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.
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.
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”.
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.
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.
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.
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.
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.
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.
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.
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.