Beauty, Happiness, Mathematics and God

I was struck by a bit of a vision today. By this, I don’t mean that I was thrown to the floor by bright lights, angelic choirs and the glory of the Beatific Vision – I’m not meant for that yet, otherwise it would have happened.

As I sat in the college chapel, my mind conjured up the image of a plane of glass in the chapel floor which raised itself up at an angle. I thought that within the glass was a picture of a cloud as one might find painted onto glass in a stained glass window. As I looked more closely, I could see that it was a galaxy, moving and swirling like a drop of milk in black tea. Of course, this is in perfect keeping with the origin of the word galaxy. It occurred to me that I was being given a window into another dimension, a vision of another multiplicity of worlds beyond the shallow horizons of my own little mind.

I do love looking at astronomical photographs. I still cannot take in that what I’m looking at is not just millions of miles away, but millions of miles in size and millions of years old. The scales are mind-boggling, and yet perhaps we forget that there are a million square millimetres in a square metre. The telescope is capable of giving us windows into the infinite, resolving angles ten-thousandths of a degree wide. Here is William Blake’s infinity in the palm of our hands.

Being a (largely failed) mathematician, my mind is always looking for the structures inherent in Creation. That’s probably high-minded of me and I do confess that praying Psalm 131 is not an easy task. Is it high minded of me to try and comprehend what’s going on? I’m hoping that it isn’t in the proviso that I understand that I will never completely understand what is going on until I have seen God face to face. Even then…

I often wonder what my students find beautiful. For teenage boys, it seems that beauty is an ill-defined affair and perhaps that’s because they lack the language for saying why something Is beautiful. They have not yet refined and honed the senses or the descriptors that will inspire them the most, but they are surely capable of appreciating beauty. In my teens, I remember being struck by the beauty of the solitary church bell and by Italian Renaissance Organ intonations. They certainly transported me beyond my own little world. Yet, I cannot be sure that my school mates had ever had that experience, and I dared not ask them. If they did, then perhaps this beauty was something that was so personal to them that even in the telling it would ruin it for them. Sometimes language destroys a dream rather than brings it into existence.

Beauty has always been something difficult to define, perhaps rightly so if the adage “beauty is in the eye of the beholder” is correct. What is certainly clear is that beauty resonates within the individual like that solitary church bell. I fail to see how it is possible for someone to register beauty without reacting to it on more than just the rational level. The resounding of that bell of beauty allows the senses of our mind to be transported to places beyond our physical senses. There is a sort of happiness here, even when one finds beauty in a sad film like Atonement. There is a happiness in one’s emotion, as if one is feeling pleasure in actually feeling something other than the background level of emotion.

Sometimes language destroys a dream. I don’t find that with mathematics, though. Mathematics doesn’t destroy dreams in my experience, largely because mathematics deals with the necessarily abstract. I’ve challenged some of my students to tell me what algebra is, but find the repeated phrase “it’s maths with letters in it” coming back to bite me. When one realises that algebra is looking at the processes of arithmetic without taking things for granted, one can see some truly wonderful structures coming into play. Algebras allow us to ask the great “what ifs” of arithmetic. What if two times three is not the same as three times two? What would happen to subtraction if we weren’t allowed to have negative numbers? What happens if I allow negative numbers to have square roots?

Algebra answers these questions by producing new algebras, rings, groups, semigroups, division rings, and quandles. Thanks to Descartes, these lead into geometry and topology in which shapes bend and warp, blow up, project and expand in so many ways, even into higher dimensions in which situations make more sense. We understand more about our universe by looking at space and time together. This gives us four dimensions, though we are hard pressed to visualise anything that might be going on in 4D. Tesseracts and hyperspheres are the province of the fiction of Robert Heinlein and his ilk, but that just shows how mathematics yields dreams rather than destroys them.

I know that mathematics passes most people by. I can sympathise as it has passed me by too. There is another old adage, “those who can, do; those who can’t, teach.” Mathematically, I fall into the latter category.  I’m not fishing for sympathy here: while I’d like to have progressed further into academic mathematics, I was simply not clever enough to do so. There is no shame in that, and perhaps it does help me appreciate that mathematics is a hard mistress to woo. I have some affinity, then, with those who struggle with the rough terrain that comes before one enters the darksome mathematical jungle. I can lead the curious to the edge of the wild but, if they can go, they must go without me.

Fractions frighten people, and negative numbers really do upset those whose experience of the world is knowing how many beans make five. How many of us know how many beans make minus five? Of course, what may well be going is in effect a category mistake, after all, negative numbers are not numbers that can count how many physical objects there are. They can count how much money someone owes, or what the temperature of liquid nitrogen is – you need a reference point for that, a zero level. However the realisation that, under the same basic rules of arithmetic that everyone knows that minus one times minus one MUST yield positive one is a moment when one can appreciate that mathematics has a strange beauty that doesn’t really intersect this world in ways we would think.

Yet mathematics goes further and gives us a glimpse of the Divine. Those of us who cannot understand how God can exist without physical space or time would do well to know that numbers do just that also. We can determine statements of truth and falsehood about numbers just like we can with physical objects, and so their existence is assured. However, they have no space, nor time. They do not exist in an area of the brain, but are objective in their presence. Numbers really do point the way to God.

Of course, there are those who would read strange numerological significances in Holy Scripture but take them too far. Seven signifies perfection, forty the nature of penitence, three the completion of the loving family. These numerological fancies are just illustrations, just stories to colour our understanding of the world around us. Those who try to force arcane meaning or significance onto numbers are missing the point. This includes physicists who try to impose ridiculous limits on mathematics by meaningless notions like adding up all the positive whole numbers and getting a negative twelfth. Forcing that kind of physical limitation on that which is not physical is as destructive as language is to the dream.

As I look at my students, my worry is that they will seek contentment in the material world. There is nothing wrong with God’s Creation as I said in Sunday’s Sermon, but to look at the world and see everything in terms of clumps of matter and worth and commodity and fashion misses the true beauty that exists in what is really there. Can they see beauty in a muddy puddle or a plastic bag blowing in the wind? If they can, then perhaps our walk together has been fruitful for all of us.