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).

Mises on The Free-Will Controversy

From Chapter 5 of Mises’ Theory and History,

Man chooses between modes of action incompatible with one another. Such decisions, says the free-will doctrine, are basically undetermined and uncaused; they are not the inevitable outcome of antecedent conditions. They are rather the display of man’s inmost disposition, the manifestation of his indelible moral freedom. This moral liberty is the essential characteristic of man, raising him to a unique position in the universe.

Determinists reject this doctrine as illusory. Man, they say, deceives himself in believing that he chooses. Something unknown to the individual directs his will. He thinks that he weighs in his mind the pros and cons of the alternatives left to his choice and then makes a decision. He fails to realize that the antecedent state of things enjoins on him a definite line of conduct and that there is no means to elude this pressure. Man does not act, he is acted upon.

Both doctrines neglect to pay due attention to the role of ideas. The choices a man makes are determined by the ideas that he adopts.

This is quite close to my own position but with a very important qualification. The choices a man makes are determined by the ideas he adopts provided he chooses to think. Mises denies that choice.

What the sciences of human action must reject is not determinism but the positivistic and panphysicalistic distortion of determinism. They stress the fact that ideas determine human action and that at least in the present state of human science it is impossible to reduce the emergence and the transformation of ideas to physical, chemical, or biological factors. It is this impossibility that constitutes the autonomy of the sciences of human action. Perhaps natural science will one day be in a position to describe the physical, chemical, and biological events. which in the body of the man Newton necessarily and inevitably produced the theory of gravitation. In the meantime, we must be content with the study of the history of ideas as a part of the sciences of human action.

The sciences of human action by no means reject determinism. The objective of history is to bring out in full relief the factors that were operative in producing a definite event. History is entirely guided by the category of cause and effect. In retrospect, there is no question of contingency. The notion of contingency as employed in dealing with human action always refers to man’s uncertainty about the future and the limitations of the specific historical understanding of future events. It refers to a limitation of the human search for knowledge, not to a condition of the universe or of some of its parts.

Having denied the choice to think, Mises treats determinism and causality as equivalent and rejects the notion of contingency for past actions. It will be interesting to see where this takes him in later chapters. One consequence is already apparant though – on his view of morality. A determinist cannot logically be a moralist and indeed Mises is not. Like Taleb, he denies the possibility of a normative science. In earlier chapters, Mises writes that the only possible judgement of human action is whether a particular means leads to a particular end. Ends cannot be judged. Adopting utilitarianism, he goes on to write about justice: “The ultimate yardstick of justice is conduciveness to the preservation of social cooperation. Conduct suited to preserve social cooperation is just, conduct detrimental to the preservation of society is unjust.”

Just goes to show how important the foundational branches of philosophy are.

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.

Intuitions and a-priori knowledge

In a comment on my post on hypotheticals, Krishnamurthy asked:

When you say “Rationality means that man must instead find principles on which to base his actions “, the question arises about how to arrive at those principles. If he cannot use his intuition, and if he cannot do the complicated expected utility maximization, then he can only arrive at the principles by evaluating the outcomes of his previous actions. But to evaluate he would need some principles to begin with. (on second thought, even to do expected utility maximization, he would need to make some evaluations). how does a human being find the principle to base his actions on ?

I hold that knowledge can never be a-priori. To see why, consider these questions

Does a digital balance know how to measure weight?
Does a computer know how to add numbers?
Does my heart know how to pump blood?
Do my eyes and brain know how to distinguish objects from each other?
Does a parrot that recites 2 + 2 = 4 know that 2 + 2 = 4?

My answer to all these questions would be no. There is no knowledge involved here. Knowledge, in the sense applicable to a human mind, involves the exercise of free will. An entity that does not have free will cannot have any knowledge. It is like a machine that does certain things because that is its nature. Since no exercise of free will can occur before one exists, knowledge cannot be a-priori.

Now consider the human mind. I believe that the mind is built with the capacity to use logic, but not with the knowledge of the laws of logic. This is a subtle point. What I am saying is that the mind has an inbuilt ability to determine whether something makes sense. But active effort is required to use this ability. And further effort is required to identify why it makes sense. Men obviously have been using logic for millenia. But it took Aristotle to identify the laws of logic. The operation of the laws of logic is part of the nature of the mind but the knowledge of the laws of logic is not. It takes active effort to grasp the laws of logic – to realize that when something “makes sense”, it is because that something is consistent with the laws of logic. The faculty that is capable of doing this grasping is reason. Man is born with the faculty of reason. But it is the use of reason that results in knowledge.

