The philosophy of the philosophy-of

A typical abstract axiomatic exposition in a subject like mathematics is extremely terse, functioning partly as a hierarchical filing system for the subject’s materiel, ensuring that nothing goes missing and content can readily be found by members of the professional community, but offering few clues to the original motivating insights and meaning of the subject.

It is the job of the philosophers (I believe), to help interpret the meaning of the contents of the abstract axiomatic filing system of mathematics, and also to help preserve the praxis and understanding of the subject between generations and also between mathematicians and non-mathematicians. The philosophy of mathematics is, then, an ongoing reinterpretive, partly pedagogical, partly popularising narrative in the margins, rather than as usually thought, a foundational monolith.

A successful philosophy of mathematics (or of art, science, ethics …) will offer an intuitive visceral approach to the subject, with suggestive concepts, metaphors, similes and viewpoints, simulating the praxis of a creative professional. A philosophy-of paraphrases, circumlocutes and metaphorises, to infuse its subject with near-praxic life.

Therefore…

• A successful philosophy of mathematics (or of art, science, ethics …) will never directly add new mathematical truths (or works of art, scientific facts, direct moral guidance…) – though it should improve the teaching of these subjects, and may assist talented creative minds to more quickly adopt a viewpoint and mantle from which praxis and discovery of new truths in them is possible.

• Philosophers-of are probably often would-be or retired practitioners-of, but who are now for whatever reason apraxic in their discipline – which is as it should be, since who better motivated to develop tools to allow non and future professionals to glimpse, and in some (rare) cases attain for themselves, the praxic viewpoint and capability of creative practitioners ?

• Practitioners-of will often view philosophers-of with suspicion, and the feeling that they add no new content to their subject (which is true) , and that they are nothing but second or third rate practitioners (which is (one guesses) often true, but is not necessarily relevant to their rating as philosophers-of).

• Contrary to intuition, while mathematics provides eternal truths, the philosophy of mathematics does not. This is because the philosophy of mathematics is concerned with the human binding and vivification of its subject – and this is a contingent activity, dependent on the intuitive ways of human thought, in each generation and in each human audience. Mathematics and science are not culturally relative, but their philosophies-of are.

• The philosophy of a subject is never finished, even should the subject it deals with eventually become completed. This is because the task of preserving and transmitting the meaning of a subject across deep time to future generations and across human space to contemporary non-specialists and students - is never finished.

• The philosophical narrative in the margins itself has marginal narrative and so on, recursively extending the subject and its philosophy as it were breadth-wise, rather than as is usually thought foundationally depth-wise. This accounts for the failure of philosophy ever to find bedrock and terminate.

Without an ongoing philosophy of mathematics (or of art, science, ethics ...) , updated and reinterpreted in each generation or two, we cannot be certain that our current terse axiomatic filing-system-like presentations of (say) abstract group theory will escape the fate of ancient Egyptian hieroglyphics - by which the language’s tokens were preserved, but meaning and worldview became completely lost. Luckily the ancient hieroglyphics had been semantically replicated into two other scripts on the Rosetta stone so their meaning could be recovered. The job of the philosophy of a subject, even of one as purely logical as mathematics, is as a marginal but vital narrative using non-specialist language, to help communicate and preserve the real meaning and insights of the subject, and not just the contents of its axiomatic filing system, across deep time and human space.

grammar

In a recent "hmmm" I suggested that breaking one's moral code - in other words "doing something out of character" - can be a disorientating experience because it is a type of existential poisoning or infection, in a way that this phrase succinctly captures - in so doing one in fact morphs into a different character, contaminated with behaviours foreign to the former good character.

A first problem with that is it is too negative - the "something out of character" could just as well have been a good thing, like being uncharacteristically kind and generous - more like an existential tonic than a poison.

A second problem with that hmmm was that it may pop up on some police profiler's scan as being a cryptic confession to having a body buried in the back yard, resulting from some recent "out of character" behaviour, that I am starting to feel guilty about. Let me say straight off that while I am often foolish, feckless and flawed - not to mention alliterative, obscure and asinine - I am basically a good, generous, hard-working, kind, non-violent person - albeit one with several bodies buried in the back yard. I think there are 2 guinea pigs (damn those rodents can wriggle ! By heck cats are good hunters !), a mouse or two and even several pet beetles that were well loved.

(Though one of the worst types of nightmare I have had - a couple of times - involved a body buried in the back yard - I never knew who and there was no memory of the act, just that I was responsible somehow...the very worst sort of nightmare, far worse than being chased by monsters, falling over cliffs etc...waking up from that dream and realising I had done no such thing was such a relief !)

If I wanted a negative example, I should have used something that is much more commonly experienced - by me and I suppose most people, at various times - which is "failure to achieve one's goal / dream". (.....to pass the exam/ get the girl/guy/bring the project in on time and budget/ be the perfect father/partner, make a fat profit etc etc)....these failures can be disorientating because also a type of existential poisoning or infection.....the self that you now find you are is not what you thought or hoped it was. And you can kind of go around in circles trying to figure out what went wrong and where things are at, but never making much progress because just re-entering the same loops again and again....the problem being the self needs to morph along a bit and catch up with events.....or not (fight or flight - the big life decision for most living things).

Another problem of course is that all this seems rather tedious and perhaps even narcissistic - sic again the police profiler's scan. Pure narcissists are extremely dangerous and evil. Clayton Weatherston (the young Dunedin university economics lecturer who stabbed his ex-girlfriend several hundred times, to death, because of some perceived slight but could simply never see beyond his own reflection in the pool...) was a classic example of such a pure narcissistic personality. I remember how stunningly this was revealed during the trial. I hazard a guess and theory that the narcissistic personality trait is more common these days than it once was. Perhaps aided and abetted by technology and commerce, which is increasingly configured for extreme personal choice and the delivery of services tailored just-so for the individual self and its foibles, so perhaps encourages this trait. (Or perhaps not - its a nice theory but perhaps a bit too nice. There are still plenty of those wonderful personalities around that appear to have not a trace of narcissism - these tend to be the most attractive personalities of all....though perhaps there are the odd rare pure zombie psychopaths who, by completely lacking any developed sense of self at all, also completely lack traits such as narcissism - but not-in-a-good-way ). I'm getting a bit long in the tooth and faded in the bloom so fortunately past the stage where too much in the way of narcissism is at all possible - though then, in these later stages of life, the danger becomes excessive self criticism and re-calculations of past failures - one has generally accumulated a few of these by this time - excessive indulgence in which is in itself a narcissistic trait, and to be avoided and weeded out.

A really robust, lucid and sensible commentator about these sorts of things was Samuel Johnson, famous for taking 10 years to write one of the first (?) dictionaries of the English language, and maybe even more famous for being the subject of "Boswell's Johnson". The biographer Boswell himself was a fascinating character - and really something of a narcissist in his youth, if you read his London Journal, which I have just done and have now started his Holland journal. At the beginning of this he suffers from a very extreme bout of depression - presaged in retrospect by bouts of deep melancholy in the London journal (...in between even more frequent bouts of cheerful rogering about the back streets and having a generally wild time with his mates). The story of the publication of these is really interesting - the London journal was written in the 1760's , but not published until 1950 !! So my copy, 1952, picked up for about 50 cents at one of Dunedin's great 24 hour book sales (can't remember if it was the Regent Theatre or Library sales, both good) is actually one of the first editions. Its a long story - but at least partly the delay was because his family did not allow publication - understandably. I loved Pepys' diary because it was a warts-and-all presentation of himself, but Boswell's journal is.....warts, gonorrhea, penis size ("....she wondered at my size, and said that if I ever took a girl's maidenhead, I would make her squeak...") and all....!!

Towards the end of the London Journal he starts spending alot of time with "The Great Man" (Johnson), and it is largely Johnson's writings and precepts ( from e.g. "The Rambler") that rescue him from what was clearly an almost fatal attack of depression, just after he went across to Holland to study law at Utrecht. (Boswell was a Scot, and finally decided to try and follow his father's footsteps into the Scottish legal system - apparently both Scots Law and Dutch law are quite different from English law, and both derive much from Roman law - hence it was common at that time for Scots to study law in Holland)

But to return to the subject - the syntax of sin - another problem with that hmmm was that I introduced a new half baked idea - "membrains" and the idea of social computation, which I am still developing - when there were already more than enough half baked ideas in there.

I should have stuck to the basic idea, which was a syntactical analyses of existential problems : the syntax of "I sin" is that there is an invariant I which sins, whereas this is both unlikely and unhelpful.

* "The giraffe stretched to reach a succulent branch"
* "The giraffe then lowered its head and turned to look in the direction of the roaring lion"

There is a basic symmetry conveyed by the syntax of these sentences - the subject, the giraffe, is an invariant substrate for the verb operators - stretching , turning of the head etc (...and mathematically these operators form a group as explained (?!) in "Operators and Morphisms....." , in this case a Lie group) : the giraffe after stretching for the branch is the same giraffe as before stretching.

Yet this need not have been the biological reality - according to Lamarck's theory giraffes obtained their long necks, and species morphed one to the other, by the morphing of individuals in their own lifetime - and he *could* have been right but for the contingent facts of evolution on this planet. In a Lamarckian world, the giraffe after the stretch is distinctly *not* the same as the giraffe before the stretch - though syntactically we would have a great deal of trouble expressing that.

And meanwhile back on the psychiatrists couch - we might say that it is largely simply syntax that forces us to be the same "self" after some intense event (failure / success/ sin / inspiration), just because we are forced to say "I came I saw I conquered/I failed/etc" - the same subject "I" throughout - yet maybe the "I" is in fact morphing or at leads needs to morph, but our syntax does not allow for this : confusion, anguish and delayed adaption to the various events of life - perhaps partly the result of a subject-verb syntax which does not reflect the Lamarckian existence we lead as selves.

Perhaps language is indeed all-enveloping, vast - all of what we are and see is language....even the scene that we see as we walk down the street, green and grey roofs and a smoking chimney and the red of the sun's sinking - is really a collection of visual nouns, verbs, participles, gerunds, phrases, sentences in an intricate unsuspected pictorial language....

Or to come at it from the other direction - we simply have a basic difficulty of grammar, if we want to express a continuous morphism of some subject such as a giraffe, or a self, along some trajectory.

We do have participles, that turn a verb into an adjective : "a smoking chimney". Then there are gerunds which turn a participle into a noun : "the red of the sun's sinking" - in this phrase the sinking is in possession of the colour red and red here is an adjective which must be attached to a noun. This is maybe getting us a little closer to what we need if we want to admit a "volatile subject" into our language - a subject which is morphed into something new every time we combine it with certain verbs in certain ways.

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)...you 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.

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.

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

is

(+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

or

c = a ^ b
e = c ^ d

(for whatever ^ we choose to define)

or

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 set....to 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 ?

Henley to Taieri Mouth

Happy New Year !

31/12/2009 New Years eve day was fine and warm if a little windy in Dunedin and I managed to negotiate and mount a most-of-the-family (16 y/o and over excepted) expedition, to walk from the end of Taieri Ferry Road towards Taieri mouth. They started off a bit reluctantly on the justifiable basis of many times bitten (by my various previous family trip schemes and paternalistically induced suffering) many more times shy - however in the end were very good company and we had a good day of it. There is a "millennium track" along this route - one of a number of these around the countryside, and its a very well maintained and enjoyable track to walk. Its about 9kms right through to Taieri mouth and the Pacific Ocean and me being me was keen to go all the way - groans and growls all around at this suggestion initially but once we got half way, to a fantastic picnic spot down by the river / estuary at a place called John Bull gully, my offer to walk back and drive the car around the long way to pick up the rest of them at the other end if they wanted to continue was taken up, with even moderate enthusiasm - the charm of this track on a nice day had done its job and generated a second wind in what is lets face it mostly a fairly sedentary and un-intrepid family !

(Watch the stinging nettle just before John Bull Gully ! - its an interesting looking plant which seems to appeal to children who then touch it and get a very big fright)



I'd been out that way a few years before with a couple of people from work, and two of my at that time pretty small children, on a morning's canoeing outing (aka paternalistically induced suffering) - however on that day it was too windy so we gave up, after a bit of nautical nonsense involving me in my canoe towing the two small children in theirs and everybody being slightly out of control. There is a bit of a wind-tunnel / venturi effect through this gorge I suspect. New Years day 1/1/2010 looked promising for having another go at canoeing this place, after enjoying walking it the day before (i.e. - to quickly capitalise on my no doubt very short lived credibility from that successful expedition.....and probably knowing me, thereby overdo it !) - fine and warm and although gales were forecast inland, I couldn't see much going on out the window on New Years Day out our way in Dunedin on the east coast, and as they were westerlies I figured the sea-breeze that would be trying to blow up the gorge from the Pacific as the morning wore on and the land heated up, might cancel out the situational westerlies.

It turned out a great day - we managed to fluke canoeing down the river , which is estuarine at this point, on an outgoing tide , my 11 y/o daughter Rosie in one canoe and I in the other. Our canoes are pretty basic - I've got an ancient one-seater fibreglass river (i.e. round-bottom, no keel or rudder) canoe I've had for years , and the other one is a lurid pink two-seat plastic canoe suitable for kids - i.e. quite wide and stable. But its amazing what you can stow away in even a small canoe (keys, camera, cell-phone, goggle and flippers in case we wanted to snorkel, picnic lunch, water bottles.....and still plenty of room). And how much speed you can make with these streamlined vessels. At this time of the year birds have been breeding - we spotted some ducklings and at one point cruised close by some large black-back seagull chicks - the chicks seemed almost the size of adult oyster-catchers. Forgot to time it but it can't have been much more than 1.5 hours, including calling in and disembarking for a picnic at good old John Bull gully from the day before, where we saw a family offloading huge amounts of picnic equipment from an outboard boat up from Taieri mouth.

After completing the trip, disembarking and climbing a hill to get cell-phone coverage at Taieri mouth, we (politely) requested backup from home - Andrew (9 y/o) was fairly keen to canoe back up the river with me, so his Mum Adele very kindly drove down from Dunedin (about 30 minutes by car) to pick up Rosie and deliver Andrew and a cup of tea. All the while, the tide had turned so Andrew and I were able to canoe up on a (according to somebody I was talking to ) very strong incoming tide - maybe it was a spring tide, I forgot to check. So we had a fairly easy trip back up-river, with me towing Andrew some of the way. On the return journey I noticed now the black-back gull nesting sites, which seem usually to be in the bush a metre or two above the water - marked out by one of the parents standing there and then taking off and squawking and dive-bombing if you loiter. Rosie and I had not noticed these on the way down - maybe because the tide was out and the parents were down on the mud-flats feeding or leading mostly-sedentary families on New Years Day family trips.

Its a surprisingly beautiful little stretch this - once you get into the gorge a bit, either on the track or by canoe, you could for all the world be on some river in the middle of the Brazilian rain forest.....though just up over the next ridge, its all sheep and dairy farms, commercial exotic (pinus radiata) forests, and towns. Oh and the jet skis and outboard boats that scream up and down there on a nice day give the true location away a bit.

The inland (Henley) end is a popular fishing spot, you can apparently catch ocean-going trout. I have a New Years resolution, to go down there after work the odd day while the days are still long and try to catch a fish. Easier resolved than done but I'm hoping I can make it happen.

actabolism in two Chambers

I was looking for a word to describe the sum total of the thoughts and actions of a person - over an hour or day, possibly week (but not a life or even a year) and thought of perhaps "actabolism", based on a mixed-up analogy with "metabolism" which Chambers defines (offline, 1976 - a big red hardback dictionary ) as "the sum total of chemical changes in living matter".

(And since I was in any case mooching about a way to interpret some meaning into all the slog of miscellaneous microscopic myopic thinking and doing as being maybe somewhat like that great complex of miscellaneous microscopic myopic biochemical metabolism - a rather disturbing mess up close but actually supporting a worthwhile homeostatic constancy if you stand back a bit. An intricate machine to provide constancy in the face of friction and adversity...so don't forget to dot the i's and cross the t's and put out the rubbish and file the weekly pointless management report because it all does have a point after all, its all part of the intricate metabolism of life - the worthwhile achievement of (more or less) homeostasis in the face of environmental, social, professional, psychological, familial and financial friction and adversity !)

Anyway - it turned out interesting to contrast that 1976 definition of "metabolism" - "the sum total of chemical changes in living matter" , with the slightly less poetic sounding online definition given by Chambers 33 years later in 2009 :

"sum of all the chemical reactions that occur within the cells of a living organism..."

So what has changed ? :

* "changes" has become the more technical and less agnostic "reactions"

* "sum total" , suggesting a complete enumeration of a collection, has become the more quantitative sounding "sum" - which vaguely suggests some sort of chemometric summation of reactions.

* ...and the lovely antique phrase "living matter" redolent of natural philosophy and the Victorians has become the more microscopically correct "cells of a living organism". (I wonder if the newer microscopically correct version is actually correct though...I would have thought a fair bit of metabolism occurs outside cells - e.g. in the action of salivary enzymes and gut acids on food....or maybe that is not counted as part of metabolism proper ?)

The other change in the online definition is that it goes on to add "...including both anabolism and catabolism of complex organic compounds", with these terms (anabolism and catabolism) hyperlinked as I have them. The offline version has both of these as separate dictionary entries but not of course inter-linked (and the entry for catabolism delegates to katabolism)

(I hadn't met these before but have met katabatic and anabatic winds, which are winds that blow down/up the slopes of mountains, as these cool or heat the air immediately above. Hence analogously katabolism breaks down compounds while anabolism builds them up. Nice words, I'm glad I met them).

Anyway getting back to "actabolism" - although Google finds it, its
clear it only makes it into their keyword index by virtue of being
an anagramatical typo' of "catabolism" - see for example
http://www.jipb.net/earticle_read.asp?id=4557 where both versions occur.

So I do actually have freedom to operate here - there is no such word as actabolism -but my derivation is not pretty and also wrong. I really (?) need to replace the "bolism" with something denoting a thought or action, not the "meta", as this prefix supplies (?) the "sum total" part of the concept. Or so I thought with my meagre etymological knowledge

This brings out another interesting contrast between the offline and online definitions of "metabolism". The offline version occurs within the context of an ordered list of terms appearing on the page, whereas the online definition occurs on its own without any context - true there are links to the related terms anabolism and catabolism, but links are completely different if not diametrically opposed, to context. The online definition occupies a whole web page - in fact it is really more fragmentary than that, just a web sentence.

In this case the ordering of the terms in the offline dictionary is quite interesting and informative - it is not strictly alphabetical, in fact when I first looked in the big red book for the word "metabolism" I couldn't find it because this word comes after "metacarpal" and "metacentre" which confused my tiny mind ! Well - thats because "metabolism" does not (apparently), despite appearances, derive from the "meta" root after all - and the offline Chambers collects all of the words derived from the "meta" root together, even if this upsets strict alphabetical ordering.

(This is explained in a note in the dictionary's preface section - "The Arrangement of Entries" :

"...Derivatives are not listed in crude alphabetical order but in a more logical form....etc etc")

Grumpy-old-luddite-bastard-time-out :

How much of that sort of informative and rich non-semantic context that you get from "Arranging the Entries", are we going to lose in the new strictly semantic electronic information infrastructure ?

Snap-value-judgement-time-out :

Are hyperlinks always all they are cracked up to be ? Is the current web too hyper' ? Can / will the web stay the way it is now or do we need / will there evolve some "Arrangement of Entries" ? Maybe it needs to become more of a "semantic manifold", with local contextual structure, density and continuity, and less of a porous semantic web, of tangled links between tiny fragments.

True , I was only looking at one of the "freebie" definitions - but even in a
full online entry - e.g. http://www.chambersreference.com/dict/external/site/main/quantity.htm (one of their marketing examples) - you still only get a single definition on a page - there is not the context of similar words to excite the pattern-finding and insight-forming parts of ones attention as you got in 1976. (Interestingly, the structure of that 2009 online page corresponds almost exactly to the design set out in "The Arrangement of Entries" note at the beginning of my big red dictionary of 33 years ago - even though there is no longer really any "arrangement of entries" needed, as there is just one per "page" !)

Well, after all that I still haven't settled on a word to denote , analogously with metabolism, the "sum total of the thoughts and actions of a person - over an hour or day, possibly week (but not a life or even a year)"

metaphysical algebra (2)

A previous blog suggested a metaphysical algebra for conceptually analysing emergence and the formula :

O = N x I

I = the relatively disordered input space of things, from which the emergence. A high dimensional space. Examples : 8 people around a table, a potential 8 way babel; 56 independent nucleons in a plasma ; several billion meaningles stimuli per second presenting themselves to our eyes, ears, taste, skin as we experience (say) a shower of rain.

O = the relatively ordered output space of emergent phenomena. A low dimensional space, viz : 1 polite conversation at the table rather than babel ; one iron atom condensing from 56 nucleons in the plasma ; one conscious subjective experience of a shower of rain.

N = that which transforms the input space into the output space. I suggested this be thought of as a space itself, with a negative dimension, and that some rough dimensional book-keeping can be done :

dim(O) ~ dim(N) + dim(I)

<=> dim(N) ~ dim(O) - dim(I)

So for example the book-keeping predicts that the thing that transforms the huge number of raw sense-stimuli elicited by a shower of rain into a simple subjective conscious experience of a shower of rain, will itself be a very complex space of high dimension. As indeed it is, being a brain with a dimensionality of perhaps 100 billion - the number of interconnected neurons it contains.

OK - so, the existence of an "output" space of low dimension is plausible. In each of the examples we can see the emergence of a coherent new level of reality, with fewer "things" (= lower dimension) than in the lower level, and exhibiting its own novel higher level laws of behaviour. For example iron atoms (once they have caught a few electrons from the plasma) have emergent behaviour such as chemical valence that is nowhere hinted at in the pedestrian lives of solitary protons and neutrons and the nuclear forces that govern their interactions.

And, OK - the existence of a distinct "lower level" "input" space of high dimension also seems plausible - for the iron atom example this is just the 56 independent nucleons in the plasma, a space with dimension at least 56. For the shower of rain it is all the billions of nerve signals elicited on our skin , eyes, ears, smell, taste by the physical effects on our bodies of the shower of rain.

But where does the space "N" come from ? Specifically - why call it a space and why assign it a negative dimension ? Perhaps there are other types of thing that N could be, that also transform a given space of high dimension into a resulting space of low dimension ?

Well it comes about mainly because I like the idea, and thinking about the idea, and seeing where it might go (which might be nowhere). Just maybe there is indeed some abstraction that can be done to find high level structure, in the daily scientific grind of explaining this and reducing that....explanation and scientific reduction as factorisation ? The formula implies an "ontologically opaque" view of things like iron atoms and consciousness - which is to say that, these things are to be regarded as new levels of coherent reality in their own right and they are not ontologically transparent - you can't just look in reductionist fashion through an irrelevant layer of iron atom behaviour and see nucleons, or an irrelevant layer of subjective experience of a rain shower and see neurons firing.

So far though that is at most merely proof-by-narcissism - "its so because I like it and it likes me". I suspect that's as far as I'll ever get - but maybe I can incrementally build a case and then one day do a summing up. Lets start with precedents - that's a good way to build a case in law.

My first precedent is the example of functional notation and spaces-of-functions in mathematics. Don't let that put you off - its very unlikely I know anything more about this than you do and in fact writing this little bit puts me in mind of finding out where this functional notation we all take for granted really came from. Since it is no less metaphysical and wacky in its own way than my "negative dimensional reducing spaces"

Take a few maths equations :

y = x + 2

y = x^2

y = x^2 + 4x + 4

y = sin(x) + cos(x) + 4x + 4

- these represent the daily calculational grind of multiplying this by that and adding it to the other to get some answer for some reason - and I would wildly guess that up until the beginning of the 19th century there was no higher level of abstraction than this. Though there was certainly an enormous level of skill in manipulating these equations.

But then somebody said "lets think of x^2 +4x +4 as consisting, actually , in a thing called a "function" operating on another thing called "x" - we will now represent the calculational rule x^2 + 4x +4 as being the formal product of a function, with x :

f(x) = x^2 + 4x + 4

and these "functions" will live in their own function space and we can take formal products of functions with other functions - in the above example , if

g(x) = x + 2

h(x) = x^2

f(x) = x^2 + 4x + 4

then f(x) = h(g(x)) so that f = hg

....standing back a bit - where on earth did that functional abstraction come from !? - its a huge innovation. You take an equation like

x^2 + 4x + 4

- which we can all understand as instructions to do some stuff on our calculators - and do an abstract factorisation of this into a thing called a function, "f", existing in some weird function space, acting on a thing called "x" , existing in a (less weird) space of numbers !? Its bizarre, only we are used to it so it seems bland.

The encouraging thing is that this apparently willful not to say whimsical piece of abstraction turned out to be very fruitful in mathematics in the long run.

(Which is not to say that my own wilful abstraction and factorisation of emergence and (soon) reductionism and scientific explanation and (after that ) certain physical processes, into formal products of spaces, and associated dimensional book-keeping, will be so productive).

An idea for a metaphysics of big and little emergences

Big Emergences :

• The emergence of life from an ambient "soup" of inanimate molecules.
• The emergence of consciousness from a cranial soup of neural networks and other brain parts that are not conscious
• Computational emergences - Cellular Automata :
  Examples : Life by John Horton Conway ; many examples in Stephen Wolfram's A new kind of science. The general idea is a simple algorithm, repeated application of which leads to totally unexpected "emergent" behaviours and patterns. There are well known examples from (computational) biology - along the lines of, simple postulated individual interactions from which emerge coherent behaviours of schools of fish, flocks of birds etc ; simple rules of growth and development which lead to emergent morphologies of coat patterns etc on animals.
• The emergence of a coherent regulated society from the (more or less) free participation of millions of individuals.

Both the big and computational emergences have a kind of implicit payload of rabbit-out-of-the-hat magic and rare privilege - events that you are unlikely to see very often and only in special staged situations, such as at the primordial birth of life, or of consciousness, or on running a computer program like Conway's Life or on occasional witness to the impressive synchrony of a flock of birds or school of fish. But they are (in my view) no different to everyday emergences...

Little Emergences :

• 8 people sitting at a table at morning tea discussing a single topic, one speaker is speaking. One topic and thread emerging from a complex of 8 different people.
• an iron atom : 56 protons and neutrons sitting at a table at morning tea discussing a single topic - how to be an iron atom. One coherent and very useful metal atom emerging from 56 unruly nuclear personalities.
• a well designed computer program interface, marshaling millions of pixels and bits of information into a simple coherent metaphorical world with which the user interacts.
  

Unifying Small, Large (and Computational) Emergences : Dimensional Reduction

My idea for a metaphysics of emergence, is a factorisation into

(1) an input factor space (I) of relatively high dimension

(2) an output product space (O) of relatively low dimension

(3) mediation of the transform from input space to output space by a complex (i.e. complicated) intermediate factor space (N) of negative dimension.

So that we may write formally as :

I x N = O

with

dim(I) + dim(N) = dim(O)

In words - an input space of very high dimension is formally composed with a negative dimensional space to yield an output space of low dimension - the dimension of the product space is the sum of the dimensions of the two factor spaces.

For a visual metaphor - we might think of the intermediate space as being like a lens, which transmits and transforms a coherent input scene into a transformed coherent output image. All three of the elements of this metaphor have themselves internally a coherent space-like structure, and with a formal composition of the lens space with the input-scene-space generating a coherent but transformed output product space.

Taking as an example a small emergence - conversation at morning tea - the input space has dimension 8, which is the dimension of the conversation space if all 8 individuals talked independently and simultaneously ; the output space has dimension 1 - the single coherent conversation that occurs. The dimensional reduction is mediated by a relatively intricate set of social mores, hierarchies, and behaviours - but which in this metaphysics we will notate and size as a negative dimensional space of a certain dimension - in this case -7.

A living creature enjoys a coherent high-level existence within a space that is quite distinct from the melee of chemical reactions that it is supported by. Furthermore by comparison with the lower substrate - it is a much simpler, lower dimensional space. High level laws of behaviour, nutrition, reproduction operate at this level. Just as in conscious life, our brains build for us an experience of a seamless space of streets, trees, flowers, colours, selfhood....that is both quite distinct from the substrate of our brain and the actual physical activity of the external world, and also vastly simpler.

But what is the point of introducing the mathematical fictions - the negative dimensional space , and the abstract composition of spaces ?

The main idea is that this analysis is philosophically useful - but also with a hope for empirical potential in that it suggests a way to predict or measure the dimensional size of these internal engines of emergence. As a philosophical guide, the analysis recommends that, where we see simple ("low dimensional") behaviour or structures emerging from a complex ("high dimensional") input substrate, then we should expect that this emergence will always be mediated by a complex intermediate structure,with a (negative) dimensional "size" almost as high as that of the input substrate.

And as an engineering guide, the analysis recommends that (for example), where we wish to engineer software interfaces or societies that are simple and coherent, we should in general expect that the internal engines of organisation that generate these interfaces and societies will be complex and messy. And vice versa we should perhaps be wary of simple and elegant internal engineering data and object models and political ideologies - according to the metaphysics-of-emergence formula, this internal elegance will be achieved at the cost of emergent societies and software interfaces that are less coherent and more complex.

So what does the complicated intermediate negative dimensional space N look like ? It is no more possible to visualise a negative dimensional space than it is to visualise other mathematical inventions such as negative numbers and potential energies : but we can say that it encapsulates a forest of dimension-reduction-engine-room internals that actually get the job of dimensional reduction done...constraints, relationships, surfaces, volumes, intersections, knottings and braiding of dimensions, dynamic censoring and suppression of dimensions...that our new metaphysics parlays into a space-like structure.

Philosophically this analysis is in contradiction to the cellular automata view - it encourages us to look for complex machinery underlying simple emergent phenomena, rather than simple machinery. (Obviously this particular antagonism will require some explication in view of the long history and many examples and proponents of cellular automata explanations of emergent phenomena).

