Theory of Everything

May 20th, 2009 • Related • Filed Under

The human mind is said to be the most complex object in the universe. Why it should possess the ability to comprehend that universe has always been considered a mystery. More recently that mystery has become even deeper with the realization that we can understand in some detail the very inner workings of some aspects of nature, that have given us no evolutionary advantage in the past. But just how far does such cognitive talent in humans reach in the understanding of a finely tuned understanding of fundamental questions? Why is it that at the level of ultimate reality we seem well disposed to comprehend the range of problems we discover, when such an understanding has no bearing on our evolution and on the talents we have required in the past to survive and flourish?

None of our sophisticated ideas seem to offer any selective advantage during our evolution, yet it is clear we have the capability to delve deep into the deepest most hidden secrets of nature. The common factor in our understanding of the universe we inhabit lies in mathematics. Nature is written in the language of math and we have an undisputed ability to discover, or some would argue invent, that language.

It was Newton who began the mathematization of reality. He produced a change in perspective probably unrivalled in the history of human ideas. It was as though he had discovered a master key to unlocking nature’s secrets and it has been proven since that with constant usage that master key just keeps opening more doors. Mathematics is the key to everything; it would seem then that any final theory from which all things can be explained will be rooted in some mathematical concept.

There are two areas of interest in the field of mathematics which can help us in the quest for ultimate truths. The first is geometry and the second is the infinite. The first is for practical purposes and the second for theoretical reasons. This article is interested in geometry.

Geometry was the king of all subjects since the Ancient Greek period, Euclid being the standard by which all knowledge would be judged for 2000 years– indeed until the time of Newton. Euclid built a geometric system based on self evident axioms which were internally consistent and each of which could be derived through pure logic from the other axioms. Euclid’s system was considered to be perfect knowledge and was understood as providing a description of the three dimensional space in which we live.

The knowledge was perfect because it was purely mathematical. All definitions within the system are based on mathematical axioms and no empirical data is required. The problem we have in the present day achieving such perfection is that there are very few problems we can study which don’t have some empirical content: For example, when we study electricity. We know much about electricity and there are whole swathes of mathematical formula we use to describe the behaviour of electric phenomena. But our knowledge is not perfect it is in fact approximate. The problem is we have to define the real world objects we are discussing prior to our mathematizing of those objects. But the objects we are attempting to define are the very objects we have under study and are attempting to understand. Electric charge is not a mathematical axiom nor can metal wires be defined in an axiomatic way.

The problem is, as we move from the perfection of a purely mathematical study and import real world objects our ability to define those real world objects creates a barrier to perfection. The aim of scientists and philosophers is to put these real world objects into a mathematical description so they can be manipulated as mathematical objects.

The enterprise has had some success during the twentieth century. Materialism has been discarded as a valid philosophy and a different ontology introduced. Electromagnetic phenomena are now described in terms of fields and lines of force, but the most striking example came with Relativity theory. Relativity is an idea which is expressed in the four geometric dimensions of space time. It is a geometric theory, not purely geometry there are still non-logical objects in there, but its success has much to do with it being mathematical, or geometric and at a distance from a mere discussion of physical objects.

Any final understanding of nature will have to emulate and go even further than relativity theory. The real world phenomena need to be describable in a way that is pure mathematics and then the relationship between all things can be described through the logic of mathematics. That mathematical description will most likely come from a geometric description.

It is uncontroversial to state that we inhabit a three dimensional space and the same can be said of our ability of awareness of that three dimensional space. The discussion begins when we try to make a case for why we should be able to understand three dimensional space. Do we discover space through experience or is the brain wired to understand space innately? This question arises in many forms in philosophy and will be a constant theme of this blog. A discussion of the nature of infinity will bring out the problem in more detail.

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