Let’s recall. There was:Let’s recall. There was: Robbie, in the TV series (Robbie, in the TV series (guess the answer)guess the answer) HAL 9000, in ……………HAL 9000, in …………… Mr. Data, in …………..Mr. Data, in ………….. Marvin the Paranoid Android, in ……………….Marvin the Paranoid Android, in ……………….

Some here will think of many more. Can machines be taught to really think? Perhaps more importantly, are we merely very complex computers? Or is “mind” more than a collection of neurons?

Quanta and ConsciousnessQuanta and Consciousness

An overview of two significant and An overview of two significant and surprising developments in 20surprising developments in 20thth-century -century science, one in physics and the other in science, one in physics and the other in mathematics. Both of these have mathematics. Both of these have implications regarding the nature of implications regarding the nature of perception and consciousness.perception and consciousness.

First, some background . . .

The development of quantum ideas

n James Clerk Maxwell

Max Planck

Niels Bohr

Louie de Broglie

Erwin Schrodinger

“If we are going to stick to this damned quantum-jumping, then I regret that I ever had anything to do with quantum theory”

Werner Heisenberg

Albert Einstein

The Bohr-Einstein debate

Bohr and Einstein: a study in contrasts.

n These are the conditions under which Einstein wrote he would agree to continue to live with Mileva in Berlin:

n A. You will see to it that: 1. My clothes and laundry are kept in good order; 2. I will be served three meals regularly in my room; 3. My bedroom and study are kept tidy, and especially that my desk is left for my use only. B. You will relinquish all personal relations with me insofar as they are not completely necessary for social reasons. Particularly, you will forgo my: 1. Staying at home with you; 2. Going out and traveling with you. C. You will obey the following points in your relations with me: 1. You will not expect any tenderness from me, nor will you offer any suggestions to me; 2. You will stop talking to me about something if I request it; 3. You will leave my bedroom or study without any back talk if I request it. D. You will undertake not to belittle me in front of our children, either through words or behavior.

n Mileva Maric left Berlin with the children shortly after this.

Bohr and his sons

n Einstein generally worked alone. In addressing a government agency after WW2, concerning what could be done with out-of-work scientists, he said in all seriousness that jobs such as “lighthouse keeper” would be ideal for many scientists.

n Bohr was a gregarious Dane who founded and built up an institute in Copenhagen. There are still physicists from all over the world who can say “I worked with Bohr” or at least their doctoral advisors did.

The “Copenhagen interpretation”

n Quantum theory is probabilistic in nature. One can calculate exactly what an outcome will be --- one can only calculate the probability of obtaining one outcome or another.

n These probabilities are contained in the “wavefunction” of the system. Before we do the observation, the wavefunction may contain many possible, overlapping outcomes. During the measurement, one of these outcomes is “selected.”

n Example: the position of an electron in an atom: the original “Bohr model” vs the “electron cloud.”

n Another example: decay of a nucleus. In a uranium atom, one can picture an alpha particle (two protons/two neutrons) bouncing back an forth against a potential barrier. Every time it hits, there is a probability that it will escape --- to tunnel through the barrier. In a certain time, we cannot predict whether the particle will escape or not, but we can predict the probability that it will escape in that time. (This would be like rolling a ball up a hill. Instead of stopping an rolling back down, there is a probability that it would disappear and reappear on the other side of the hill.)

n (More on this later)

n A central part of quantum theory is the Heisenberg Uncertainty Principle, which puts certain limits on our possible knowledge of the state of a quantum system. In a nutshell, it states that we cannot know two “conjugate variables” to arbitrary precision at the same time. For example, we cannot know the position and the velocity of a particle at the same time; there must be an uncertainty in our measurement. To wit,

n (Delta)x times (Delta)v >= Planck’s constant

Einstein’s take on all this

n (From a letter to Max Born, 1926): “Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.”

