Monday, April 26, 2010

Membrains, rocks and spiders

Real living things have a membrane-bound existence, with the membrane separating the content of self from its complement, non-self. Though its a permeable membrane.

Innovative fictional living things often do not lead a membrane-bound existence - in fact surprisingly often not. Think of SkyNet from the Terminator movies - spontaneously coming into existence when the defense network achieved consciousness - a single amorphous network-based life form with no clear physical boundary between self and not self.

Or Fred Hoyle's Black Cloud.

Or the machine intelligences of the Matrix.

Or HAL, distributed over the space-ship subsystems in 2001

Star Trek had its share of weird distributed cloud-like intelligences.

But I can't think of any real substantial living thing that does not lead a membrane bound existence.

Furthermore, not only physically, but psychologically we can establish our existence as a distinct individual self, only if there is some abstract membrane that distinguishes....opinions we hold from those we do not, tastes that are ours from those that are not, actions that we would perform from those we would reject on moral grounds. So that our sentient existence is also a kind of membrane-bound existence. And I suppose with the points on the membrane boundary itself being indeterminate....."do I like that or do I not - I am really not sure ?" - these are the boundary layer opinions, tastes, moral actions.

Hence the sense of disorientation and bleakness that one experiences after a major failure - say, a betrayal of one's moral code - it is like an existential infection or poisoning - we have allowed some foreign action to cross the membrane between self and not-self. That boundary between self and not self is redrawn and weakened ever so slightly, every time some action or event or run of misfortune crosses it.

Hence also perhaps the psychopathic behaviour generally attributed to distributed sentiences such as HAL , SkyNet, etc - these non-membrane-bound beings ultimately have no vested interest in choosing to do some actions but rejecting others - as we do because these distinctions define us as "selves". In a sense these machines have no existence as selves in the way we membrane-bound beings do, so it is not possible for them to truly act morally - ultimately they will execute a moral random walk. ("They will execute a random walk....?" - it is not even clear there is a "them" to which the "they" in that phrase refers ! HAL and Skynet are stochastic processes that sometimes appear to act in an intelligent way....despite claims to the contrary from some AI / computational neuroscience hard-liners, we true membrane-bound living things are not that ! We have membrains !)

Its useful to remember that we are membrane-bound sentiences, at 5:00am in the morning when you wake up and "see the darkness" (ref Johnny Cash) know the kind of occasional 5:00am thing ....the general and very-clearly-in-the-darkness-of-pre-dawn fragility of one's prospects and position, the probable fundamental absurdity of much of the day to come, the real silliness of some plan that seemed extremely cunning as you thought about it drifting off to sleep the night before...

The thing is - for a membrane-bound sentience, whats outside the membrane doesn't really have to make sense, in fact lets face it , it definitely doesn't make sense. Our only job is to keep the bilge-pumps working across that membrane, pumping out whatever chaos we can, and pumping in whatever of the orderly good stuff we can - just making a difference at the margin, maintaining just enough of that voltage difference across the membrane which keeps things ticking along for ourselves and our pet rock and the big spider which lives a good life snug and dry in our mailbox thanks to us.

And its no mystery that we wake up today at 5:00am disappointed that we have no more idea of a decent solution to our problems, than we had at 5:00am yesterday morning, or the morning before.....despite all the progress we thought we made on each of those days, and how confident and snug we felt as we drifted off to sleep the night before, with several cunning plans and rationlisations seemingly securely in hand. That's because there is no final solution, and even though we pump out the bilges each day, the membrane leaks so we'll need to pump them out again tomorrow.

Whereas for a non-membrane-bound sentience - like HAL, or Skynet - or maybe slightly ourselves when we are younger and the world is also the oyster and we still have tidy minds that expect things to make sense - where there is less or no clear distinction between self and non-self, then either everything has to make sense and be resolved, or nothing can be - potentially the whole world has to be pumped dry and cleaned up and made to make sense, because there is a less well defined existential membrane available to distinguish self from non-self and other-selves, but which when stronger allows us to just concentrate on the limited task of pumping up just enough membrane voltage that we need and no more; keeping a bit of the chaos out and the order in, just enough for ourselves and a rock and a spider or two.

