logic> vagueness

Causality demands some sort of beginning. For anything to happen, something must have been there to get things going in the first place.

This seems a perfectly reasonable view to take. It only becomes a problem when we ask the big questions about what came before there was anything? Before the Universe was created there was what?

There are three usual answers on this. The first is that whatever exists must have originally sprung out of nothing. In the beginning was a void.

Although to really count as a nothingness, there must have been less than a void as that is an empty hole. A void implies a structure of space and time, a concrete set of dimensions, within which a “nothing” can be found to exist.

The second standard answer is that if springing into being of out nothing makes no sense, then whatever exists must be eternal. So while other kinds of things may appear to have beginnings and endings, reality itself for some reason enjoys an infinite and uncreated existence.

endlessly self-creating worlds A third possible answer is circular – a self-closing closed loop of causality. Imagine a ring of  gods each creating the god in front and being created by the one behind. A circle is finite yet also infinite.

A few modern cosmological tales, like the spawning multiverse or cyclic big bangs and big crunches have this kind of logic. Though at the end of the day, there still seems to be a valid question about where the whole ring of creating gods sprang from?  Or the eternally collapsing and exploding universe.

Circularity solves nothing. It reduces back to same old binary choice between a creation from nothing and no creation that appear to exhaust all logical possibility. These two choices are the asymmetric dichotomy, the mutually exclusive extremes. What else could we say? Either there was a clean jump into being that had no cause. Or there was no jump and just an eternal existence of everything.

And whichever alternative we choose, there is still no cause explaining why it ought to be so. We are still left with the question of why a something should exist when a nothing would seem to be so much simpler?

Causality is about the inherent logic of a sequence of observed events. But causality cannot say anything about why a sequence would exist, or get created if it didn’t already exist.

How do we get out of this bind? What is the organic alternative?

Think again and you will see that we have been assuming that a beginning state must be crisp. It must be definite, certain, sure. It is not a mixed state or a confused state but pure and absolute. Either we start with absolutely nothing and then imagine a sudden creation act. Or we have eternal existence of all that there is – an absolute everythingness that is equally definite in the way it was never an “event”, a causal occurence.

vagueness is unformed potential This kind of crispness can be opposed by the idea of vagueness. A state of unformed and immaterial potential. A kind of plastic fog out of which more substantial shapes swim into concrete existence. It would be a state which is both an everything and a nothing. But in a vague way.

Vagueness is an ancient idea. Yet it is pretty unfamiliar to most people. Let’s look at the usual kind of things people say about vagueness and then define it more carefully as the monadic basis to a dichotomous organic logic.

ontic vs semantic vagueness

A first essential distinction is between semantic vagueness and ontic vagueness. One is about the vagueness of words. The other is the claim that reality itself is grounded in vagueness – which is what we are arguing here.

Semantic vagueness is a reasonably trivial idea. It is usual here to cite the classic example of the Sorites paradox, a logic puzzle attributed to Eubulides of Miletus

Sorites comes from the Greek word soros meaning a heap. Eubulides asked would you say a single grain of wheat was a heap? No? Well, what about two grains? Or perhaps three? At some point, there must surely be enough to make a heap, but where exactly do we draw the line?

Another version of this riddle is the falakros or bald man. Would we describe a person with one hair on his head as bald? Yes? Then what about two or three strands? Again where is the line crossed between bald and hairy?

Quite clearly the issue here is semantic. The real world is always in some definite state. A heap is crisply a certain size. A man has some exact number of hairs on his head. It is just that we have not invented words to distinguish these finer shades of difference.

The philosopher Bertrand Russell (1872-1970)  – who had an axe to grind because he was seeking a fixed ground for the formalisation of all maths and logic – is widely taken to have proven that all vagueness is merely semantic. The imprecision is in our mental representation of the world and to believe anything else is to commit “the fallacy of verbalism”.

smudged photos could be vaguely anyone Russell  gave the example of a smudged photographic plate. It might be a picture of Brown or Jones or Robinson – we cannot really be sure. But we can see that the vagueness lies in the representation. Out in the real world there will be a real person.

It is the same with our words, images, thoughts or even scientific models. The representations may be smudged but we should always believe that the world itself is definite, capable of being measured with complete accuracy if so desired.

Ontic vagueness by contrast says the physical world itself can be in some ill-determined or undecided state. Semantic vagueness is about epistemology – about the limits of what we can know. Ontology is about what actually is, whether are making an effort to know it or not.

Now right here we need to say that the organic position is that a system such as a universe is in fact constructed out of a kind of “knowing”. In some deep sense, the universe self-organises into crisp being by an act of self-observation – a bootstrapping process we can call pan-semiosis. So the distinction between epistemology and ontology is not so clean-cut. Or rather we could say that reality is epistemic. The knower and the known arise together as the one thing.

But before getting any deeper into that, let’s talk a bit about reasons for suspecting that physical reality might be vague.

quantum vagueness

A good reason to suggest reality is vague is the strange causality of the quantum realm.

Quantum states are states of undetermined potential that need to be collapsed by an “act of observation”. A particle exists as a wavefunction – a band of possibilities. Then the world as a system must decohere this vagueness to some crisp classical actuality.

a position~momentum trade off And of course the world can never completely “know” a particle. There is a yo-yo trade-off for dichotomous kinds of information, like position and momentum. An increase in certainty about one property creates increasing uncertainty – or vagueness – about the other. Pin an electron down to some almost exact spot and its speed and direction become almost completely unknown.

A number of theorists have started to use the term vagueness to describe a state of quantum potential.

The ordinary idea of probability is that it is the collapse of a collection of microstates. Crisp variety. When you spin a roulette wheel the outcome already exists as one of a set of pockets on the wheel. The ball bounces randomly and makes its choice, so determining the system’s macrostate.

But vagueness would be different in that there is no pre-existing variety from which to make a selection. No set of hidden variables. Both the pocket that gets chosen and all the pockets skipped over - the other 36 pockets that make up the spinning wheel, the global system that does the “observing” - would evolve from a common ground of vagueness.

vagueness is both the small and the large

Are you starting to get a sense of what vagueness is about?

It is a raw potential that lacks any form of definite existence. There are a whole list of things which it is not – but to which it must be the precursor state. It is neither large nor small; neither orderly nor disorderly; neither general nor specific; neither discrete nor continuous; neither a substance nor a form. Instead it is a potential to divide towards these opposing outcomes, these crisply dichotomous alternatives.

In normal logic, our instinct is to look for some definite starting place and so we look for something that is fundamental – something that is as small and stabily existent as possible. A collection of uncuttable atoms or indivisible units of spacetime. We know that largeness is an effect rather than a cause, something that gets built up from fundamental parts.

Well vagueness challenges that. It dissolves the small along with the large. Inside vagueness, there is no crisp variety, only vague possibility. When vagueness then gives rise to something definite, it must produce both the small and the large, the crisp figure and the crisp ground.

vagueness, haziness, indecision and fuzziness

We need to develop vagueness as a mathematical idea, a new kind of abstraction that captures some essential truth about reality. So let us look a little closer at how philosophers and mathematicians have treated vagueness in recent history.

Gottlob Frege, the father of modern symbolic logic,  pointed out in the late 1800s that mechanical logic depends on concepts having utterly crisp boundaries – exact meaning – otherwise the results of combining elements became uncertain. If 1 means only vaguely 1, then you could only be vaguely sure that 1+1 = 2.

Aristotle, of course, long ago set down the law of the excluded middle. And noted that sometimes middles can’t be excluded, leaving them vague.

But anyway, Frege decided the problem was semantic. Everyday language simply lacked the necessary precision and so logicians needed to create their own more precise technical language for making completely crisp statements.

Most other prominent thinkers, like Russell and Wittgenstein, followed this lead. An exception was the Harvard philosopher, Charles Sanders Peirce.

CS Peirce Peirce was undoubtedly a difficult chap. He managed to screw up his academic career and spent many years in the wilderness writing reams of unpublished notes. But we can celebrate him as a pioneer of modern organicism. He saw all reality as semantic – a process of interpretance or semiotic. A self-knowing into existence. And his views on vagueness seem much like the ones we advance here.

But back to the mainstream. In the 1930s, another famous philosopher and logician, Max Black distinguished vagueness from ambiguity, generality, and indeterminacy, though of course he still defined vagueness as semantic. Other scholars such as Kortabiński, Adjukiewicz and Fleck were drawn to discuss vagueness without adding much.

In the 1950s, the geometer, Karl Menger, talked about a geometry based on vague objects – variable lumps rather than crisp Euclidean points - which he called “ensembles flous” or hazy sets.

Around the same time, Post, Tarski, Knuth and Lukasiewicz played around with logics that allowed indecision. The middle ground between two crisply defined alternatives was only somewhat or loosely excluded.

Then in the 1960s, Lotfi Zadeh popularised the idea of fuzzy sets in which the excluded middle was represented as an actual spectrum of possibility. Things could have graded membership of a set or class of events.  This eventually became a big deal because it offered a way of doing computing when objects were semantically ill-defined. It also led to a lot of new talk about vagueness even though it did not shed any particular light on the matter.

There are still other recent movements such as the rise of Bayesian probability and paraconsistent logic that hinge on the question of why things may be uncertain or ill-determined.

The main difference here is that is that my notion of vagueness is completely tied to the idea of the asymmetric dichotomy as the engine of causality. That is, I can tell you what vagueness is only by reference to what appears later to emerge out of it. There does not seem much point in trying to begin with a crisp definition of vagueness as a state of raw potential would be completely murky. But we should be able to model it by virtue of the fruits it generates.

modelling vagueness

To recap, any model of causality would seem to need some ground from which events can proceed. Something must exist – even prior to existence itself.

Traditional logic faces two equally unsatisfactory choices here. Either everything must magically spring out of nothing. Or else everything is eternal. It is uncreated and has no beginning.

which came first? the chicken of the vagueness? Organicism offers a third choice. Everything begins in a state of vagueness, of raw potential. This vagueness is both an everything and a nothing. It hovers somewhere between complete existence and complete non-existence. So it a prior kind of thing, hence a ground for causal development. But it is quite unlike our usual notion of something that exists – or doesn’t exist.

We now need an actual model of vagueness. We want some formal idea that captures its essence and predicts is actions.

An obvious difficulty is that a model is by definition something completely crisp! Vague modelling would give us vague answers. What we want is solid, specified, idea that is not like vagueness – it is not a simulation – but which formally relates to vagueness in a way that gives us a crisp impression of it. Crank the handle of the model and it will spit out the right kinds of observables.

the precursor principle

A first step to defining vagueness is that it must be composed of whatever later comes out of it. This is the precursor argument.

It seems a pretty solid principle. If vagueness is to be the causal ground, the capacity to generate whatever emerges must have been orginally “in it” otherwise why bother with vagueness at all? It had to have the potential, the precursors, if something more definitely developed could emerge.

So in a world of cats and cherry trees, there must have been a vagueness from which such things sprang.

Let’s consider the cherry tree. A cherry stone is kind of a vague thing. It is relatively unspecified as yet. Stick it in the ground and water it. It may grow up in many particular ways depending on soil, disease, competition, accidents of weather. The cherry stone has the necessary precursors to create many different fates.

But a cherry stone is a one particular stone. We can then generalise and say that there is a greater vagueness in the potentials of bios, of life. There was some ancestral population of trees that had the potential to become many kinds of cherry species. Some ancient population of cells that had the potential to become every kind of existing lifeform. And also all the extinct, future and “might have been” species. Generalising further, there is the potential for life in the material universe, full of entropy gradients hoping for dissipative structures to speed their degradation. This potential may have given rise to all sorts of alien life on all sorts of planets.

