored today.

CT deals with discontinuities, which are moments of sudden change. For example, if you take a thin card and press it at its sides, it will do very little for quite a while, then suddenly crumple or flip into a different shape, and letting go of the card will not lead to its return to anything like the flat form it had before, although it will tend to spring back a little. The same applies to a snapping rubber band under tension and a host of other situations, such as the epileptic seizures I mentioned just now, although one would hope in this last case that the brain can in fact fairly quickly return to a more organised state. Unfortunately this is rarely not so, in which case it becomes a medical emergency.

The programme’s title, ‘Happy Catastrophe’, is interesting. When we use the word in English, and it is of course a Greek word, we generally mean something negative. The Greek word, “καταστροφη”, consists of the words “κατα”, meaning “down”, and “στρεφειν” – to turn, in other words a “downturn”, and with the usual connotations of falling does indeed have negative connotations. The word was prominently used in drama, where it referred to the fourth and final part of a play, after protasis, epitasis and katastasis. We’re familiar with it today through tragedy, but in fact it also applied to comedy, and in that setting it referred to a happy ending such as a wedding. Hence our own usage has become predominantly negative, but for some time I attempted to use it with a more neutral connotation, which in fact makes the word a lot more useful, although it can be confusing and we don’t really have control over the meaning of words, particularly when we lack something like L’Académie Française. There were two types of catastrophe, whether happy or otherwise, in Greek drama. In a simple catastrophe, there’s simply a transition from dramatic events to a quieter set of circumstances without any change in character, unravelling or revelation. Complex catastrophes involve sudden discoveries by the character or sudden changes in fortune which are feasible and upon which the plot depends, rather than being a deus ex machina. In a way, simple catastrophes occupy one side of the graph whereas complex ones occupy the other. This is what I mean:

Taken from here. Will be removed on request.

A simple catastrophe can be thought of as a movement across the steady slope on the left hand side of this graph. It descends into repose without anything huge happening. I don’t know what examples there are of this but to be honest they sound a bit boring. Complex catastrophes, on the other hand, are movements along the right hand side of the graph and involve events “falling off a cliff” in such a way that they permanently change things. This graph is of course the “cusp catastrophe”. It makes me wonder what the variable labelled as “u” is in drama. ‘Œdipus Rex‘ definitely occupies the right hand side – it has a low value of u, whatever that might be. It’s also important to remember that if you turn this graph upside down, you more or less have the same graph, and that therefore comedies are also catastrophic in nature. ‘Much Ado About Nothing’ is just as catastrophic as ‘Œdipus Rex‘, but in a positive way.

Incidentally, in what I’ve just said I can’t help but be reminded of this:

Can you usefully take a quantitative approach to literature? In a way the answer is a definite “yes”, because for instance you could look at repetition of certain words and phrases or the prosody or rhyme scheme of a particular poem, but in general it does have a bad rap. But I can’t help noticing that when John Keating gets the pupils to rip out the introduction to ‘Understanding Poetry’, it is a catastrophic event, and of course later in the film there are other incidents more deserving of the word, but there’s no going back once the introduction has been ripped out, as the end of the film illustrates.

The cusp catastrophe graph looks like the kind of shape you’d get if you held a thin sheet of metal horizontally and bent it towards or away from you. This is because that situation is in fact a catastrophe with two control dimensions and one behaviour dimension. The buckling which occurs on one side of the sheet is dramatically greater than on the other. This now sounds like an engineering or metallurgy issue, but can be used for drama, as with the 1951 film ‘No Highway In The Sky’, which involves the catastrophic failure of aircraft in this way. In this case the behaviour axis involves the plane falling out of the sky and killing everyone, although there’s another catastrophe where Theodore Honey deliberately damages a plane to prevent it taking off and killing the occupants:

I’ve mentioned control and behaviour dimensions, or axes, without really explaining what they are. To elaborate, it makes sense to consider the simplest possible models, including non-catastrophic ones, which have two dimensions. A section of a two-dimensional line graph can have a number of shapes relevant to CT. It can be a slope, a trough, a peak or a fold. Except for the slope, these are all the same basic shape. With a fold, the shape is like a C rather than a U or an “n”. This means that as the control variable increases, the behaviour of the system can either become more dramatic or less so, to choose one possible label for a variable, but will be stuck in that trend unless the other variable reduces considerably. Or, it can be reflected along the Y axis and will be stuck in a trend unless that variable increases a lot. This is the “zone of inaccessibility” and can be shown in several other examples.

There are substances whose melting points are not the same as their freezing points. That is, if a solid of this nature is heated, it will melt at a particular point, but if the resultant liquid is then cooled, it may need to be made colder than the temperature at which it melted to solidify. I seem to remember that cocoa butter does this, but there are many examples. Similarly, when tuning in an analogue radio with a manual tuner, one can find a station, then tune up past it and then find that it seems to be on a lower frequency than one previously found it when twisting the knob back again. These are examples of the kind of behaviour which is modelled in the overhang found in the cusp catastrophe. A value can increase smoothly until it leaps to a higher value if another value is high, but can also stay on the lower surface, and likewise can stay on the higher surface until it is lower than when it initially leapt up. I have a feeling that tidiness is like this. It takes more effort to tidy something up in one big go than it appears to when one does it bit by bit, and then it slips down into untidiness more easily.

