The grail
machine: one ©
2003 by Rolf Mifflin
The
grail machine: One
Temporal propositions and
the solution to the Gödelian paradox
by Rolf
Mifflin
Abstract: Here I present an extension to symbolic
logic - the temporal propositions, as well as the intuition on which they are
based. This extended logic allows
statements that emulate the entire breadth of human thought. As an example of their utility, I will present
language in this new formalism that exceeds the limitations of Gödel’s Second
Incompleteness Theorem. I will also
introduce a few of the scientific and philosophical ramifications of these
propositions and make initial suggestions regarding a symbolic theory through
which to formally and completely express our own minds and artificial minds.
Table of contents
1: Metaphysics and mathematics
4: Atomic, spatial and temporal
propositions
6: Solution to the Gödelian paradox
7: Consciousness, sensation and free will
8: SuperDeterminism and the transphysical problem
1:
Metaphysics and mathematics
We know the sensible world in which we are enveloped by our intuitions, that company of guiding urges granted by nature and the long history of our predecessors. They tug us one way when we thought to go another, whisper in our ear what we had forgotten, and seize our heart when we might falter. It is only natural that we seek to know them as keenly as possible, and so know our world keenly, too.
The parsing of intuition into its components has been the
business of philosophy, and for my argument two areas of philosophy are most
salient: metaphysics, the intuitions into which the physical world is embedded,
and mathematics, those same intuitions stripped of their physical
connotations. In these two places are
found the most fundamental statements of the physical world as they are so far
constructible.
The foundation of modern mathematics (as it is stated in
ZF, for example) is a fusion of two kinds of stripped intuition, atomic
intuition and spatial intuition. This
is an uncommon way of introducing the structure of mathematics, but it allows
the neat expression of the lacking third intuition, the temporal intuition.
In the early years of the 20th century, that great
proponent of atomic propositions, Ludwig Wittgenstein, made a number of
complaints about the adoption of certain logic by his contemporaries. Die
Theorie der Klassen ist in der Mathematik ganz überflüssig, he wrote. – In mathematics, the theory of classes has no
function. (Tractatus Logico-Philosophicus 6.031) Those very few statements of Wittgenstein's that are no longer
relevant, like this one, are directed at the spatial form in mathematics, which
a staunch atomist did not see as important.
It took some time for these two opposed intuitions to be merged into the
single theory we have today, and now the opportunity has come for the third to
be incorporated into modern formalism.
I wrote the third
intuition, but temporal prepositions may not be the only additional concept
necessary to complete formal symbolics.
Other intuitions may be required in addition to the temporal, depending,
for instance, on how we ultimately account for causation in both its
Hamiltonian and Quantum Mechanical forms.
I will treat that issue later on.
For now, I mention it to remind us that whatever formal systems we
construct, they may be but partly more complete than any preceding systems, and
still be surrounded by voids in need of deliberate exploration.
To quickly show the utility of a mechanism that might
otherwise seem more curious or might be misunderstood, I will begin directly with
a useful result, the circumvention of the restrictions that Gödel’s Second
Incompleteness Theorem puts on formal systems.
The Theorem tells us, in a nutshell, that a mathematician can deduce
more from a formal system than that formal system can deduce for itself. This leads to the conclusion that the human
mind can not be explained as a formal system.
This is troubling, in turn, because there were no known physical processes
that were not explained as formal systems until the arrival of Quantum Mechanics. The existence of Quantum Mechanics, as well
as the observed structure of time and a number of human phenomena like free
will and emotion, all strongly motivate the mode of logic proposed.
But these statements will become clearer. I will first work through the Theorem,
highlighting those aspects most salient to my discussion, and then work through
to the solution and on to some of its attending implications. As the addition I purpose is foundational,
and so does not require a fine understanding of the heights of mathematical
logic, my presentation of Gödel’s Theorem will not be too daunting...
