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).

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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.

Tradeoffs as Debt

From the latest post in Rico Mariani’s series on the history of Visual Studio

Debt is a great way to think about trade-offs: every time you make a choice that isn’t right in the long term, any short-cut, accumulates some debt.  Any bug you choose to defer, that’s debt.  Some debt you should write-off, that fix just isn’t happening, some you should address, but always you should be aware that you’ll have to deal with it sooner or later, and it may as well be sooner.

Book Review: Fooled by Randomness

I chanced upon Fooled by Randomness – The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb at a friend’s place and took the time to read it. Having a bit of a financial background – I work in a company that did some financial modeling before I joined it – I had heard of Taleb and was curious. Besides, I want to understand probability better than I currently do – I mean philosophically, not mathematically – and the title was attractive.

The book is divided in three parts. Part I starts off with a long and rather boring story of two traders – a rash, ignorant and over-confident John and a conservative Nero. John succeeds for a time – purely through luck – makes a lot of money and then blows up – market slang for losing more money than you thought possible. Nero remains risk-averse and makes a steady amount but suffers snubs from people like John before being vindicated. The reason for including this story is primarily to show how large a role randomness plays in the markets. Taleb also comments on the fact that Nero suffered emotionally from the snubs by people who made more money than him though he always knew himself to be better. Taleb says that this shows that the rational mind cannot prevent us from experiencing irrational emotions. Taleb then discusses an “accounting method” by which a dentist is much richer than a lottery winner. If one were to consider all the “paths” that the dentist’s life could take, there would not be much variation in the money he makes and the “average” would be close to what he makes in any particular “path”. If one considers all the paths that the lottery winner’s life could take, the average would be much lower than the money he makes on the winning path. This notion should seem familiar to anyone with a knowledge of Monte-Carlo simulations but I had not seen anyone putting it so explicitly. Taleb then goes on to discuss the difference between noise and significant information and how noise can affect perceptions in short timescales. He also discusses the dangers in fitting models to historical data. This is interrupted by an unexpected attack on Hegel’s pseudo-scientific philosophy that draws on Alan Sokal’s famous hoax. Taleb then talks of rare events, how their existence makes the difference between the median and the mean important and how most people including statisticians often unwittingly ignore this difference. He then talks briefly about Bacon, Hume and Popper in relation to the problem of induction and the difficulty of induction in the presence of rare events.

Part II deals with various biases in the perception and evaluation of events and outcomes in areas where randomness plays a major role. He draws on work by Kahneman and Tversky – which I am not even remotely familiar with – to claim that in dealing with uncertainty, our minds adopt certain heuristics/biases that are blind to reason (Prospect theory, Affect heuristic, Hindsight bias, Belief in the law of small numbers, Two systems of reasoning and Overconfidence). While it is easy to see how a person with no understanding of probability theory could be misled in the many examples Taleb gives, it is difficult to believe that people trained in probability would also be misled.

Part III deals with Taleb’s interpretation of stoicism as the solution to living in a world with so much uncertainty. Taleb writes that we should accept that we are incapable of making our emotions rational and attempt to behave with dignity in all circumstances. He writes that stoicism should not mean a stiff upper lip and a banishment of emotions but an acceptance of emotions and the uncertainties of life with the focus being on the process rather than the outcome. This part is titled Wax in my ears in a reference to the story of Odysseus and the Sirens. Taleb writes that he knows that he is not as great as Odysseus and instead of tying himself, he chooses to have wax in his ears. That is, he chooses to accept that his emotions will always be fooled by randomness and the only solution is to avoid situations where he might encounter such emotions (by not listening to the news or not tracking prices of assets on a moment-by-moment basis etc).

Overall, several anecdotes in the book are mildly entertaining, but intellectually, there is very little that I gained from the book. I agree with a lot of Taleb’s views on the role of luck in the markets and the inadequacy or even meaninglessness of most financial models, but I had already reached these views before reading Taleb and frankly I don’t think they merit a significant part of a book. These views can be easily expressed in a few pages – perhaps I will write a post myself. Taleb does not provide any definition of probability – something that I had hoped for – apart from the following excerpt. Taleb’s style is quite disconnected and the numerous back and forward references are irritating, especially since the references are hardly convincing. For example in the following excerpt he refers to something in Chapter 3, but there is no convincing arguement there, not even a hint.

Ask your local mathematician to define probability, he would most probably show you how to compute it. As we saw in Chapter 3 on probabilistic introspection, probability is not about the odds, but about the belief in the existence of an alternative outcome, cause, or motive. Recall that mathematics is a tool to meditate, not compute. Again, let us go back to the elders for more guidance – for probabilities were always considered by them as nothing beyond a subjective, and fluid, measure of beliefs.