Monday, April 12, 2010

Avalanche Thermodynamics

Some days, if there's just the right resonance, you can, just for a little while, be better than you really are - like an electron quantum tunneling through an energy barrier it can't climb over, you can tunnel through the personality barrier, even if only for one day.

Friday late afternoon 9th April, there wasn't a breath of wind on lake Wanaka. I checked - through squinting intensely peering eyes in the twilight I observe the tops of the tallest poplar trees : the most flimsy autumn nearly-fallen, most delicate leaves on the very tallest tips do not move a millimetre. Stillness in a vast space like that has a kind of re-scaling effect - the lake becomes an intimate pond, everything seems within reach. (Wind is like a 5th dimension, another degree of separation between things - between people so they can't hear each other; making journeys longer, harder, rougher, bumpier. And usually scaling along with the other dimensions (big things like lakes have big winds, small things like glasses of water have tiny winds) - so everything seems oddly sort of closer together, in a big space, when there is no wind).

What luck ! - a big golden delicious friendly autumn anti-cyclone over the whole country - blue and gold skies, no wind. And the autumn sun never heats things up enough to kick off the huge convections and sea breeze fronts of summer, so the wind doesn't blow up strong and annoying in the late morning and afternoon like it does in the heat of January, in central Otago.

Adele and I had been quietly scheming for months to walk up the Rob Roy glacier track, from the Matukituki valley, just the two of us, the first time out and about without children for 18 years ! - thanks to a relative's visit and a little bit of begging (for three days children-and-house-sitting) and the miracle of my being organised enough to successfully apply for an annual leave day for the Friday (application made two whole days before the trip) ! That track has been a family tradition for nearly ten years but....Adele had never quite made it onto any of the various permutations of kids, grandmother and myself that had previously walked it (in retrospect...mistakes were made...) - so time to make amends. The first time I went up there was in 2000, with my daughter than aged 6 and I had no idea that at the end of the track was a stunning hanging glacier, above a beautiful alpine valley - I had just chosen that wiggly line off the map as it was about the right length and gradient for us - and not *too* wiggly. About an hour into the walk we became alarmed by rumbling sounds - almost making us turn back. After pausing for awhile to see what events these rumbles might presage - avalanches in our general direction ? time portals opening ? volcanoes erupting ? - and observing nothing, we continued on to the end and were stunned to find the cause - a deep blue hanging glacier far above us and the valley, large bits of which were falling off and crashing down into the valley's base, every 20 minutes or so.

So the plan was - drive up from Dunedin on Friday, walk the walk Saturday, drive back Sunday.

We left as early as we could on Friday morning - about 10:00am. There's nothing quite as good as setting sail on the first day of (what one feels, rightly or wrongly, is a deserved) holiday, with a holiday-beginner's determination and confidence that no stone of relishment, small delight or amusement will be left unturned - no opportunity for a laugh or potential point of interest will be overlooked. So first stop was Milton -we never ever usually stop there, but this was now an official road-trip and diversions were our business - for a take-away coffee on the sun-drenched terraces of the war memorial and a chat to a friendly local who said gidday - turns out she was raised on the slopes of Sandymount on the Otago peninsula, when there was a school still there, and she and Adele talked about "The White Masai" and other books they both had read, and she corrected us on a point of local history - Larnach of Larnach's castle fame did not benefit from the estate of his wife's family as much as we had thought, since it seems her step-father siphoned most of it off. And a look through the antique shops - Adele recognised the proprietor of one of them as a regular at the auctions in Dunedin. I guess, buy in town and retail from low-rent premises in small towns on the main highways North and South from Dunedin, with lots of wheel-traffic (the small towns just to the North of Dunedin also boast well appointed antique retailers - Milton is on the road south). (Thats something that always starts to unsettle me on a holiday - seeing the ingenuity with which other people are making a living, and contrasting it with my own plodding, and so the charter of my holiday starts to unravel and existential anxiety returns...but not today, I'm tunneling the barrier...)

So then on , turn right , and to Lawrence - we *do* always stop there, but usually just for the donut from the corner store. This time we stroll as well.

Next stop Alexandra (Roxborough is skipped on the way out but we stop for autumn fruit on the way back - peaches, plums, apples) - the road trip is official now, we buy a couple of CD's to play on the CD player (having a car that has one is still a novelty !) : Johnny Cash ; Sinead OConner. Plus a new gold pan (and a small shovel) as I forgot to pack my rusty old pan and on the return journey I want to try a spot I noticed a few weeks back at Beaumont, on the Clutha - a rocky river reef with potholes in it is exposed as the river is low and I wonder if the potholes may tend to trap heavier sediment - e.g. iron and gold (yep - later I do get some very small gold flakes, and lots of black iron sands, from the sediment at the bottom of one of these - after much very patient swirling and eluting and dissecting out the small gold flakes with a sharp pointy kitchen knife back at home). And stroll and admire the cedars and the old black and white photos of the long gone arbor day crowd who planted them, of which even the little kids must all be long gone now.

On again - to Cromwell, another stop to enjoy the autumn sun and then the final leg, skirting along the valley to Wanaka. We finally arrive at 5:00pm - thats a 7 hour journey but apparently some people can do it in 3.5, though I've never done better than 5. But in time to snag a good motel room before the "No Vacancy" sign goes up. We hurry to get down to the lake to walk in the sun just before it dips behind the mountains. Friday late afternoon 9th April, there wasn't a breath of wind on lake Wanaka...

Well Saturday was a Perfect Day, a Perfect-Day-and-I'm-Glad-I-Spent-It-With-You type of Perfect Day, blue sky and gold sun and still still. Away by 9:00am and off up the road to the Matukituki valley, past Mt Roy (thats another really good day walk but a bit of a gut buster - but the view at the top is incredible. Do it on a hot day, take plenty of water, and when you get back to the bottom, put your foot down and get to the lake as fast as you can and dive in before you cool off. Aaahhhhh.), onto the gravel road and through all the fords carefully - its the driest I've ever seen , only one had water flowing to wet the tyres.

I've got some new advanced technology for our walks now - a butane canister and a little shiny metal tripod thing that looks like some small and mischievous critter out of the Transformers movies - after only half an hour on the job of walking up that track we set down on a grassy spot overlooking the river for a totally undeserved rest and I masterfully if not majestically, wrangle the technology and boil the billy, for a cup of coffee - luxury !

There have been some big changes on the Rob Roy glacier track since I was there last - but not where you would expect ! The ridiculously dangerous looking landslides - the ones which look like somebody just pressed "pause" at a particularly exciting and fast moving part of their descent - such as the one with what looks like a completely unsupported whole floor of a large building projecting out into space over your head without any visible means of support - "don't whisper too loud or it will come down" - are exactly as they have been for the past 10 years, presumably protected by some sort of spatio-temporal anomaly.

Yet there are other parts where something very scary indeed has happened to quite small insignificant streams - in one section a torrent of assorted boulders has ripped across the track and through the trees since I was last there - an assortment of all shapes and sizes - from huge down to....large, fairly big, big, sizable, soccer-ball size, tennis-ball size, golf-ball size - and quite a few marbles. And if you pause to look, its not just chaos - there are some intriguing patterns, like the dry-stone-wall we notice by the side of the track. We look carefully and can see what has happened - the wall runs along a line of 4 trees, with a few metres between each tree. The gaps have been large enough to let most of the rocks through - until finally a metres-long boulder got stuck at an angle - still leaving gaps but smaller now, so the filter can now trap metre sized boulders - and the metre boulders leave smaller gaps again, so the wall will start to collect half-metre boulders, etc etc - each new addition caught by this filter further alters the properties of the filter, so stacking up what looks like an orderly straight wall of boulders , size-sorted with the biggest at the bottom. And then the occasional stone placed apparently by hand at the top of the wall, or some other unlikely place, where you feel sure it should have rolled off, or could not have got there by chance in the first place - but you can imagine how these represent the peak of the energy of the flood - these placements have had just the right amount of energy to make it to where they are and no further - and probably at the peak of the flood, so nothing more with that peak energy came through to knock them off. And then also, in such a torrent, the total energy will be distributed thoroughly among the millions of rocks, so that it will be able to explore the unlikeliest destinations and architectural constructions - just because there is almost certainly at least one rock with just the right amount of energy, to end up placed *just so*" at the tippety top of wherever you might care to nominate. In thermodynamic terms, the "temperature" of that rock fall was probably quite high.

(Its actually interesting to think about an avalanche like that in thermodynamic terms, and consider defining the "temperature" of the fall, in the sense of the distribution of the energies of all of the millions of rocks coming down. To recap on what "temperature" is - consider your cup of tea - the kinetic energies of all of the tea molecules in there, at any instant of time, ranges from zero for a very few that are stationary (just for that instant, by chance), up through lots in the mid range, on up to a few very energetic molecules moving very fast. And the hotter your cup of tea, the more that the tea molecules spread themselves out to occupy all of the energy levels available. "Temperature" is a parameter that describes the shape of this distribution, of how many tea molecules there are at each level of energy. Applying that to the rock slide, and thinking about the distribution of energy levels of all of the rocks, we must have a similar situation - at any instant there will be a very few rocks temporarily stationary, with zero kinetic energy (e.g. they have just collided and bounced back off another larger rock) ; many in the midrange, on up to (presumably) a very few very energetic rocks. So there should be an "avalanche temperature", in the sense of one or more parameters that describe the shape of this distribution of how many rocks there are at each energy level. Maybe the most interesting (and frightening) prediction of that thermodynamic way of thinking, is that there will be the very occasional extremely energetic boulder, heavy and moving very fast - much faster than the group average velocity of the avalanche as a whole - and the higher the "temperature" of the avalanche, the more of these there will be - though even in a "low temperature" avalanche, very occasionally you will see one. Maybe this accounts for the peculiar danger and almost bizarre unpredictability of mountain floods and avalanches - where people are swept away and killed by apparently docile babbling brooks that suddenly run amok after a bit of rain, in every-day spots you and the kids have been to lots of times : the idea is that we are actually here experiencing the micro-behaviour of large assemblies that usually you have to be the size of a molecule to experience - it is as though we have been shrunk down to the size of a tea molecule and experience directly the energies and collisions and unpredictability of individual tea molecules - rather than the smoothed out average behaviour we see from our large-scale vantage point. Hmmm - the thermodynamics of rocks slides - must look into it...).

Another change on the track is that there's some serious upgrading going on, using a small digger evidently dropped in by helicopter - looks like its become very popular in the last 10 years and all those feet are gradually wearing down the track base leaving a tangle of exposed tree root trip-wires - so its being widened and built back up, and in some places shifted.

Its a pleasant walk - Adele doesn't find it too bad, and I'm tunneling the barrier, being better than I really am - relishing the time, not thinking about work or wishing for something better or remembering something worse...

The most amazing change is up the top at the end of the track. What used to be a wide fairly flat grassy beautiful alpine meadow, is now devastation, like a bomb has gone off in a quarry pit. (We later learn this was all caused by a night's rain only two weeks ago - the Saturday night of the Warbirds-over-Wanaka weekend). We can see how high the water level was via debris caught in the lower branches of bushes, and flattened grasses. Amazingly the long-drop dunny they have up there survived - though apparently they had to dig it out, it was half buried. (Thank goodness it wasn't flushed in the process !).

(I reflect that if I had been caught out up there in the rain and had to pick the safest place to pitch my tent, I would have pitched it up the top in that meadow, right under the bomb. Yet further down where the creek roars and foams and looks ferocious would have been , counter intuitively, much safer. Stay well away from mountains when its raining ! Else you will likely experience thermodynamic rescaling, and what its like to be a tea molecule in a hot cuppa)

On the way back we did a reckless river crossing of the Matukituki river rather than go over the swing bridge, just for the hell of it. It was really a bit too deep, we shouldn't have done that.

