Wednesday, July 29, 2009

Stephen Wolfram and Cellular Automata.

I recently watched this YouTube video featuring Stephen Wolfram expounding his theory of cellular automata. I have tinkered around with cellular automata in the past myself, but it seems that Wolfram is trying to develop the notion into the ultimate killer science. Looking around on the internet, however, it seems that not everybody is impressed with Wolfram’s lack of modesty and unwillingness to give credit. For myself I was surprised that in his lecture Wolfram didn’t at least make some mention of researchers like Kolmogorov and Gregory Chaitin. But this totalizing kind of theorist often goes together with a large ego. I hear rumor the Benoit Mandelbrot was of a similar type.

Nevertheless, the lecture was fascinating stuff. Wolfram believes that in order for our science to make further progress, especially in the area of complexity, new kinds of primitive are required. Needless to say, Wolfram believes that cellular automata are those primitives. He showed us how a variety of objects that cellular automate are capable of generating, from the complexity of high disorder through patterns that looked like particle collisions, to spatial geometries that had the properties of gravitational curvature. Of particular interest to me were Wolfram’s references to touring automata which build up structure by weaving a single thread of time, like a kind of continuous scan line, through reality. But he pointed out a problem with touring automata; the reality they generate may be dependent on the order in which the automata visits various locations. Hence a class of automata is required that is insensitive to the update order and Wolfram referred to these kinds of automata as “causally invariant”. He claimed that causally invariant automata give rise to relativistic like consequences. But what about quantum mechanics and the non-locality of the EPR experiment? How do strictly local automata deal with this? I believe Wolfram has been challenged on this point, but during his lecture he made passing references to extra dimensional links between particles which could cater for this. Wolfram believes that cellular automata provide a very literal digital simulation of reality and this leads him to declare that in his view time and space are very different things, perhaps mirroring the distinction between memory space and computational steps.

On the face of it, it all looks rather impressive, although Wolframs treatment of the EPR experiment does look to me a little contrived. Wolfram expressed the belief that he thought real physics to be digital and it seems that he really does believe that cellular automata capture some kind of reality about the universe. But just what is the nature of this correspondence? It is clear that standard Turing machines can simulate a large class of reality and yet we don’t conclude that the ontology of the cosmos is based on Turing machines. The question, then, is this: Is Wolfram simply propounding just another model of computation, like Turing machines, recursive functions, counter programs etc, or does cosmic ontology closely correspond to cellular automata? Wolfram is in fact facing the same question that I faced when I wrote my book on Gravity and Quantum Non-Linearity: Having arrived at some interesting results using the particular model I employed I conceded that: “The work in this book may be little more than a novel way of simulating quantum mechanics and relativity for the price of one”. But then I went on to say: “All theories are, I believe, simulations, but the crucial question is, do those simulations have a life of their own in as much as artifacts of their construction anticipate aspects of reality as yet unknown?” I then tried to show how my proposal did make predictions, and that is precisely what Wolfram must also do: Can he show how cellular automata have an ontological reality that distinguishes them from a mere computation technique? Providing predictions is another point on which Wolfram has been challenged.

Reality aside, Wolfram’s model of computation, perhaps because it is very visual, does seem to bring out some results in the theory of computation very clearly. The work of people like Kolmogorov, Chaitin and others have made us aware of the notion that any given computational task has a lower limit on its computational complexity in terms of the minimum resources (such as memory space, initial program string length and number of computational steps) required to carry it out. Hence, given a particular computational task these resources cannot be indefinitely reduced and thus a task has an irreducible minimum of required resources. This idea is, I think, equivalent to Wolfram’s notion of computational irreducibility, a notion that comes out very clearly in Wolfram’s lecture when he says that a process is computationally irreducible if the only way to find out its result is by running it. Computational irreducibility follows (presumably) when a task’s computational resources are at an absolute minimum, in which case it is not possible to find an analytical solution that arrives at the same result any quicker or easier by using less resources. From this follows Godel’s and Turing’s undecidability theorems: if a process is computationally irreducible and the only way we are going to find out if it stops is by running it, then its halting point may be too far off for it to be humanly possible to make the halting question decidable. Wolfram may not be original here, but I personally found Wolfram’s visualization of Godel and Turing helpful.