The “EPR” paradoxn To illustrate deficiencies in the quantum theory, Einstein

came up with many “gedanken experiments.” Bohr always came up with resolutions to Einstein’s proposed contradictions, but Einstein doggedly kept at it. In 1934, in one of his rarely co-authored papers, Einstein issued one last challenge. This was the famous Einstein, Podolsky, and Rosen (EPR) paper. It caused quite a stir.

n Excerpt from a letter by Wolfgang Pauli to Heisenberg: “Einstein has once again expressed himself publicly on quantum mechanics. . . .(together with Podolsky and Rosen --- no good company, by the way). As is well known, every time that happens it is a catastrophe.”

n The original EPR paper dealt with the linear momentum (mass times velocity) of two particles which interact but are then separated. A fundamental tenet of the paper is that after the particles separate, there is a “local reality” associated with each. The idea that one particle could affect the other, say, when on opposite sides of the solar system seems preposterous. In this way, EPR seemed to “get around” the limitations of Heisenberg uncertainty.

n This time, Bohr did not have an iron-clad comeback. He eventually said that “the trend of their argumentation. . . does not seem to me to meet the actual situation with which we are faced in atomic physics.” As lame as this was, most physicists seemed to buy his arguments, gave a sigh of relief, and went back to “real” work.

Bohr talked of his arguments with Einstein until the day of his death in 1962. He had countered every attack on the theory as if it had been a personal one. The issue then lay more or less dormant for many years.

EntanglementEntanglementn In 1952, David Bohm changed the setting of the In 1952, David Bohm changed the setting of the

EPR paper in a way that made the issues more clear EPR paper in a way that made the issues more clear and concise. He reduced the problem to two and concise. He reduced the problem to two particles and only one variable for each: the spin or particles and only one variable for each: the spin or polarization. He also championed the notion of polarization. He also championed the notion of “hidden variables” which provide a complete “hidden variables” which provide a complete picture of quantum reality.picture of quantum reality.

n John Bell was a researcher at CERN (a high-energy John Bell was a researcher at CERN (a high-energy facility in Geneva), and in his “spare time” worked facility in Geneva), and in his “spare time” worked on the deeper issues of quantum theory. In the mid on the deeper issues of quantum theory. In the mid sixties, he published two ground-breaking papers. sixties, he published two ground-breaking papers. Bell’s Theorem, as it was called, provided a means Bell’s Theorem, as it was called, provided a means for real experiments to test alternatives to quantum for real experiments to test alternatives to quantum ideas.ideas.

John Bell and his wife John Bell and his wife Mary, Mary,

also a physicist.also a physicist.n David BohmDavid Bohm

Bell’s TheoremBell’s Theoremn John Bell knew that Einstein and colleagues were John Bell knew that Einstein and colleagues were

partly correct: the “EPR paradox” was no paradox partly correct: the “EPR paradox” was no paradox at all. What was wrong was their insistence on at all. What was wrong was their insistence on “local reality” --- that the total, “mixed” wave “local reality” --- that the total, “mixed” wave function could not extend across large regions of function could not extend across large regions of space.space.

n Thus, Bell viewed two alternatives: (1) Quantum Thus, Bell viewed two alternatives: (1) Quantum theory is right, or (2) local realistic models are theory is right, or (2) local realistic models are right. But both cannot be right.right. But both cannot be right.

n Bell produced a Bell produced a mathematical theoremmathematical theorem containing containing certain inequalities. He suggested that if his certain inequalities. He suggested that if his inequalities could be violated by experimental inequalities could be violated by experimental tests, it would provide evidence in favor of tests, it would provide evidence in favor of orthodox (Copenhagen) quantum theory.orthodox (Copenhagen) quantum theory.

Tests of Bell’s TheoremTests of Bell’s Theorem

n With apologies I With apologies I am leaving out the am leaving out the work of Shimony, work of Shimony, Clauser, Horne, Clauser, Horne, Aravind, Zeilinger, Aravind, Zeilinger, and others.and others.

n The most The most convincing tests of convincing tests of Bell’s inequalities Bell’s inequalities has been done by has been done by Alain Aspect (in Alain Aspect (in France).France).

Aspect’s experimentsAspect’s experiments

n We conclude that “hidden variables” or other forms We conclude that “hidden variables” or other forms of local reality are NOT correct. “Spooky action at of local reality are NOT correct. “Spooky action at a distance” (to use Einstein’s description) correctly a distance” (to use Einstein’s description) correctly describes quantum systems. describes quantum systems.

n By the way, the notion of entanglement is By the way, the notion of entanglement is absolutely fundamentalabsolutely fundamental to the development of to the development of quantum computing, a hot topic these days.quantum computing, a hot topic these days.

Now, back to some “old” stuff……Now, back to some “old” stuff……

The role of the “Observer”The role of the “Observer”

Schrodinger’s cat.Schrodinger’s cat.

The Von Neumann-London-Bauer argumentThe Von Neumann-London-Bauer argument

The probabilities that are computed in quantum The probabilities that are computed in quantum mechanics are probabilities of mechanics are probabilities of outcomes of outcomes of measurements.measurements.

