Growing up, down and sideways, and George Clooney.

Ginkgo seedling 1
Ginkgo seedling 1 (Photo credit: Wikipedia)

Often a plant seed will end up in a place that is not particularly suitable for it. In particular it may end up in a gravel path or similar where there is little real soil. Or it may end up in sand which drains quickly and may, if near the sea, contain high amounts of salt.

In such an environment it may grow stunted or may be deformed. For instance Bonsai trees are kept in a small container and kept relative short of nutrients so that stay small and become gnarled and twisted. They may even have their roots trimmed.

Bonsai IMG 6396
Bonsai IMG 6396 (Photo credit: Wikipedia)

One can imagine a society of concerned individuals fighting against the sustained torture of the trees treated in such a manner, but strangely, I’ve never heard of one. Maybe it is because trees can’t scream?

All members of a species have the same genetic make up, the same genotype. All individuals grow in much the same way, to produce similar adult individuals. This is termed the phenotype.

Genetics diagram: Punnett square describing on...
Genetics diagram: Punnett square describing one of Mendel’s crosses, between parents that are heterozygous for the purple/white color alleles. Category:Punnett squares (Photo credit: Wikipedia)

There may be sexual dimorphism, where the female of the species differs from the male of the species, but in most ways, all members of one sex are pretty similar to one another. I am not too dissimilar from George Clooney. My wife is much like Angelina Jolie.

Of course individuals are not identical. I’m slightly taller than George, for instance. This difference can be genetic, or it may be environmental. My genes may be the cause of the difference, or maybe the environment when we were growing up has slightly affected our growth. Our good looks are almost certainly genetic.

English: George Clooney at the Toronto Interna...
English: George Clooney at the Toronto International Film Festival 2011. (Photo credit: Wikipedia)

Sometimes a plant grows in a particular way in one environment will look completely different in another environment. Also a young specimen of a plant may look different from a mature specimen of the same species. Lanceword (Pseudopanax crassifolius) has a juvenile form so different from the mature form that it was initially thought to be two different species.

The environmental effect on the phenotype or expressed shape can be seen in genetically identical twins. One would expect their phenotypes to be identical at all ages, however, while “identical twins” look very very similar there are detectable differences.

English: Comparison of typical zygote developm...
English: Comparison of typical zygote development in monozygotic identical and dizygotic twins. (Photo credit: Wikipedia)

For instance if one twin had suffered a serious illness at a critical stage of growth, then their adult sizes may be significantly different. If one twin had a rich diet and the other twin a restricted diet that also might affect their sizes and expectations of longevity. Scientists can tell a lot about the processes of growth and development by studying genetically identical twins.

A more subtle variation in the phenotype can be seen when populations are considered rather than individuals. A population of moths that lives on darker surfaces may tend to be darker in colour than the same species that lives on lighter surfaces. Since this effect happens slowly, over many generations, it appears to be a genetic change or shift. However this change to the genotype is at a lower level than the species as the lighter moths and the darker ones can interbreed.

A black-bodied peppered moth (Biston betularia...
A black-bodied peppered moth (Biston betularia f. carbonaria) in the Ahlenmoor, a hill moor in northern Lower Saxony, Germany. (Photo credit: Wikipedia)

Some plants look completely different if grown in different environments. The weed that grows in gravel may look completely different from the weed that grows a metre away in a more favourable environment. It’s as if a switch has been thrown which turns on a totally different way of “building” the plant, as it may well be something like that.

If the genome of the organisation is a “program” to “build” the plant, it is perfectly feasible that a lack of resources at an early stage in the plant’s life might well kick in a different path in the developmental process from the path that it would take if resources were abundant.

if (abundant_resources == true)

then build_good_version

else build_poor_version

This is a simple branching process in a computer program, but the process is almost certainly a lot more complex in real life. However the principle is sound, I believe.

English: Capsella bursa-pastoris, Brassicaceae...
English: Capsella bursa-pastoris, Brassicaceae, Shepherd’s Purse, flowers and fuits; Karlsruhe, Germany. The fresh aerial parts of the blooming plant are used in homeopathy as remedy: Capsella bursa-pastoris (Thaspi.) Deutsch: Capsella bursa-pastoris, Brassicaceae, Gewöhnliches Hirtentäschel, Blüten und Früchte; Karlsruhe, Deutschland. Die frischen, oberirdischen Teile der blühenden Pflanzen werden in der Homöopathie als Arzneimittel verwendet: Capsella bursa-pastoris (Thaspi.) (Photo credit: Wikipedia)

A simple iterative process can be used to generate complex shapes that look a lot like real plants. Minor changes to the process can cause significant changes to the end results. Tall thin shapes can morph into shorter bushier ones with a few changes to fixed numbers (constants) in the iterative process.

