★★☆☆☆ Knocking down a straw man

The Open Universe: An Argument for Indeterminism from the PostScript to the Logic of Scientific Discovery

Karl Popper

I read The Open Universe between 1987 and 1990 -- I know that, because I remember why I read it. I was a postdoc in Cambridge during those years and complained to one of my colleagues that philosophers, despite their vaunted claims to be the princes of logic, are often not very good at it -- their arguments are often logically flawed. My colleague told me that if I read Karl Popper's The Open Universe I would find the logic flawless. I did, and I didn't. I pointed out some of the logical flaws to my colleague, and he agreed I had a point.

The Open Universe is Popper's argument against determinism. Now, if you're hoping to read an open-minded discussion of the determinism/nondeterminism question, this is not the place to look. It is clear right from the start that Popper despises the very thought of determinism, and sets out in this book to demolish it.

Personally, I am not very interested, because I think the determinism/nondeterminism question is vacuous. No one can tell you in any concrete way how a deterministic universe would be different from a nondeterministic one. I mean, it is completely clear as a practical matter that we can't predict the future perfectly -- I don't know what weather we will have a year from today -- but that we can sometimes make predictions that have a good chance of becoming true. I have lived in Canada eight years, and I have learned that if Environment Canada says we're going to have a huge snowstorm in the next day or two, a snowstorm will in fact materialize.

Recognizing this problem, Popper begins by setting out in detail how he will define determinism. And it is here that his plan becomes clear. He defines determinism in so strong and strict a way that it is impossible for his criteria to be satisfied. Then he writes a book in which he shows that failure in detail.

I'll give one example. Popper argues correctly that the motions of a system of three bodies moving according to Newtonian gravitation cannot be predicted into the indefinite future. He is correct about this -- Henri Poincaré showed it in 1887 -- it was the beginning of the field now known as Chaos Theory. The three-body problem shows what we now call the "Butterfly Effect", extreme sensitivity to initial conditions, such that even the smallest change eventually leads to completely unpredictable results.

Popper apparently fails to realize that this is not a statement about physics -- it is a statement about mathematics. We now know that much simpler systems than the three-body problem show the Butterfly Effect. Perhaps the simplest is the logistic map: x ← cx(1-x), where x is a number between 0 and 1 and c a constant between 0 and 4. If c is a rational number (i.e., an integer fraction) and x is also rational, then all the subsequent x's are rational and known with perfect precision, and any number of them can be calculated on a computer in a finite and completely predictable way. Yet, by Popper's definition, this mathematical system is not deterministic.

It is an absurd definition.

Now, understand, my complaint is not that Popper doesn't know about chaos or the logistic map. Popper wrote The Open Universe in 1951-1956 (although it wasn't published until 1982), and the logistic map became an active object of study in the 1970's. My complaint is that he defined determinism is such a way as to make his criteria impossible to satisfy.

In The Open Universe, Popper convincingly refutes a position that absolutely no serious person believes.

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