The observer is outside the system. He intervenes in the The observer is outside the system. He intervenes in the system by making a measurement. The observer’s system by making a measurement. The observer’s intervention takes one out of the realm of the intervention takes one out of the realm of the hypothetical and into the realm of the actual.hypothetical and into the realm of the actual.

One might think we could give a complete mathematical One might think we could give a complete mathematical description of not only the experimental devices but of description of not only the experimental devices but of the observer herself, at least in principle.the observer herself, at least in principle.

But this cannot be done!But this cannot be done!

If we could describe by the mathematics of quantum If we could describe by the mathematics of quantum theory everything that happened in a measurement, theory everything that happened in a measurement, even up to the point where a definite outcome was even up to the point where a definite outcome was obtained by an observer, then the math would have to obtained by an observer, then the math would have to tell us what the outcome was. But this cannot happen, tell us what the outcome was. But this cannot happen, for in quantum theory the math will yield only for in quantum theory the math will yield only probabilities.probabilities.

In short, the mathematical descriptions of the physical In short, the mathematical descriptions of the physical world given to us by quantum theory presuppose the world given to us by quantum theory presuppose the existence of observers who lie outside those existence of observers who lie outside those mathematical descriptions. And the theory works.mathematical descriptions. And the theory works.

What about the line between the “system” and the What about the line between the “system” and the “observer”?“observer”?

Godel’s theoremGodel’s theorem

§ Kurt Godel was an Kurt Godel was an Austrian logician and a Austrian logician and a good friend of Einstein.good friend of Einstein.

§ (In the movie “IQ”, (In the movie “IQ”, Walter Matthau played Walter Matthau played Einstein and Lou Einstein and Lou Jacobi played Godel. Jacobi played Godel. Meg Ryan played Meg Ryan played Einstein’s niece.)Einstein’s niece.)

§ Godel proved that in any consistent formal Godel proved that in any consistent formal mathematical system (in which one can at least do mathematical system (in which one can at least do arithmetic and simple logic), there are arithmetical arithmetic and simple logic), there are arithmetical statements which can neither be proved nor disproved statements which can neither be proved nor disproved using the rules of that system using the rules of that system but which are but which are nevertheless true statements. nevertheless true statements.

§ These are called “formally undecidable propositions” of These are called “formally undecidable propositions” of that system.that system.

§ Moreover, Godel showed how to find, in any particular Moreover, Godel showed how to find, in any particular consistent formal system, how to actually find one of its consistent formal system, how to actually find one of its formally undecidable-but-true propositions.formally undecidable-but-true propositions.

The theorem - 1931

§ If If FF is any consistent formal system that is any consistent formal system that contains logic and arithmetic, Godel showed contains logic and arithmetic, Godel showed how to find a statement in arithmetic, which we how to find a statement in arithmetic, which we may call G(may call G(FF), that is neither provable nor ), that is neither provable nor disprovable using the rules of disprovable using the rules of FF. He further . He further showed that G(showed that G(FF) is nevertheless a TRUE ) is nevertheless a TRUE arithmetical statement.arithmetical statement.

§ This can be applied to computer programs. For This can be applied to computer programs. For a computer program a computer program PP that is known to be that is known to be consistent, one can find a statement in consistent, one can find a statement in arithmetic, G(arithmetic, G(PP), that cannot be proven nor ), that cannot be proven nor disproven by that program. And one can show disproven by that program. And one can show that G(that G(PP) is a true statement.) is a true statement.

The Lucas-Penrose argument.The Lucas-Penrose argument.

§ In 1961, John R. Lucas, a philosopher at Oxford U., set In 1961, John R. Lucas, a philosopher at Oxford U., set forth an argument based on Godel’s Theorem, to the forth an argument based on Godel’s Theorem, to the effect that the human mind cannot be a computer effect that the human mind cannot be a computer program.program.

§ Roger Penrose, the widely-known mathematician and Roger Penrose, the widely-known mathematician and physicist, revived Lucas’ argument in the late 80’s. His physicist, revived Lucas’ argument in the late 80’s. His book book The Emperor’s New MindThe Emperor’s New Mind was published in 1989. was published in 1989. In answer to the large amount of criticism it provoked, In answer to the large amount of criticism it provoked, Penrose published a second book, Penrose published a second book, Shadows of the Shadows of the Mind. Mind.