On the subject of computational equivalence Wolfram is less compelling. Simple processes can produce the high complexity of disorder, but simple processes, which are relatively small in number, merely sample a very small part of the huge class of complex and disordered configurations. The only way of reaching this huge class is to start with complexity. Accordingly I am not sure I know what Wolfram means when he talks about a “maxing out” of complexity. Complexity of process seems to have no upper limit. Perhaps I do not understand Wolfram aright but it seems to me that there is a huge platonic realm out there of highly complex algorithms, whose rules are beyond our ken, algorithms whose computational resources are too great to be humanly manageable. Let’s just thank God that much of our world, seems readily reducible too simple computations.

Towards the end of the video, during the question time, Wolfram appeared not to directly answer a query about just what light his work throws on the metastable states of life, structures that make extensive use of cybernetics in order to respond to an open ended world of random perturbations. He did at one point show us a picture of a cellular automaton run that generates rare conditions whereby a particular pattern starts to take over the available space. But all this was worked out in Wolfram’s strictly deterministic world. If our world has an open endedness about in that it is perturbed by a complexity that is not reducible to simple algorithms, then it is anybody’s guess what kind of perturbations could arise to undermine the “determinism” of such patterns.

Wolfram’s concepts certainly cover a lot of ground but perhaps not everything. Although Wolfram may have found a useful tool of visualization and understanding, some aspects of our world, if they should exist, may not be naturally amenable to his computational model; for example, non-locality, algorithmic irreducibility, and a cosmos where there is parallelism in time.

Thursday, July 23, 2009

Darwin Bicentenary Part 25: Challenging a Spinful View of Evolution

In a relatively obvious sense the universe has direction; the laws of physics, along with the boundary condition imposed on the universe, are such that the cosmos is in thermodynamic disequilibrium. This results in the well known temporal directionality which means that the time line does not look the same in both directions. It is also clear that there is a biological arrow of time; there is a kind of morphological disequilibrium in that an increasing fraction of cosmic matter, on Earth at least, has slowly become locked up in organized (and complex) biological structures. Morphological disequilibrium may, in fact, be an aspect of thermodynamic disequilibrium as I will shortly suggest.

One thing is very clear: In terms of what is absolutely possible the organized complexity of living structures are highly unrepresentative configurations. Something, therefore, has considerably enhanced the chances of living structures coming about, and if this enhancement is not down to the direct action of design by intelligence (as postulated by the ID theorists) then the only other alternative currently on the agenda is that the probability of life has been considerably raised by the appropriate physical package of laws and boundary conditions, a package that must severely restrict the degrees of freedom available to matter.

If we imagine a system where probability was evenly spread over every conceivable possibility, then probability would permeate the space of possibility as an extremely thin “vapor”. Because the class of living structures has such a low statistical weight, then under these conditions there would be no realistic chance of life making an appearance. However, if evolution has occurred then the cosmic physical regime must so constrain what is possible that it forces this very thin vapour to “condense” into a fibrous network of very thin fibrils, fibrils in which are embedded the structures of life. Moreover, these structures must have sufficient near neighbor relations within the strands of the fiber to allow evolutionary diffusion to migrate from structure to structure in a quasi continuous way. It is important to understand that this conjectured fibrous network doesn’t exist in any tangible sense but only in a platonic mathematical sense in as much as it is an abstract structure implicit in the laws of physics. It would classify as a kind of chaotic complexity arising out of mathematically simple laws.

If this fibrous network has a mathematical and real existence it means that evolution is, ironically, an outcome of the second law of thermodynamics. This follows because thermodynamic disequilibrium ensures that matter, gas like, strives to fill the volume of possible states open to it; if that volume includes the states of life as a significantly large fraction of what is possible then diffusing matter will very likely find those states. But, and this is the big but, this dynamic only has a realistic chance of generating the highly unrepresentative structures of life if the physical regime has considerably suppressed the overwhelming number of non-life configurations, leaving an abstract fibril network that severely limits what states matter can visit.

As an aside, here, let me acknowledge that the ID theorists have a right to challenge the mathematical and evidential basis for this conjectured morphological network of fibers. As far as I am concerned the ID community’s challenge may be valid and that is why I’m carefully scrutinizing their work. However, if evolution has occurred in the way currently understood then the morphological platonic fibril structure is a necessary condition of evolution.

However, proceeding under the assumption that evolution has occurred, then it is clear that the morphological network effectively gives evolutionary diffusion direction in as much as there are preferred directions of development, directions weighted by the underlying physical laws. It’s a bit like a hand powered railroad car operated by a drunk unsure in which direction to move, east or west. Although there may be no preference by the drunk driver whether to move east or west, there is clearly a directionality here determined by the railway track. Of all the directions in which movement can proceed the railway restricts motion only to an east-west directionality. In a similar sense evolution certainly has directionality; the laws of physics must provide a limited set of directions in which random change can proceed, otherwise there will be no chance of evolution.

The foregoing preamble was necessary in order to comment on this post by Larry Moran where he posts a video discussion between Robert Wright and Daniel Dennet on the subject of direction and purpose in evolution. Larry says:

Watch Robert Wright and Daniel Dennet discuss direction and purpose in evolution.

Now imagine what the discussion would look like if they really understood the important role of chance and accident in evolution and, instead of humans, they used lobsters, ginkgo trees, shiitake mushrooms, rotifers, and cyanobacteria as examples of modern evolved species with three billion years worth of ancestors.

Even worse, think about the octopus. Is there any sane person who would point to the existence of those eight-legged slimeballs as evidence that evolution must have a direction and a purpose?

The directionless and purposeless of evolution is one of Larry’s big themes. However, his continual emphasis on the important role of chance and accident in evolution entirely misses the point. Chance and accident only have an effective role in evolution given the sort of highly organized fibril structure I have been talking about. This structure, if it exists, must be implicit in the laws of physics and so limit the degrees of freedom of matter that evolution attains a realistic probability. In and of themselves chance and accident are useless and to over sell the role of chance and accident is to put an entirely wrong spin on evolution. This spin is a distraction from the remarkable fact that if evolution as we know it is to work then a very constrained directionality must be imposed by the physical regime.

At first sight a dithering drunk operating a hand powered rail car may seem to be directionless, but when we become aware of the outer frame which includes the very limited degrees of freedom imposed by the track then it is clear that directionality is the name of the game. For reasons of his own Larry Moran finds it difficult to give cognizance to the outer frame of laws and boundary conditions which impose directionality on the universe. On the question of whether evolution has purpose I will not comment here, neither will I comment on whether or not the complex pathways of a network of “rail tracks” of evolutionary development can be encoded in few lines of physical equations. But one thing is clear; evolution requires matter to have such limited degrees of freedom that directionality effectively exists.

Monday, July 20, 2009

Wednesday, July 15, 2009

Calling All Epistemological Stalinists

Dr William Feelgood, cognitive psychologist, writes…

Dear Jo,

Mr. Timothy V Reeves has asked me to consider your case as you appear to be poorly adjusted to your world, epistemologically speaking.

As I think Mr. Reeves has made as clear as mud, science is unable to progress on certain questions - in particular if the description of our world won’t yield to simple statistical or mathematical treatment. However, that much of our world is regular and statistical enough to be science friendly is testified by technological advances as you point out. And yet we have no known reason why this “science friendliness” should be so and have no obvious way of turning science on itself on order to scrutinize and throw light on this meta-question. We have no proof that our world has any logical obligation to be amenable to science or that it always will be so. In this sense we are at the mercy of providence.

Historians, of course, have to face “science unfriendliness” all the time, and much of their subject cannot be reduced beyond large swathes of debatable text and dates to remember. String theorists are working hard to reduce even the contingencies of history to a full mathematical theory, but as yet that is barely light at the end of a very, very long tunnel, and in any case their theories hold out little prospect of a logically closed self sufficient cosmos that explains itself.

In the meantime we have to come to terms with the fact that the map of knowledge will forever have blank areas beyond the known world. There are two ways of reacting to this 1. The fearful superstitious reaction of populating the unknown areas with the fanciful monsters and beings of our nightmares Viz: “Here be dragons” (or “little grey aliens” as it is today) 2. The equally fearful reaction of declaring that what is not known or what is not amenable to science simply doesn’t exist.

In order to come to terms with the unknown, I prescribe some simple measures. Firstly say to yourself daily such things as “I don’t know. We don’t know. I may never know. We may never know”. I also advise you to read some of Arthur C Clarke’s books dealing with the mysteries of our world. (Caveat: never get involved with the occult). Also worth reading is Nicholas Nassim Taleb’s book “The Black Swan”. (Taleb is a paradoxical blend of great intellectual arrogance and preciousness and yet exemplary epistemic humility)

Once a position of epistemic humility is fostered one is then ready to receive with great joy discoveries that fortuitously fall into one’s lap. The realization finally dawns that all knowledge is in fact a providential revelation, and an unwarranted, perhaps even undeserved, gift.

The alternative is a foolish, crass and naive truimphalistic “clockwork” scientism based on an irrational and unselfaware faith that it is our unalienable and logical right to receive knowledge in a in cosmos that owes us an epistemological living. That our world can hatch some nasty surprises for us, along with the many good things that are not in our power to receive at will, is then a well deserved and chastening lesson. However, if we adopt epistemic humility we will find ourselves much better adjusted for the mysteries of reality as they truly are. The way is then set for renewed epistemological attitudes that acknowledge where our epistemological bread is buttered. Then with joy, humble hope and expectation we can say. “I may yet know. We may yet know. One day I may know. One day we may know”.

Monday, July 13, 2009

The Only Way of Knowing? Who Says?

What's that mustached guy think he's doing in that church?

No sooner had I juxtaposed “anti-theist” with “crypto-fascists” than I saw this entry on Larry Moran’s anti-theist blog. As far as Larry is concerned religion is not just non-science, but does in fact conflict with science; anyone who claims there is no conflict is derisively referred to as an “accommodationist”. I suspect there is little point in trying to engage Larry on questions related to aseity, ontological complexity, epistemological tractability, meta science and the question of the excluded middle between law and disorder. However, I would be the first to admit that science is the only epistemology in the public domain with social authority. But I certainly wouldn’t claim that its formal law court type methods are the only way of acquiring knowledge, especially when it comes to the vexed questions of world views and source ontology. With the latter methods are far more informal and proprietary. What I find sinister is that Larry seems to go as far as suggesting that formal science is the only way of knowing, else why would he get so uptight about extra-curricular claims to knowledge? True, science is the only publicly authoritative way of knowing just as the courts are the only authoritative way of settling legal cases. But like other anti-theists Larry tries to draw a line round the whole of rationality rather than drawing a line within it. His theory of necessary conflict thus sets science’s authority on a collision course with private knowledge. This is blatant intellectual hegemony. What makes this doubly sinister is that science’s public domain authority is being used to apply a kind of intellectual duress on people, perhaps not unlike what was once seen in the Soviet Union. In this context science’s authority has the potential to be abused as a tool to transgress human rights.

All this makes my rather tongue in cheek remark about “crypto-fascists” just a little too close to the mark for comfort.

Monsters From The Id

During the writing of the last blog entry and posting it on Network Norwich I indulged in a little mild trolling by bringing the anti-theists into close proximity with the term “crypto-fascists”. Sure enough I got a response on Network Norwich thus:

Crypto fascists? Closed ended epistemology? Closed ended ontology? Above all a closed mind? My dear Timothy V Reeves, where on earth did you get all this nonsense? And if you listen and read "the likes of Dawkins et al" properly, you will certainly come to a rather different conclusion. That is if you open YOUR mind.

This is very reminiscent of my blog post here where I indulged in a little trolling by referring to the anti-theist add campaign as “Probably the worst poster campaign in the world”. This provoked the following emotional riposte:

This is probably the worst blog entry in the world. You seem to be obsessed with the word 'probably' but in your struggle to appear erudite and witty you come across as a muddled-thinker and a bit neurotic, actually.

Unfortunately these gut reactions haven’t so far proved to be the starting point of any constructive discussions of the big issues concerned. The tapping into deep seated emotions gets a response but it is perhaps is a little hazardous.

Sunday, July 12, 2009

Anti-theism, Anti-Liberalism and Fascism

Crypto Fascists?

Here is my response triggered by someone who posted a thread entry on Network Norwich and Norfolk here.

Continuing this rather bizarre interleaving of topic threads:

Hi Mike_v_02

Let’s get back to basics Mike and forget God, for the moment; if such is possible for you.

The explanatory objects used by the physical sciences are primarily just two; namely “laws” and “statistics”. These objects are only descriptive in nature; that is their logical type. When using them to describe physical phenomena a certain amount of logical compression of descriptive content takes place, but it is no more profound than that.

This descriptive role ensures that the road of science has an in principle terminus; science halts if and when finite scientific description ends in a complete description. That this point is implicitly understood amongst physicists is evidenced by the search for a “Final Theory”. If (a big “if”) and when a full and complete description has been reached, then attempts to push the boat out any further leads to a tortoise like regression whereby one effectively describes the descriptions of the descriptions and so on without any real content being added. (Clearly we are using the tortoise scenario metaphorically here and not literally!)

Science only works if there is a threshold level of complexity in the world: This is because its descriptive role depends on and presumes the existence of pattern, and pattern posits a relatedness of one item to another. Thus, scientific explanation is meaningless in an imaginary cosmos consisting of, say, only a few bits of data.

Standard scientific description, then, requires some kind of a-priori complexity to be meaningful. It is also clear that an even greater level of a-priori complexity is required if we are expecting to find some kind of “self explanation” (that is aseity). The two basic explanatory objects of science (law and statistics) are simply not logically complex enough to provide self explanation. Science deals with finite objects, and baring “tortoises all the way down” type explanations, standard physical science terminates in axiomatic brute facts.

In order to circumvent the “tortoises all the way down” effect we have to ask ourselves if self explanation (=aseity), is an intelligible concept, a concept we are not going to find in the merely descriptive world of law and statistics. So what kind of object do you think would have the property of aseity? Can we, in fact, conceive such an object? Again forget gods, china teapots, tortoises, and try and come to this question afresh. Be prepared to face down your demons and avoid a state of nervous denial that may stem from a fear of what might pop out of the ontological woodwork….

The above questions require one to take a meta view of science, a view that actually introduces a level of logical circularity in that one has to use a kind of science to probe the nature of science. One of the few persons who seems to have understood that this meta perspective immediately creates difficult issues that need to be dealt with and who is trying to tackle them from a NON-THEISTIC angle seems to be Paul Davies. One DOES NOT have to be a theist to appreciate the mystery of aseity or see that bog standard “law and disorder” science doesn’t provide ultimate solutions, if indeed there is a meaningful problem here. Positing the aseity of an a-priori complex deity is just one attempt to tackle the problem. Some anti-theists seem to be in state of denial about these meta issues because it seems to make them jittery about what’s coming next. (BTW James Knight has alerted to me to the “anti-theist” category, a category of proactive protagonist who seems to be a very different fish from the plain atheist. Anti-theists seem to have deep subliminal fascination with God; but then so has the devil!)

Given that science attempts to describe the world using the simple objects of law and statistics we can’t expect science to provide any deeper answer to “why” than the description of mechanism. But just as the descriptive explanations of science require a threshold level of complexity to work, deeper “whys” require yet another layer of given complexity to be meaningfully asked. E.G. the standard question “why” is often a loaded term that presumes all the complexity and trappings of a-priori personality and intentionality to be meaningful.

Even if religion is merely some kind of mythologically expressive structure invented by humanity as it attempts to come to terms with its predicament, it is clear that the orbiting teapot makes no serious point here; it trivializes world view analysis and sheds little light on the anthropological role of a religion in the life of a culture as that culture attempts to make sense of its place in the cosmos.

Finally forget the concept of simple “evidence”. Such evidence really only comes into its own for the simple objects of physical science like springs and moving weights, fields and matter; objects that are amenable to the formal procedures of science. Like the law courts formal science puts its evidences and verdicts in the public domain, but its methods only seem to work for a sub class of cases. Some cases (like alien big cats, Jack the Ripper et al) remain enigmatic and scientifically intractable, although that doesn’t stop one having private opinions based on one’s proprietary experience of the world which, of course, doesn’t classify as authoritative public domain knowledge. What deeply worries me about the anti-theist agenda is that in the final analysis it is a fascist agenda that is anti-liberal; it seeks to attempt to mobilize a contrived scientific authority in order to outlaw religious people and proprietary domain knowledge. Shades of the Soviet gulags and KGB persecution of Jehovah’s witnesses etc come alarmingly to my mind when I listen to the likes of Dawkins et al.

