Can we really know everything? - Mathematics

Sometime in February 2011 I explored numerous ideas which previously I couldn’t pay attention to. One of them is the idea that science will explain everything in the near future. It seemed cool at first but later I disagreed with it. All our science is based on models. Einstein explains it best: ‘One thing I have learned in a long life: that all our science, measured against reality is primitive and childlike – and yet it is the most precious thing we have.’

It is through the scientific method we make great leap into the realm of knowledge and comprehend more about nature and our place in it however it is itself ‘primitive and childlike’. It limits us to the models and what they predict. Between the reality and our understanding sits Statistics – The Mathematical theory of ignorance.  It’s true that our understanding of our universe deepens everyday however this does not mean that we will know everything one day. I’ve gone somewhere deeper into mathematics to show why I think we cannot know everything one day or any day. Bear with me.

As the year progressed forward my interest in mathematics was getting profound. My interest developed in things that I could never think of just a few months back. This started to happen so much so that one day as I was sitting in the garden of my college I thought that mathematical objects do not really exist in reality. This thought was penetrating. Further on this it occurred to me that these objects are just representations of something more magnificent.

On the day of my C3 exam I was engaged into thinking something other than thinking about my exam or revising for it. Yes I was thinking about mathematics although C3 itself is a mathematics exam. This time my mind was orbiting around the fact that every proof has an assumption. Expanding on this I thought that the most fundamental theorem must also have an assumption. Therefore it means that we may know something is true but we cannot be certain.

So if we have a H axiom we would end up with a different mathematical statement to when we have a J axiom. This means that in reality mathematics does not exist. All the laws of mathematics are a reflection of one mathematical statement. All the mathematical objects are just representations to express those mathematical statements that come from the one mathematical statement.

Another way of imagining this is by thinking about words. Words themselves are representations of their meanings. Meanings are the real deal. When we are communicating, we are sending meanings through the medium of words.

I can describe it best by imagining different frames of references. If we look at an object from one point, the image we would see is the mathematics we have today but if we see it from another angle, we would see a different image of the same object perhaps a different type of mathematics. This also shows that mathematical statements are relativistic.

All the laws of nature, laws of physics, laws of mathematics and laws that we do not know of yet are nothing but a reflection of this one object. This is why our predictions forecast valid and precise outcomes when we do calculations on natural phenomenon. Definition of a law may actually require another blog post since they are relativistic.

Another idea, an essential extension to my idea is incompleteness theorem. After the book Principia Mathematica was written, Alfred North Whitehead and Bertrand Russell had the desire to work on completeness theorem. The idea is that you can start anywhere in mathematics and prove everything else from it and you wouldn’t need any axioms. (An axiom is a statement that is assumed to be true.)

Gödel came along and proved that this exact thing cannot be done. His Incompleteness theorem underlines the fact that we cannot start anywhere without axioms. In other words we always have to make assumptions. It also means that we cannot know whether a problem is solvable. We can keep working on it for 1000s of years yet still don’t know how close we are to the solution. This is where faith comes into mathematics. Though there is faith, it’s not a blind faith because maths is consistent with nature. It is this consistency which gives maths the power it has today.

In Summary

We cannot prove everything. We know specific things are true but cannot know them for certain. We have to start with axioms build our mathematics on it and eliminate the axioms. First, our mathematics is based on axioms, secondly, we cannot know whether we can solve a certain problem, in this case it would be of eliminating axioms. When these two statements stand, how can we happily say that we will know everything. We just cannot have all equations and have all knowledge before us.

Originally I wrote this in 2011, however didn’t get time & place to share it on the web. If you have read this article fully and got as far as this, please comment, let me know how much you agree or disagree and share it.