Studying Pure Mathematics
8th August 2022: Galois Theory, Incompleteness and the Existence of God
Many years ago I decided that as a seeker I needed to spread the net wider and become a more powerful person both professionally and personally. How I did this is an interesting example of the incidental effects of doing something seemingly unrelated to the eventual result.
I sat in an old school classroom in north London grappling with the intricacies of Galois Theory. This is an obscure area of abstract algebra that connects Field Theory and Group Theory and simplifies the solution to complex problems. This is fine as long as you assume that you understand the theory to start with or that you understand the problem to be solved or even the solution arrived at!
What On Earth is it All For?
As a simple example of what I am talking about consider the following taken from The Wikipedia article on Galois Theory,
One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals—the Abel–Ruffini theorem. This is due to the fact that for n > 4 the symmetric group Sn contains a simple, non-cyclic, normal subgroup, namely the alternating group An.
I could spend the rest of this letter translating this into simpler english and attempting to clarify what it is all about. There would be no point, though, because virtually all the readers would still not understand what it is saying. This goes beyond abstract algebra being another language, it strays into what on earth is it all for.
I found, thirty years ago, that I loved grappling with problems such as this simply because they were examples of complex logic that could be understood and could become clear. At my wedding, a few years ago, my brother, in his speech, said of me, “He studies maths and he actually enjoys it…”
I spent a year tussling with Galois Theory and eventually nailed it. I came to understand it and was able to use it. Now it has drifted back into the hidden part of my brain that retains the stuff I no longer need or use. The quotation above is as obtuse to me as it is to you, the only difference being that I know I could understand it again and enjoy the process of getting there.
The point, though, is what this tussle brought beyond the sense of achievement in understand the maths. There were two revelations which transformed my approach to the world and myself.
This is Enjoyable Stuff
The first revelation was that this was enjoyable stuff that was worth coming to grips with. We all know that arithmetic is important. If we could not add up there would be untold problems when out shopping, for example, but obscure maths, what is that for?
My tutor for this course on Galois Theory was a woman who worked at the Government Communication Headquarters (GCHQ) in England. It is where the British spy on the world from. It is the home of communications monitoring by the Secret Service in the UK and is vital to our security as well as the security of our allies, such as the USA. It is a fascinating place full of advanced technology and advanced theory. She worked there because of her skill with high level obscure abstract mathematics. I have no idea of the details of her work—she was not allowed to talk to us about that—but I have no doubt that it went far beyond simple code breaking.
She loved this stuff, she earned her living using it and she worked to make the western allies secure against all manner of enemies, internal and external. This was important to her and, most significantly, to the country. Maths, even obscure maths, can be, and is, important.
It had started for me as a way of improving my calculating skills as a lighting designer, so that I could advance in my profession, and had become a fascinating hobby. Now I realised that its importance stretched way beyond the idea of it being either a means of improving my skill or the idea of it being an interest or hobby. That was a dramatic shift for me.
The second revelation was that it took me into the areas of thought that had been occupying me for as long as I could remember. It took me into ways of finding the answers I had been seeking.
I discovered that delving into the depths of obscure pure maths opened the deepest question of our existence. Understanding maths enabled the understanding of philosophy, religion and obscure existential questions. This was mind-blowing for me and helped me to realise that I entered this world for a deep and long-lasting reason.
Let me try and explain how I got here without getting too technical. Bear with me because a little obscure maths is involved and is necessary.
The reason the course was being run by the Open University, where I studied as a mature student and where I received a first class honours degree in mainly mathematics, was so that we could understand how to prove Godel’s two incompleteness theorems.
Gödel’s two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent, assuming it is indeed consistent. These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial.
Stanford Encyclopedia of Philosophy
OK, so that makes no sense to you, but trust me after a years of Galois Theory understanding this is like falling off a log.
Things We Cannot Understand or Resolve
One of the first consequences of the publication of the proofs to Godel’s theorems in the 1930s was the acceptance by a mathematician called David Hilbert that one of the threads of his life’s work was no longer achievable. Simply put, that was the attempt to show that all knowledge was capable of being organised and categorised. This conclusion caused Hilbert to throw away many years of effort as he was forced to accept that Godel had proved the impossibility of his task.
Godel showed, in a simple sense, that nothing could be completed or seen as an organised whole. There will always be areas that we cannot understand or resolve. The world, and the ideas it contains, thereby cease to be some mechanistic device that we occupy as part of a grand design.
Sometimes quite fantastic conclusions are drawn from Gödel’s theorems. It has been even suggested that Gödel’s theorems, if not exactly prove, at least give strong support for mysticism or the existence of God.
Stanford Encyclopedia of Philosophy
Do not worry if you still do not understand what I am talking about, that is completely understandable. The important point is that, for me, an apparent logical device—mathematics—opened up the big questions of existence for consideration. Understanding this esoteric world enabled me to start delving deeper into questions that had been plaguing me and gave me a method of coming closer to solutions.
What these solutions were is not for now or for this discussion. What is important is that I stumbled on something vital for my sanity while not looking for it. Understanding this opened up my future.
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