Knowledge Limited for Dummies….Didactics.

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Bertrand Russell with Alfred North Whitehead, in the Principia Mathematica aimed to demonstrate that “all pure mathematics follows from purely logical premises and uses only concepts defined in logical terms.” Its goal was to provide a formalized logic for all mathematics, to develop the full structure of mathematics where every premise could be proved from a clear set of initial axioms.

Russell observed of the dense and demanding work, “I used to know of only six people who had read the later parts of the book. Three of those were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated.” The complex mathematical symbols of the manuscript required it to be written by hand, and its sheer size – when it was finally ready for the publisher, Russell had to hire a panel truck to send it off – made it impossible to copy. Russell recounted that “every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up.”

Momentous though it was, the greatest achievement of Principia Mathematica was realized two decades after its completion when it provided the fodder for the metamathematical enterprises of an Austrian, Kurt Gödel. Although Gödel did face the risk of being liquidated by Hitler (therefore fleeing to the Institute of Advanced Studies at Princeton), he was neither a Pole nor a Texan. In 1931, he wrote a treatise entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which demonstrated that the goal Russell and Whitehead had so single-mindedly pursued was unattainable.

The flavor of Gödel’s basic argument can be captured in the contradictions contained in a schoolboy’s brainteaser. A sheet of paper has the words “The statement on the other side of this paper is true” written on one side and “The statement on the other side of this paper is false” on the reverse. The conflict isn’t resolvable. Or, even more trivially, a statement like; “This statement is unprovable.” You cannot prove the statement is true, because doing so would contradict it. If you prove the statement is false, then that means its converse is true – it is provable – which again is a contradiction.

The key point of contradiction for these two examples is that they are self-referential. This same sort of self-referentiality is the keystone of Gödel’s proof, where he uses statements that imbed other statements within them. This problem did not totally escape Russell and Whitehead. By the end of 1901, Russell had completed the first round of writing Principia Mathematica and thought he was in the homestretch, but was increasingly beset by these sorts of apparently simple-minded contradictions falling in the path of his goal. He wrote that “it seemed unworthy of a grown man to spend his time on such trivialities, but . . . trivial or not, the matter was a challenge.” Attempts to address the challenge extended the development of Principia Mathematica by nearly a decade.

Yet Russell and Whitehead had, after all that effort, missed the central point. Like granite outcroppings piercing through a bed of moss, these apparently trivial contradictions were rooted in the core of mathematics and logic, and were only the most readily manifest examples of a limit to our ability to structure formal mathematical systems. Just four years before Gödel had defined the limits of our ability to conquer the intellectual world of mathematics and logic with the publication of his Undecidability Theorem, the German physicist Werner Heisenberg’s celebrated Uncertainty Principle had delineated the limits of inquiry into the physical world, thereby undoing the efforts of another celebrated intellect, the great mathematician Pierre-Simon Laplace. In the early 1800s Laplace had worked extensively to demonstrate the purely mechanical and predictable nature of planetary motion. He later extended this theory to the interaction of molecules. In the Laplacean view, molecules are just as subject to the laws of physical mechanics as the planets are. In theory, if we knew the position and velocity of each molecule, we could trace its path as it interacted with other molecules, and trace the course of the physical universe at the most fundamental level. Laplace envisioned a world of ever more precise prediction, where the laws of physical mechanics would be able to forecast nature in increasing detail and ever further into the future, a world where “the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule.”

What Gödel did to the work of Russell and Whitehead, Heisenberg did to Laplace’s concept of causality. The Uncertainty Principle, though broadly applied and draped in metaphysical context, is a well-defined and elegantly simple statement of physical reality – namely, the combined accuracy of a measurement of an electron’s location and its momentum cannot vary far from a fixed value. The reason for this, viewed from the standpoint of classical physics, is that accurately measuring the position of an electron requires illuminating the electron with light of a very short wavelength. But the shorter the wavelength the greater the amount of energy that hits the electron, and the greater the energy hitting the electron the greater the impact on its velocity.

