Let me clear up another mystery first – where have I been all this time? It’s not so much that I have been too busy to write, but have been embroiled in financial considerations. You may have read in my previous post I have had to take this year out from my personal studies to complete my CFA Level 1 – I passed this in January, and have found myself in the wilderness of not being enrolled to study anything I love. As I have said before, from a credibility perspective I don’t like to dwell on too many matters I have not studied; but I observed my financial mathematics didn’t have a terribly broad platform on this blog. This caused me to go quiet for a while as I ran out of things to say, but I am back.

I am registered to complete all of my second year of my second degree, along with third year studies in complex analysis this coming autumn. In preparation for that I have begun studying two beautiful books;

• The Road to Reality: A Complete Guide to the Laws of the Universe, Rodger Penrose, in the hope of resharpening a broad overview of physics; and
• Complex Analysis: A Hitchhiker’s Guide to the Plane, Ian Stewart, a textbook that seems to be horribly underrated on Amazon that saw me good in my first undergraduate.

As such, you can expect random high level physics posts and more technical complex analysis posts until my studies start again, and I hope to post with a little more depth.

Today I thought I would share with you something interesting I read in Penrose’s book – right near the start about three “realms” that live around us and interact, which brings three mysteries. It raises some interesting questions around how the world should be viewed.

The physical world 

This world is easy to understand, it is the world of things which take a Physical existence. You would fit into this category, as would a rock or a table. There is a deeper philosophical question around what a physical object actually is, but leave that for those who like to get tangled up in all of that nonsense. For us a physical object is an object in the sense that immediately springs to mind – the sense which is inherent in our everyday experiences of what physically exists.

The mental world

The Mental world extends beyond the physical, and includes a whole host of things which are not present in the physical world. Anger is an example – it does not manifest itself in a physical form in the true sense, but it is of course present in the mental world. You know what it is and you know when you feel it – it is within most people’s emotional repertoire.

Platonic mathematical world

The Platonic Mathematical world is probably the most contentious of the three worlds, but makes reference to the fact that mathematical entities do not belong in space or time, they are eternal and unchanging. A square for example, lives in this world – you cannot construct a perfect square in the physical world. You can however imagine one, which means it also lives in the mental world – we will discuss the links later. There is a sort of assumption that the mathematical world exists and is eternal, we merely borrow items from this realm and return them when we are done.

There are other more subtle considerations as to what the platonic mathematical world actually means – for example It may be considered a realm of everything which is mathematically true, allowing for mathematical entities to exist even if they cannot all be proven from a consistent set of axioms. I don’t want to get my hands dirty with the philosophy – it’s not my bag and does not interest be greatly.

Penrose’s prejudiced view

The above title is not be being dismissive of the great mathematician, this is how he himself labels his view, which is presented in a diagram below, with the three mysteries numbered. We will discuss them one by one.

1. The physical world is described in its entirety by the Platonic world of mathematics. This is a large claim, but a belief I certainly hold – that there exists timeless mathematics which can be drawn upon to explain the workings of the entire physical world. Ideas like this, or course, destroy notions of free choice and alike but I think I am OK with that. Only a small portion of the Platonic world of mathematics is required to explain the entire physical world – this makes perfect sense. There are a lot of mathematical beings which exist purely in the mathematical world, we need not imply that every single piece of mathematics explains something in the physical world. It is however interesting, how more and more pure mathematics concepts are finding application in the physical world – applied mathematics seems to grow along with our understanding of the universe.
2. The mental world is fully and totally contained in a small section of the physical world. This makes perfect sense, but you won’t like it if you believe in certain religious notions. Everything that comprises the mental world, namely brains, products of the physical world – they are made of stuff, just like everything else. This makes sense to me, I don’t view my consciousness to be anything more than the product of physical building blocks. But clearly not everything physical is mental – such as a stone which has , so a small section of the physical world encompasses the mental world.
3. Finally, a small section of the mental world encompasses the Platonic mathematical world – that is to say anything which is capable of being expressed in “true” mathematical terms can be held in the mental state, by some being or another, but that there are things in the mental state beyond the Platonic mathematical – earlier we used the example of anger, but generally any emotion works.

This arrives us at an interesting predicament – if all of the platonic world is contained within some of the mental world, all of the mental world is contained is some of the physical world and all of the physical world is contained within some of the Platonic something is broken! This would break a basic transitivity law!

Penrose points out that there are many possible explanations for this – most interestingly perhaps these worlds are not as distinct as we are making out in this representation, which means the question of one being contained within another is not a proper question. If there is more overlap than we have drawn in the above diagram, where is it? Or maybe it has been drawn wrong…

The non-prejudiced view 

Like any good and open scientific mind, Penrose presents and considers the alternative view which does embody his own personal views on the world.

Here some alternations have been made:

1. We have allowed for the mental world containing things which are beyond the physical world – this will appease those who have a more spiritual view on life.
2. Not all of the platonic world is capable of being contained in the mental world – this will upset those who like me, think we have the theoretical potential to decipher all.
3. Not all of the physical world can be explained by the Platonic mathematical world – again this goes against the grain to any mathematician or scientist, who will like to believe that the world can be depicted through some continually refined mathematical model.

What is the right answer? At this stage nobody knows. You cannot, for example, tell me mathematics which is incapable of entering the mental state, because in order to do so you have negated your point. I favour Penrose’s view over the one just presented, but I do accept the other opinions.

Of course, there could be a problem with the very way this question is constructed – but none the less it’s good fun to play around.