The phenotype of a plant of a particular species will be similar in all individuals. If an individual has leaves, stem, flowers of a particular sort then the phenotype of an individual in a different (eg poorer) environment, will most likely have similar parts, though some differences will be obvious.

POOR SOIL, DAMAGED BY ACCUMULATED SALT, IS EXA...
POOR SOIL, DAMAGED BY ACCUMULATED SALT, IS EXAMINED BY MEXICAN FARMER, GILBERTO BUITIERREZ BANAGA, NEAR MEXICALI… – NARA – 549083 (Photo credit: Wikipedia)

Maybe the stem into of being long and flexible, it may be a lot shorter and stiffer. Maybe the leaves will be a lot thicker and fleshier in the poor environment plant and may therefore be able to retain water which will be scarcer in the poor environment. Perhaps the flower will be more robust in the harsher environment.

One would expect such variations of phenotype, the poor and the rich, to be implicit in the genome if the wider environment is patchy, with areas of rich soil mixed up with areas where the soil is poor. Otherwise, the ability of the genome to be expressed in multiple ways would likely be bred out of the population, as nature always goes for the simpler rather than the more complex.

English: Edge of a ditch on a gravelly, lime-r...
English: Edge of a ditch on a gravelly, lime-rich soil at eastern Jutland, near the Kattegat. Dansk: Grøftekant på gruset, kalkrig jordbund i Djursland, nær Kattegatkysten. (Photo credit: Wikipedia)

The flexibility of the genome is something that the organism benefits from its whole life. For example there are some fish which live in groups of one male and several females. If the single male is killed by another fish, an octopus or a human being, one of the females will change sex and become male, taking the place of the missing male fish.

I’ll not speculate on how that happens in detail, but it seems that it must be implicit in the genome. The trigger is the absence of the male fish, but how the “genetic program” detects this, I don’t know, but once is does it transforms the largest female into a male, presumably by triggering changes in the genetic organs. That’s bound to be a complex process.

Male and female Gold Molly. Watch the Gonopodi...
Male and female Gold Molly. Watch the Gonopodium of the left fish. Its the male. Left is the female fish. (Photo credit: Wikipedia)

The central idea in this post is that the genome is much like a computer program and that the environmental influences are like the parameters to such a program. This is probably an over simplification in many ways, but by considering it as a program can explain why the same genome can produce such different individuals.

A computer program can be controlled by inputs while it is running, and similarly the environment can shape an organism while it is growing and after it has reached maturity. The idea of organism as computer controlled machine is not new, but I like to bring it out and have another look at it now and again.


Embed from Getty Images

The Solstice Again

The Sun rising over Stonehenge on the morning ...
The Sun rising over Stonehenge on the morning of the summer solstice (21st June 2005). A crowd of between 14,000 and 19,000 people greeted the sun as it rose at 04:58 BST. (Photo credit: Wikipedia)

Today is the summer solstice in the Southern Hemisphere, the time when the sun is furthest south in the sky and hence at its highest. From here on in, the days get shorter as we slide back towards winter.

In the Northern Hemisphere,  it is of course the winter solstice, and those living there can expect the days to lengthen, as they move towards summer. Today is the Northern Hemisphere’s shortest day.

English: Daisy Rock's "solstice gap"...
English: Daisy Rock’s “solstice gap”” This shows the gap in the rock along which the sunset is viewed on the longest day. (Photo credit: Wikipedia)

Seasonal lag means that we can look forward to the warmest months of the year after the solstice, and those unfortunate enough to live in the Northern Hemisphere can look forward to a couple of their coldest months before things start to warm up.

I read somewhere that winter months are the months when people tend to put on weight and this was attributed to the fact that in winter, in the coldest weather people tend to exercise less and eat more. The reduced exercise is attributed to the tendency to stay home in the warm, by the fireside to avoid the often hostile weather.