What is true in the subatomic sphere ends up being true – though with rapidly diminishing significance – for the macroscopic. Nothing can be measured with complete precision as to both location and velocity because the act of measuring alters the physical properties. The idea that if we know the present we can calculate the future was proven invalid – not because of a shortcoming in our knowledge of mechanics, but because the premise that we can perfectly know the present was proven wrong. These limits to measurement imply limits to prediction. After all, if we cannot know even the present with complete certainty, we cannot unfailingly predict the future. It was with this in mind that Heisenberg, ecstatic about his yet-to-be-published paper, exclaimed, “I think I have refuted the law of causality.”

The epistemological extrapolation of Heisenberg’s work was that the root of the problem was man – or, more precisely, man’s examination of nature, which inevitably impacts the natural phenomena under examination so that the phenomena cannot be objectively understood. Heisenberg’s principle was not something that was inherent in nature; it came from man’s examination of nature, from man becoming part of the experiment. (So in a way the Uncertainty Principle, like Gödel’s Undecidability Proposition, rested on self-referentiality.) While it did not directly refute Einstein’s assertion against the statistical nature of the predictions of quantum mechanics that “God does not play dice with the universe,” it did show that if there were a law of causality in nature, no one but God would ever be able to apply it. The implications of Heisenberg’s Uncertainty Principle were recognized immediately, and it became a simple metaphor reaching beyond quantum mechanics to the broader world.

This metaphor extends neatly into the world of financial markets. In the purely mechanistic universe of classical physics, we could apply Newtonian laws to project the future course of nature, if only we knew the location and velocity of every particle. In the world of finance, the elementary particles are the financial assets. In a purely mechanistic financial world, if we knew the position each investor has in each asset and the ability and willingness of liquidity providers to take on those assets in the event of a forced liquidation, we would be able to understand the market’s vulnerability. We would have an early-warning system for crises. We would know which firms are subject to a liquidity cycle, and which events might trigger that cycle. We would know which markets are being overrun by speculative traders, and thereby anticipate tactical correlations and shifts in the financial habitat. The randomness of nature and economic cycles might remain beyond our grasp, but the primary cause of market crisis, and the part of market crisis that is of our own making, would be firmly in hand.

The first step toward the Laplacean goal of complete knowledge is the advocacy by certain financial market regulators to increase the transparency of positions. Politically, that would be a difficult sell – as would any kind of increase in regulatory control. Practically, it wouldn’t work. Just as the atomic world turned out to be more complex than Laplace conceived, the financial world may be similarly complex and not reducible to a simple causality. The problems with position disclosure are many. Some financial instruments are complex and difficult to price, so it is impossible to measure precisely the risk exposure. Similarly, in hedge positions a slight error in the transmission of one part, or asynchronous pricing of the various legs of the strategy, will grossly misstate the total exposure. Indeed, the problems and inaccuracies in using position information to assess risk are exemplified by the fact that major investment banking firms choose to use summary statistics rather than position-by-position analysis for their firmwide risk management despite having enormous resources and computational power at their disposal.

Perhaps more importantly, position transparency also has implications for the efficient functioning of the financial markets beyond the practical problems involved in its implementation. The problems in the examination of elementary particles in the financial world are the same as in the physical world: Beyond the inherent randomness and complexity of the systems, there are simply limits to what we can know. To say that we do not know something is as much a challenge as it is a statement of the state of our knowledge. If we do not know something, that presumes that either it is not worth knowing or it is something that will be studied and eventually revealed. It is the hubris of man that all things are discoverable. But for all the progress that has been made, perhaps even more exciting than the rolling back of the boundaries of our knowledge is the identification of realms that can never be explored. A sign in Einstein’s Princeton office read, “Not everything that counts can be counted, and not everything that can be counted counts.”

