Jonathan P. Seldin
Department of Mathematics and Computer Science
University of Lethbridge
Lethbridge, Alberta, Canada
jonathan.seldin@uleth.ca
http://people.uleth.ca/
jonathan.seldin/
Thomas Kuhn [Kuh62] presents a view of the history of science as a succession of ``paradigms'' which are incomemnsurable with each other. Karl Popper [Pop70] attacks this view as being relativistic and denying that there is objective scientific truth.2 Popper seems to be saying that in order for science to be objectively true, every two scientific theories must be comparable in the sense that there is some testable statement true in one and false in the other. In taking this position, Popper is making a claim for a kind of completeness that is ruled out for theories strong enough to be ``interesting'' by Gödel's Theorem.
One way of looking at Gödel's Theorem is that it says that for any theory strong enough to be ``interesting''3 it is impossible to give any sound and complete axiomatization. Here, an axiomatization does not necessarily have to be finite; it is sufficient if it gives an effectively computable test for correctness of any supposed proof. Hence, what Gödel's Theorem really says is that there is no sound and complete specification of the truth conditions of the theory. This applies, in particular, to the theory of arithmetic.
On the other hand, in order to accept the proof of Gödel's Theorem, we need to believe in the truths of arithmetic.4
Gödel's Theorem implies that interesting scientific theories are incomplete. Suppose that we want to compare two scientific theories, T1 and T2. To make the comparison, we would need a third theory, T, which is strong enough to permit us to talk about both T1 and T2. But by Gödel's Theorem, T is also incomplete. Thus, the comparison of T1 and T2 cannot be complete enough to avoid what Kuhn called ``local incommensurability'' in [Kuh83]. But just as Gödel's Theorem does not imply that the truth of arithmetic is subjective, so this incommensurability of scientific theories does not imply that scientific truth is subjective.
Despite Popper's later change of mind, it appears that he failed to understand just how close to Kuhn's views of science his own views were. Kuhn himself [Kuh70] points this out,5 especially with regard to the analysis of particular historical episodes in the history of science. I believe that the root of this failure of understanding is his failure to appreciate how Gödel's Theorem implies incommensurability of theories. Once this is understood, I think we can say that, roughly,
Gödel's Theorem also has something to tell us about the views expressed by Paul Feyerabend in [Fey75].6 Feyerabend thinks that he is arguing that there is no such thing as a scientific method. But when his arguments are stripped of rhetoric, what is left is a catalog of instances in which various descriptions of scientific methodology fail to correspond with the way science has actually progressed. Feyerabend seems to be arguing that the absence of a sound and complete description of the scientific method means that there is no such method. But this is not the conclusion we draw from Gödel's Theorem: the theorem does not say that there are no truth conditions in arithmetic, but only that we cannot give a sound and complete axiomatic description of them. Feyerabend may have produced some evidence that we can never soundly and completely describe the scientific method, but he has no evidence whatever that there is no such method.7
At this point some readers may wonder what we can say to those who really deny the value or truth of science. I think there is an answer to them, but it is not the same kind of answer that Popper was seeking. The answer I have in mind does not try to prove the truth of every assertion of the accepted scientific paradigms. Instead, it concentrates on those questions that are of interest to legislative bodies and courts. Legislative bodies and courts are interested in the scientific evidence for their conclusions. And I think that the refereeing procedure that is standard in the best scientific journals is the procedure that generates the most evidence for or against the assertions involved.8 This gives us a basis for saying that there is an objective basis for these assertions. Of course, at a later time, the evidence may change with regard to any particular question, but at the time the question is being considered, a consensus of the literature by specialists on the matter in question in refereed scientific journals is the best evidence available on that question.9 This is the way we can refer to science as embodying objective truth. There are questions that arise in science that are never considered by courts or legislative bodies: an example is the question that arose in the early days of the special theory of relativity, when there was an alternative theory due to H. A. Lorenz based on classical physics which made exactly the same predictions as Einstein's theory. Einstein's arguments for his approach did not constitute the same kind of evidence from the point of view of courts and legislative bodies as, say, the view that AIDS is caused by the HIV virus. But then these arguments of Einstein were on a subject unlikely to get before a court or legislative body. It is at the point at which science touches public policy through courts and legislative bodies that we are best able to talk about the objective reality of science.10