Brooklyn Museum - Fireside Companion - Platt P...
Brooklyn Museum – Fireside Companion – Platt Powell Ryder – overall (Photo credit: Wikipedia)

And the eating more is because, well, what else is there to do but eat, when you are trapped by the weather. Our ancestors used to use up all the reserves that they had laid up for just this occasion, the hams and preserves, dried fruit and root vegetables and so on.

When the summer solstice happens, the weather is warmer and better, so people can get out an exercise, and, for our ancestors at least, agriculture kept them on the move, and the aim was to replenish the stores for the winter months, hence an emphasis on growing rather than eating. Besides, most crops would not be ready for harvesting.


Embed from Getty Images

The winter solstice is one candidate for the start of the year. It marks a definite point in the cycle of the year. It’s after the solstice (a few months after the solstice) that things start growing again. The summer solstice is probably not a good choice as things are humming along then, ploughing and planting, growing and nurturing so it doesn’t really fit as the start of the year.

The spring or vernal equinox falls in March, around the 21st in the Northern Hemisphere. This is also a candidate for the start of the year, but to my mind, it is too late. Winter is tailing off at that time, things are starting to grow and because of the seasonal lag, it’s the start of spring. The year, are I see it, is already under way.


Embed from Getty Images

Interestingly our fiscal year ends on 31st March. This is the date used by individuals to account for tax obligations. In many countries using the Gregorian calendar, the fiscal year ends on 31st December and almost aligns with the (winter) solstice based year. Other countries which use other calendars have fiscal years which relate to the local calendar.

As I have said the summer solstice in the Southern Hemisphere falls on 21st December (in most years). It is an astronomical point in time, not a whole day and can happen on 20th December. In the decade from 2010 to 2020 it falls on the 20th on three occasions.


Embed from Getty Images

The summer solstice, Christmas Day, and the official 1st January New Year Day all fall within just over a week of each other. There is good reason to suspect historical links between these days, and there is much debate on the actual historical relationship between these events.

It is often said that early Christians adopted the winter solstice in the Northern Hemisphere, to squeeze out or replace a pagan celebration at that time. This may or may not be the case (or it may be partially true), but what is evident is that many cultures outside of the Tropics of Cancer and Capricorn celebrate a festival at around the time of the solstice.

World map with the intertropical zone highligh...
World map with the intertropical zone highlighted in red. (Photo credit: Wikipedia)

Between the two Tropics the sun is overhead twice in a year while the sun reaches a southerly point at the time of the southern solstice (winter in the north and summer in the south) and a northerly point at the time of the northern solstice, the hottest time occurs when the sun is overhead. This divides the year into unequal parts in these latitudes.

The climate of these regions is dependant on local conditions, such as whether or not the region is close to an ocean or is in the middle of a continent, and many tropical areas have wet and dry seasons, typically of unequal extents. One example know to many people outside the tropics is the monsoon season when a regions rainfall may predominantly happen.


Embed from Getty Images

On the Arctic and Antarctic circle, at the solstices the sun just grazes the horizon at the summer solstice and the day lasts 24 hours. At the winter solstices the sun just barely reaches the horizon and the night lasts 24 hours. Closer to the poles the number of sunless days or days with the sum always above the horizon increase. At the poles the sun is below the horizon for three months and above it for three months. (I hope this is correct. I did research this a little, but I am not 100% sure).

Interestingly, I learnt recently that the sunset will continue to become later for the next few weeks. The reason for this according to the linked article is because we have tied our clocks to 24 hours exactly and the day is not exactly 24 hours long. Not only is it not exactly 24 hours, but its length varies during the year. In Wellington the sunset goes out to around 3 minutes to 9 and doesn’t dip below that time until 7th January 2015.

English: Sunrise at Winter Solstice (December ...
English: Sunrise at Winter Solstice (December 21, 2006 at 8 a.m.) as viewed through the doorway half way up Maiden Tower (Photo credit: Wikipedia)

(December data here, January data here).

While looking up these numbers I noticed that the day length in Auckland is nearly half an hour shorter up there. Also sunset is about a quarter of an hour later in Wellington meaning that when the summer weather finally arrives we will have an extra 14 minutes to enjoy the balmy evenings. That’s yet another reason to prefer Wellington over Auckland! We have more time to celebrate the solstice.

English: Wellington Harbour (New Zealand) view
English: Wellington Harbour (New Zealand) view (Photo credit: Wikipedia)

[Darn! I completed this on Monday but forgot to publish it. Better late than never, I guess!]

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.


Embed from Getty Images

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.