The only thing that I got from the book is a reminder that I need to formulate more completely a proper alternative to Popper’s scepticism.

Scepticism and Morality

I ended my last post with the statement that sceptics cannot take ideas – particularly moral ideas seriously. Here is an excerpt from the book Fooled by Randomness – The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb that serves as an illustration of my point.

Current thinking presents the two following polarized versions of man, with little shades in between. On the one hand there is your local college English professor; your great-aunt Irma, who never married and liberally delivers sermons; your how-to-reach-happiness-in-twenty-steps and how-to-become-a-better-person-in-a-week book writer. It is called the Utopian vision, associated with Rosseau, Godwin, Condorcet, Thomas Paine, and conventional normative economists (of the kind to ask you to make rational choices because that is what is deemed good for you), etc. They believe in reason and rationality – that we should overcome cultural impediments on our way to becoming a better human race – thinking we can control our nature at will and transform it by mere edict in order to attain, among other things, happiness and rationality. Basically this category would include those who think that the cure for obesity is to inform people that they should be healthy.

On the other hand there is the Tragic Vision of humankind that believes in the existence of inherent limitations and flaws in the way we think and act and requires an acknowledgement of this fact as a basis for any individual and collective action. This category of people includes Karl Popper (falsification and distrust of intellectual “answers”, actually of anyone who is confident that he knows anything with certainty), Friedrich Hayek and Milton Friedman (suspicious of governments), Adam Smith (intention of man), Herbert Simon (bounded rationality), Amos Tversky and Daniel Kahneman (heuristics and biases), the speculator George Soros, etc. The most neglected one is the misunderstood philosopher Charles Sanders Pierce, who was born a hundred years too early (he coined the term scientific “fallibilism” in opposition to Papal infallibility). Needless to say that the ideas of this book fall squarely into the Tragic category: We are faulty and there is no need to bother trying to correct these flaws. We are so defective and so mismatched to our environment that we can just work around these flaws. I am convinced of that after spending almost all my adult and professional years in a fierce fight between my brain (not Fooled by Randomness) and my emotions (completely Fooled by Randomness) in which the only success I’ve had is in going around my emotions rather than rationalizing them. Perhaps ridding ourselves of our humanity is not in the works; we need wily tricks, not some grandiose moralizing help. As an empiricist (actually a sceptical empiricist) I despise the moralizers beyond anything on this planet: I still wonder why they blindly believe in ineffectual methods. Delivering advice assumes that our cognitive apparatus rather than our emotional machinery exerts some meaningful control over our actions. We will see how modern behavioral science shows this to be completely untrue.
(emphasis mine)

To which I will only say: If our cognitive apparatus exerts no meaningful control over our actions, isn’t Taleb wasting his time writing a book? He should be composing music instead.

Interpreting History and Sceptcism

In an email exchange regarding an article in The Hindu regarding secularization and modernization, a friend (call him X) commented: “As far as the article goes ….. I didn’t like it as much. More like the author already has some conclusions and wants to write something to highlight those conclusions.”

Indeed the author already has some conclusions or rather an interpretation of history. The same could be said of my post regarding the same article. Why is that bad? Could it even be otherwise? Anyone who is sufficiently interested in a subject to write about it will and should have an interpretation of the relevant historical events. Having an opinion/conclusion/interpretation is not bad. Not having the honesty to revise ones ideas if one finds facts that contradict them is. In fact, forming tentative hypotheses and refining/correcting them as one encounters new facts is the proper method to deal with anything that has a large scope. 

As an example of this, I am a software application developer. My work involves building upon an enormous amount of previous work of which I know very little. I do not know much about how the hardware on which my application runs. I do not know much about the network over which it communicates. I do not know about all the intricacies and design details of the software libraries that I use. Yet I need and have a mental model for all these things. The model is better in some areas than in others but it is not complete and will probably never be. But the fact that I might never be able to have a complete mental model does not mean that I should try not to have a model at all.

Of course, the example is not fully analogous. It is not possible to test an interpretation of history in the way that it is possible to test a model of computer hardware or software. But the necessity for interpreting history remains. History is the only place where we can actually see ideas in action. It is the only empirical source for validating ideas.

The attitude of discounting something because the author seems to have firm opinions smacks of scepticism. In the same email, X also wrote “I think that religion in not the center of the universe  for most”. Scepticism seems to me to be very common among the non-religious. But scepticism is an intellectual dead end. It transforms philosophy from a tool for living well to a game of no consequence. If one believes that one should never form firm ideas, one cannot take ideas (especially moral ideas) seriously. (I will provide an example of this in my next post.) And that is not conducive for a good life.

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