The behavioral analogue to the Uncertainty Principle is obvious. There are many psychological inhibitions that lead people to behave differently when they are observed than when they are not. For traders it is a simple matter of dollars and cents that will lead them to behave differently when their trades are open to scrutiny. Beneficial though it may be for the liquidity demander and the investor, for the liquidity supplier trans- parency is bad. The liquidity supplier does not intend to hold the position for a long time, like the typical liquidity demander might. Like a market maker, the liquidity supplier will come back to the market to sell off the position – ideally when there is another investor who needs liquidity on the other side of the market. If other traders know the liquidity supplier’s positions, they will logically infer that there is a good likelihood these positions shortly will be put into the market. The other traders will be loath to be the first ones on the other side of these trades, or will demand more of a price concession if they do trade, knowing the overhang that remains in the market.

This means that increased transparency will reduce the amount of liquidity provided for any given change in prices. This is by no means a hypothetical argument. Frequently, even in the most liquid markets, broker-dealer market makers (liquidity providers) use brokers to enter their market bids rather than entering the market directly in order to preserve their anonymity.

The more information we extract to divine the behavior of traders and the resulting implications for the markets, the more the traders will alter their behavior. The paradox is that to understand and anticipate market crises, we must know positions, but knowing and acting on positions will itself generate a feedback into the market. This feedback often will reduce liquidity, making our observations less valuable and possibly contributing to a market crisis. Or, in rare instances, the observer/feedback loop could be manipulated to amass fortunes.

One might argue that the physical limits of knowledge asserted by Heisenberg’s Uncertainty Principle are critical for subatomic physics, but perhaps they are really just a curiosity for those dwelling in the macroscopic realm of the financial markets. We cannot measure an electron precisely, but certainly we still can “kind of know” the present, and if so, then we should be able to “pretty much” predict the future. Causality might be approximate, but if we can get it right to within a few wavelengths of light, that still ought to do the trick. The mathematical system may be demonstrably incomplete, and the world might not be pinned down on the fringes, but for all practical purposes the world can be known.

Unfortunately, while “almost” might work for horseshoes and hand grenades, 30 years after Gödel and Heisenberg yet a third limitation of our knowledge was in the wings, a limitation that would close the door on any attempt to block out the implications of microscopic uncertainty on predictability in our macroscopic world. Based on observations made by Edward Lorenz in the early 1960s and popularized by the so-called butterfly effect – the fanciful notion that the beating wings of a butterfly could change the predictions of an otherwise perfect weather forecasting system – this limitation arises because in some important cases immeasurably small errors can compound over time to limit prediction in the larger scale. Half a century after the limits of measurement and thus of physical knowledge were demonstrated by Heisenberg in the world of quantum mechanics, Lorenz piled on a result that showed how microscopic errors could propagate to have a stultifying impact in nonlinear dynamic systems. This limitation could come into the forefront only with the dawning of the computer age, because it is manifested in the subtle errors of computational accuracy.

The essence of the butterfly effect is that small perturbations can have large repercussions in massive, random forces such as weather. Edward Lorenz was testing and tweaking a model of weather dynamics on a rudimentary vacuum-tube computer. The program was based on a small system of simultaneous equations, but seemed to provide an inkling into the variability of weather patterns. At one point in his work, Lorenz decided to examine in more detail one of the solutions he had generated. To save time, rather than starting the run over from the beginning, he picked some intermediate conditions that had been printed out by the computer and used those as the new starting point. The values he typed in were the same as the values held in the original simulation at that point, so the results the simulation generated from that point forward should have been the same as in the original; after all, the computer was doing exactly the same operations. What he found was that as the simulated weather pattern progressed, the results of the new run diverged, first very slightly and then more and more markedly, from those of the first run. After a point, the new path followed a course that appeared totally unrelated to the original one, even though they had started at the same place.

Lorenz at first thought there was a computer glitch, but as he investigated further, he discovered the basis of a limit to knowledge that rivaled that of Heisenberg and Gödel. The problem was that the numbers he had used to restart the simulation had been reentered based on his printout from the earlier run, and the printout rounded the values to three decimal places while the computer carried the values to six decimal places. This rounding, clearly insignificant at first, promulgated a slight error in the next-round results, and this error grew with each new iteration of the program as it moved the simulation of the weather forward in time. The error doubled every four simulated days, so that after a few months the solutions were going their own separate ways. The slightest of changes in the initial conditions had traced out a wholly different pattern of weather.