Freeman Dyson wrote an essay entitled "Why is Life So Complicated". He meant, why is the machinery of life so complicated. He writes "It seems to be true, both in the world of cellular chemistry and in the world of ecology, that homeostatic mechanisms have a tendency to become complicated rather than simple"

My answer would be - the machinery of life is complicated because

(1) the non-living world (I) with which it interacts is complicated : dim(I) is large

(2) life itself (O) is - almost tautologically - simple : dim(O) is low

(3) the machinery of life (N) is therefore mathematically required by the metaphysical emergence formula to be complex since

dim(N) = dim(O) - dim (I)

Life is complicated because there is a higher level metaphysics governing emergence, and for the same reasons that societies are complicated, the laws of nuclear physics that make iron atoms possible are complicated etc.

The simplicity of life itself - as opposed to the machinery that supports life - is perhaps the key part of this equation. One of the striking things about a living creature is its coherence and unity of purpose - how so many intricate parts mesh into a coherent whole that....swims, runs, searches for food, mates....has a being. But this is another way of saying that life itself - the end product of all of the machinery - exists in a rather low dimensional space. Indeed one way of explicating "being" is that it is a space of dimension 1 - this is the dimension of the thread of existence that we envisage for ourselves, stretching linearly back into the past and into the future.

Reciprocal Spaces and Negative Dimensionality

The logarithm function (among other things) measures (approximately) the lengths of numbers expressed in our standard place-value notation - for example (base 10 logs rounded to the nearest whole number) :

4 : length = 1 log = 1
12 : length = 2 log = 1
90 : length = 2 log = 2
100 : length = 3 log = 2
150 : length = 3 log = 2
446 : length = 3 log = 3
.
.
165462454 : length = 9 log = 8


( ....this off the reciprocal space / negative dimensionality topic but....I wonder what are the bounds on the lengths of the names that can be given to numbers, under different possible naming schemes.

There is the trivial naming scheme whereby the names of numbers are the same length as the numbers themselves, so that the length of the name of a number increases identically and linearly as the number it names :

* = 1
** = 2
*** = 3
etc

And easy to construct names whose length increases as, say, the square of a given number :

* = 1

**
** = 2

***
***
*** = 3
etc

- but what about the more useful schemes, whereby the length of the name of a number increases sub-linearly ?

Roman Numerals can be more compact

C = 100
M = 1000

....or less

XXXIV = 34

....but the Roman system is "non deterministic" - yes it is able to compress orders of magnitude to a single symbol as with C and M , but there is no automatic compact Roman symbol for 1,000,000 unless and until we intervene and assign one.

So then a question - is there a deterministic naming system for integers, under which the length of the name of a number N increases at less than log(N) ? )

(see for interest the Berry paradox relating to lengths of names of numbers - http://en.wikipedia.org/wiki/Berry_paradox )


But to return to the topic - there are some parallels between the log function and the concept of dimensionality.

• the dimensionality of a space gives the length of the "names" of points in that space, in a similar way in which the log of an integer gives the length of the name of that integer. For example points on a 2-D plain have names like (1,3) , of length 2 ; points in a 3-D space have names like (3,4,8) , of length 3 etc
  
  
• When we multiply numbers we add their logarithms ; and when we "multiply" spaces - i.e. take a cross product (aka cartesian product, direct product) - we add their dimensions. So for example, the dimension of the Cartesian plane is two , the sum of the dimensionality of the x and y axes that are "multiplied together" to create the plane.

We can use this parallel to motivate an interpretation of negative dimensionality, by considering what is the dimensional analog of a negative logarithm, via these correspondences.

Negative logarithms are obtained from positive reciprocal numbers.

Log(10) = 1
Log(1/10) = -1

This is so since the log of 1 is zero , and we must have

log(10) + log(1/10) = log ( 10 * 1/10) = log (1) = 0

This might suggest that :

• a space with negative dimensionality is in some sense perhaps a "small" space , just as a number with a negative logarithm is a smallish number
  
  
• a space S with negative dimensionality -D is in some sense a reciprocal space in that , if T is another space with dimension +D , it seems that we should have, operationally, and as with the logarithm analogy :
  
  dim ( S X T )
  = dim(S) + dim(T)
  = D - D
  = 0
  

So without further ado (there is far too much ado in this blog as it is ! ) I will take this as my working concept of a negative dimensional space : I will call it a reciprocal space , meaning that on cross multiplication with a positive dimensional space it yields (in some strange black-box way yet to be specified) a product space whose dimension is the dimension of the positive space, less the negative dimension of the reciprocal space - and if these dimensions are equal, the product space is a zero dimensional point.

The obvious fact that we cannot yet specify what actually goes on in this multiplication, and that we cannot visualise what a negative dimensional reciprocal space looks like, need not worry us for the time being. It is similarly impossible to visualise a negative length (and I do not count the metaphor of oppositely directed lengths as such a visualisation - this is just a model of a negative number (see also Roger Penrose on negative numbers, page 65 in "The Road to Reality") - yet we can still discover how negative numbers should behave, and find a use for them. (I would argue that fractional numbers are equally abstract and it is in fact impossible to visualise 1/2, but that is for another blog !)

So far this is all pretty harmless. Next blog will leave planet earth entirely and start to see reciprocal spaces everywhere - the brain as a reciprocal space ; consciousness as the lower dimensional product of this reciprocal space, with the very high dimensional space of the flux of experience and sensation - thus "explaining" certain aspects of our conscious experience. And making a few predictions though probably not testable.

(That will be pretty harmless as well , apart from the small carbon footprint made by the disk space used up in the post)

(I am dimly aware of previous characterisations of negative dimensional spaces - there is a fractal one from Mandelbrot I briefly encountered - but its good to just go Sunday driving without a map with these things sometimes - its of course impossible you will actually discover part of the countryside that hasn't been mapped already (you have to be an intellectual mountaineer for that , which I am not), but you might get to see some interesting places you would not otherwise have seen had you been better prepared ! )

(And a "reciprocal space" in crystallography is a Fourier transform - I do not mean anything like a Fourier tansform in my use of this term however)

The sea the sand the wind and your foot meet at two points (the dimensionality of paddling is -4)

How many different things can meet in one place ?

...I found myself wondering, brain no doubt starved of oxygen, three quarters of the way up the long haul by foot and bike from the city (Dunedin) to Roslyn, so I could enjoy the ride along Highgate and zoom down to home in Mornington. For some reason I was quite excited by the question and have often idled around it since then - that was about three years ago I think !

So for an example, the sea and the air meet at the ocean surface ; the earth and the air meet on the dry ground surface ; the earth and the sea meet on the sea-bed surface - with each intersection we lose a dimension, so volumes (3D) of sea, air and earth intersect at surfaces (2D).

And since surfaces (2D) intersect along lines (1D) , so then these three vast volumes of sea, earth and atmosphere all finally intersect together along a single meandering thin line - the tide-line of the sea on the sand - on an outgoing tide on a gently sloping sandy beach you can see the linear traces of this final 1 dimensional intersection of these three vast volumes contouring along the beach, marking the successively retreating limit of the pour of each wave up the sand.

So if you paddle your foot in the tide , half in the water and half out, you can add your foot to this grand intersection of earth, sea and sky and reduce the dimensionality of the final intersection to zero - two distinct points where earth, sea, sky and your foot all meet - there's one point on each side of your foot, down near the sole where the edge of the sea meets the sand meets the air in a long line to thread your foot. (Points having dimensionality zero)

Now this concept of how many things can meet in one place lacks (among other things) a clear definition , so I wanted to give it a name in the hope that a groove of clearer meaning might be worn down by usage. I decided to avoid a derivation from words like convergence, confluence, intersection etc , because these have a spatial / geometric sense, when in some cases the "one place" and the "many things" are not going to be particularly spatial. "Cardinality" is a word that can mean "how many" , but is also extensible to more abstract senses of the size of a collection, so I started with this word. And since in some cases in the animate world, the number of things that can be brought into one place is very large - they "crowd-in", I decided on the term "crowdinality" - an as yet unclaimed term, according to Google.

As well as the crowdinality of paddling, I also wanted to mention the crowdinality of puns (usually 2) ; the crowdinality of maps (4, by the 4 colour theorem ?); the crowdinality of stories and movies (the higher the crowdinality the better the story. The movie "O brother where art thou" has a high crowdinality on many levels (most of which I completely missed until I read the wiki page !)) ; crowdinality as an explication of consciousness (more later); crowdinality as one of the hallmarks and prerequisites of creativity (more later); the crowdinality of computers (low - around 2 ) as compared with brains (high - in the hundreds of thousands if not millions) ; the crowdinality of the nucleus of an iron atom at the center of a big old star (usually 56 , i.e. its the number of protons and neutrons crowded together and spending life as a single nuclear unit, and the most commom isotope of Fe has 26 protons and 30 neutrons); the crowdinality of a neutron ( 3 , due to its mutually intersected 3 quarks , 1 Up and 2 Down ) ; the crowdinality of a proton (also 3 quarks , 2 up and 1 down); the crowdinality of a scientific paper (the higher the better. "An Alternative Menaquinone Biosynthetic Pathway Operating in Microorganisms" , Tomoshige Hiratsuka et al. was a beauty I came across recently. I am a very lowly bioinformatics foot soldier's foot soldier by trade and I loved the intersection in one study of a bit of bioinformatics with a whole lot of other threads to yield a genuine new discovery ); the crowdinality of a sentence - rather low from the viewpoint of logic formalisms, which considers only the syntax and semantics of the symbols in the sentence - but very high according to recent alternative analyses , such as provided by "situation theory", which explicitly introduce into the analysis the context within which language is conducted (for some examples - including how this type of analysis resolves the famous and ancient Liar paradox - see "Goodby Descartes", by Keith Devlin); the high crowdinality of molecular complexes such as spliceosomes and signalling cascades in the world of molecular biology; the vast crowdinality of a richly synapsed neuron inside a brain ; crowdinality as an approach to the explanation of emergence.

Regarding creativity - A Gardiner in "The Princeton Companion to Mathematics" describes the "delight in a double-edged strategy, which points in two directions at once...[and]...has much in common with the pleasures we derive from....puns and double entendres". Gardiner goes on to describe how Koestler showed how scientific and literary creativity often flows from the identification and exploitation of "double meanings with a built-in tension". Koestler called them bisociations.

In his book "The Space Between Our Ears : How the brain represents visual space " , Michael Morgan has the picture "The Death of Marat" (http://en.wikipedia.org/wiki/The_Death_of_Marat ) with the tart caption "The only writer on consciousness that got what he deserved". (The writing referred to is Marat's "Philosophical Essay on Man (1772), in which he apparently theorises about the mind). Nice shot.

That said, point taken and duly cautioned and all that, I do have a distinct phenomenological vision of consciousness as consisting in the topmost teetering single neurological summit point , of the highest peak in a vast cerebral mountain range of intricately wavering peaks, each peak the final intersect of a huge cast of buttressing slopes of supporting neurological modules, memories , current sensations, that intersect in ascending ridges , cols and cirques of semi-thought, which in turn finally conspire in a single zero dimensional point of maximum crowdinality. (There you go I've done it , and I'll do some more , knife me pink and call me Marat !). Well actually, considering the time dimension, lets call that a one-dimensional peak of maximum crowdinality, the stream of consciousness.

(And there seemed to be some useful predictions to be made from this view, such as that other animals will lead a conscious life of some richness, differing in degree (crowdinality) but not in kind from our human kind ; that in our conscious life, which consists essentially of an intersection point, the higher the crowdinality of that point the richer will be our experience - the more and wider learning and engagement and current passing conversation and sensation we bring to that intersection point, the higher will be the peak - the prediction is that techniques like meditation and others involving the removal of stimuli actually lead to a lower rather than higher level of consciousness. Not that that lower level of consciousness is necessarily unpleasant or unworthy of pursuit for its restorative power - just that it is not in itself deeper or more meaningful than a more engaged and busy level of consciousness)

But the new term - "crowdinality"- had a few problems. Firstly it was ugly; secondly it offended ontological parsimony which should always be respected both in thought and prose style - in other words, preferably, invent no new things either deliberately, or accidentally via long winded reifications ; and finally - the new term has itself low crowdinality - there are not enough different ideas meeting in this one place to justify the creation of this new word.

In order to increase the crowdinality of the concept of crowdinality, and also to remove the need for a totally new word, I decided to attempt to intersect this concept with another idea I have idled and addled over from time to time - the idea of negative dimension. The claim will be that the arena of subjective experience is an example of a negative dimensional space, and that here is the source of the conceptual difficulties we have when trying to understand consciousness and subjective experience using analytic tools and ideas based, as they are, on positive dimensional mathematical spaces.

So now the phenomenological vision of consciousness is similar but inverted by the negative dimensional space interpretation - it is the bottom-most gurgling neurological gully trap, of the lowest ravine in a vastly deeply dissected cerebral canyon of intricately carved and banded gorges, each ravine the final lowest intersecting foot of a huge cast of ascending slopes of supervening neurological modules, memories , current sensations, that intersect in descending scallops, anti-cols and anti-cirques of demi-thought, which in turn finally conspire in a precipitous tomo of maximum negative dimensionality !

I'll provide an attempt at a supporting characterisation of negative dimensional spaces in the next blog.

Its Good To Try New Things - Hormesis And The Advice Theorem

The advice theorem says the following :

"All advice is good advice, because there is some course of action , between the advised course and its complete opposite, which must with mathematical certainty lead to the best possible outcome".

An imaginary example shows how application of the advice theorem could save your life.

Suppose you are suffering from scurvy because you are eating only trace fruit and veg and are completely ignorant of the requirement for dietary vitamin C, and a well meaning but (as) ignorant friend advises you that your ill health is caused by the presence in your diet of small amounts of fruit and veg - you will be fine, your friend advises, if you switch to a diet consisting solely of corned beef from a tin.

Luckily you are in possession of the advice theorem. Applying the theorem to the advice you have received, you appreciate that whatever the merits of this advice, the fact is that the best possible outcome will be achieved with a diet somewhere between all corned beef, and the complete opposite of that - say, all fruit and veg - i.e. somewhere along the new dietary axis implied by your friend's advice.

Unfortunately the theorem is unable to help with the problem of choosing a point along that axis - but commonsense suggests that the optimum point is more likely to be an interior one, rather than at either end - there are vastly more interior points than there are boundary points (just two) - so rather than a diet of all corned beef or all fruit and veg, you decide to introduce a moderate amount of fruit and veg into your diet so as to operate at an interior point of the new advisory axis, rather than at the "all corned beef" end point suggested by your friend. Within days your health is improving - your friend's incorrect advice, moderated by the advice theorem, has saved your life.

The only prerequisites for application of the advice theorem, are that :

1. You are able to define one or more axes of action implied by the advice - so that you can identify the two extremes within which both the actual course of action you take, and the optimum outcome, must occur - i.e. you can identify a course of action which is in some sense the complete opposite of the one advised. Obviously there will generally be no unique "completely opposite" course of action - but it doesn't matter , there will be some optimum point on whatever axis you choose. Of course , some axes will be more productive than others - but any axis you choose must contain some course of action which will result in some zero or greater improvement to your current situation.

2. You are able to rank at least notionally the outcomes of actions on a numeric scale such that the concept of a maximum is meaningful.

Provided these two conditions are met, then the advice theorem may be pictured as a graph of outcomes, with the vertical axis being how good the outcome, and the horizontal axis being the course of action taken intermediate between that recommended and its opposite.

Then - if for example the graph is a horizontal straight line, it does not matter which course you take. And there will be some graphs where one of the end points *is* the best outcome. And some with a hump - the optimum in the middle , and some a wiggly line and the optimum is just somewhere along there. However - for all possible graphs , it is the case that at least one of the courses of action along the axis *must* achieve the maximum possible outcome.

I discovered the theorem late last year, while driving back from Omarama to Dunedin after a weekend away with the kids in a tent, and swimming in the Ahuriri river and having a look at the World Gliding Grand Prix. As I drove back down towards Kurow, I reflected somewhat soberly, amid that somewhat sober landscape, on a bunch of advice I had dished out to a colleague a week before, and wondered whether in fact the complete opposite of my recommendations might not be the best course.

Then it hit me - whatever the true situation, my advice had at least some value in that it created for my colleague a new axis of action - consisting of all courses of action between what I recommended and its complete opposite, and that somewhere along that axis there must surely be a point which would achieve the best possible outcome. On my return to Dunedin I communicated by email the exciting news to my colleague - my advice could be shown mathematically to guarantee the best possible outcome....though it may need a bit of titration to find the optimal point, between following it to the letter, and doing the complete opposite. (Shortly afterwards - on Boxing day actually - my entire family came down with Salmonella, which made for a miserable Christmas and New Year. )

I was just reading an interesting article about something called "hormesis" in the New Scientist magazine today (9 August issue). I experienced an odd sensation of anti-deja-vu....I have never seen this before ! Which is indeed odd for such an apparently basic idea. So - this *is* homeopathy , right, under a different name ? (Not that I have anything against homeopathy - I learned from Mandy's blog that she consults a homeopath, and she's a really clever bastard so it can't be complete bollocks !)

But also - I *have* seen this before - its nothing other than my advice theorem : almost anything is good for you , its just the dose that you have to get right - but that is at least in part a simple mathematical tautology, rather than being biologically meaningful.

So my advice is - always apply the advice theorem to any advice you receive. (Unfortunately this leads to a still to be resolved paradox - "the advice paradox" : should we apply the advice theorem to the advice to always apply the advice theorem ? If we choose not to apply the advice theorem to this advice, then this implies we accept without qualification the advice to always apply the advice theorem, which contradicts the assumption that we did not apply it. This suggests that it is impossible not to apply the advice theorem to this advice - yet in that case it is impossible to apply the theorem, which requires us to be able to not apply the theorem)

But....pointless paradoxes aside - the advice theorem is a wonderfully liberating thing for advice-giving busy-bodies like myself - we can go forth and dish out our hot air with promiscuous abandon - just so long as we also hand out the antidote - a pamphlet describing the advice theorem (with suitable warnings not to try applying the theorem to advice relating to the theorem itself as serious injury may result)

And we should always every minute of our lives try to find novel axes of action and titrate our way up to the optimal point along them. Since - its a simple mathematical fact that its good to try new things.