Embed from Getty Images

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.


Embed from Getty Images

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.


Embed from Getty Images

 

 

How do I get from A to B?

Ngaio Tree; Português: Mulateira. Portimão, Po...
Ngaio Tree; Português: Mulateira. Portimão, Portugal. (Photo credit: Wikipedia)

We had reason to visit another suburb today. It wasn’t until I was sitting waiting for some traffic lights to change that I thought about how I was navigating from home to destination.

We just got into the car and drove there. I didn’t consider the route in advance, and it seemed that I just pointed the car and we got there. Obviously I knew the way, as we had been there or through there a number of times in the past. But I didn’t have the destination in mind from start to finish, at least not consciously. I’m not sure that I had it in the forefront of my mind at all.

English: Driving Route 40 to El Chalten was pu...
English: Driving Route 40 to El Chalten was pure driving pleasure. (Photo credit: Wikipedia)

I knew that it was in that direction though, and that did not leave a lot of route options. I did have a general feeling that I should go south, in this instance and that really only leaves two options, the back road, or the motorway. The back road is a lot prettier!

I made the choice to take the back road but it was not, as I said, at the forefront of my mind, as I was doing other things at the time, like finding my keys, my phone, my wallet and these things occupied the forefront, while the decision about which route to take was more background.

English: Mind the dip Looking down the road is...
English: Mind the dip Looking down the road is a hidden dip. The farmers are busy with the harvest while the weather stays dry. (Photo credit: Wikipedia)

So the route was chosen more of less in the background, but not subconsciously. Much the same process happened on the way there, and at each junction or turning point, I didn’t have to consider at the front of my mind which direction I should drive. I just did it. Some part of my mind knew that to get to our destination I had to turn right, or go straight on or whatever.

This is good because the front of my mind was doing the driving, keeping the car on line, signalling, accelerating or braking, keeping us safe on the road. Except that it wasn’t right at the front mind, since I was also talking to my wife about various things. Christmas things from memory.

English: Two motorcycle trailing off the brake...
English: Two motorcycle trailing off the brakes through Tooele Turn at Miller Motorsports Park. Rider on the white bike is Warren Rose, Rider of the green bike is Dave Palazzolo (Photo credit: Wikipedia)

I’ve been driving for many years and I’m confident that if needed the driving part of my mind can instantly oust the things currently in my mind should the unexpected happen. Many year ago, when I used to smoke, I was driving with a friend and an emergency happened. When it was over I realised that I was no longer holding my cigarette. Meantime my friend was scrabbling between his legs where my cigarette had ended up when the driving part of my mind grabbed precedence and the cigarette holding part was temporarily ousted.

The route planning part of my mind would not suddenly get control like that, fortunately. That would be highly dangerous. I could if I had wanted have brought the route planning part of my mind to the front, but it wouldn’t say much except “turn left at the next junction”.

Turn Left, Turn Right
Turn Left, Turn Right (Photo credit: Wikipedia)

I have on occasion made a navigating mistake. I’m going to A and the route to B is the same in part. Suddenly I realise that I have missed a junction and will have to backtrack. It seems that the route finding part of my mind spends much of the time dozing and checks in only infrequently, sometimes missing the turning point or ritually following a more usual route.

It also seems that the information it keeps is like an instruction to take an action at each decision point rather than the whole route from home to destination as well as a general direction, less well specified. GPS guidance systems seem to work this way too in that they instruct you to take an action at each junction without setting out the whole route each time.


Embed from Getty Images

The model of the mind that I’ve used above, of various parts of the mind at various levels of “forefrontness” or consciousness is nothing new. The need to make a part of the mind the one at the top of the conscious levels, suddenly as a result of a danger, or selectively by choice, as in route following reminds me of the way that computer programs

Computers have several methods for navigating through programs and reacting to things that happen when they are running. One big part is called “handling errors”. Dividing by zero is an error and if the computer reaches a point where it has to divide by zero something needs to be done. The program can report the error and gracefully stop, or it can take some action to fix the error and then carry on.

English: A Texas Instruments TI-86 graphing ca...
English: A Texas Instruments TI-86 graphing calculator displaying an error message, indicating that the user or a running program has attempted to divide by zero. (Photo credit: Wikipedia)

Computers handle error by means of “interrupts”. Whether the errors is software (eg divide by zero) or caused by hardware connected to it (eg input/output errors) the computer stops what it is doing and runs a bit of program that handle the errors by sending a message or fixing things up. The bits of program that were running are suspended and after the error is handled the bits that were running may be given back control.

