Required background: none
The two main topics of this blog – physics and mathematics – are often introduced in textbooks and courses without much regard for their underlying assumptions. The guiding principles are quite easy to state, although the ramifications are very deep. The importance of these principles cannot be overestimated.
Mathematics relies on a set of propositions commonly called axioms. Strict logical arguments, with no further assumptions, are then made to prove the truth of every statement in mathematics. As such, every truth we know about mathematics relies ultimately on the original axioms. Obviously, we don’t want to assume anything more than we absolutely have to in order to be able to prove things, so one goal of pure mathematics is to find the smallest set of axioms from which it is possible to derive everything else.
There is one snag in this beautiful idea, and it was discovered by mathematician Kurt Gödel in 1931. Known as Gödel’s incompleteness theorems, they state that no matter what axioms you choose, there will always be some statements within your axiomatic system that are true, but which you cannot prove to be true. Put another way, Gödel is saying that you can never discover all the truths in any mathematical system simply by starting with the axioms and applying logical deduction.
Despite this flaw, one key thing to notice about mathematics is that it is independent of anything in the real world. We can invent whatever axioms we like (as long as they don’t introduce logical impossibilities like self-contradiction) and start proving things from them. Even though the incompleteness theorem says that we will never be able to discover all truths in our system, those truths we do discover are logically consistent and do not depend on the outside world.
Physics is quite a different beast, for its entire subject matter is this real world which was of no consequence to mathematics. Despite this, the methods of physics are eerily similar to those of mathematics, for the following reason.
Where mathematics begins with a set of axioms, physics begins with some assumptions about the nature of the real world. Once these assumptions are cast into mathematical form, we can apply the methods of mathematics to derive consequences of the assumptions. The mathematics will tell us that these consequences are logically true, given our assumptions. However, physics then requires us to take these logical consequences and see if they correctly describe the real world. If they do, then it is evidence that our original assumptions were valid, and we can continue to derive further consequences from them. If at some stage we discover that the logical facts we derive from our assumptions do not match what we find in the real world, then we realize that our original assumptions must have been wrong. We therefore need to make some different assumptions, derive some more consequences, and check their validity with the real world.
Thus physics operates like mathematics, but with reality checks thrown in. When studying physics it is very important to realize exactly what assumptions are being made at each stage, and this is something that is often glossed over in textbooks.
There is one final point that arises from this little discussion. We’ve seen that mathematics is a self-contained logical system which is true independently of what happens in the real world. However, this same mathematics actually does provide a logical description of the way things in the real world work. It is fairly obvious that there is no particular reason why this should be so, and in fact it remains an unresolved problem at the very core of science. Put simply: why is the universe understandable at all?