The grail machine: one © 2003 by Rolf Mifflin

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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

2: The Gödelian argument

3: Oracles and pervasion

4: Atomic, spatial and temporal propositions

5: Grail machines

6: Solution to the Gödelian paradox

7: Consciousness, sensation and free will

8: SuperDeterminism and the transphysical problem

9: Conclusion

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...

2: The Gödelian argument

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:

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.

3: Oracles and pervasion

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.]

5: Grail machines

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, s,

for all possible fully Resolved grail machines:

(xv) If s halts, then Ŧr does not halt.

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.

9: Conclusion

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.