While no one has succeeded in refuting the Lucas-While no one has succeeded in refuting the Lucas-Penrose argument, it has not changed many mindsPenrose argument, it has not changed many minds..

The argumentThe argument§ Suppose someone shows me a computer program, Suppose someone shows me a computer program, PP, that , that

has the ability to do simple arithmetic and logic. I know this has the ability to do simple arithmetic and logic. I know this program to be consistent, and I know all the rules by which program to be consistent, and I know all the rules by which it operates. Then as Godel proved, I can find a statement in it operates. Then as Godel proved, I can find a statement in arithmetic (call it G(arithmetic (call it G(PP)) that the program )) that the program PP cannot prove (or cannot prove (or disprove.) But following Godel’s reasoning, I can show disprove.) But following Godel’s reasoning, I can show G(G(PP) to be a true statement of arithmetic. ) to be a true statement of arithmetic.

§ So far, no big deal. The programmer could modify the So far, no big deal. The programmer could modify the program so that it can also prove G(program so that it can also prove G(PP). But I know all the ). But I know all the new rules, too, so I can find a new statement which is true new rules, too, so I can find a new statement which is true but which cannot be proven or disproven by the new but which cannot be proven or disproven by the new program. Again the programmer could improve the program. Again the programmer could improve the program, and we can keep playing this game, with me program, and we can keep playing this game, with me always “outwitting” the new programs. However….always “outwitting” the new programs. However….

§ Suppose I myself AM a computer program: call me Suppose I myself AM a computer program: call me HH (for human). When I prove things, it is just by some (for human). When I prove things, it is just by some computer program running in my brain. And now computer program running in my brain. And now suppose I am shown that program, learning in complete suppose I am shown that program, learning in complete detail how detail how HH works. Then assuming I know works. Then assuming I know HH to be a to be a consistent program, I can construct a statement in consistent program, I can construct a statement in arithmetic, call it G(arithmetic, call it G(HH), that cannot be proven or ), that cannot be proven or disproven by disproven by HH, but which I, using Godel’s reasoning, , but which I, using Godel’s reasoning, can show to be true. can show to be true.

§ Contradiction: it is impossible for Contradiction: it is impossible for HH to be unable to to be unable to prove a result that I am able to prove, because prove a result that I am able to prove, because HH is me! is me!

§ (Thanks to Stephen Barr for this concise description of (Thanks to Stephen Barr for this concise description of the Lucas-Penrose argument.)the Lucas-Penrose argument.)

Our assumptions were:Our assumptions were:

§ (a) I am a computer program.(a) I am a computer program.§ (b) I know that the program is consistent.(b) I know that the program is consistent.§ (c) I can learn the structure of the program in complete (c) I can learn the structure of the program in complete

detail.detail.§ (d) I have the ability to go through Godel’s “steps.”(d) I have the ability to go through Godel’s “steps.”

A materialist has “escape routes” by denying (b), A materialist has “escape routes” by denying (b), (c), or (d) instead of (a).(c), or (d) instead of (a).

In conclusionIn conclusionn Herrmann Weyl, one of the great mathematicians and Herrmann Weyl, one of the great mathematicians and

physicists of the twentieth century, wrote in 1931: physicists of the twentieth century, wrote in 1931: n ““We may say that there exists a world, causally We may say that there exists a world, causally

closed and determined by precise laws, but . . . the closed and determined by precise laws, but . . . the new insight which modern [quantum] physics affords new insight which modern [quantum] physics affords opens several ways of reconciling personal freedom opens several ways of reconciling personal freedom with natural law. with natural law. It would be premature, however, to It would be premature, however, to propose a definite and complete solution of this propose a definite and complete solution of this problem. One of the great differences between the problem. One of the great differences between the scientist and the impatient philosopher is that the scientist and the impatient philosopher is that the scientist bides his time.scientist bides his time. We must await the further We must await the further development of science, perhaps for centuries, development of science, perhaps for centuries, perhaps for thousands of years, before we can design perhaps for thousands of years, before we can design a true and detailed picture of the interwoven texture a true and detailed picture of the interwoven texture of Matter, Life, and Soul. But the old classical of Matter, Life, and Soul. But the old classical determinism of Hobbes and Laplace need not determinism of Hobbes and Laplace need not oppress us longer.”oppress us longer.”

PostscriptPostscript

n The answer is NOT “42”.The answer is NOT “42”.

(With apologies to Douglas Adams, who wrote The Hitchhikers Guide to the Galaxy.)