When it comes to the far more complex and intractable objects of “world view” ontologies, the epistemological procedures are very seat of the pants, and proprietary. The resultant ontologies are very narrative intense and cannot be expressed using law and statistics only. In this connection it is better to think in terms of huge sense making mental structures that attempt to embed a very wide experience of life into a grand narrative. These structures are created informally and on the hoof. It’s an exciting form of “joining the dots” with all the risks of being wrong that that entails. That’s what humans do, even when the data samples (=dots) are thin on the ground. But remember this; a search engine only needs a few search key words to sift out a few web pages from millions, so in principle a few dots may be enough.

One of my greatest fears is that the anti-theists are in actual fact crypto fascists with a closed ended epistemology, a closed ended ontology and above all a closed mind. They will not stop until they have made their insecurity everybody’s insecurity and brought to an end the exciting project of the world view analysis of an open ended world.

c Timothy V Reeves (NOTE Copyright. Beware plagiarizers)

Wednesday, July 08, 2009

What is Consciousness?

Somebody once asked the above question and here was my briefest of possible replies:

For me the subject of consciousness generates major category conundrums. In one sense our whole world is consciousness, (or better ‘conscious cognition’): From intimate internal experiences, through perceptions revolving round the senses, to elaborate theoretical artifacts that interpret those perceptions; all these things are part of 'the consciousness experience' and we can only connect with the putative 'world beyond' via the medium of conscious cognition. Conscious cognition is not a thing that takes its place amongst the many observed things of the world; that is, it is not simply another category that exists side by side with things like cars, atoms, and animals, but rather it is a kind of super category that embraces all things: How can we point to consciousness when it is consciousness that’s doing the pointing? How can you define consciousness when it is consciousness that is doing the defining? How can we declare consciousness doesn’t exist when it is consciousness that is declaring it? How can consciousness detect itself when it is consciousness that is doing the detection? How can one observe consciousness when it is consciousness that is doing the observing?

And yet paradoxically consciousness itself delivers the conviction that consciousness is not all that there is; conscious cognition itself suggests that there is a world beyond whose elements, if rightly juxtaposed, succeeds in ensconcing consciousness. So consciousness may see its ‘personal self’ as an assembly of impersonal elements thus producing that familiar philosophical tension between personality and the ‘plain elemental things’ of which conscious cognition seems to consist.

I have never been keen on dualism myself, but it is difficult to get away from consciousness’s partitioning of the world into the ‘us’ of conscious beings and the ‘it’ of the world of elementals beyond sentience. What is primary then? Us or it? In an attempt to resolve the dualist dilemma, the best of a bad bunch of solutions is to think of consciousness like a programming language whose compiler is written in terms of that self same programming language; like a self describing programming language consciousness can be described in terms of its own theoretical artifacts, artifacts that are too simple to classify as having ‘personality’ or consciousness - for example such things as atoms, fields, neurons, computation or whatever. Consciousness describes itself in terms of its consciousness of unconscious elementals. The question of which is primary, ‘us or it’, doesn’t then arise; ‘us and it’ can never be logically separated, since one requires the other.

Now that’s what I call a philosophical dodge – declare the problem of dualism as unintelligible and therefore null and void.

The above expresses my lifelong slant toward a kind of “cognitive” positivism; that is, my opinion that a universe of elemental bits and pieces is unintelligible without the context of a-priori sensing, perceiving and thinking mind(s). If the cosmos consisted only of one bit of data then the descriptive burden of science would be at its lowest and there would be very little to explain. But if only one bit exists there is nothing else to relate that bit to and “explanation” is impossible, because explanation can only happen in relation to and within the context of other preexisting stuff in which the single bit is embedded. In short, the simple “one bit cosmos” is meaningless. Moreover, we will certainly not find aseity in one bit or even in a cosmos that is a collection of elemental bits; elementals are far too simple for that. The foregoing considerations suggest to me that some highly complex context must have an a-priori existence in order to lend meaning to the concept of explanation and in turn to give meaning to that which is elemental.

Sunday, July 05, 2009

A Note on Nomenclature

This post by Larry Moran gives a good indication of differences in the way atheist and ID communities insult one another: Atheists tend to use expletives and accusations of feeble mindedness, whereas ID supporters, with all the seriousness of the religious mentality, have the know how and gravitas of purpose to implicate their opponents in unforgivable sins - like the holocaust for example. Expletives add nothing to the atheist argument, although use of the holocaust, if anything, effectively subtracts from the ID argument when we simply remember the word "Inquisition".