Intrigued by his chance observation, Lorenz wrote an article entitled “Deterministic Nonperiodic Flow,” which stated that “nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states.” Translation: Long-range weather forecasting is worthless. For his application in the narrow scientific discipline of weather prediction, this meant that no matter how precise the starting measurements of weather conditions, there was a limit after which the residual imprecision would lead to unpredictable results, so that “long-range forecasting of specific weather conditions would be impossible.” And since this occurred in a very simple laboratory model of weather dynamics, it could only be worse in the more complex equations that would be needed to properly reflect the weather. Lorenz discovered the principle that would emerge over time into the field of chaos theory, where a deterministic system generated with simple nonlinear dynamics unravels into an unrepeated and apparently random path.

The simplicity of the dynamic system Lorenz had used suggests a far-reaching result: Because we cannot measure without some error (harking back to Heisenberg), for many dynamic systems our forecast errors will grow to the point that even an approximation will be out of our hands. We can run a purely mechanistic system that is designed with well-defined and apparently well-behaved equations, and it will move over time in ways that cannot be predicted and, indeed, that appear to be random. This gets us to Santa Fe.

The principal conceptual thread running through the Santa Fe research asks how apparently simple systems, like that discovered by Lorenz, can produce rich and complex results. Its method of analysis in some respects runs in the opposite direction of the usual path of scientific inquiry. Rather than taking the complexity of the world and distilling simplifying truths from it, the Santa Fe Institute builds a virtual world governed by simple equations that when unleashed explode into results that generate unexpected levels of complexity.

In economics and finance, institute’s agenda was to create artificial markets with traders and investors who followed simple and reasonable rules of behavior and to see what would happen. Some of the traders built into the model were trend followers, others bought or sold based on the difference between the market price and perceived value, and yet others traded at random times in response to liquidity needs. The simulations then printed out the paths of prices for the various market instruments. Qualitatively, these paths displayed all the richness and variation we observe in actual markets, replete with occasional bubbles and crashes. The exercises did not produce positive results for predicting or explaining market behavior, but they did illustrate that it is not hard to create a market that looks on the surface an awful lot like a real one, and to do so with actors who are following very simple rules. The mantra is that simple systems can give rise to complex, even unpredictable dynamics, an interesting converse to the point that much of the complexity of our world can – with suitable assumptions – be made to appear simple, summarized with concise physical laws and equations.

The systems explored by Lorenz were deterministic. They were governed definitively and exclusively by a set of equations where the value in every period could be unambiguously and precisely determined based on the values of the previous period. And the systems were not very complex. By contrast, whatever the set of equations are that might be divined to govern the financial world, they are not simple and, furthermore, they are not deterministic. There are random shocks from political and economic events and from the shifting preferences and attitudes of the actors. If we cannot hope to know the course of the deterministic systems like fluid mechanics, then no level of detail will allow us to forecast the long-term course of the financial world, buffeted as it is by the vagaries of the economy and the whims of psychology.

Vector Representations and Why Would They Deviate From Projective Geometry? Note Quote.

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There is, of course, a definite reason why von Neumann used the mathematical structure of a complex Hilbert space for the formalization of quantum mechanics, but this reason is much less profound than it is for Riemann geometry and general relativity. The reason is that Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics turned out to be equivalent, the first being a formalization of the new mechanics making use of l2, the set of all square summable complex sequences, and the second making use of L2(R3), the set of all square integrable complex functions of three real variables. The two spaces l2 and L2(R3) are canonical examples of a complex Hilbert space. This means that Heisenberg and Schrödinger were working already in a complex Hilbert space, when they formulated matrix mechanics and wave mechanics, without being aware of it. This made it a straightforward choice for von Neumann to propose a formulation of quantum mechanics in an abstract complex Hilbert space, reducing matrix mechanics and wave mechanics to two possible specific representations.

