July 15, 2011
The logic of science-5: The problem of incompleteness
(For previous posts in this series, see here.)
As I discussed in the previous post in this series, our inability to show that an axiomatic system is consistent (i.e., free of contradictions as would be evidenced by the ability to prove two theorems each of which contradicted the other) is not the only problem. Godel also showed that such systems are also necessarily incomplete. In other words, for all systems of interest, there will always be some truths of that system that cannot be proven as theorems using only the axioms and rules of that system. So the tantalizing goal that one day we might be able to develop a system in which every true statement can be proven to be true also turns out to be a mirage. Neither completeness nor consistency is attainable.
Belief in god depends upon ignorance for its very existence and some religious people have seized on Godel's theorem to try and argue that 'god exists' is one of these true statements that cannot be proved. This is a misunderstanding of what Godel proved but is typical of attempts by religious people who seize upon and use important results in science and mathematics (especially those that impose some limits to knowledge, such as the uncertainty principle) to justify the unjustifiable.
The fact is that you cannot simply assert that any proposition you choose belongs in that niche that Godel discovered. The true yet unprovable statements have to be constructed within that particular system to meet certain criteria and are thus dependent on the axioms used, and a statement that is true but unprovable in one system need not be so in another one. Simply by adding a single new axiom to a system, statements that were formerly unprovable cease to be so while new true but unprovable statements emerge. Whenever religious people invoke Godel's theorem (or the uncertainty principle or information theory) in support of their beliefs, you should be on your guard and investigate if what they say is actually what the science says.
So what can we do in the face of Godel's implacable conclusion that we cannot construct an axiomatic system in which the theorems are both complete and consistent? At this point, pure mathematicians and scientists part company. The former have basically decided that they are not concerned with the truth or falsity of their theorems (and hence of the axioms) but only with whether the conclusions they arrive at (the theorems) are the necessary logical conclusions of their chosen axioms and rules of logic. Even a statement such as '2+2=4', which most people might regard as a universal truth that cannot be denied, is seen by them as merely the consequence of certain starting assumptions, and one cannot assign any absolute truth value to it. So pure mathematicians concern themselves with the rigor of proofs, not with whether the theorems resulting from them have any meaning that could be related to truth in the empirical world. Mathematical proofs have become disconnected from absolute truth claims.
For the scientist dealing with the empirical world, however, questions of truth remain paramount. It matters greatly to them whether some result or conclusion is true or not. While the methods of proofs that have been developed in mathematics are used extensively in science, scientists have had to look elsewhere other than proofs to try and establish the truth or falsity of propositions. And that 'elsewhere' lies with empirical data or the 'real world' as some like to call it. This is where the notion of evidence plays an essential role in science. So in mathematics while the statement '2+2=4' is simply a theorem based on a particular set of axioms, in science its empirical truth or falsity of it has to be judged by how well real objects (apples, chairs, etc.) conform to it.
This dependence on data raises a problem similar to that of the consistency problem in mathematics that Godel highlighted. We can see if '2+2=4' is true for many sets of objects by bringing the actual objects in and counting them but we obviously cannot do so for everything in the universe. So how can we know that this result holds all the time, that it is a universal truth? Such a concern may well seem manifestly overblown for a simple and transparent assertion like '2+2=4' but many (if not most) results in science are not obviously and universally true and so they can be challenged. For example, for a long time the tobacco industry challenged the conclusion that smoking causes cancer by pointing out that there exist some smokers who do not get cancer.
So however much the data we obtain supports some proposition, how can we be sure that there does not exist some undiscovered data that will refute it? This does not mean that we cannot be definitive in science. But the justification of scientific conclusions depends upon a line of reasoning that is different from those involving direct proofs, as will be seen in subsequent posts.
Next: The logic of science and the logic of law