The mind seems to work in a similar way. When an emergency arises the current part of the mind that is at the forefront gets suspended and the emergency is handled by another part of the mind. A pedestrian steps into the road and you react by standing on the brakes “before you know it” as the saying goes. As soon as the emergency is over, the conscious mind takes over again.

a short .gif of the Taiwanese animated pedestr...
a short .gif of the Taiwanese animated pedestrian road crossing sign (Photo credit: Wikipedia)

You do indeed react “before you know it”, one might say instinctively. But humans have not been driving cars for much more than one hundred years, so it appears that the reaction is not instinctive in itself, but is an instinctive reaction to a danger that has been learned. We seem to have this fast reaction to events which is instinctively based but can be applied to learned situations, which is much more flexible than hard-wired instincts would be.

So, pondering on how I get from A to B has led me to conjecture that there are parts of the mind which are forefront in our minds and other parts which are not directly in the forefront but which can be brought to the forefront in an instant, when an event happens. It is evident that these parts are only partially backgrounded as the mind as a whole has some aware of the location at the time, but they do act semi-autonomously, that is until the pedestrian steps out onto the roadway.


Embed from Getty Images

Evidently there are parts of the mind that are less foregrounded and more backgrounded. When the part of the mind that is concerned with driving wants to signal or change gear, another part of the mind which controls the arms and legs wakes up and make the limbs move as needed.

I’ve spoken above as if all these different levels are discrete states, but I think it more likely that is a continuum from the foreground of the mind to the background or a least the series of levels of consciousness are close enough togerther to appear so.  The mind is a complex and wonderful thing.


Embed from Getty Images

[Comment: After finishing this post I went looking for other discussions of the same topic. I first found this Wikipedia article which has the issues mentioned in the article’s header. Interestingly the implication in the article is that there is a single level of consciousness at any one time. This I do not agree with. Another article I found was a little better, I feel, but only because it acknowledges that several levels may be active at the same time, but divides them into three levels with well defined scopes. I feel that it is a lot more complex than that, with all sorts of sections of the mind at all sorts of levels being active at the same time. Neither article deals with the issue of one section of the mind apparently seizing the highest level when an external event triggers it.]

Dis-Continuum

English: The Clump looking from the Redhouse
English: The Clump looking from the Redhouse (Photo credit: Wikipedia)

Where ever one looks, things mostly seem to be in lumps or clumps of matter. We live on a lump of matter, one of a number of lumps of matter orbiting an even bigger lump of matter. We look into the sky when the bigger lump of matter is conveniently on the other side of our lump of matter and we see evidence of other lumps of matter similar to the lump of matter that our lump of matter orbits.

We see stars, in short, which poetically speaking float in a void empty of matter. We can see that these stars are not evenly distributed and that they gather together in clumps which we call galaxies. Actually stars seem to clump together in smaller clumps such as the Local Cluster of a dozen or so stars, and most galaxies have arms or other features that show structure at all levels.

Ancient Galaxy Cluster Still Producing Stars
Ancient Galaxy Cluster Still Producing Stars (Photo credit: Wikipedia)

The galaxies, which we can see between the much closer stars of our own galaxy, also appear to be clustered together in clumps, and the clumps seem to be clumped together. Of course, the ultimate clump is the Universe itself, but at all levels the Universe appears to have structure, to be organised, to be formed of lumps and clumps, variously shaped into loops, whorls, sheets, arms, rings, bubbles, and so on.

OK, but in the other direction, towards the smaller rather than the larger, our planet has various systems, weather, orogenic, natural, social and evolutionary. All sorts of systems at all levels, from global scope to the scope of the smallest element.


Embed from Getty Images

In other personal worlds, below the level our interactions with our families, we have all the systems that make up our own bodies. The system that circulates our blood, the system that processes our food, the system that maintains our multiple systems in a state homeostasis.

That is, not a steady state, but a state where all the individual systems self-adjust so that the larger system does not descend into a state of chaos, leading to a disruption of the larger whole. Death.