In order to discuss the limitations of computation we will
need as general a statement as possible of what a computation is. All computations can be thought of as a list
of instructions in a formal language, or, slightly more generally, as a list of
symbols in a formal language. The
formal language tells us what the symbols mean. The computation tells us what symbols to use and in what
order. The formal language can be
thought of as a device called a Turing machine that carries out these
computation in much the way a computer carries out a program; I will symbolize
the Turing machine by T and give it an integer index i telling what computation
it is executing: Ti. We can
give each computation a unique integer based on its contents. For instance, if the formal language we are
using has ten symbols, we can give the one symbol computations the numbers 1
through 10, the two symbol computations the numbers 11 through 110, etc. (This
scheme gives numbers to all the senseless computations as well as to all the
sensible ones, but there is no loss.)
So we have, in general, a countable number of Turing machines working on
different computations:
(i)
T0 , T1 ,
T2 , T3 ... Ti ... (Turing machines)
It is natural to think of each computation as a list of
binary digits, like a computer program, as it is natural to think of the Turing
machine as a computer. In fact, general
purpose computers were largely built as physical versions of the idealized
devices conceived of by Turing, Church, Post, and others. The extended logic I will begin presenting
in the next section will also immediately suggests physical mechanisms for emulating
its behavior. But first, we must
carefully pare away some metaphors that are brought by the analogy of a
physical computer but that are separate from the logical ideal of the Turing
machine. Later discussion can easily
become confused by misunderstandings born at this level, so I will be specific
about these metaphoric issues early.
A Turing machine operates by executing its symbols and
thereby producing a list of internal states.
This suggests the flow of time, and in a computer there literally is
such a flow, a computer carries out its computations to the ticking of a clock
and each instruction or batch of instructions requires a block of time to
execute. The sequential internal states
of the machine occur at sequential instants in time. There is no flow of time for Turing machines. Turing machines are carefully divorced from
physical actions that are not purely logical, such as the flow of time. Mathematics is intuition divorced from the
physical.
What matters is that the operation of a Turing machine is
fully defined and fully explicable.
What that list of internal states may actually be is not always
important, but the fact that it exists, and could be described exactly, is
essential.
For Gödelian purposes one distinction between two different
types of Turing machines is especially important. Some Turing machines are said to halt, others are said not to
halt. If a Turing machine does not halt
then its full explication is an infinitely long list of internal states,
otherwise its full explication is finite.
This, in particular, suggests the flow of time, suggesting that a Turing
machine that does not halt requires an infinite amount of time to operate and
so its full statement must be unknowable.
But even an infinite number of steps is perfectly well defined and
perfectly explicable, the fact that the full list of internal states exists is
everything. The identification of
machines that do not halt as being such machines is the central problem of the Gödelian
argument.
The central problem is, specifically, determining whether a
Turing machine will not-halt and doing so in a finite number of steps. That is, can a process that halts determine
whether another process does not halt?
There are mechanical evaluating procedures that can be used to examine
Turing machines. These are themselves
computations, like the Turing machines.
I will call them Turing evaluators and symbolize them as 
. A Turing evaluator
examines a Turing machine, using its (the evaluator's) internal procedures to
determine whether the Turing machine does-not-halt. The evaluator halts if the
machine it is evaluating does not halt.
(If the evaluator is incapable of ascertaining the behavior of the
machine it is examining, it will not, itself, halt. It will operates forever, metaphorically, processing a problem
beyond its powers to evaluate.)
This may be easier to understand in symbolics. As Turing evaluators are computations in a formal language, there are an integer number of them, similar to the Turing machines:
(ii) 
, 
, 
, 
… 
… (Turing evaluators)
If the 
evaluator is
operating on the Ti machine, call it 
. Think of the
computation i as encoded into 
's interior to make it 
. Then 
, by its identification as a Turing evaluator, satisfies the
statement:
(iii) If 
halts, then Ti
does not halt.
Turing evaluators are not only similar to Turing machines;
their relationship is closer. Since the
list of Turing machines includes every possible computational machine, it must
include all the Turing evaluators as well, so Turing evaluators are, in fact,
Turing machines:
(iv) 
=Tm
(v) Therefore, if Tm halts, then Ti does not halt.
