What is Reality?

Reality is the name used by philosophers for God, after the name has been depleted of all fanatism of all organized religions, so that the highly emotional content that goes with the mention of God can be avoided during a rational discussion. Many scientists believe in what is called pantheism, the view that the only part of reality that we can rationally investigate is the enormously complex working of nature that is visible to us. These scientists carry out their investigations using mathematics as their tool. They fabricate theories like electromagnetic theory, group theory, and derive all their theorems from a set of axioms pertinent to the theory.

Metamathematics. Since the mathematical derivations are known to be mechanical, it becomes clear that the most important part of any theory is the set of axioms that define it. When we investigate the axioms and the derivation rules of a theory, the study is called metamathematics.

From all this one might conclude that the best that humans can do is to restrict themselves to metamathematics, however, this is easier said than done. Unfortunately, we have been attempting to do this at least for the last two millennia, but our violent history shows that we are unable to do it. There seem to be something in the human soul which pines for a realm even beyond metamathematics. Obviously, we cannot rationally comment about it here.

We are not out of trouble even if we restrict ourselves to metamathematics. If you ask a mathematician to show that 2+3=5, he we will ask you to count five apples and give two to Tom and three to Dick and notice that there is none left. If you ask him to justify commutative law of multiplication, he will tell you that you will need a dozen apples whether you distribute them between four children three each or three children four each. The situation becomes more complicated when he tells you that a four dimensional cube has 32 edges. When pressed to explain he will ask you to take the projection of a four dimensional cube on plain paper and count the number of edges in the projection. The situation becomes impossible when he tells you that the volume of a four dimensional sphere is ((\pi^2)/2)r^4. When pressed he will explain that you need to know quite a few fundamental concepts of mathematics to understand and accept this formula.

Mathematics as an Axiomatic Theory. So, what do we make out of all this? The basic facts here can be explained without introducing complications. The whole of mathematics can be considered as a theory with its set of axioms and derivation rules. It is believed that these set of axioms do not have hidden contradictions in it, since it has worked well for us for the last two thousand years. Any derivation we make without violating rules of mathematical logic is supposed to give us a theorem which we can trust. The volume of the four dimensional sphere given above is such a derivation.

With all that we have said so far, we are still not out of trouble. Even though the natural numbers form an infinite set, it turns out that they are just not enough to count the number of points in a unit interval or explain the space beyond the stars. Defining the concept of powersets and cardinals we can generate bigger and bigger infinities indefinitely, eventually finding out that there is no biggest cardinal. With the proliferation of cardinals in set theory it has not been possible to order them in a sensible fashion. The continuum hypothesis of Cantor attempts to solve this problem in a reasonable way.

Symbol Manipulation: the toy of homo sapiens. I consider symbol manipulation as the highest achievement of our civilization. To answer the question, what is reality, I would say that it is the set of derivations we make with well chosen symbols and axioms of a theory. If the symbols have been created with insight and the axioms have been chosen carefully, we can be sure that our derivations will be meaningful and represent some facts of reality. A conspicuous example of symbol manipulation is provided by the prediction of radiation by Maxwell. Another example is provided by the derivation of bending of light rays by Einstein. The developments in mathematical logic shows that we will never be able to comprehend reality in toto. Philosophers get uptight and uneasy when they find that they have reached this impossible situation after all their painstaking studies. To shift his problem to the common man, Bertrand Russell asks the question: What should a true democrat do, when the majority insists that they do not want democracy? Attempting to find a solution to the problem, vedanta philosophy represents the inexplicable Brahman (Reality) by the symbol OM, and asks the vedantist to make a resounding utterance of OM in helpless supplication. Recently, this method of investigation seems to have gained some popularity around the world.

For more of my opinions regarding reality, see the presentation The Mathematical Universe in a Nutshell.