Vagueness is quite simply defined by whatever is seen to have come out of it. The argument above suggests that it is merely the more general which in turn can generate a great variety of particulars. Some actually happen, many more could have just potentially happened.

Vagueness is something like the general, the universal, that is the generating ground to the particular or the specific. The global to the local almost. But reducing vagueness to generality would be making a mistake.

vagueness dichotomises What we are constructing here is an organic logic based on asymmetric dichotomisation or symmetry-breaking. The product of the process is a scale dichotomy. The crisp result is the production of the local and the global. Or the particular and the general, the substance and the form, the atom and the void, the matter and the mind. So vagueness is something else beyond what it produces. If there exists the general and the particular – a general universal possibility for tree-ness and some actual gnarled, diseased, cherry tree in our back garden – then vagueness is a third kind of state, the realm of precursors to both the generality and the particularity.

Let’s compare this a moment to Plato’s famous realm of perfect forms. Plato’s world contained one perfect example of everything – both the general and the particular. There was one ideal cat, one ideal cherry tree. And also one ideal “goodness” and other general qualities. The physical world then produced an actual smeary variety of these entities as it tried to stuff its chora into these ideal moulds. Reality was a bunch of shoddy replicas.

So Plato separated things into insubstantial forms and formless substance. A perfect asymmetric dichotomy. Or at least it would be if it recognised scale. And also if it put both the forms and the chora in the future, rather than in the past of the cycle of development. In my organic logic, substance and form are limit states and so – by the precursor argument – both would have to be the final crisp outcome of a shared arc of development that started in some vaguer realm of potential. If we see two dichotomous extremes in existence, then we can assert that they were both in their together, confused and undifferentiated, in the original state of vagueness.

Both aspects of any dichotomy will have been equally present in vagueness. Or equally vague at one time.

This will turn out to be a principle with strong logical consequences. For instance, it is usual to think that the Universe is ruled by the second law of thermodynamics. Everything began (somehow) in a primal state of order and has been proceeding irreversibly towards a state of disorder. We look around us now and find two things – order and disorder – then try to tell a mechanically logical story of how one must have been the necessary precursor state to the other,

Organic logic says that wherever we now find two crisp and complementary alternatives – mutually exclusive possibilities – then we must feel the urge to tell a different tale. Both would have arisen by separation out of a common vagueness. And so vagueness would be composed – by the precursor argument – of the potential for both these outcomes. To be dichotomised in just this strict fashion.

Therefore if order and disorder are properly complementary and mutually exclusive concepts, they must have both been equally present at the “true beginning”. So, oh dear!, the second law would have to be rewritten to make sense – to be logical – from the viewpoint of an organicist.

Again, something like a second law is just a model. So we are not attempting to rewrite reality but add a second complementary model to the existing mechanical one.

But anyway, the precursor argument is one method of modelling what is “inside” the realm of vagueness.

vagueness as symmetry

As will be argued elsewhere, there are many dichotomies but one dichotomisation logic. This core logic is arrived at by generalising. We boil down a lot of particular dichotomies until we distil one essential dichotomy.

This essential dichotomy is a dichotomisation based on scale. It has a dimensional description. And dimensionality becomes dichotomised by differentiating (and integrating!) across scale. There is a separation, and a thorough mixing.

A dichotomy based on scale is asymmetric. Marked or lop-sided in some fashion. Its symmetry is broken. There is the large and it is marked by the small. There is the slow and it is marked by the quick. There is nearly everything and it is marked by almost nothing.

The dichotomised is the realm of the crisply developed. It is where things get to when they have happened. So vagueness must be the opposite of whatever the crisp happens to be. It is the undeveloped, the unmarked, the not yet dichotomised. Or to put it more simply, it is the realm of the symmetric. A state of unbroken or entire symmetry.

vagueness can be modelled by symmetry So vagueness can be modelled as symmetry. Which is useful as there is a huge amount of mathematics based on the idea of perfect symmetry and the kinds of breakings that are possible. Symmetry is already the basis of physics and the source of many powerful principles such as Noether’s theorem.

But note. We are not saying vagueness IS symmetry. Symmetry is just the convenient model.

This does stack up neatly. Symmetry and asymmetry are complementary opposites. They are themselves a dichotomy. So if scale asymmetry is a suitable model of crisp reality, then scale symmetry ought to be a suitable model of vague reality.

We are assuming reality is a development from the vague to the crisp. We then construct a model (and models are always crisp) that parallels this outer reality as best we can. Symmetry would describe the vagueness and asymmetry the crispness. Dichotomisation is the process that moves reality from vague possibility to crisply developed existence.

actual symmetries of our universe

What is symmetry then? It is when a change makes no observable difference. You can move but it looks just the same as if you had stayed still.

The universe has three basic symmetries – which as Noether’s theorem asserts, creates a set of inertias or conservation laws.

three symmetries A body can spin and each full revolution will bring it back to where it started. Hence spin is a conserved property and an inertia. The total quantity of spin in the universe cannot be changed, just redistributed. And any particular body can spin freely. Once set spinning (if we ignore the decelerating effects of any friction, a body can spin at the same rate forever.

This is rotational symmetry. Another is translational symmetry. You can move three paces left and then three paces right and you will be back where you started, as if you never shifted. This symmetry gives us a conservation of momentum and inertial motion. A body can travel freely forever at the same speed in a straight line.

A third inertia is the result of symmetry in time. It could be called the inertia of existence. Fundamental particles look the same backwards or forwards in time. The total mass and energy are conserved even when it gets redistributed in decays or interactions. So in this sense they freely exist, or persist.

Of course, it is another matter to say a particle actually can travel backwards in time. But for now the point is that the standard laws of physics are based on the assumption of such a symmetry. And symmetry is tightly tied to a family of ideas such as dimensionality, free actions or inertias, meaningless or unobservable change (where a change can end up not looking like a change), and the consequent conservation or persistence of the “changee”.

Why do I even mention Noether’s principle and the actual symmetries of the universe here? Well, we need to generalise the notion of symmetry to see what it might mean as a model of vagueness.

A symmetry turns out to be a dimension – a direction of free travel in some sense. There is a location and its possible motions. A dimension is formed by the dichotomy of conserved location~inertial motion in fact. Two ways of freely staying the same. Whatever is at a spot, stays at the spot even when it moves. And a spot that has been set moving will continue moving in that way forever until it is interfered with – “observed” in some fashion.

All this goes back to the mechanics of Galileo and Newton. They divided the world – dichotomised it! – into inertias and accelerations. Certain kinds of motions happened freely because they stuck to the hidden gridlines of the fabric of spacetime. Other kinds involved actual observed changes because they deviated from the straight and narrow. For a moving or spinning body to go faster or slower requires some external push – an accelerative force. The universe was in fact a system with three or four inertial dimensions. And any actions that then took place “across” several dimensions had to be paid for by some form of energetic accounting. 

Einstein of course later completed the story by showing how spacetime really was a system. The local inertial freedoms were really part of a global context of contraints.

symmetries of the Infinoverse

The universe has a set of actual symmetries. Rotation, translation and “existence”. Symmetries are somehow the same as dimensions – degrees of freedom, seams of inertial action. Directions in which things can happen without seeming interference with the wider world.

When we count up the number of directions in which things can go we find that there are three. The universe is 3D. Well it is 4D if we also treat time as a dimension in which things travel. And everything travels in time even if its just stands still. So we could say that our dimensional system divides along the dichotomy of locations and motions, stasis and change. Three dimensions serve to measure the location of things and a fourth ticks off the intervals over which they change position.