Now, although I have not presented the argument is enough
detail for it to be especially obvious, we have a great deal of freedom in the
way we number the Turing machines. We
can choose to construct a numbering system so that:
(vi) m=i
(This clever trick is from a mathematician, Georg Cantor,
to whom we owe a great deal of set theory.)
(vii) If Tm
halts, then Tm does not halt.
We can deduce immediately from this self-reference:
(viii) Tm
does not halt.
The surprise here is that we have a piece of information
that the evaluating procedure could not deduce. Since the procedure does not halt, Tm can not
determine whether Tm does not halt, but a mathematician executing
this proof can deduce so. This suggests
that the operation of the mathematician's mental processes can not be described
as a Turing machine. The assumption
that the mind is a Turing machine descends eventually into contradiction. But it had been an ideal of science that
every physical process would be describable in formal language and so would be
equivalent to a Turing machines.
Gödelian Incompleteness opens a certain unsettling hole in mathematical
thought.
There is one more curious aspect of the Gödelian argument
to mention, one which will direct us towards the solution and a clearer
understanding of formalism. If we
postulate a new machine, one that can answer the halting problem through some
undisclosed but always accurate procedure, the problem recreates itself.
Assume a new machine, that when fed the index of a Turing
machine returns a True or False, telling whether that Turing machine halts or
not. Then build a analogy to the Turing
machine that uses the new machine as a subroutine; we call this device an
Oracle machine and give it an index like we gave an index to the Turing
machines. Continue the argument, which
Oracles halt and which don't? The
situation is identical to that of the original Incompleteness argument. A mathematician can deduce more than an
Oracle machine itself can determine.
We can, furthermore, make 2nd-order Oracles and
repeat. We can repeat on to nth-order
Oracles. We can also claim that each
bit in the binary representation of a Turing machine is itself produced by a
subOracle and work our way down to mth-order subOracles. None of these gymnastics will recast the
problems into a soluble form.
This illuminates the first step towards understanding the
Gödelian paradox. Wherever the solution
lies, it must pervade. Wherever the
solution is thought to lie, we can rewrite our formalism so that it will appear
in the simplest symbols. It must
appear in that basal level, in the string of Trues and Falses that describe the
Turing machine, in the Trues and Falses themselves. From that base it pervades through all the nth-order Oracle and
mth-order subOracle machines. But what
modification can be made to the most fundamentals symbols of logic?
4: Atomic, spatial and temporal
propositions
The philosophical basis for modern symbolic logic is the
atomic proposition, a statement that is True or False. The world is considered to be a great
structure of interpenetrating atomic propositions. I have made the claim that modern logic is based on two kinds of
intuitions, and will say here that there are two kinds of propositions that
reflect these intuitions, atomic propositions and spatial propositions. Problems had with the non-constructive
axioms, for instance, are not always owing to their non-constructibility, but
are often due to their mixture with the Axiom of Infinity, which shifts one's
thinking from the atomic to the spatial.
The two modes of thinking can be difficult to reconcile and that
difficulty is often mistaken for something it is not.
There is a third variety of proposition suggested by this
claim: the temporal proposition. The
separation between these three intuitions is fundamental and not merely a
convenience, it transforms our models of nature and suggests more completely
the structure of the mind and physics.
That structure pervades from within mathematics and within
metaphysics. Metaphysics and
mathematics are the same ideal in two forms and both imply the universe in
itself. From the interactions and the
overlapping of these three intuitions within mathematics we can begin to educe
the foundational structure of the physical world.
Let me introduce the temporal proposition through an
example. There is a experimental device
in quantum physics called a Stern-Gerlach apparatus. In one experiment, spin-1/2 particles are shot through a magnetic
field in the device. The particles swerve either to the left or to the right
and land in one of two detectors set to catch them. The curious thing about these experiments is that each particle,
when the particles are prepared properly, will travel to both machines, on two
different paths, simultaneously. Only
when one of the two entangled paths reaches the detector will measurement
determine which of the two detectors the particle was actually bound for. Until then, the particle was bound for both.
Consider the following propositions:
(ix) The next
particle will be detected on the right.
(x) The next
particle will be detected on the left.
