Computability and Free Will

In this post I will draw on a proof from Roger Penrose’s book Shadows of the mind that I think is important to the free will issue. The proof goes like this.

Consider an algorithm that takes a single positive integer as an input. Depending on the input and the algorithm itself, either the algorithm terminates in a finite time or it does not. In the first case the algorithm is said to stop.

Define a mapping from the set of natural numbers to the set of algorithms (of the kind above). A1, A2, A3 … Let Ai(n) denote the ith such algorithm operating on the input n.

Let B be an algorithm that also takes a single input, such that B(n) stops if it determines that An(n) does not stop. Since B is an algorithm of the same class of algorithms (taking a single input), it exists at some location in the list of As. Let B = Am. That is, Am(n) stops if An(n) does not stop

Consider the operation of B (= Am) with the input m. B(m) = Am(m) stops if Am(m) does not stop. That is, the task of B(m) is to stop if it is able to determine that it itself does not stop. If B(m) stops, we have a contradiction. Therefore B(m) does not stop. Therefore B is unable to determine that its own operation on the input m does not stop.

Now suppose that B represents all human understanding of algorithms that can be expressed as an algorithm. All of this algorithmic understanding is unable to detemine that B(m) does not stop. But we as humans are able to determine that B(m) does not stop.

Therefore atleast some aspect of human understanding is non-algorithmic. (Or in other words, the human mind can solve some problems that are not computable)

Penrose intended this proof (which parallels Godel’s proof of the incompleteness theorem) to debunk the claims of strong AI (artificial intelligence) that the human mind works by just “running” a higly evolved and complex algorithm. He explicitly steered clear of taking any position on the issue of determinism. And with good reason. Computability is not the same as determinism. A non-computable process can still be fully deterministic.

Most people who deny free will do so because they cannot reconcile free will with present day science. But Penrose’s proof conclusively demonstrates that present-day science (all the physical theories widely accepted in physics today are computable)  is incapable of explaining human understanding. It is not just that present day science does not have a theory of the mind. It cannot have a theory of the mind even in principle. Penrose argues that we need a non-computable theory in physics.

While this is not a proof of free will (without a scientific breakthrough, I don’t see how the existence of free will can be proved), it destroys the most common arguement against free will – the success of present day science (physics in particular).

The scope of free will

I was debating the issue of anarchy and was directed to this article (pdf) by Prof. Moshe Kroy which highlights some fundamental differences between Rand’s philosophy and Rothbard’s including the scope of free will.

According to Rand’s theory of human freedom, man’s only fundamental freedom, the sole domain in which he is capable of being a “first cause”, the only realm where he can exercise absolutely unpre-determined choice, is his own consciousness. Man’s basic choice is between identifying the facts of reality through an act of consciousness, and evading the knowledge of these facts. This freedom does not extend to man’s decisions and actions: Your decisions and actions are the necessary product of your values and premises, Rand claims.

Rothbard’s theory of man, however, assumes another dimension of freedom in man: the freedom to make decisions, to originate action. For Rothbard values, and their hierarchy, are not the product of perception alone, though, clearly, his writing implies that awareness of the facts is highly relevant to your choice of values.

In Rand’s words (also look at the related concept of focus)

That which you call your soul or spirit is your consciousness, and that which you call “free will” is your mind’s freedom to think or not, the only will you have, your only freedom, the choice that controls all the choices you make and determines your life and your character.

In any hour and issue of his life, man is free to think or to evade that effort. Thinking requires a state of full, focused awareness. The act of focusing one’s consciousness is volitional. Man can focus his mind to a full, active, purposefully directed awareness of reality—or he can unfocus it and let himself drift in a semiconscious daze, merely reacting to any chance stimulus of the immediate moment, at the mercy of his undirected sensory-perceptual mechanism and of any random, associational connections it might happen to make.

Psychologically, the choice “to think or not” is the choice “to focus or not.”
(Emphasis mine)

I am not quoting Rothbard because I have not read his works. Aristotle The Geek has directed me to this article by Rothbard but the focus of that article is more on refuting determinism than the scope of free will.

How does one decide which theory is correct? The existence of a choice to focus or not (ranging from no focus to full focus) is immediately available to introspection. When I am solving a difficult problem, I am consciously choosing to be fully focused. When I try to go to sleep, I consciously suspend my focus. But does free will extend beyond that? Am I free to choose the object of my focus or the subject of my thoughts? Am I free to choose the outcome of my thoughts? I don’t think so.

Firstly, there are occasions when I get distracted. This is an indication – though not a proof – that I lack control over the object of my focus. There are occasions when I want to stop thinking about something but cannot. This is an indication – again not a proof – that I lack control over the subject of my thoughts. There exist such things as mental habits and character. These concepts would surely be meaningless if I were free to choose the outcome of my thoughts.

Secondly (and less importantly), one can apply Occam’s razor. The freedom of choice to focus is necessary to explain human behavior. It is also sufficient. Why assume a greater freedom without evidence – especially when free will sits uncomfortably with known physical theories? And until we discover physical theories that can explain free will, I don’t think this issue can be proved either way.

On these grounds, I agree with Rand’s position.

