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### July 29, 2011

#### The logic of science-9: Can scientific theories be proven true?

(For previous posts in this series, see here.)

In mathematics, the standard method of proving something is to start with the axioms and then apply the rules of logic to arrive at a theorem. In science, the parallel exercise is to start with a basic theory that consists of a set of fundamental entities and the laws or principles that are assumed to apply to them (all of which serve as the scientific analogues of axioms) and then apply the rules of logic and the techniques of mathematics to arrive at conclusions. For example, in physics one might start with the Schrodinger equation and the laws of electrodynamics and a system consisting of a proton and electron having specific properties (mass, electric charge, and so on) and use mathematics to arrive at properties of the hydrogen atom, such as its energy levels, emission and absorption spectra, chemical properties, etc. In biology, one might start with the theory of evolution by natural selection and see how it applies to a given set of entities such as genes, cells, or larger organisms.

The kinds of results obtained in science using these methods are not referred to as theorems but as *predictions*. In addition to the mathematical ideas of axioms, logic, and proof, in science we are also dealing with the empirical world and this gives us another tool for determining the validity of our conclusions, and that is data. This data usually comes either in the form of observations for those situations where conditions cannot be repeated (as is sometimes the case in astronomy, evolution, and geology) but more commonly is in the form of experimental data that is repeatable under controlled conditions. The comparison of these predictions with experimental data or observations is what enables us to draw conclusions in science.

It is here that we run into problems with the idea of truth in science. While we can compare a specific prediction with experimental data and see if the prediction holds up or not, what we are usually more interested in is the more basic question of whether the underlying theory that was used to arrive at the prediction is true. The real power of science comes from its theories because it is those that determine the framework in which science is practiced. So determining whether a theory is true is of prime importance in science, much more so than the question of whether any specific prediction is borne out. While we may be able to directly measure the properties of the entities that enter into our theory (like the mass and charge of particles), we cannot directly test the laws and theories under which those particles operate and show them to be true. Since we cannot treat the basic theory as an axiom whose truth can be established independently, this means that the predictions we make do not have the status of theorems and so cannot be considered *a priori* true. All we have are the consequences of applying the theory to a given set of entities, i.e., its predictions, and the comparisons of those predictions with data. *The results of these comparisons are the things that constitute evidence in science*.

So what can we infer about the truth or falsity of a theory using such evidence? For example, if we find evidence that supports a proposition, does that mean that the proposition is necessarily true? Conversely, if we find evidence that contradicts a proposition, does that mean that the proposition is necessarily false?

To take the first case, if a prediction agrees with the results of an experiment, does that mean that the underlying theory is true? It is not that simple. The logic of science does not permit us to make that kind of strong inference. After all, any reasonably sophisticated theory allows for a large (and usually infinite) number of predictions. Only a few of those may be amenable to direct comparison with experiment. The fact that those few agree does not give us the luxury of inferring that any future experiments will also agree, a well known difficulty known as the *problem of induction*. So at best, those successful predictions will serve as evidence in support of our theory and suggest that it is not obviously wrong, but that is about it. The greater the preponderance of evidence in support of a theory, the more confident we are about its validity, but we never reach a stage where we can unequivocally assert that a theory has been proven true.

So we arrive at a situation in science that is analogous to that in mathematics with Godel's theorem, in that the goal of being able to create a system such that we can find the true theories of science turns out to be illusory.

Next: Can we prove a scientific theory to be false?

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