From cs240@cs.rochester.edu Tue Nov 4 17:53:54 2003 Date: Tue, 4 Nov 2003 17:53:04 -0500 From: cs240 To: tetreaul@vienna.cs.rochester.edu <b>Erin Kimball: Assignment 1</b>

Erin Kimball

cs240: Assignment 1

ek007j@mail.rochester.edu

September 11, 2003

 

The Emperor^Òs Overstated Mind

 

 

The Précis of The Emperor^Òs New Mind by Roger Penrose tackles a prevalent question in the world of artificial intelligence and cognitive sciences of whether or not the brain can be modeled as a computer in the formal definition that is accepted today.  At first it seems to be a rigorous article presenting a stance of opposition with thorough explanations of the theories that Roger Penrose further explains in his book.  However, under closer inspection, many of his arguments are vague or inconclusive, indicating a heavier focus on the quantity of arguments rather than on their quality.  Specifically, Penrose^Òs dependence on theories bridging the gap between quantum mechanics and classical physics that have not been deduced yet, assertion that intelligence and consciousness can not be algorithmic, and implications from the results of Gödel^Òs theorem are main points of the article that are inherently weak.

 

After complex numbers, Turing machines, and computable numbers are explained, Penrose brings up Gödel^Òs theorem, which showed that in ^Óany consistent formal system that is broad enough to contain arithmetic and the normal rules of logical procedure one can explicitly construct well-defined mathematical statements that are not provable^×nor are their negations provable^×using the rules laid down in that formal system.^Ô (Penrose, 648)  Since it is possible for us as humans to see that this is true using our intuition and understanding, and since there are similarities between algorithms and formal systems, mathematical truth cannot be an algorithmic issue.  This all leads to Penrose^Òs conclusion that there must be something going on within a human^Òs intelligence that is nonalgorithmic, or outside a formal system, and thus supports the opposition to the question of the brain^Òs computational limits.  The hole in this argument lies in the fact that humans do not have to perform some intelligent process of reasoning or even have to invoke consciousness to know that Gödel sentences are true.  What is required is that we have ^Óa built-in faith that mathematical systems are consistent.^Ô (Chalmers, 658)  Such a ^Ófaith^Ô could be reasonably programmed in a computer, and thus becomes computable, contrary to what Penrose is trying to prove.  Of course, this conclusion depends on the validity of the rest of mathematics as we know it, an issue that Penrose never presents an argument against, but rather supports with another point.

 

Penrose offers a consideration in conjunction with Gödel^Òs theorem.  ^ÓIf the first mathematician is prepared to accept all the axioms and rules of procedure of a particular formal system as giving only true propositions, then he must also be prepared to accept its Gödel proposition as describing a true proposition.  It would be exactly the same for the second mathematician.^Ô (Penrose, 653)  He implies that when a mathematician discovers a proof of some statement that is sound and without error, the communication of the truth between other mathematicians is quick and easy (Boolos, 655).  Again nothing in mentioned against the conclusions of the mathematicians being accepted as true, as opposed to being true only if the rest of mathematics is also true.  Penrose is obviously comfortable with relying on faith in this area, yet when discussing the physical mechanisms of the brain, he deviates from the accepted theories.

 

Another one of the main arguments in the presentation of the brain as an object beyond the realm of computers is the speculation that the physical processes occurring at the lowest levels might not be algorithmically specifiable.  In science today there is a decent understanding of the behavior of particles and other discreet entities at the quantum level.  Yet the very laws that make up the area of quantum mechanics disagree with much of what is taken to be true in the classical physics that describe the macroscopic world.  One of the main issues today is the contradiction between quantum theory and Einstein^Òs theory of relativity.  Many people believe that the current ideas of how particles on the scale of electrons behave are inaccurate and still others believe that Einstein^Òs theory is just plain wrong.  Penrose, however, believes that there is a middle ground between the micro- and macro-scopic worlds that is just waiting to be discovered.  Without presenting any theories as to what kind of laws might govern this ^Óno-man^Òs land^Ô, he goes on to propose that those very laws are probably the ones that will describe the phenomena occurring in the neurons of the brain and furthermore, that they are nonalgorithmic.  To successfully refute the belief of many neurologists and cognitive experts, one would hope that more concrete evidence would be provided, but this is not the case in Penrose^Òs article, making for a very unconvincing argument.  The question of computability arises in many contexts throughout the paper, none of which are much more satisfying.

 

One of those contexts is the formal definition of what is or isn^Òt computable, in other words, what can or can not be done with a Turing machine.  Penrose provides the classic example of a non-computable problem, known as the halting problem, which is the question of deciding whether or not a given Turing machine, when given a specific input, will ever come to a halt.  There is in fact no algorithm that may be followed to solve this problem in general, as Penrose points out, so the ability of humans to use insights to determine whether a certain Turing machine will halt allows us to outdo any algorithm that may attempt to solve the problem.  The weakness in this point stems from Penrose^Òs definition of Turing machines as being able to simulate any ^Ócomputational process whatsoever (that operates with finite discrete quantities).^Ô (Penrose, 646)  As discussed earlier, Penrose believes that the brain^Òs processes may not be describable with the current laws governing ^Ófinite discreet quantities^Ô, so the possibility arises that the signals the brains receives and outputs may be infinite and continuous, and therefore, the functionality of the brain might be able to be described by more powerful models of computation than the Turing machine (Breuel, 657).  Again the issues involving the precise workings of the brain are not fully understood so the arguments for or against using the Turing machine as a model of the brain are only speculative and will remain so until concrete evidence is produced.

 

In all, Roger Penrose presents many interesting ideas that support the notion of humanity as an entity in a class above anything that requires a silicon chip.  However, a collection of ideas describes the article more accurately than a persuasive argument.  Whether presenting the implications of Gödel^Òs theorem, the inadequacy of theories to describe the physical phenomena occurring in the brain, or the model of the Turing machine to determine algorithmic processes, the paper fails to give concrete evidence to support the conclusions and thus proves to be unconvincing.  As more studies are done and advancements made in the areas of physics and cognitive sciences, many of Penrose^Òs ideas may be confirmed, but his ideas alone are not sufficient to contribute to the answer of whether or not the brain is merely a biological computer.

 

 

 

References:

 

1.       Boolos, George.  ^ÓOn seeing the truth of the Gödel sentence.^Ô Behavioral and Brain Sciences 13 (1990):    655-656.

 

2.       Breuel, Thomas M.  ^ÓAI and the Turing model of computation.^Ô  Behavioral and Brain Sciences 13 (1990):  657.

 

3.       Chalmers, David J. ^ÓComputing the Thinkable.^Ô  Behavioral and Brain Sciences 13 (1990):    658-659. 

 

4.       Penrose, Roger. ^ÓPrécis of The Emperor^Òs New Mind: Concerning computers, minds, and the laws of physics.  Behavioral and Brain Sciences 13 (1990): 643-655.