One problem with the Hilbert space representation was known from the start. A (pure) state of a quantum entity is represented by a unit vector or ray of the complex Hilbert space, and not by a vector. Indeed vectors contained in the same ray represent the same state or one has to renormalize the vector that represents the state after it has been changed in one way or another. It is well known that if rays of a vector space are called points and two dimensional subspaces of this vector space are called lines, the set of points and lines corresponding in this way to a vector space, form a projective geometry. What we just remarked about the unit vector or ray representing the state of the quantum entity means that in some way the projective geometry corresponding to the complex Hilbert space represents more intrinsically the physics of the quantum world as does the Hilbert space itself. This state of affairs is revealed explicitly in the dynamics of quantum entities, that is built by using group representations, and one has to consider projective representations, which are representations in the corresponding projective geometry, and not vector representations.

Representation in the Philosophy of Science.

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The concept of representation has gained momentum in the philosophy of science. The simplest concept of representation conceivable is expressed by the following dyadic predicate: structure S(HeB) represents HeB. Steven French defended that to represent something in science is the same as to have a model for it, where models are set-structures; then ‘representation’ and ‘model’ become synonyms and so do ‘to represent’ and ‘to model’. Nevertheless, this simplest conception was quickly thrown overboard as too simple by amongst others Ronald Giere, who replaced this dyadic predicate with a quadratic predicate to express a more involved concept of representation:

Scientist S uses model S to represent being B for purpose P,

where ‘model’ can here be identified with ‘structure’. Another step was set by Bas van Fraassen. As early as 1994, in his contribution to J. Hilgevoord’s Physics and our View of the World, Van Fraassen brought Nelson Goodman’s distinction between representation-of and representation-as — drawn in his seminal Languages of Art – to bear on science; he went on to argue that all representation in science is representation-as. We represent a Helium atom in a uniform magnetic field as a set-theoretical wave-mechanical structure S(HeB). In his new tome Scientific Representation, Van Fraassen has moved essentially to a hexadic predicate to express the most fundamental and most involved concept of representation to date:

Repr(S, V, S, B, F, P) ,

which reads: subject or scientist S is V -ing artefact S to represent B as an F for purpose P. Example: In the 1920ies, Heisenberg (S) constructed (V) a mathematical object (S) to represent a Helium atom (B) as a wave-mechanical structure (F) to calculate its electro-magnetic spectrum (P). We concentrate on the following triadic predicate, which is derived from the fundamental hexadic one:

ReprAs(S, B, F) iff ∃S, ∃V, ∃P : Repr(S, V, A, B, F, P)

which reads: abstract object S represents being B as an F, so that F(S).

Giere, Van Fraassen and contemporaries are not the first to include manifestations of human agency in their analysis of models and representation in science. A little more than most half a century ago, Peter Achinstein expounded the following as a characteristic of models in science:

A theoretical model is treated as an approximation useful for certain purposes. (…) The value of a given model, therefore, can be judged from different though related viewpoints: how well it serves the purposes for which it is eimployed, and the completeness and accuracy of the representation it proposes. (…) To propose something as a model of X is to suggest it as way of representing X which provides at least some approximation of the actual situation; moreover, it is to admit the possibility of alternative representations useful for different purposes.

One year later, M.W. Wartofsky explicitly proposed, during the Annual Meeting of the American Philosophical Association, Western Division, Philadelphia, 1966, to consider a model as a genus of representation, to take in that representation involves “relevant respects for relevant for purposes”, and to consider “the modelling relation triadically in this way: M(S,x,y), where S takes x as a model of y”.20 Two years later, in 1968, Wartofsky wrote in his essay ‘Telos and Technique: Models as Modes of Action’ the following:

In this sense, models are embodiments of purpose and, at the same time, instruments for carrying out such purposes. Let me attempt to clarify this idea. No entity is a model of anything simply by virtue of looking like, or being like, that thing. Anything is like anything else in an infinite number of respects and certainly in some specifiable respect; thus, if I like, I may take anything as a model of anything else, as long as I can specify the respect in which I take it. There is no restriction on this. Thus an array of teacups, for example, may be take as a model for the employment of infantry battalions, and matchsticks as models of mu-mesons, there being some properties that any of these things share with the others. But when we choose something to be a model, we choose it with some end in view, even when that end in view is simply to aid the imagination or the understanding. In the most trivial cases, then, the model is already normative and telic. It is normative in that is chosen to represent abstractly only certain features of the thing we model, not everything all at once, but those features we take to be important or significant or valuable. The model is telic in that significance and value can exist only with respect to some end in view or purpose that the model serves.