More recently in physics the idea has grown that everything can be accounted for in terms of symmetries or dimensions. Fundamental particles and all their properties are really complex knots of dimensionality. A particle like an electron is not a zero-dimensional point but a higher dimensional shape like a loop of string or a curved surface, a brane.

So string theory suggest there is another kind of dichotomy, one between the four fully unfurled dimensions that make the global stage of spacetime and the six curled up dimensions which make the point-like objects that occupy the various locations in spacetime.

As it happens, the total of 10 dimensions for the actual universe (three for space, one for time, six for the strings) seems to be the result of a further dichotomisation. The mathematics of symmetry – or rather, symmetry-breaking – suggest that the total tally may be 11. This is known as M-theory. And M-theory has a number of puzzling dualities where the same mathemathics appears to have a “large” and a “small” interpretation.

This is a little by-the-by for the moment perhaps. But it is worth pointing out that standard physics uses symmetry-breaking to explain dimensionality. And that while the total number of crisply-existent dimensions may have increased from the four of Einstein to the 10 or 11 of string theory, our universe can be said to have remarkably few dimensions all things considered. It we were to pick some number randomly between zero and an infinity of dimensions, the actual number making up our universe seems tellingly close to one end of a spectrum of possibilities.

Anyway now we want to generalise the notion of symmetry and dimensionality to set up a suitable model of vagueness – a realm of unbroken and unlimited potential.

So let’s imagine an infinoverse. A realm with an infinity of directions. Or effectively a continuity of dimensionality.

What would it be like inside the infinoverse? It would seem that in such a place you would be pointing in absolutely every direction at once and thus in effect pointing in no particular direction at all. With no dimension broken off from any other, you could not select one to be seen to be moving in, and all the others to appear to be standing still. Every step in one direction would be a step in all directions.

paddling a boat in the fog Being a completely symmetrical realm, any kind of action or motion would produce no visible change. Much like paddling a boat in a still fog. Nothing ever gets nearer, nothing ever gets further away.

And the same would apply to the notion of time in the infinoverse. Existing for next to no time would be much the same as existing for nearly forever. You would have no landmarks to measure a passing of time and so a minute might be a second or a millenium as far as you could tell.

So an infinitely symmetric realm, one with an unbroken dimensionality, would be a very vague place indeed. It would seem anything could happen (or not happen) and there would be no visible difference. There exists every degree of freedom and so in some sense everything would be happening. The infinoverse would be busting with activity. It would not be a dead emptiness. But nothing definite would be happening.

Think of a block of marble, In some sense a solid block of marble contains the potential for every statue that ever existed. It is an everythingness of statutary. But in its unbroken state, it is not a something – some actual statue.

So symmetry does seem to make a good model of vagueness. Symmetry does suffer from the fact that it suggests rather too concrete a substrate for reality. There is a something – an infinite bunch of dimensions, a large slab of marble – waiting to have its symmetry broken, its existence constrained. Whittlled down to some lesser number of actual dimensions or some actual famous work of art like Rodin’s The Kiss. But for the sake of modelling, the infinoverse behaves like a primal vagueness in many essential ways.

from vagueness to dichotomies

So we have assembled a few tools with which to make some sense of this idea of vagueness. It can be modelled in the language of symmetry. And what must be reduced to featureless symmetry are the asymmetries that seem to pop out of it – the various dichotomies.

The main dichotomies we will keep returning to are local~global, substance~form, atom~void, location~motion, matter~mind, particular~general. All these are scale asymmetries even if it is not immediate apparent. Each is about something small within something large, a focus within a context, a figure marking a ground.

There is a kind of conical relationship here. The dichotomies start jumbled up together in vagueness. Location and motion, substance and form, atom and void, figure and ground – there is a lack of scale and so the potentials are as yet indistinguishable. Then there is the division. Scale grows and we have the dichotomy fanning out to make a world with fast developing distinctions.

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