Computability and Comprehension – Is Science About Prediction?By: Bruce
Science is the process of how we use reason to find patterns in reality and then to explain them in finite explanations of reality that allow us to represent reality via processes that are computable.
In my last post, I introduced David Deutsch’s book, The Fabric of Reality. Deutsch’s main interest is in understanding – and by that he means understanding everything. Deutsch believes that understanding something is to have an accurate explanation of it and that this, in turn, serves as a sort of algorithmic compression of all observational data.
Deutsch’s point of view falls under what we might call Scientific Realism. It’s the idea that science is not just about coming up with clever predictions about the world, but rather it’s about discovering reality’s true nature and comprehending it.
The alternative point of view I throw under the umbrella of Positivism. Positivism comes in many forms. Physicist and author Stephen Hawking describes positivism like this:
Any sound scientific theory… should in my opinion be based on the most workable philosophy of science: the positivist approach put forward by Karl Popper and others. According to this way of thinking, a scientific theory is a mathematical model that describes and codifies the observations we make. A good theory will describe a large range of phenomena on the basis of a few simple postulates and will make definitive predictions that can be tested.
If one takes the positivist position, as I do, one cannot say what time actually is. All one can do is describe what has been found to be a very good mathematical model for time and say what predictions it makes. (The Universe In a Nutshell, p. 31) 
When stated this way, Positivism sounds rather innocuous. Isn’t it just a truism that our science never really uncovers reality? We know there are often multiple interpretations of the same physical phenomenon. For example, in most cases Newtonian Physics and General Relativity both give exactly the same predictions. Not until you start to approach the speed of light do they diverge. And in the case where the two views do diverge, we favor General Relativity because of its better predictive power. But does that really mean we should pretend that the universe is curved even though common sense tells us it’s not?
In other words does it really matter if science finds some ultimate view of reality? Isn’t science really just about making predictions?
Look at our definition of what science is above. Think of Positivism as concentrating on the ‘computational’ aspect of science and Scientific Realists as concentrating on the explanation side of science.
Deutsch is in the ‘ultimate view of reality’ category, for he believes that science is solely about trying to explain and therefore comprehending reality. The computational aspect just falls out as a matter of course once you understand something.
For even in purely practical applications, the explanatory power of a theory is paramount and it’s predictive power only supplementary. (The Fabric of Reality, p. 4)
To prove his point of view, Deutsch suggests a thought experiment. Pretend that aliens give us poor humans a magic box, an ‘oracle’ so to speak, that can “predict the outcome of any possible experiment, but provides no explanations.” (The Fabric of Reality, p. 4) In theory this should be a Positivist’s dream. Since we only care about the predictive power of science, we now no longer need science because we can literally predict anything.
The problem Deutsch points out is that we still don’t know what experiments to have the oracle predict for us. So it does us no good to have it without first using science to decide what to try to predict. Deutsch goes on to say:
If we gave [the oracle] the design of a spaceship, and the details of a proposed test flight, it could tell us how the spaceship would perform on such a flight. But it could not design the spaceship for us in the first place. And even if it predicted that the spaceship we had designed would explode on take-off, it could not tell us how to prevent such an explosion. (The Fabric of Reality, p. 4)
So it would seem that Scientific Positivism is fatally flawed. “Prediction – even perfect, universal prediction – is simply no substitute for explanation. …the oracle would not be replacing theories at all: it would be replacing experiments.” (p. 5)
So there is a deep relationship between explanation and computability. But ‘computability’ is subservient, so to speak, to explanation itself. But doesn’t this just make sense? If you comprehend something, like what PI is, you can figure out how to calculate it. But if you know how to calculate it only, you don’t really understand what it is or what to do with it.
Criteria for an “Explanation”
But this does give us a sort of criteria for what is or isn’t an ‘explanation.’ Explanations are deeply related to ‘computable algorithms.’ That is to say, there is no such thing as an explanation that doesn’t also have an attached algorithm. (Though there might be an algorithm that has no attached explanation.) At first this might seem a bit uncomfortable. Is it really true that all patterns in nature are explainable and thus algorithmic? We don’t actually know and can never know that for certain. But we can know that the alternative – that some things in nature can’t be explained at all – is unacceptable.  If we were ever confronted by an unexplainable pattern in nature we would never give up trying to understand it via explanation. Unless we could somehow explain why the pattern is unexplainable (via a proof/algorithm, of course), we’d forever insist that it must be explainable and keep seeking for that explanation. 
Can you think of any ‘explanations’ that don’t include an algorithm?
Can you think of any algorithms that don’t include an explanation?
For those that are familiar with Omega already (see note 2 and link) what significance is there that we can define a number that we can’t compute, but that we can compute that we can’t compute it? (Did you follow that?)
Deutsch is hard on “Positivism” because he feels it downplays explanation in favor of prediction. Is this the only way to interpret Positivism? Or is Deutsch being too harsh here?
If you owned an alien artifact that could predict any scientific question, what questions would you ask it? (Note: You have to ask it scientifically specific questions. It won’t understand abstract or general questions.)
 It’s interesting that Hawking gives Karl Popper credit for Positivism because Deutsch gives Popper credit for Scientific Realism. After reading Popper, I’m not surprised that both schools of thought trace themselves to Popper. However, Popper claims he is not a Positivist unless you stretch the term well beyond the way people normally understand it. (Myth of the Framework, p. 75)
In a future post, I’ll consider Hawking more recent arguments that there is no “ultimate reality” to find in the first place and therefore Positivism actually represents reality better than Scientific Realism.
 Gregory Chaitin’s article “The Limits of Reason” challenges the idea that all things are explainable in mathematics and implies that this might also be true for physics. Within the scope of this one post, I can’t tackle everything he says other than this tiny nod to him. Needless to say, even if that is true, unless it’s something like Chaitin’s “omega number” where we can actually explain why we can’t compute it and compute that we can’t we’d have no basis for no longer seeking an explanation. In other words, we can’t compute his ‘omega number’ directly, but we can explain what this non-computable number is via an algorithm (Godel’s theorem and the halting problem are essentially the same theorem in different forms and are both algorithms.) Furthermore, we can come up with algorithms to figure out what finite portions of “omega” are. So it’s not strictly true that ‘omega’ can’t be computed but only that there is no hope of computing it in it’s infinity. In any case, ‘omega’ is a carefully tailored exception to what I am suggesting. It’s a non-computable number that we can still explain via an algorithm. So I’m not sure ‘omega’ is, strictly speaking, unexplainable.
 Chaitin uses the Goldbach conjecture as an example of a truth (maybe) without a reason. I note here that mathematicians continue to try to find a proof for or against it, unable to accept that there isn’t one. And this is in mathematics, not physics. Oy!