Further, during the 1950ies and 1960ies the role of analogies, besides that of models, was much discussed among philosophers of science (Hesse, Achinstein, Girill, Nagel, Braithwaite, Wartofsky).

On the basis of the general concept of representation, we can echo Wartofsky by asserting that almost anything can represent everything for someone for some purpose. In scientific representations, representans and representandum will share some features, but not all features, because to represent is neither to mirror nor to copy. Realists, a-realists and anti-realists will all agree that ReprAs(S, B, F) is true only if on the basis of F(S) one can save all phenomena that being B gives rise to, i.e. one can calculate or accommodate all measurement results obtained from observing B or experimenting with B. Whilst for structural empiricists like Van Fraassen this is also sufficient, for StrR it is not. StrR will want to add that structure S of type F ‘is realised’, that S of type F truly is the structure of being B or refers to B, so that also F(B). StrR will want to order the representations of being B that scientists have constructed during the course of history as approaching the one and only true structure of B, its structure an sich, the Kantian regulative ideal of StrR. But this talk of truth and reference, of beings and structures an sich, is in dissonance with the concept of representation-as.

Some being B can be represented as many other things and all the ensuing representations are all hunky-dory if each one serves some purpose of some subject. When the concept of representation-as is taken as pivotal to make sense of science, then the sort of ‘perspectivalism’ that Giere advocates is more in consonance with the ensuing view of science than realism is. Giere attempts to hammer a weak variety of realism into his ‘perspectivalism’: all perspectives are perspectives on one and the same reality and from every perspective something is said that can be interpreted realistically: in certain respects the representans resembles its representandum to certain degrees. A single unified picture of the world is however not to be had. Nancy Cartwright’s dappled world seems more near to Giere’s residence of patchwork realism. A unified picture of the physical world that realists dream of is completely out of the picture here. With friends like that, realism needs no enemies.

There is prima facie a way, however, for realists to express themselves in terms of representation, as follows. First, fix the purpose P to be: to describe the world as it is. When this fixed purpose leaves a variety of representations on the table, then choose the representation that is empirically superior, that is, that performs best in terms of describing the phenomena, because the phenomena are part of the world. This can be established objectively. When this still leaves more than one representation on the table, which thus save the phenomena equally well, choose the one that best explains the phenomena. In this context, Van Fraassen mentions the many interpretations of QM: each one constitutes a different representation of the same beings, or of only the same observable beings (phenomena), their similarities notwithstanding. Do all these interpre- tations provide equally good explanations? This can be established objectively too, but every judgment here will depend on which view of explanation is employed. Suppose we are left with a single structure A, of type G. Then we assert that ‘G(B)’ is true. When this ‘G’ predicates structure to B, we still need to know what ‘structure’ literally means in order to know what it is that we attribute to B, of what A is that B instantiates, and, even more important, we need to know this for our descriptivist account of reference, which realists need in order to be realists. Yes, we now have arrived where we were at the end of the previous two Sections. We conclude that this way for realists, to express themselves in terms of representation, is a dead end. The concept of representation is not going to help them.

The need for substantive accounts of truth and reference fade away as soon as one adopts a view of science that takes the concept of representation-as as its pivotal concept. Fundamentally different kinds of mathematical structure, set-theoretical and category-theoretical, can then easily be accommodated. They are ‘only representations’. That is moving away from realism, StrR included, dissolving rather than solving the problem for StrR of clarifying its Central Claim of what it means to say that being B is or has structure S — ‘dissolved’, because ‘is or has’ is replaced with ‘is represented-as’. Realism wants to know what B is, not only how it can be represented for someone who wants to do something for some purpose. When we take it for granted that StrR needs substantive accounts of truth and reference, more specifically a descriptivist account of reference and then an account of truth by means of reference, then a characterisation of structure as directly as possible, without committing one to a profusion of abstract objects, is mandatory.

The Characterisation of Structure

Roger Penrose and Artificial Intelligence: Revenance from the Archives and the Archaic.

Let us have a look at Penrose and his criticisms of strong AI, and does he come out as a winner. His Emperor’s New Mind: Concerning Computers, Minds, and The Laws of Physics

sets out to deal a death blow to the project of strong AI. Even while showing humility, like in saying,

My point of view is an unconventional among physicists and is consequently one which is unlikely to be adopted, at present, by computer scientists or physiologists,

he is merely stressing on his speculative musings. Penrosian arguments ala Searle, are definitely opinionated, in making assertions like a conscious mind cannot work like a computer. He grants the possibility of artificial machines coming into existence, and even superseding humans (1), but at every moment remains convinced that algorithmic machines are doomed to subservience. Penrose’s arguments proceed through showing that human intelligence cannot be implemented by any Turing machine equivalent computer, and human mind as not algorithmically based that could capture the Turing machine equivalent. He is even sympathetic to Searle’s Chinese Room argument, despite showing some reservations against its conclusions.

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The speculative nature of his arguments question people as devices which compute that a Turing machine cannot, despite physical laws that allow such a construction of a device as a difficult venture. This is where his quantum theory sets in, with U and R (Unitary process and Reduction process respectively) acting on quantum states that help describe a quantum system. His U and R processes and the states they act upon are not only independent of observers, but at the same time real, thus branding him as a realist. What happens is an interpolation that occurs between Unitary Process and Reductive Process, a new procedure that essentially contains a non-algorithmic element takes shape, which effectuates a future that cannot be computable based on the present, even though it could be determined that way. This radically new concept which is applied to space-time is mapped onto the depths of brain’s structure, and for Penrose, the speculative possibility occurs in what he terms the Phenomenon of Brain Plasticity. As he says,

Somewhere within the depths of the brain, as yet unknown cells are to be found of single quantum sensitivity, such that synapses becoming activate or deactivated through the growth of contraction of dendritic spines…could be governed by something like the processes involved in quasi-crystal growth. Thus, not just one of the possible alternative arrangements is tried out, but vast numbers, all superposed in complex linear superposition.

From the above, it is deduced that the impact is only on the conscious mind, whereas the unconscious mind is left to do with algorithmic computationality. Why is this important for Penrose is, since, as a mathematician believes in the mathematical ideas as populating an ideal Platonic world, and which in turn is accessible only via the intellect. And harking back to the non-locality principle within quantum theory, it is clear that true intellect requires consciousness, and the mathematician’s conscious mind has a direct route to truth. In the meanwhile, there is a position in “many-worlds” (2) view that supersedes Penrose’s quantum realist one. This position rejects the Reduction Process in favor of Unitary Process, by terming the former as a mere illusion. Penrose shows his reservations against this view, as for him, a theory of consciousness needs to be in place prior to “many-worlds” view, and before the latter view could be squared with what one actually observes. Penrose is quite amazed at how many AI reviewers and researchers embrace the “many-worlds” hypothesis, and mocks at them, for their reasons being better supportive of validating AI project. In short, Penrose’s criticism of strong AI is based on the project’s assertion that consciousness can emerge by a complex system of algorithms, whereas for the thinker, a great many things humans involve in are intrinsically non-algorithmic in nature. For Penrose, a system can be deterministic without being algorithmic. He even uses the Turing’s halting theorem (3) to demonstrate the possibility of replication of consciousness. In a public lecture in Kolkata on the 4th of January 2011 (4), Penrose had this to say,

There are many things in Physics which are yet unknown. Unless we unravel them, we cannot think of creating real artificial intelligence. It cannot be achieved through the present system of computing which is based on mathematical algorithm. We need to be able to replicate human consciousness, which, I believe, is possible through physics and quantum mechanics. The good news is that recent experiments indicate that it is possible.