It was the best day I've had in years.

Tuesday, April 6, 2010

Groups and Symmetry, Operators and Morphisms

We have repeatedly stressed the over-arching role of symmetry in modern physics. The systematic study of symmetry falls under the heading of "group theory" for the mathematician.

- John D Barrow, "Theories of Everything" (1990 - since updated and re-released as "New Theories of Everything")

I've been trying for awhile to understand this connection between groups and symmetry.

Mathematical groups were (and still are) (sadly) blandly (i.e. axiomatically) defined in textbooks, as consisting of a set of objects, and a binary operation that combines pairs of these objects to make product objects also in the set - for example the objects could be numbers and the operation could be addition : 2 + 3 = 5. One of the elements of the set is a distinguished one, called the identity - for example, "0" ; and for each element there is a unique partner called its inverse, and when these two are combined they yield the identity. For example 3 + -3 = 0. And the associative law holds when we write statements that combine in turn 3 or more objects. For example 2 + 3 + 4 = (2 + 3) + 4 = 2 + (3 + 4).

While seeming plausible if somewhat pointless, as a conceptual cartoon highlighting some basic properties shared by addition, multiplication and similar algebraic processes, there was no hint that these "groups" had anything especially to do with symmetry, or any clue about why they might be so important not just in mathematics but beyond, in physics and maybe even further afield.

I'm a barking mad collector of introductory maths books, so lets have a look at how a few of these present groups, and how they explain the relationship with symmetry :

"Introduction to the Theory of Finite Groups" by Walter Lederman , 1961, - a lucid down-to-earth favourite, but the word "symmetry" does not even appear in the text at all as far as I and the index can see.

On the other hand another '60s text "A Brief Survey of Modern Algebra" by Garrett Birkhoff and Saunders MacLane, from 1965, begins its chapter on Groups with...

"1. Symmetries of the Square"

"...the idea of "symmetry" is familiar to every educated person. But fewer people realize that there is a consequential algebra of symmetry....".

Moving on to the 1970's - Herstein's "Topics in Algebra" (1975) describes groups as "one of the fundamental building blocks for the subject today called abstract algebra" - but does not mention symmetry.

And then a more modern introductory textbook - "Contemporary Abstract Algebra" (2002)by Joseph A. Gallian. This has very many examples of geometric symmetry in the chapter "Introduction to Groups", starting with the symmetries of the square as did Birkhoff and MacLane - i.e. reflections and rotations that map a square onto itself.

Finally, the Princeton Companion to Mathematics (2008), introduces groups in a section headed "Four Important Algebraic Structures" :

"...wherever symmetries appear, structures known as groups follow close behind".

So of these 5 books, 3 explicitly make the link from symmetry to group theory - but none actually makes the opposite link : is it also the case that wherever structures known as groups appear, symmetries follow close behind ? Are "symmetry" and "group theory" coextensive ?

One answer seems to be provided by "Representation Theory", and the answer is "yes" - wherever groups appear, symmetries do follow close behind. This is because for any group, "Group Representation Theory" shows how the elements of the group can be interpreted ("represented") as matrix operators in a linear algebra on a vector space, in such a way that the multiplication of these matrices precisely mirrors the composition of the elements of the group. And these linear matrix operators are just reflections, rotations etc. In a sense the group provides the syntax of symmetry - the internal structure of it - and the representation of the group as a set of matrices provides the semantics by which we see the group structure at large, as rotations , reflections etc, of actual spatial figures.

(It seems to me that even in introductory text books, it would be an idea to demystify this relationship between groups and symmetry by making at least a passing reference to the enterprise of representation theory)

My bit of mathematical phenomenology

However it seems to me that considerations of symmetry in algebra arise at a more fundamental level, than is suggested by the examples such as geometrical symmetries of the the square, as provided by the technology of group representation theory.

A bit of phenomenological navel gazing (see below) at what is going on as soon as we write down any sentence in algebra, reveals implicit assumptions of underlying symmetry, and that the group concept is part of the fabric of algebra, rather then being a garment built from that fabric, as suggested by the textbook axiomatisation.

And I conclude that groups are essentially about composition of "operators" - and the basic concept of an operator is a very interesting and powerful one - a thoroughgoing "operator phenomenology" is an intellectually stimulating way of thinking not just in mathematics and physics but beyond.

A simple example from arithmetic - the operator-phenomenological recasting of

2 + 3 = 5


(+2) o (+3) = (+5)

(where I am using the o symbol to mean composition of operators)

- i.e. , "the plus two operator composed with the plus three operator gives the plus 5 operator". The "group operation" is not really addition , or multiplication, or rotation - it is always composition. It is the members of the group - operators - that increment and decrement, multiply, rotate.

The (on reflection) fascinating, distinctive , unexpected thing about "operators" is that they can be studied and manipulated independently of any individual subject or situation being operated on - they completely depart their substrate (numbers to be incremented and decremented, squares to be rotated...) and we study their internal interactions without any reference to a substrate. This "operator phenomenology" is obscured in the usual examples of groups such as integers under addition - because numbers are not obviously operators. It is much more obvious in group examples such as symmetries of a square, where the elements of the group are quite clearly operators - rotation and reflection- acting on a symmetrical substrate - squares, hexagons, circles etc.

Now for the humble phenomenological navel gazing that leads to these conclusions.

Algebraic sentences such as

c = a + b
e = c + d


c = a ^ b
e = c ^ d

(for whatever ^ we choose to define)


e = a * b * d

by their very nature describe a process that has a subtle inherent symmetry - that is , the process involves a transformation, that leaves some fundamental aspect of the transformed substrate unchanged (- a reasonable definition of symmetry).

The symmetry inherent in these statements, consists in the fact that the products of the process are identically qualified to be inputs to further processes - i.e. simply, we can take the product of the process "a" + "b", and feed it into a new process of adding "d". While the process of adding "b" to "a" does transform "a" into something else, that "something else" is precisely concordant with "a" in the way that it may be further transformed exactly in the way that "a" was.

(....of course this is exactly what algebra is. As Herstein puts it, to operate algebraically is to "combine two elements of a obtain a third element of the set". This seems to be a fairly trivial and obvious thing - but is actually a much more exotic and unlikely activity than it appears to be at first sight - it is extremely rare if not completely unknown, to see anything behaving algebraically in the natural world. Two atoms do not (normally) combine to make another atom (except in a fusion reaction) - they combine to make something else - a molecule. Language does not in general behave algebraically - two words do not combine to form another word, they combine to form a phrase, which is a different type of thing)

So - there is an inherent symmetry as soon as we do any algebra at all, and indeed we will find that the sorts of real world processes that may be usefully modelled by these types of algebraic statements, do posses clear symmetries not possessed by substrates that may not be so modelled.

Thus if "a" is a 10 degree clockwise rotation of a hoop about its center, and "b" is a 20 degree clockwise rotation, we can let c be (say) a 30 degree clockwise rotation and this working model makes perfect sense as a realisation of these algebraic statements, precisely because of the rotational symmetry of a hoop - a hoop rotated about its center is just as good a place from which to embark on further rotations, as it was at the start.

By contrast if, say, we let "a" = a bolt and "b" = a nut and try to express composing these algebraically : "c=a+b" == "the nut screwed onto the bolt", this does not work since we cannot go on to write c+d - there is no symmetry associated with this interpretation - the product of the two elements is different to either of them - there is no symmetry in the substrate available here, to permit algebra to be done.

But there is a refinement we need to make.....the symmetry that our phenomenological navel gazing perceives in the fabric of any algebraic sentence, may not necessarily be a global symmetry, preserving the same substrate no matter which operators are involved and allowing us to compose any two elements to obtain a third - it may only be a local symmetry allowing us to compose some terms. When we can combine any two terms , then these terms are truly operators, and there is a global symmetry ; by contrast when only some compositions of terms makes sense, then these terms are what we might call morphisms, and we are not doing algebra, though we may use formulations that look algebraic.

Consider for example the morphisms depicted by the diagram :

* -- a --> *
^ |
| \ b
e c |
| \ | |
| - . v
*<-- d -- *

- we may write as above

c = a + b
(i.e. c is the directed diagonal from upper left to lower right, and gives the same result as "a" followed by "b")
e = c + d

but we cannot write a + d - these two morphisms cannot be combined.

(And we cannot write a + a - a morphism cannot in general be composed with itself - unless it is the loop-back morphism that goes back to its starting point.)

This is because the symmetry inherent in these sentences is now just a local symmetry - the transformation a + b takes us to a place that looks locally the same as the place we started from - a point with incoming arrow and outgoing arrows - and from which we can embark on some further morphisms, but not any morphism. This is because there is not a global symmetry inherent in these sentences - when we zoom out and look at the whole picture, a + b has moved us to somewhere that looks different to the place we started from.

So there is this fundamental thing about operators , that distinguishes them from morphisms, and it is that thing that gives groups their distinctive nature, and that explicates the connection of groups with symmetry : an operator does not care where it starts from or where it sends something to , whereas a morphism does. Another way of putting it is that in a system of operators, such as a group, any substrate that one may provide for these operators to operate on - e.g. a square under rotation and reflection, a number system under addition and subtraction - must be mapped identically back to itself in some sense by any and all operators, simply because the result of any operation must be indistinguishably suitable as a jumping off point for further operations by any other operator.

"Where algebraic sentences are written then operators are not far behind"

"Where operators appear then symmetry and groups are extremely close"

I feel that the relationship between groups and symmetry is a little deeper and more interesting - less accidental - than is suggested by a cursory reading of the usual group axiomatics. And something of the flavour of the idea of a group is lost when we insist on the fully abstract set-based axiomatisation, and refuse to call the elements of the set "operators".

A few interesting (to me) questions come up from this :

* Is a system of morphisms almost a system of operators ? - can we incrementally augment a system of morphisms - i.e. , a directed graph - attaining a series of intermediate structures , which concludes in a fully globally symmetric system of operators forming a group ?

* Is there any graph-like diagram that can be used to depict a group ?

* What sorts of systems of operators are there, that are not groups ?

* Can we develop in logic an operator-based system of predication ?

...Since - I claim that the symmetry inherent in writing down algebraic composition of operators, consists essentially in the way the results of the composition may in turn be composed with other operators - as though there is some substrate being operated on that remains invariant under the operation, and so is identically available for further operation.

...and one way to think of this would be , as though the operators were predicates - "blue", "heavy", "sharp", and then the invariant substrate is the subject being described.

...and so an operator-based formulation of predication then would look something like

Blue(Heavy(Sharp(x))), rather than the predicate calculus formulation

Blue(x) and Heavy(x) and Sharp(x)

...and we may compose operators to obtain other operators in an interesting way - for example (maybe)

Heavy(Light(x)) = Does Not Exist(x)

* Physics tends to involve systems of operators; biology systems of morphisms ? (I'm thinking of networks of gene regulation etc). That is why Biology is "less symmetrical" than Physics ?

* Quantum physics tends to involve systems of operators, groups and much symmetry ; General Relativity systems of morphisms, without a group structure, and little symmetry ?

* Computer programming languages typically allow us to write sentences in various algebras, i.e. operator based ( - e.g. all languages support Boolean algebra ; most linear algebra , over various fields - real , and sometimes complex). Could we conceive of languages that are augmented to allow us to calculate with morphisms rather than operators ? Would this be useful for biology and bioinformatics ? (for example for doing graph transformations and calculations)

* Appetites and the self are fundamentally morphism-like things ; ethics and universal values are fundamentally operator-like things - "good", "evil" - that may be composed and studied independently of the self ? Can fundamental mathematical concepts inform moral philosophy - for example is there an ethics group, that summarises the pattern of interaction between fundamental operator-like moral terms ?