Adding a dimension clearly results in three-dimensional graphs, and again there are a certain number of these. Incidentally, before I go on I want to point out that CT graphs only focus on a narrow range of variables where something interesting is occurring, and are therefore small portions of potentially infinite graphs. The two-dimensional “fold” catastrophe could easily diverge to an ever-increasing but smooth extent along its control axis, even to infinity. Also, in illustrating these graphs the section can be a small map of a much larger landscape, such as one including peaks and basins or mountains and valleys. It’s just that the distinctive shapes can be broken down in this way.

Three-dimensional graphs could just be extensions of two-dimensional ones, so for example a valley could just be long and not do much interesting in the Z-axis, so all the types still exist in three and more dimensions and are not cancelled out by the new ones, but each added dimension does introduce additional graphs. In the three-dimensional case, X and Z can be the controls and Y the behaviour, which makes the surfaces more relatable as they’re more like topographical features. There’s the slope which rises diagonally to the axes, the peak, what I’m going to call the “crater”, which is a dent in a surface, and two less familiar shapes, the col and the cusp. I want to mention the col even though it isn’t catastrophic, because it’s less well-known or easy to relate to than the others.

A col is a gap between two peaks. These are often nameless locations, although passes are cols. They occur also in air pressure patterns, where there’s a low-pressure point between two high pressure weather systems. There’s also the saddle:

Saddles differ from cols in continuing to curve away in both directions, concave on one side and convex on the other. A col is the central point of a saddle according to one definition.

The cusp is crucially different from all of these because it has a kind of asymmetry to it along one axis, although it also is rotationally symmetrical in that turning it 180° around the axis labelled u in the earlier graph, assuming it’s aligned correctly, will lead the same shape. This mixture of asymmetry and symmetry doesn’t apply to the other shapes and the cusp is the only discontinuous shape involved.

These shapes appeal to the eye, and it’s been said that CT is particularly visual. It shares this feature with many fractals and the Mandelbrot Set, and in this respect serves as a kind of herald to those later, particularly visually appealing, mathematical excursions. It also has a kind of universalising tendency, which despite its name has been described as modelling rather than a theory. Calling it a theory is a bit like counting two legs on a person and seeing that there are two stars in a binary star system and calling that “integer theory”. It’s more that this kind of model can be applied to natural phenomena, and as seen above with the illustration of catastrophes as a dramatic device, also in the social sciences and humanities. The issue of their beauty may be similar to the beauty of regular fractals and the Mandelbrot Set, in that certain features echo the characteristics of being a product of the Universe, which is who we are in one respect.

There are a total of seven graphs, according to CT, which can between them be used to model all discontinuities. These are: the fold, cusp, butterfly, swallowtail, hyperbolic umbilic, parabolic umbilic and elliptic umbilic. The hyperbolic umbilic is illustrated at the start of this post, where it comprises the upper part of the image. Because it’s a five-dimensional shape, the illustration isn’t exactly what it “looks” like, but is in fact what’s known as the bifurcation set of the hyperbolic umbilic. This is a projection of the shapes which are discontinuous in the graph. In the case of the cusp, this is a kind of curved V-shape extending to infinity or the edge of the graph, like a kind of shadow cast by illumination on a transparent model, or alternatively, and this is more important than it might seem, the kind of light reflected by illuminating a smooth metallic version. The bifurcation set of an hyperbolic umbilic is like two superimposed half-pipes at a shallow angle to each other semicircular in cross-section at opposite ends smoothly becoming curved V-shapes at the other. That probably isn’t very clear. It has two behaviour dimensions rather than one, and three control dimensions. Umbilics are points on locally spherical surfaces, and hyperbolic ones have just one ridge line passing through the point in question, which if I’ve described the above clearly means the point of intersection between the two half-pipes. It’s interesting to contemplate what it would be like to skateboard around the bifurcation set of an hyperbolic umbilic.

The other two umbilics are the parabolic and elliptic. Elliptic umbilics have three control and two behaviour dimensions and the bifurcation set looks like a cross-sectionally curved triangular prism pinched smoothly to a point at the centre, which is the three ridge points passing through the umbilic point. Finally, the parabolic umbilic is six-dimensional, with four control and two behavioural dimensions, making it particularly hard to visualise as even the bifurcation set has four dimensions, but are transitional between hyperbolic and elliptic umbilics, with two ridges, one of which is singular. Visualised using the fourth dimension as time, running in one direction the bifurcation of a parabolic umbilic looks like a shrinking paper plane crashing through the fold in a sheet of paper folded into a V-shape while another V-folded paper shape at the bottom is flattening out and bowing outward.