There is an apparent shift in Penrosean ideas via what he calls “correct quantum gravity”, which argues for the rational processes of the mind to be completely algorithmic and probably standing a bright chance to be duplicated by a sufficiently complex computing system. As he quoted from the same lecture in Kolkata,

A few years back, scientists at NASA had contemplated sending intelligent robots to space and sought my inputs. Even though we are still unable to create some device with genuine feelings and understanding, the question remains a disturbing one. Is it ethical to leave a machine which has consciousness in some faraway planet or in space? Honestly, we haven’t reached that stage yet. Having said that, I must add it may not be too far away, either. It is certainly a possibility.

Penrose does meet up with some sympathizers for his view, but his fellow-travelers do not tread along with him for a long distance. For example, in an interview with Sander Olson, Vernor Vinge, despite showing some reluctance to Penrose’s position, accepts that physical aspects of mind, or especially the quantum effects have not been studied in greater detail, but these quantum effects would simply be another thing to be learned with artifacts. Vinge does speculate on other paradigms that could be equally utilized for AI research hitting speed, rather than confining oneself to computer departments to bank on their progress. His speculations (5) have some parallel to what Penrose and Searle would hint at, albeit occasionally. Most of the work in AI could benefit, if AI, neural nets are closely connected to biological life. Rather than banking upon modeling and understanding of biological life with computers, if composite systems relying on biological life for guidance, or for providing features we do not understand quite well as yet to be implemented within the hardware, could be fathomed and made a reality, the program of AI would undoubtedly push the pedal to accelerate. There would probably be no disagreeing with what Aaron Saenz, Senior Editor of singularityhub.com said (6),

Artificial General Intelligence is one of the Holy Grails of science because it is almost mythical in its promise: not a system that simply learns, but one that reaches and exceeds our own kind of intelligence. A truly new form of advanced life. There are many brilliant people trying to find it. Each of these AI researchers have their own approach, their own expectations and their own history of failures and a precious few successes. The products you see on the market today are narrow AI-machines that have very limited ability to learn. As Scott Brown said, “today’s I technology is so primitive that much of the cleverness goes towards inventing business models that do not require good algorithms to succeed.” We’re in the infantile stages of AI. If that. Maybe the fetal stages.

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(1) This is quite apocalyptic sounding like the singularity notion of Ray Kurzweil, which is an extremely disruptive, world-altering event that has the potentiality of forever changing the course of human history. The extermination of humanity by violent machines is not impossible, since there would be no sharp distinctions between men and machines due to the existence of cybernetically enhanced humans and uploaded humans.

(2) “Many-worlds” view was first put forward by Hugh Everett in 1957. According to this view, evolution of state vector regarded realistically, is always governed by deterministic Unitary Process, while Reduction Process remains totally absent from such an evolutionary process. The interesting ramifications of this view are putting conscious observers at the center of the considerations, thus proving the basic assumption that quantum states corresponding to distinct conscious experiences have to be orthogonal (Simon 2009). On the technical side, ‘Orthogonal’ according to quantum mechanics is: two eigenstates of a Hermitian operator, ψm and ψn, are orthogonal if they correspond to different eigenvalues. This means, in Dirac notation, that < ψm | ψn > = 0 unless ψm and ψn correspond to the same eigenvalue. This follows from the fact that Schrödinger’s equation is a Sturm–Liouville equation (in Schrödinger’s formulation) or that observables are given by hermitian operators (in Heisenberg’s formulation).

(3) Halting problem is a decisional problem in computability theory, and is stated as: Given a description of a program, decide whether the program finishes running or continues to run, and will thereby run forever. Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. In a way, the halting problem is undecidable over Turing machines.

(4) Penrose, R. AI may soon become reality. Public lecture delivered in Kolkata on the 4th of Jan 2011. <http://timesofindia.indiatimes.com/city/kolkata-/AI-may-soon-become-reality- Penrose/articleshow/7219700.cms>

(5) Olson, S. Interview with Vernor Vinge in Nanotech.Biz <http://www.nanotech.biz/i.php?id=01_16_09&gt;

(6) Saenz, A. Will Vicarious Systems’ Silicon Valley Pedigree Help it Build AI? in singularityhub.com <http://singularityhub.com/2011/02/03/will-vicarious-systems-silicon-valley-pedigree-help-it-build-agi/&gt;