One of these statements will be True and the other will be
False, but we won't know which is which until the actual measurement
occurs. Classically, in accordance with
the idealizations of atomic propositions, logic considers one of these to be
True and the other False and the fact that we don't know until the measurement
event happens is considered a subjective distinction. Modern experimental evidence shows us, however, that we must
consider both statements to be True to explain more complex observations, while
still recognizing that they are contradictory.
In order for our systems of logic to better reflect the
observed universe we must expand atomic propositions.
The new component that forms the center of a temporal proposition
is the Unresolved truth-state, a truth-state analogous to the two classical
states, True and False. Unresolved
states are evaluated in two modes:
(xi) In the Future or Unresolved mode, an Unresolved truth-state is interpreted as a True
state in one of two distinct parallel structures and a False state in the other
structure, neither of which is preferred, although both exist.
(xii) In the Past or Resolved mode, an Unresolved truth-state is replaced by either a
True state or a False state. These
states are completely indistinguishable from a True or a False that did not
arise via Resolution. An Unresolved
truth-state becomes a True truth-state or a False truth-state.
At this level of discussion, I will not consider
probability. There is no statement as
to whether the two outcomes are of equal probability. That is a matter for extended formal languages to define. Here we assert only the existence of the two
particular peculiar states. As with
discussions of Turing machines, existence is all.
The two statements above, (ix) and (x), both look to the
future; they are statements presented in the Future mode. They are in the Unresolved state, but notice
the two statements do not incorporate two Unresolved logic variables. As they depend on entangled events, they are
a single Unresolved variable seen from two perspectives. From this we see that not-Unresolved is the
same as Unresolved, ~U=U.
[If you prefer, it is reasonable to imagine four logic
states instead, connected by two processes: an Unresolved/True state that
Resolution turns into a logical True, and an Unresolved/False state that
Resolution turns into a logical False.
Then, no choice between states seems to occur, in closer accord with
ordinary formal procedures. This may
seem more palatable to atomic thinking and it is equivalent to the method just
presented. The important point is that
Unresolved/True and Unresolved/False are completely indistinguishable without
Resolution, and so are indistinguishable from unmodified Unresolved truth-states.]
Now we have the equipment to solve Gödel’s paradox and
begin building formal models of the human mind. A grail machine is a Turing machine that includes Unresolved
truth-states in its structure; I will symbolize it as Ŧ. All Turing
machines are fully Resolved grail machines.
(But not vice versa.) All Turing
machines are also trivial grail machines, that is, grail machines with no
internal Unresolved truth-states.
To show how to proceed in arguments concerning grail machines,
I will present two specific examples.
Consider the grail machine ŦA:
(xiii) ŦA is two Turing
machines connected by a switch containing an Unresolved variable. If the state of the switch is True the
machine turns to a Turing machine that halts; if the state is False the machine
turns to a Turing machine that does not halt.
Does ŦA halt? Once it has
Resolved it either does or does not. An
evaluator that can handle either branch of the grail machine will tell you
whether it halts. The Gödelian argument
can be immediately constructed around this machine. Notice, however, that there is no one-to-one correspondence
between Turing machines and Unresolved grail machines.
Consider another grail machine, ŦB:
(xiv) To begin
with, ŦB prints out a zero.
Then it consults an Unresolved truth-state. If the truth-state is False, the machine halts. If the truth-state is True, the machine
repeats its procedure, consulting a new Unresolved truth-state, and so on.
Does this machine halt?
There is one case where it never halts and an infinity of cases where it
does. (אo cases where
it does. אo is the
smallest of the infinities, being the number of distinct integers.) The machine prints out anywhere from one to
אo zeros.
In order to evaluate this machine (assume it is fully
Resolved) another machine must evaluate an infinite series of truth-states
sequentially. The evaluator can never
halt if it is to ascertain that ŦB
does not halt. We can not construct
step (iii) for this machine. The
general solution begins to peep through.
6: Solution to the Gödelian paradox
It seems, perhaps, that we can use grail machines to
evaluate other grail machines, and thereby build the Gödelian argument around
these new machines as we did with the Oracles.
Let us try. First, notice that
we are no longer dealing with an integer number of Turing machines, Ti,
but with a real number of fully Resolved grail machines, Ŧr. To see this,
notice that the most general Unresolved grail machine is an infinitely long
list of Unresolved truth-states.
Resolution transforms this into an infinitely long list of Trues and
Falses, which is the same as the binary representation of a real number, an
infinitely long list of zeros and ones.
Analogous to our original procedure, we attempt to gather
grail evaluators, 
(xv) If 
This fails immediately.
A real number of distinct machine can not be evaluated in an integer
number of steps. The real numbers are
too dense to find associable evaluators for every possible machine index. To circumvent Gödelian restrictions, any
system with properties like those of the human proof-maker must, therefore, be
described as one of these non-evaluable grail machines, what we might call
irrational grail machines.
The human mind is an irrational grail machine. Multiple minds acting in concert are such an
irrational machine. All society is such
a machine. Nature around us is such a
machine. The universe itself is such a
machine.
What Gödel’s Second Incompleteness Theorem reminds us is
that the logical Future is richer in information than the logical Past, richer
in a certain variety of seemingly spontaneous information. The fact that we can identify the physical
future with the logical Future, and likewise the Past, was simply the reason
for choosing those names.
With these realizations, a great number of human mental
qualities become expressible in turns as solid as geometry. To deal with an open-ended future the mind
must incorporate open-ended strategies; later I will refer to these strategies
as Emotions. With these realizations,
other curious qualities of the physical universe also become eminently
expressible. The past is always a
Turing machine and completely describable, while the future is fundamentally
too dense in information. This will
allow the better scientific definition of the past and of the future.
7: Consciousness, sensation and
free will
I have said this development will allow the construction of
formal models of the human mind. From
there we may proceed immediately to practical mechanism in silico or other mediums.
These aspects will be the province of later essays, but here I will
introduce some of the implications as well as some basic arguments regarding
perception and the mind.
Consciousness is best represented as a set of inward
directed sense organs, the inner ear, for instance, that listens to each of our
own internal monologues, or the various body senses tied to our emotions and to
the maintenance of homeostasis. The
problem of what consciousness might be is precisely the problem of what our
physical sensations are. Why do our
sensations seem more than that which an inanimate object, like a rock or an
ocean wave, might feel?
Our awareness is the result of two commingled processes:
sensory data arriving in the brain and the resolution in the brain of
Unresolved logic states. The resolution
of those logic states is the quality of sensation. Sensation is the resolution of information by the passage of
time. We are lit up by the arrival of
information from the future. We are lit
from within. As a light bulb filament
sprays photons ejected by the passage of an electrical current, our minds light
with the transformation of the multi-ordered future into the single-ordered
past. We are ourselves illuminated by the resolution of information. Sensation, and consciousness, are an
information process that appears latently everywhere in the quantum world
combined with structures of potential information within us. The universe thinks empty thoughts
everywhere and we have the extended structure needed to make these empty
thoughts our own thoughts.
Free Will is the active expression of the mind, and
identical to the processes of sensation and consciousness. Where sensation is inward directed, using
sensory data to illuminate the sensing regions of the mind, Free Will is
outward directed, illuminating the physically expressive regions of the
mind. Free Will is an inescapable
necessity for sensation, for consciousness, for awareness; it is the
fundamental mechanism of thought itself.
8: SuperDeterminism and the
transphysical problem
I have made no mention of the probabilities associated with
state-reduction in Quantum Mechanics. I
only mentioned Quantum Mechanics to provide a physical example of the phenomena
in need of clear explanation. How
Unresolved truth-states become Resolved is not formally important; the probabilities themselves are unnecessary
to explain the logical structure of time, although they are mechanically
necessary to explain observations.
The mind is not deterministic, in that its present state
does not determine its future states, but only limits them to a certain field
of possibilities. The universe is,
likewise, semi-deterministic, mixing its
present state with information embedded in the future as that
information arrives. But we can not
claim with surety that, simply because that information lies in our own future,
it has not been generated by some process as exacting as Hamilton's
principle. If the information appears
to us as purely stochastic, we can only claim it depends on nothing we can
observe. Something that is apparently
stochastic might be instead the result of an exacting but orthogonal process.
If there is an exacting process that determines Quantum
Mechanical events and if it operates in a realm inaccessible to our
experiments, we can not differentiate it from a purely stochastic one. But a covert exacting theory, a theory that
explains state-reduction as the result of non-probabilistic, but hidden,
process, might satisfy our need for harmony or parsimony in Nature.
SuperDeterminism is the idea that the universe is
determined by two confluent processes of causation, one clearly deterministic,
the other apparently stochastic. (Two
processes identical to what Kant called, in the equivalents of his age, the
Sublime and the Beautiful.) SuperDeterminism
does not claim that the second process is necessarily either a hidden exacting
one or a completely probabilistic one.
It claims that the two interpretations are indistinguishable, and the
difference irrelevant. The universe is,
at most, covertly deterministic. To
explain the universe, determinism can not be adopted because it assumes too
much structure, covert determinism can only be tolerated because it says
nothing of relevance, and SuperDeterminism is only preferred because it asserts
unknowability.
This unknowability is the transphysical problem. Time may be explicable as the intersection
of two purely atomic (metaphysically atomic) processes, but one of these two
processes may be utterly meaningless to ourselves or utterly
undiscoverable. Wittgenstein may have
been right, that everything is atomic, but that fact may be unknowable and that
unknowability may make the universe what it is.
But then, at some moment in the future, perhaps remote
beside the duration of our own lives, some experimenter may stumble upon a
symmetry that determines the evolution of the universe exactly, exposing
causation as a grand and elegant thing of two identical halves, or as something
even stranger, and so transform our certain unknowability into something more mysterious.
This, by the way, is exactly the predicament a grail
machine finds itself in, and that a Turing machine never can. Its final resolution may exist somewhere in
the future, but unknowably so. It may take an infinite amount of time to
evaluate causation and this grail-emulating problem. That we are grail machines appears buried in every root of our
thought, emotion and philosophy.
The history of thought is less a history of invention than
a history of unification. The language
each age invents to explain the sensible world skirts the essential. That what we think right now has been
thought on numberless occasions before
by countless minds is less significant than our realization that two
dissimilar thoughts are truly the same.
In the Critique of Judgment, Kant divides our understanding
into two forms, one of the Beautiful and one of the Sublime. These forms are precisely echoed in our
modern understanding of the physical world; they arise precisely from the two
orthogonal modes of causation. Every
physical object or process can be divided into an aspect driven by energetic
causation, which one age calls the principle of least-action while another says
the Sublime, and an aspect driven by quantum causation, which one age calls
state-reduction while another says the Beautiful.
Temporal propositions show us how to draw wider and wider
reaches of thought into the folds of formalism, showing us that what we thought
was a division is no division at all.
They lead towards clearer considerations of everything from space-time
and the structure of causation to Darwinian and superDarwinian theories. We are lead to clearer divisions in
philosophy, and to better tacks into the knowledge of everything from the
object-in-itself to the moral sciences.
Most immediately, though, we are lead into considerations
for the understanding and symbolizing of the mind in its general form, from
which we will be able to proceed to its construction in a variety of
mediums. The next step on this path
will be the introduction of Free Will as a constructible phenomenon. Free Will, as I will define it, sits at the
heart of every variety of mental information processing. It is identical to that curious living
sensation we have in our minds, that thing demanding its difference from any
mechanical process around it, demanding it is more. And it is correct...
Some suggested reading:
Penrose, Roger. Shadows of the Mind. Oxford
University Press. New York, 1994.
Penrose, Roger. The Large, the Small and the Human Mind. Cambridge University Press. Cambridge, UK, 2000.
Hofstadter, D. Gödel, Escher, Bach: An Eternal Golden
Braid. Basic Books. New York, 1979.
Wittgenstein, Ludwig. German with English trans. by Ogden,
C. K. Tractatus Logico-Philosophicus. Routledge. London, 1999.
Just, Winfried & Weese, Martin. Discovering Modern
Set Theory: I. American Mathematical Society. Providence, Rhode Island,
1998.