The other two are the rather less awkwardly-named butterfly and swallowtail. The former is interestingly named because of the butterfly effect, but is not more closely linked to that than the others. It’s five-dimensional, with four control dimensions and one behaviour dimension, and has been used to model eating disorders. It looks odd, even reduced to three dimensions, which effectively destroys its usefulness but enables one to work out what it’s doing, as it looks like a cusp catastrophe with three cusps linked in a kind of triangle. That is, a triangle can be drawn between the three points where the cusps split off from the smooth side, but that triangle isn’t oriented in three-dimensional space unless the butterfly is rotated in such a way that most of it is in hyperspace.

The swallowtail catastrophe is so named because a mathematician was trying to describe it to a blind person, who responded that it sounded like a swallowtail, which it does. It’s merely four-dimensional and its bifurcation set looks like a swallowtail at one end with a U-shape above it with the tail diminishing into the U halfway along. This has one behaviour dimension and three of control. Salvador Dalí’s last painting, if it was his, in 1983, was based on this graph, and was entitled “The Swallow’s Tail”:

This is a cross-section of the bifurcation set with some extra bits added. The monoline S shape is a cross-section of the cusp catastrophe. Dalí described CT as “the most beautiful æsthetic theory in the world”. The artist used to kind of “riff” on scientific theories in an artistic way, using them as inspiration without necessarily understanding them in an analytical way. He also included a formula describing the swallowtail in his 1983 painting linked here entitled ‘El rapte topològic d’Europa. Homenatge a René Thom’. The last few years of his life are controversial because it’s alleged that he was made to sign canvases by his carers which would later be used to paint forgeries, and the above painting may not be his because his hands were said to be too shaky for him to draw such a line, which brings Britney Spears to my mind. After completing this painting, if he did, Dalí tried to enter a state of suspended animation through fasting and died five years later, soon after giving the visiting Juan Carlos a drawing entitled ‘The Head Of Europa’.

One way of looking at these graphs is to see the compartments as representing different stable states. Hence the six “cells” of the parabolic umbilic plus the seventh open space nearby are each conditions some systems can enter if there are four main factors determining their behaviour, which can in turn be described in terms of two factors. The same can be applied to the others.

I mentioned Dalí’s tremor making his creation of ‘The Swallowtail’ questionable, but in fact tremor and noise are not likely to disturb the behaviour of catastrophes. They’re quite stable in this respect, which calls into question the often-quoted explanation as to why they’re now so seldom modelled in this way being that not many systems can be adequately described with so few variables. This property is accompanied by what are called “attractors”, which CT has in common with Chaos Theory. An attractor is a set of states a system tends to drift towards, or in this case jump towards. Each one of the cells I mentioned just now is an attractor. After having got there, the system will tend to continue to be at least somewhat like that. It occurs to me in fact that limerence could be modelled in this way. It’s easy to get fixated on someone but it can be a lot harder to get over them. That, then, would be literally an attractor: a person one finds attractive. This suggests it would be fruitful to work out which control variables are involved, since in certain crucial circumstances, people do end up suffering from long-term limerence. However, discussing it and other psychological models in this way raises the question of positivism, which can be criticised on the grounds of reductivism.

You may or may not have heard of Gartree Prison, which was well-known for its helicopter escape in 1987. I have two personal connections to Gartree. One is that it ended up housing the bloke who abducted me in 1989 and the other is that one of my tutors on the herbalism course was married to a Gartree prison guard. Rather startlingly, Gartree prison disturbances were modelled using CT, more specifically the cusp catastrophe. This makes for a significant case study of the application of CT to social phenomena. When this was done, CT was riding on a wave of popularity triggered by the ‘Horizon’ broadcast and was possibly quite immature in its development, although as a modelling method it dates back to Edwardian times, the modelling having been published in 1976. The control variables seem to have been tension and alienation, which were assessed quantitatively, an approach which seems quite vague. They were based on governor applications, inmates requesting segregation, staff absenteeism, welfare visits and inmates in the punishment cells, and the shape of the graph seems to have been derived using a method which, it’s said, could have been made to fit almost any data set. There may have been an issue in the dominant connotations of the word “catastrophe” here, because it tends to be interpreted as negative and would perhaps consequently tend to lead to applications of the theory to model negatively-perceived events such as prison riots. It might also have been used by the prison service to make its operations and management appear more scientific than it actually was. And in any case, scientific management is widely regarded as a bad thing, at least for workers, as it’s seen as leading to redundancy, monotonous work, exploitation of workers, and from the management side expensive to implement, time-consuming and leading to a deterioration in quality. This could have implications for the situation inside prisons, as they are also workplaces for the staff and sometimes also for prisoners, so simply making the measures required might impair the function of the institution.

This could be applied more widely to other institutions such as mental hospitals and schools. For instance, if it successfully predicted grades in a school and also ways of manipulating variables in order to get those grades onto a higher tier of the graph, it wouldn’t necessarily improve less quantifiable measures of school performance. Likewise, a similar approach might lead to higher “cure” rates in a mental hospital, but that would only be in terms of particular paradigms of “abnormal” behaviour. Could it be applied to increase the quality of poetry? Maybe it could. Maybe J Evans-Pritchard would be able to measure the greatness of the poetry output by all these “cured” psychotics and high-achieving school-leavers with his scale. Or, maybe we just like to imagine that we aren’t reducible in such a way to a few variables and graphs, but maybe we’re wrong about that.

The modelling here, and in the other two as far as I know fictional examples I gave (the mental health one is less fictional than one might think), is applied to systems which depend on many assumptions about how society should be. For instance, assuming the prison study was valid, it might still fail to show anything because prisons of that kind are constrained by social factors always to be on one side of the cusp, and whereas manipulating the variables beyond that range is theoretically possible, doing so would not be possible given factors like level of public funding, policy regarding responses to crime and the nature of the buildings used. Then again, maybe we do want an entirely evidence-based set of policies. I would personally prefer that. It’s called socialism.

In a realm entirely outside the question of social policy, meditation, states of consciousness or mental illness, catastrophe graphs turn up in another rather surprising place: caustics. Caustics are projections of light rays reflected or refracted by a reflective or transparent medium onto a surface. I mentioned previously that a model of a cusp catastrophe could be made of mirror-like reflective material and be illuminated, and such a situation could lead to the projection of a caustic onto a flat screen. Caustics are the kind of light pattern you see when you look down into a clean, empty mug into which sunlight is shining, and they alter their shape and size according to the angle of incidence. They can also be seen in the dappling effect on a sandy seabed of waves on a sunny day. They can also have a kind of three-dimensional appearance, and in the teacup case they seem to look rather like a swallowtail bifurcation set, but in three dimensions in each case. Moving the cup leads to a different section of the graph. Caustics are odd because they’re always sharp and it isn’t clear what’s so special about the area they illuminate as opposed to its surroundings. They’ve also historically been problematic in computer graphics because depicting them accurately is computationally intensive, so in CGI they tend to be more decorative than realistic. It would be interesting to know whether catastrophe theory could simplify or has ever been used to generate caustics in computer images. Moreover, it would also be interesting to know if images of three-dimensional slices of higher-dimensional CT graphs could be accurately generated using three-dimensional reflective surfaces to generate their caustics.

A major question remains. Why don’t we hear so much about CT nowadays when it was so popular forty-odd years ago? An answer might be found in an illustration from herbalism, and at this point I shall intrepidly venture onto the territory of one of my other blogs. It’s been noted that herbal prescriptions with an odd number of remedies tend to be more successful than those with an even number. This needs to be restrained in various ways. For instance, it doesn’t mean that an even-numbered ℞ can be made more effective by omitting one of the herbs or adding one which is not relevant to the patient’s needs. I hypothesised that the reason for this was that an odd-numbered prescription could be modelled in terms of relative doses using catastrophe theory, whereas an even-numbered ℞ couldn’t. However, there are a number of problems with this which can be extended to other situations. The herbs here are presumed to be the control dimensions of the graph. A fold catastrophe has one control dimension, a cusp two, a swallowtail three, a butterfly four, a hyperbolic umbilic three, parabolic four and elliptic three, so the number of remedies would seem to have to be three or one if this is to hold true. In fact ℞s tend to have five or seven remedies, if one is in the low number of remedies in high doses as am I, because I feel the high number of remedies in low doses is beginning to look like homeopathy. Hence it can’t be applied to most herbal prescriptions other than simples, and there would have to be something which makes the fold, swallowtail and hyperbolic and elliptic umbilics distinctive in terms of their efficacy, which may be true but I’m not sure about that. But there’s a bigger problem which applies more widely. Herbs are not single remedies. They generally include a large number of different compounds with various effects on each other and physiology. Thus it seems implausible to apply catastrophe theory to herbalism, and this can be broadened out into biology more generally, since in most biological situations the number of control dimensions would be too high for CT to be relevant.

CT is still applicable to engineering and physics, but its intended target, the inexact sciences such as sociology, psychology and ecology, is rather more slippery. It does still happen, for instance in modelling the population dynamics of aphids via the butterfly catastrophe (it would have to be named after an insect – presumably the swallowtail is useful for modelling bird migration), but there really do seem to be too many variables and the smoothing effect initially claimed doesn’t seem to hold. That said, the formulæ used to generate the graphs are quite simple, and this could lend them to use in computer games, both in generating caustics on the graphics side and the likes of political and social interactions in games like Sim City.

Our Shadow Twins

There more or less have to be parallel universes because this Universe is “fine-tuned”. The alternative would seem to be to require a Creator, and although there is a Creator, or rather a Sustainer because God is not within time, nothing in the Universe should be allowed to imply or suggest that there is one as that would be a “God Of The Gaps”.

I should probably explain fine tuning. There are certain constants governing the relative strengths of the four known forces in the Universe which, if they varied even slightly, would make rocky planets and life as we know it impossible. Examples are as follows:

  • Electromagnetism is a sextillion (long scale) times stronger than gravity. If it were much smaller, the Universe would have collapsed in on itself before the stars could have formed.
  • When deuterium nuclei fuse to form stable helium-4, the nucleus loses 7% of its mass. If it lost 6%, only hydrogen would exist, and if it lost 8%, all the nuclei in the Universe would’ve fused together within a fraction of a second of the Big Bang and there would be no atomic matter at all. That said, that is quite a large range, determined by the strong nuclear force.
  • If dark energy was slightly stronger compared to gravity, stars would not be able to form because they’d be ripped apart by the expansion of space. If it was slightly weaker, the Universe would’ve collapsed by now.
  • If other than three spatial dimensions were extensive (there are others, which are however very small and so don’t influence this), there would be problems with the weakening of gravity at a given distance which would again either cause collapse or make it impossible for stars to form.

There are several other examples, but taking these together is enough to illustrate the issue, because the improbabilities multiply, and some of them even seem to be part of an infinite range of possibilities, usually very boring ones because they either involve the Universe collapsing in on itself almost immediately after the Big Bang or merely consisting of hydrogen atoms thinly spread throughout space. The situation we actually find is fantastically improbable because of this. It’s also been suggested that the specific existence of the element carbon is suspiciously unlikely, and water is also such an unusual compound that it too is unlikely, but the details of these involve once again the strengths of the strong nuclear force in the case of carbon and that of electromagnetism in the case of water. There is presumably a version of water in a parallel universe which is still H2O but is a gas at well below its current freezing point and contracts when it freezes, is not a good solvent and so forth. In fact probably most versions of water are like that. Likewise, and this is I admit a very sloppy calculation because several different forces are involved in holding atomic nuclei together and they don’t obey the inverse square law, the last element to have stable isotopes is bismuth, and if the strong nuclear force was forty percent weaker, stable carbon atoms could not exist and carbon-based life would be impossible.

Because of all this stuff, some theistic religious people believe that there must be a God. However, there’s a problem, or rather several problems, with that argument. Firstly, even if it does entail a creator, it fails to entail a God like the one in the Bible, Qur’an or whatever. Secondly, the Universe which actually exists is almost completely empty and life seems to be a mere detail, possibly on only one planet and even if widespread it would still only have come into existence on the tiny grains in a vast void. In fact, this almost completely empty void may be a clue to the nature of reality. What we’re confronted with when we look into the night sky is unimaginably enormous distances between stars, whose visible examples are unsuitable for life as we know it, organised into galaxies which are also separated by relatively much smaller distances and organised into clusters forming a kind of “foamy” arrangement around enormous voids like bubbles. Only occasionally are the conditions suitable for the concentration of nuclear matter, and even more seldom do rocky globes form. When we consider Earth, we realise how special she is, but that exceptional nature is contingent on the fact that we are here in the first place to do the considering. The anthropic principle says the same is true of the Universe: there are plenty of other universes but they don’t have any life or observers in them. Ergo, there are parallel universes. The alternatives seem to be enforced belief in a Creator with a capital C or a multiverse, and that multiverse would likewise consist almost entirely of empty universes which have either already ceased to exist or contain only widely space hydrogen atoms and perhaps molecules floating in otherwise empty space. Although I’m theist, I choose the latter.

The question then arises of how inevitable anything is. Alternate history usually depends on PODs, Points of Divergence, such as Hitler dying before coming to power or JFK not being assassinated, and from a macroscopic level it seems entirely plausible that Henry Tandey could have decided to shoot Hitler on 28th September 1918 or that Lee Harvey Oswald missed his target on 22nd November 1963. But in fact these PODs are only apparent. Free will is probably illusory, there’s a whole chain of unknown events influencing those moments and for all we know that chain of cause and effect stretches all the way back to the beginning of the Universe. It will undoubtedly be the case that slight variations in physical constants do indeed lead to differences in the universe, but what we imagine is easily possible could turn out to be completely impossible. The question of whether this is true depends on chaos theory and quantum physics.

I’ll take chaos theory first. This is the whole butterfly effect thing. It was found at some point that computer programs written to forecast the weather gave completely different results depending on how many decimal places the data input to them were calculated to. Given the very many decimal places involved before one hits the Planck length, Planck time and so forth, which amounts to the fixed “resolution” of the Universe, which can’t be calculated anyway because so many perfect instruments would be involved that they’d nudge the weather in a particular direction, there seems to be only a weak connection between cause and effect, and for all we know, as David Hume asserted, none at all. If science is supposed to be based only on what can be observed, cause and effect can’t be and therefore it’s problematic including it in science at all, which rather undermines the whole of science. That said, it does still seem that in principle cause and effect often operate deterministically. You can’t usually expect to jump off the roof of a skyscraper and not fall to your very probable death. Maybe the improbabilities are smoothed out by the arbitrary nature of the universe on a small scale. I don’t think chaos theory is a very promising reason to posit that things could have been different.

Quantum mechanics is another matter entirely. There are no hidden variables. That is, if a radioactive atom is observed, there is no way to predict when it will decay into an atom of another element and that isn’t just because we’re unable to observe processes going on at a sufficiently small scale, but because there simply is no causal chain involved at all. All that can be done is to predict that half of a sample of carbon-14, for example, will have decayed in 5 730 years, give or take forty years, and that prediction only approaches fifty percent with the increase of the size of the sample. However, these are acausal processes. There is absolutely no chain of events other than the formation of the atom leading up to its destruction. It just happens.