Recollect the time when you learnt the truth table for “p AND q” where “p” and “q” are propositions. How did you grasp that the truth table was correct? I did so by substituting actual propositions for “p” and “q” and verifying the values in the truth table. This indicates that knowledge of the truth table was not a-priori but the ability to verify particular propositions was. The truth-table was <i>induced</i> from the ability to verify particular propositions. More importantly, this also indicates that in the absence of any particular propositions, I could not have induced the truth table for “p AND q”. This is another reason that knowledge cannot be a-priori.

The ability to understand and evaluate propositions and to induce principles is inbuilt. If you want to call this ability intuition (I call it reason), I have no problem accepting the validity of intuitions, provided effort is made to express the result of this “intuition” in terms of the laws of logic, observations and any other principles one has already validated. But I don’t think this is what anybody means by intuition. For example, the Merriam Webster dictionary defines intuition as
1: quick and ready insight
2 a: immediate apprehension or cognition b: knowledge or conviction gained by intuition c: the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
Note the parts I have emphasized. They all indicate that intuition is knowledge achieved without active effort and without the use of reason.

Does this answer your question?

Why should values be agent-relative?

Heumer’s critique of Ayn Rand’s “The Objectivist Ethics” begins with

…premise 1 [Value is agent-relative; things can only be valuable for particular entities] begs the question.
One of the central groups of opponents Rand is facing is people who believe in absolute value, and not just agent-relative value. The absolutist view is that it is possible for some things to be good, simply, or in an absolute sense; whereas agent-relativists think that things can only be good for or relative to certain individuals, and that what is good relative to one individual need not be good relative to another. (N.B., this should not be confused with what are commonly called “moral relativism” and “cultural relativism.”)
Rand bases her ethics on the agent-relative position, but she offers no argument for it, only a bald assertion.

Why indeed should value be agent relative? The answer lies in Rand’s claim that

The first question that has to be answered, as a precondition of any attempt to define, to judge or to accept any specific system of ethics, is: Why does man need a code of values?
(emphasis in original)

The clue to the answer is in the question “Why”. If X is an absolute value (not just agent-relative), one can ask “Why is X an absolute value?”. There are three possible responses:

a) God said so:
This is unacceptable to me since the existence of God is an arbitrary claim and I do not want to elaborate further in this post (especially since this is not Heumer’s response).

b) It is self-evident:
It is impossible to argue with someone who really means this. If a particular proposition seems self-evident to him and it does not seem self-evident to me, how do we argue? I can only say that apart from the axioms in metaphysics and epistemology (‘Something exists’, ‘I am conscious’, ‘entities have identity’ and ‘I have free will’) no proposition which is not a direct observation is self-evident to me. For example, ‘the sky is blue’ is self-evident because it is a direct observation. But I have no sense organ that senses value – no sense organ that tells me that the sky is valuable. If someone who gives this answers (It is self-evident) really means it, then we have fundamentally different natures – so that I do not even know what he means when he says ‘It is self-evident that X is an absolute value’. But I don’t believe this. To see why consider just one example of a self-evident value that Heumer cites – justice. Heumer claims that it is self-evident that “It is unjust to punish a person for a crime he didn’t commit.” It is obvious that different people have widely varying understandings of the concepts ‘crime’ and ‘justice’. For example, I don’t think it is just to punish or compensate people for crimes that their ancestors committed or suffered. But the whole concept of affirmative action depends on doing just that. Therefore even if the affirmative action advocates say that they believe it is unjust to punish a person for a crime he didn’t commit, they mean something very different from what I understand by it. The only way to say that justice is a self-evident value is to broaden the concept so much that it becomes useless. 

c) The question why is not appropriate:
There are some things about which really does not make any sense to ask why. For example, it makes no sense to ask “Why does anything exist at all?”. Is the question “Why is X an absolute value?” like that, atleast for some X? If so, then the why immediately turns into a how – “How do you know that X is an absolute value?” One answer to this could be that it is self-evident, but I have already dismissed that. Another answer could be that it is axiomatic (like the four axiomatic propositions I stated above). But just claiming that something is an axiom is not sufficient. Even axioms have to be validated. An axiom can be validated by assuming that it is not true and then looking at the implications. If the axiom is true, one immediately reaches a contradiction or an absurdity. I won’t actually demonstrate this for the four axioms I stated. Anyone should be able to see that this is so. For what X does the proposition “X is not an absolute value” lead to a contradiction or an absurdity? I know of no such X (and none of the supposedly self-evident principles that Heumer cites – more on them in another post – indicate the existence of any such X). Anyway it is not my task to prove that no such X exists. Obviously I cannot (just like the existence of God). It is upto someone who believes that such X exist to identify them. Even one example would be enough.

But if one actually wants to answer the ‘why’, one will have to say something of the form “because it [verb] [noun]” where noun is some purpose. And only agents can have purposes. Atleast I cannot think of any other way to answer the ‘why’. If one does give the “because it [verb] [noun]” answer, then X is a value relative to the agent who has the particular purpose and not in any absolute sense.

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