The main pathways of metabolism in humans, sho...
The main pathways of metabolism in humans, showing all metabolites that account for >1% of an excreted dose. ;Legend PNU-142300, accounts for ~10% of excreted dose at PNU-142586, accounts for ~45% of excreted dose at steady state PNU-173558, accounts for ~3.3% of excreted dose at steady state (Photo credit: Wikipedia)

By and large most systems in our environment are made up of molecules, which are in turn made up of atoms. Atoms are a convenient stopping point on the scale from very large to very small. They are pretty “well defined”, in that they are a very strong concept.

Atoms are rarely found solo. They are sociable critters. They form relationships with other atoms, but some atoms are more sociable than others, forming multiple bonds with other atoms. Some are more promiscuous than others, changing partners frequently.


Embed from Getty Images

These relationships are called molecules, and range from simple to complex, containing from two or three atoms, to millions of atoms. The really large molecules can be broken down to smaller sub-molecules which are linked repeatedly to make up the complex molecules.

To rise higher up the scale for a moment, these molecules, large and small are organised into cells, which are essentially factories for making identical or nearly identical copies of themselves. The differences are necessary to make cells into muscles or organs and other functional features, and cells that make bones and sinews and other structural parts of a body.

A section of DNA; the sequence of the plate-li...
A section of DNA; the sequence of the plate-like units (nucleotides) in the center carries information. (Photo credit: Wikipedia)

As I said, atoms are a convenient stopping point. Every atom of an element is identical at least in its base state. It may lose or gain electrons in a “relationship” or molecule, but basically it is the same as any other element of the same sort.

Each atom consists of a nucleus and surrounding electrons, a model which some people liken to a solar system. There are similarities, but there are also differences (which I won’t go into in this post). The nucleus consists a mix of protons and neutrons. While the number neutrons may vary, they don’t significantly affect the chemical properties of the atom, which makes all atoms of an element effectively the same.

An early, outdated representation of an atom, ...
An early, outdated representation of an atom, with nucleus and electrons described as well-localized particles on well-localized orbits. (Photo credit: Wikipedia)

Each component of an atom is made up of smaller particles called “elementary” particles, although they may not be fundamentally elementary. At this level we reach the blurry level of quantum physics where a particle has an imprecise definition and an imprecise location in macroscopic terms.

Having travelled from the largest to the smallest, I’m now going to talk mathematics. I’ll link back to physics at the end.

Nucleus
Nucleus (Photo credit: Wikipedia)

We are all familiar with counting. One, two, three and so on. These concepts are the atoms of the mathematical world. They can be built up into complex structures, much like atoms can be built into molecules, organelles, cells, tissues and organs. (The analogy is far from perfect. I can think of several ways that it breaks down).

Below the “atomic” level of the integers is the “elementary” level of the rational numbers, what most people would recognise as fractions. Interestingly between any two rational numbers, you can find other rational numbers. These are very roughly equivalent to the elementary particles. Very roughly.

Half of the Hadron Calorimeter
Half of the Hadron Calorimeter (Photo credit: Wikipedia)

One might think that these would exhaust the list of types of numbers, but below (in a sense) the rational numbers is the level of the real numbers. While many of the real numbers are also rational numbers, the majority of the real numbers ate not rational numbers.

The level of the real numbers is also known as the level of the continuum. A continuum implies a line has no gaps, as in a line drawn with a pencil. If the line is made up of dots, no matter how small, it doesn’t represent a continuum.

Qunatum dots delivered by ccp
Qunatum dots delivered by ccp (Photo credit: Wikipedia)

A line made up of atoms is not a continuum, nor is a line of elementary particles. While scientists have found ever more fundamental particles, the line has apparently ended with quarks. Quantum physics seems to indicate that nature, at the lowest level, is discrete, or, to loop back to the start of this post, lumpy. There doesn’t seem to be a level of the continuum in nature.

That leaves us with two options. Either there is no level of the continuum in nature and nature is fundamentally lumpy, or the apparent indication of quantum physics that nature is lumpy is wrong.

Pineapple Lumps (240g size)
Pineapple Lumps (240g size) (Photo credit: Wikipedia)

It’s hard to believe that a lumpy universe would permit the concept of the continuum. If the nature of things is discrete, it’s hard to see how one could consider a smooth continuous thing. It’s like considering chess, which fundamentally defines a discontinuous world, where a playing piece is in a particular square and a square contains a playing piece or not.

It’s a weak argument, but the fact that we can conceive the concept of a continuum hints that the universe may be fundamentally continuous, in spite of quantum physics’ indications that it is not continuous.


Embed from Getty Images