Hence there are two contrary factors involved in the nature of parallel universes. On the one hand, there is the causal chain stretching back to the Big Bang, and on the other there are acausal events associated with quantum events. The question then arises of whether the Big Bang itself, or its immediate aftermath, was strongly associated with such events. It could be that things have always been different or that all significant events in our own history can be traced back to quantum events after the beginning of the Universe. All that can be said confidently is that if a known chain of events can be traced back to a quantum event, there are parallel universes where this turned out differently.

A fairly trivial example is the issue of the discoveries of technetium/masurium and astatine/alabamine. The actual names of these elements are technetium and astatine. Neither have stable isotopes. The reason their names could have been different is that they weren’t discovered for sure when they were first apparently identified. In 1925, German chemists bombarded a mineral called columbite with an electron beam and appeared to detect a faint X-ray signature of what would be element 43. Later researchers, however, could not replicate this experiment and consequently, although it was named masurium, it was still considered undiscovered. If a greater number of atoms in the sample of this element had not decayed in the attempt to reproduce the result, element 43 would have been confirmed and would have been named masurium. Likewise, with alabamine, scientists at Alabama Polytechnic believed they had discovered the missing halogen which belonged under iodine in the periodic table in 1931, but their method was found to be invalid. The case of alabamine is slightly different, which I’ll go into in a moment. But because of the method of its discovery, there undoubtedly is a parallel universe in which technetium is called “masurium”. That’s a real place.

The case of astatine is slightly different. Astatine is only a couple of nucleons too heavy to be a stable element. Using the same rough and ready calculations as I did with carbon, for there to be a stable isotope of astatine the strong nuclear force would only have to be 0.08% stronger than it is. This may be the wrong figure but the principle is the same: it would only have to be a hairsbreadth stronger than it is “here” in our timeline for stable astatine to exist. In such a situation, polonium would also have a stable isotope and therefore would be less dangerous and would not have been used to poison Aleksandr Litvinenko. This, however, is a minor detail because probably it would just mean francium would’ve been used instead.

The two scenarios are therefore two different ways alternate histories could happen. In one, the Universe has been different since the Big Bang, astatine is a stable element and Litvinenko was poisoned using francium instead of polonium. In the other, its timeline and ours forked in 1925 and is probably practically identical to our own with the exception that technetium is called masurium.

This brings me to the Mandela Effect. Nowadays, most people seem to have reached the conclusion that the Mandela Effect is only accepted by cranks, and I would agree that there’s a lot of noise in the signal, but in the masurium/technetium example we have a real live Mandela Effect which is present in the scientific community that pivots on an acausal principle. This is inside the establishment, although it looks very different to a typical ME. For this reason, I will continue to maintain that parallel timelines are a valid explanation for some MEs. That’s it: that’s all I’m going to say about this for now because I know it’s generally considered crazy and you’re going to think I’ve gone to Nubicuculia if I go on.

There have been attempts to set up quantum lotteries. Although these are successful, as far as I know there are no serious lotteries using this principle. This is a pity, because if there were they’d amount to real forks in history set off by quantum events. As it stands, the only examples I can think of which involve genuine quantum forks other than masurium/technetium are very improbable, although there are guaranteed to be timelines where this happened. For instance, radioactivity was first discovered when Ernest Rutherford left a piece of the mineral pitchblende next to a photographic plate in a drawer and discovered it was blackened. If this hadn’t happened, radioactivity would have taken longer to be discovered. However, the only way in which that could have happened is if the number of atoms decaying was so small that it wasn’t enough to influence the emulsion on the plate, and considering the amount of substance involved, it’s very improbable. That said, somewhere out there such a timeline does exist. There’s presumably a timeline where radioactivity has yet to be discovered, which would leave a lot of mysteries about the Universe, such as how stars work or how old this planet is. There would be no radiotherapy, the Second World War would not have ended in the way it did, there would be no atomic batteries or nuclear power stations, no Cold War and so on. It is a fantastically improbable universe. But it does exist out there somewhere, and is a very different world. Even the people who live in it don’t understand it, because a big piece of the puzzle is missing. However, radioactivity can be discovered at any time. History is teetering on a knife edge in this world.

The question now arises of who we are. If a POD has occurred after our conception in any parallel universe, are we the same people? My ME explanation requires transworld identity, because I believe memories are transferred between universes when the brain is in an unusual state such as a stroke, seizure or coma. Transworld identity is the belief that an object can exist in more than one possible world, including the actual world (and here the world “actual” really just means “this” and “actual world” means “here”). The alternative theory is that counterparts exist in other possible worlds but that they’re not the same thing. David Lewis holds this, for example. It’s feasible that most people would hold that one is the same person if a POD takes place after conception, or perhaps birth, rather than before it. If they believe in the transmigration of souls, they would almost certainly hold that it doesn’t require a POD to take place that late because they would already claim that someone is the same person living a life in another time and place. If they also accepted that karma existed, different circumstances regarding conception might lead to that soul entering a different body and this could mean that the “same” person could be different in many ways in another possible world, being born in the Congo rather than Canada, in the rainforest rather than Vancouver, and so forth. This is someone else’s belief system rather than mine.

Even so, I do have something in common with people who believe in reincarnation: I don’t actually believe personal identity depends on karyotype. Here’s why. If it turns out that someone has a genetic disorder, they and the people close to them would tend to wish that they had never acquired that disorder rather than wishing they were someone else. These are two different things. Therefore, we don’t identify with our genes and our identity doesn’t depend on having been conceived in a particular way. Nor does it depend on the specific substance of our bodies, because if our parents, particularly our pregnant mothers, had eaten a different diet (such as the potatoes on one side of the field rather than the other, not miso instead of yeast extract or something), it wouldn’t make us different people unless it had a major influence on our development, and possibly not even then. What does that leave? There is no soul, so it isn’t that. Nor is it our genes. Nor is it the substance of our bodies. The answer, I think, is that we are socially defined, both passively and actively. In one sense we are the “software” running on the “hardware” of our bodies, although the metaphor of the brain as computer shouldn’t be pushed too far and it’s important to be aware that other parts of our bodies, such as the endocrine system and the nerves in our digestive system, also form a supervenience base for our psyches. It’s difficult to know how close our brains are to computers and how relevant this is to our identities. In another sense, we are externally defined. For instance, we have the legal concept of “next of kin”, which formalises a custom which already exists in social life: we are siblings, offspring, parents and so forth. Therefore, in a parallel universe where a child whose genetic makeup is rather different from this one, has a different temperament and so on, could still be the eldest daughter, have the same name, same birthdate and so forth, and is arguably the same person. In particular, she might not have the leukæmia which killed her in another universe, because at no point was that leukæmia something anyone in the family owned psychologically: it was a disease attacking her, an outsider enemy. I presume this is how many people with cancer approach their illness, but maybe I’m wrong. But that disease could be in her genome.

I don’t know enough detail about how ionising radiation interacts with DNA to be sure about this, and I should probably know more, but I would expect cosmic rays, which are nuclei and protons raining down onto Earth’s surface at near-light speed, to be to some extent the product of nuclear decay and to some degree interact with the molecules in question in such a way as to change the isotope of specific atoms. The existence of radiation in the environment on this planet, whether or not it results from human activity, would certainly be non-deterministic in nature, although the actual presence of that radiation is only technically not so. That is, it’s possible for a scenario as described above with Rutherford’s pitchblende failing to be sufficiently radioactive to influence his photographic plate to occur, but its probability is infinitesimal. Hence there is an element of pure luck involved in mutation which means that it is possible for minor phenotypical differences between members of the same species in parallel worlds to occur, though only to the extent that this doesn’t influence their fitness to survive, although this does also mean there are extinctions which occurred in one world but not another. However, there is another aspect to identity which suggests the “shadow people” I referred to in the title.

It’s widely known that ordinary human body cells each have two pairs of chromosomes which are reduced to one pair in gametes via meiosis:

Overview of Meiosis
20 June 2016
Own work
– slightly cropped

It should be noted that the four daughter nuclei in this process are complementary to each other. The one at the top is a perfect counterpart to the one at the bottom and the two in the middle are counterparts of each other. Therefore, for either of the gametes which led to the cell line associated with who we are, there is a complementary alternative. This means there are at least four possible versions of each of us, even assuming the copying process goes without a hitch, which incidentally it never does. For instance, for a White blue-eyed fair-haired child whose mother is White with brown eyes and dark hair and whose father is White with blue eyes and fair hair, there is another potential version who is perfectly complementary, and two more versions who are partly complementary, because different gametes united. These gametes will have existed at some point, and they might even produce a viable child in the case of fraternal twins. These complementary people probably do occasionally exist in the same world. I would estimate that this occurs in one pair of about 500 million fraternal twins. Since in a population of eight thousand million there are around 350 million twins, there’s an even chance that somewhere out there today, this situation exists, and there have probably been about six or seven such pairs in the whole of human history, which by the way emphasises the fact that there are a lot of people around today. But in any case, we all have these shadow people, which brings me to the illustration at the top of the post.

This is a fairly famous gender-swapped version of post-war Prime MInisters of the United Kingdom, which notably has only two men because there have only been two female PMs. The counterparts in question here would usually have different karyotypes. That is, if you are yourself XX, your shadow twin would be XY and therefore usually male. The main situation where they wouldn’t be, incidentally, is complete androgen insensitivity – this is not about trans issues at all right now. However, although we do tend to focus quite strongly on gender as part of identity, there would also be lots of other traits which would differ. We have two children, one of whom resembles one parent quite closely and the other of whom resembles the other. I presume this is because dominant traits from one gamete are more strongly expressed in one than the other. Their shadow twins would be the other way round, which means that they would look very like their siblings, just in a different birth order. Their eyes would also be a different colour. My own shadow twin would still have blue eyes, but also straighter hair. I say that, but the popularly understood traits said to be inherited by single alleles are often not, such as eye colour. There’s also another sex-related issue. Two of the intersex conditions are referred to as Klinefelter’s and Turner Syndrome. The former is XXY and the later just one X chromosome with no counterpart. These two conditions are therefore complementary and a Turner person’s shadow twin would be XXY and vice versa. There’s also chimerism. Some people would be reverse chimeras of their twins, for instance they would be largely cell line A with some of cell line B, but their shadow twin would be largely cell line B with some of cell line A.

It’s also true that every generation of a lineage produces only a quarter of these potential individuals. This means that there are also sixteen possible parents involved, and the number rapidly becomes extremely large. This brings home how unlikely it is that any of us were ever born. Just focussing on the perfect complements, the probability that every person in the world today was their shadow twin is of the order of four to the power of eight thousand million to one. Although this is very improbable, it’s far more likely than the situation I described with the discovery or otherwise of radioactivity.

At this point it becomes clear that there is an issue with the nature of probability. Rutherford’s discovery is genuinely probabilistic and acausal. It could “just happen”, and there’s no need for an explanation. It isn’t so clear that the shadow twin situation could simply happen because there definitely seems to be a deterministic thread running through the whole of meiosis and fertilisation. This raises the question of the nature of probability. Probability is sometimes seen as simply a measure of the frequency of occurrences, so for example half the time a coin comes up heads and half tails, so it has a 50/50 probability of coming up either way. This is an empirical approach, as it’s simply based on observation. The other approach is based on rational degree of belief. For all we know, a coin tossed on a particular occasion might come down heads or it might come down tails, and there is no known reason to prefer one outcome over the other. However, there is in fact a cause, each time, for it landing the way it does, presumably to do with how forcefully it was flipped, the angle, air currents and tiny differences between individual coins which make them slightly unfair. For instance, I believe it’s slightly more likely that a coin will land heads up because I think the tails side is slightly heavier and will tend to weigh the coin down, and I tested this once and found the coin I was tossing was heads up sixty-four times out of a hundred. This helps confirm the hypothesis but doesn’t prove it. Ultimately, there may be two kinds of probability, one deterministic and one not, but the deterministic version of probability could stem from the initial conditions of the Big Bang and therefore not be ultimately so. Incidentally, using possible worlds semantics makes it difficult to use certain terminology. For instance, the world “probably” then comes to mean “in most possible worlds”, in other words something like “usually”. This gets confusing when referring to the theory itself. For instance, I can’t say “most parallel universes have always been separate” because I would then be effectively saying “in most possible worlds, most possible worlds have always been separate”. It could even be that this leads to a contradiction which refutes the theory of parallel universes, and that’s pretty serious because it starts to look like proof for the existence of some kind of First Cause and supports theism or deism to a limited extent.

I am now going to make one of these odd-sounding statements. Namely, “it’s possible that shadow twins exist in other universes”. This could be expanded as saying “there are some possible worlds in which there are some possible worlds where there are shadow twins.” This sounds peculiar, and makes it sound like there are two levels of possible worlds, on whose higher level lies the idea that there are a vast number of arrays of further vast numbers of parallel universes. Using the “rational degree of belief” view of probability, this can be restated as “for all I know, there exist possible worlds where shadow twins exist”. If this is so, it’s possible to imagine the following situation. There is a parallel universe where every representative of a final generation of humans is their shadow twin. In fact there would be several. This uses the criterion of childlessness to select the set of people involved. There’s also the question of whichever cohort includes you. You have a shadow twin, and depending on whether you have descendants you are either in the final generation or one of its recent predecessors.

Getting back to the prime minister picture, these are not photographs of a common type of parallel universe. Not only would the individuals concerned look different besides their gender, and also probably have different personalities, but also these are photographs from a matriarchal society, and quite an odd one at that because the political system of the United Kingdom is otherwise very similar, with Eton, Oxbridge and so forth putting these people in the same positions. In reality, most of the people depicted in the picture would not have become Prime Minister at all because they would have different histories based on their gender. This picture asks us to believe that a woman, Winston Churchill’s shadow twin, would have become PM in 1940, only twenty-two years after Constance Markievicz, which is hard to imagine. Their lives would probably have been much more like those of their sisters, assuming they had any, than their lives in this world. The idea of shadow twins constitutes an interesting thought experiment regarding the nature of gender roles and the patriarchy.

Finally, I’m going to revisit the fringe theory of the Mandela Effect. If there really are shadow twins who are to some extent a sex- or gender-swapped version of oneself in parallel universes, this could sometimes have an interesting consequence which is similar to the idea of a soul of one gender in the body of another as an explanation for gender identity issues. My explanation for hardcore MEs is that individual experiences and memories occasionally get transferred into brains in parallel universes when the brain enters an unusual state. If this happened often enough with a shadow twin, the person concerned could conceivably end up with a different gender identity. However, this suggests that we all go around constantly thinking to ourselves something like “I am a man opening this door” or “I am a woman picking this apple”, when of course we do nothing of the sort. Also, it’s quite an outlandish explanation compared to something much simpler and more easily testable such as chimerism or CAG repeat sequences on the AR gene. Hence I’m going to put that out there, note its similarity to the dubious idea that there are not only souls but also that those souls are gendered, and acknowledge that believing in non-psychological explanations of MEs at all is widely considered dubious. But I do wonder sometimes.