How is this relevant to anarchy? I will deal with that in a separate post.

Science and philosophy – 2

I just finished reading “dreams of a final theory” by Steven Weinberg. The book is somewhat less technical than I had expected. Despite having a chapter titled “Against Philosophy”, Weinberg deals with several issues that have more to do with the methodology of science than with its content, such as reductionism, “aesthetics” in science, positivism, belief in God etc. Overall, the book is quite enjoyable. I am interested in developing a better understanding of several of the issues Weinberg discusses, so this book has given me a lot of starting material.

To me, the most interesting arguement that Weinberg makes is that “aesthetics” has played a significant role in the formulation as well as the acceptance of several of the theories that have been developed in the last century. I had encountered similar claims before but had not taken them very seriously. But Weinberg makes his case quite convincingly. He argues that validating a theory by means of experiment is not as simple as it may seem. There can be any number of reasons for an anomaly in experimenal results. In judging whether a theory may be valid, whether it is worth trying to validate, physicists necessarily rely on “aesthetic” judgements. As an example, he argues that physicists were more or less certain of Einstein’s theory of general relativity before it was conclusively validated by experiment. As another example, he argues that physicists were sceptical of the theory of quantum electro-dynamics although it was in agreement with experimental results because some calculations based on it involved “ugly” infinities in intermediate steps. He writes that part of this “beauty” lies in simplicity – not the simplicity of the equations but of ideas. Another part of this beauty lies in what he calls logical isolation or rigidity. For example, he writes that no one has yet found a way to make a small modification in the principles of quantum mechanics without destroying the theory altogether. It would make little difference in Newton’s inverse square law of gravitational force if the exponent were changed to 2.01 instead of 2, but even the introduction of a small non-linear term in the linear equations of quantum mechanics produces nonsensical results. Such a theory does not explains why it should be correct but it explains why it cannot be just a little wrong. Concluding a chapter titled “Beautiful Theories” Weinberg writes

We believe that, if we ask why the world is the way it is and then ask why that answer is the way it is, at the end of this chain of explanations we shall find a few simple principles of compelling beauty… the beauty of present theories is an anticipation, a premonition, of the beauty of the final theory. And in any case, we would not accept any theory as final unless it were beautiful. (emphasis mine)

The last sentence in that excerpt sums up most of what Weinberg is saying about beauty. If one keeps asking why as Weinberg does, there will come a point where one will have to stop. How does one decide what that point is? For Weinberg that point will have been reached when we have a simple and logically isolated theory that “explains” everything including the values of what we call universal constants.

In my next post in this series, I will try to present my own thoughts on what it means for a theory to explain something and on beauty.

Science and philosophy – 1

A couple of days back, in a comment on this video, I wrote

Unfortunately because of the philosophy of logical positivism, most scientists today are afraid or unwilling to accept the necessity of metaphysics at all.

Today, I started reading a book “dreams of a final theory” by Steven Weinberg. Early on in the book, he writes

Speaking of a final theory, a thousand questions and clarifications crowd into the mind. What do we mean by one scientific principle ‘explaining’ another? How do we know that there is a common starting point for all such explanations? Will we ever discover that point? How close are we now? What will the final theory be like? What will it say about life and consciousness? And, when we have our final theory, what will happen to science and to the human spirit?

A few pages later, explaining what the pursuit of a final theory means, he starts with “Chalk is white. Why?” Some answers (light, wavelengths etc.) and then another “why”. Some more answers (atomic structure, energy levels etc.) and then another “why” and so on until he comes to a point where he does not know the answers. Finally, he writes,

Even so, it is a tricky business to say exactly what one is doing when one answers such a question. Fortunately, it is not really necessary.

and then, a few paragraphs later,

Ludwig Wittgenstein, denying even the possibility of explaining any fact on the basis of any other fact, warned that ‘at the basis of the whole modern view of the world lies the illusion that the so-called laws of nature are the explanations of natural phenomena.’ Such warnings leave me cold. To tell a physicist that the laws of nature are not explanations of natural phenomena is like telling a tiger stalking prey that all flesh is grass. The fact that we scientists do not know how to state in a way that philosophers would approve what it is that we are doing in searching for scientific explanations does not mean that we are not doing something worthwhile. We could use help from professional philosophers in understanding what it is that we are doing, but with or without their help we shall keep at it.

As an engineer, I can certainly sympathize with what Weinberg is saying. Philosophers such as Wittgenstein have betrayed science (and philosophy). But that is not a good enough reason for scientists to abandon philosophy. Proper science needs a solid base in philosophy. I dare say that without an explicit (philosophical) understanding of what “searching for scienific explanations” means, scientists will not discover a final theory. I think science has reached a level (based on some reading of quantum mechanics) from where it cannot proceed without answering some fundamental questions that are not just scientific.

Anyhow this is a fascinating subject, and I will return to it often. I have been fascinated by this subject for about an year now, and have been reading some books off and on when I find time. So far I have not written on this subject, because I do not like to write unless I have a clear idea of what I am writing. But it seems unlikely that I will find answers to questions like this without a lot of thinking. So I intend to think by putting my thoughts on this blog instead of doing it before going to sleep.

%d bloggers like this: