The Grail Machine One

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Temporal propositions and the solution to the Gdelian 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 Gdels 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.


<p>The grail machine: one 2003 by Rolf Mifflin</p> <p>Return to Googol Room</p> <p>The grail machine: OneTemporal propositions and the solution to the Gdelian paradoxby Rolf Mifflin</p> <p>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 Gdels 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.</p> <p>Table of contents 1: Metaphysics and mathematics 2: The Gdelian argument 3: Oracles and pervasion 4: Atomic, spatial and temporal propositions 5: Grail machines 6: Solution to the Gdelian paradox 7: Consciousness, sensation and free will 8: SuperDeterminism and the transphysical problem 9: Conclusion</p> <p>1: Metaphysics and mathematics</p> <p>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</p> <p>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 berflssig, he wrote. In mathematics, the theory of classes has no function. (Tractatus LogicoPhilosophicus 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 Gdels 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</p> <p>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 Gdels Theorem will not be too daunting...</p> <p>2: The Gdelian argument</p> <p>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:</p> <p>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</p> <p>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 Gdelian 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 Gdelian 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</p> <p>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</p> <p>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:</p> <p>(ii)</p> <p>,</p> <p>,</p> <p>,</p> <p> (Turing evaluators)</p> <p>If the</p> <p>evaluator is operating on the Ti machine, call it 's interior to make it . Then</p> <p>. Think of the , by its identification</p> <p>computation i as encoded into</p> <p>as a Turing evaluator, satisfies the statement:</p> <p>(iii) If</p> <p>halts, then Ti does not halt.</p> <p>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:</p> <p>(iv)</p> <p>=Tm</p> <p>(v) Therefore, if Tm halts, then Ti does not halt.</p> <p>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:</p> <p>(vi) m=i</p> <p>(This clever trick is from a mathematician, Georg Cantor, to whom we owe a great deal of set theory.)</p> <p>(vii) If Tm halts, then Tm does not halt. We can deduce immediately from this self-reference:</p> <p>(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. Gdelian Incompleteness opens a certain unsettling hole in mathematical thought.</p> <p>3: Oracles and pervasion There is one more curious aspect of the Gdelian argument to mention, one which will direct us towards the solution and a clearer understanding of formalism. If we</p> <p>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 Gdelian 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?</p> <p>4: Atomic, spatial and temporal propositions</p> <p>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 nonconstructibility, but are often due to their mixture with the Axiom of Infinity, which shifts</p> <p>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 b...</p>