The necessity of God and mathematics
Regular Network Norwich and Norfolk columnist James Knight presents a different way to look at reality where only God and mathematics qualify as having ‘necessary existence’.
I’d like to give you something to consider – something you may not have considered before. Just about every part of reality that we interface with through perception or conception is made up of analogies, metaphors and symbolic expressions. It sounds absurd, but when we think about it carefully, we find it’s true.
Whether we are describing the physical world, relations of ideas, or constructing images to convey some wisdom, we are never far from analogies, metaphors and symbolic expressions. Almost all of our engagements with reality are descriptive terms used by virtue of our being humans – and these terms are formulated on the basis that we are physical beings trying to describe a physical world.
With that in mind, I think our understanding of reality can be sorted into two distinct classifications; things of contingency and things of necessity. If something is contingent it means that it could have been different and that it does not qualify as having necessary existence. If something has necessity then it does qualify as having necessary existence. Naturally, descriptive terms used by humans to describe the physical world are contingent facts, which means everything about the physical world is contingent, not a necessity, because no laws of logic would be broken if the physical world didn’t exist.
What, then, if anything, does have a necessary existence? Given what I’ve said, once we begin to trim away the contingent things in the hope of leaving only things that have a necessary existence, I find we can even trim away the physical reality around which we employ all our empirical considerations. As a Christian, my gut feeling is that after all the trimming is done, the only things we’re going to find left in our qualification for necessary existence are God and mathematics. All the rest is, in a certain sense, fiction – but not just any old fiction – it is a fictional reality that, for now, is a precursory disquisition to a dénouement that we are at present only tapping into.
As has been said before by St Paul – it is an image of a reality with which we have not yet come ‘face to face’. Our most primary sampling of it thus far has been the Incarnation, and the consequent spiritual legacies that have emerged since, and are still playing out now, and will do henceforward.
Here’s why I say ‘God and mathematics’ are the only things that have necessary existence. I think God has necessary existence, for the obvious reason that Christians believe Him to be the timeless, boundless, omnipotent and omniscient Creator of everything. And I think mathematics has necessary existence because it makes no sense to talk of a supreme Divine mind without that mind having mathematical reality as an inherent and inextricable part of its cognition. To be a Divine mind (or any mind actually) involves notions of quantity that necessitate some kind of mathematical framework. Even a young child can’t even engage in the most basic thinking without an underpinning of numbers, arithmetic and quantity.
Those Christians that want to tell you that mathematics is a mere human construct designed to deal with physical reality, too easily forget that quantity is implicit in God’s very own personhood. After all, we are told that God is triune in nature – He is the tri-aspectual Father, Son and Holy Spirit, which is 3 persons, 1 God. So whichever way we cut the cloth, it seems to me that numbers are inextricable from God’s personality, which is why I say both God’s mind (and, I assume, all its qualities) and mathematics have necessary existence.
This can bring about a new perspective to our empirical endeavours – not just in how we think of God and mathematics, but in how we think of the things that only have a contingent existence too. So, for example, science is a fictional reality in the same qualitative sense that poetry, literature, theology and art are fictional realities. But, as indicated, by ‘fictional’ I do not mean ‘non-factual’, I mean fictional in the sense that science is only one branch on a huge tree of human conceptions that involve objects that are real to the human mind in ways that they are not real in the reality ‘out there’ beyond the mind. In other words, it is a new perspective where fiction and fact intersect in an embrace, as all the objects we convey in reductionist science (rocks, sand, water, atoms, protons, etc) are as they are because they are projected onto our minds.
If our mental conceptions are a tool box full of a variety of tools suitable for different tasks, then science is one kind of creative tool for understanding reality, and mathematics is another, poetry another, and theology another, and so forth (often with overlap between them). Just as in physics the question of whether quantum-scale objects are particles (metaphors), waves (also metaphors) or equations for probability amplitudes (numbers) depends on the kind of question we are asking, this also applies to our other considerations of reality too, where each tool (science, mathematics, poetry, theology, etc) is ideally suited to enable specific enquiries related to different aspects of reality.
Analogy, metaphor and the symbolic
What do I mean when I say that our reality is largely made up of analogies, metaphors and symbolic expressions? Well, we know that physical objects like trees, rocks and buildings are made up of smaller component parts – but once we get down to the full reality of those smaller component parts when we engage with the quantum world, we find they are made up of, or only describable with, metaphors and analogies or numbers (or more accurately, what those numbers represent). This is evidenced by the fact that if we don't describe them with numbers and concomitant equations, we are forced to revert back to the macroscopic world of metaphor and analogy to describe them - particles, waves, forces, position, momentum, and so forth – which, as we’ve seen, are extrapolated from descriptions of everyday life.
That is to say, physical reality is in the eye of the beholder, it's just that humans are one kind of beholder that can only describe things in terms that are implicitly human. Any sense of the physical is bound up in the fact that we evolved in the realm of the physical mechanism of natural selection, so our neural network is implicitly physical, which makes our engagement with reality implicitly and explicitly physical. It is due to this human-centred limitation that external reality to us is almost entirely expressed in terms of the metaphorical, analogical and symbolical.
What, then, makes an analogy so special? If you think about the key components of an analogy – it involves a combination of ideas or impressions already in our memory and a fresh experience or idea that needs organising into analogical form. When you hear that half the politicians have abandoned a political party in times of duress after a controversial bill on immigration, you might analogise it with an impression of ‘rats leaving a sinking ship’. When the poet George Herbert instructs us to forgive others by informing us that “He who cannot forgive breaks the bridge over which he himself must one day cross” we take what is a complex set of factors and compress them into one simple image. When we ‘taste’ victory in sport or war or politics, and when we are ‘moved’ by a story, and when we feel the ‘full force’ of our rival’s anger, we are compressing real life events into abstract images (even in using the term ‘compressing’ I still find myself in the world of the metaphorical).
The brilliance of the mind’s ability to construct a mental effigy of reality in terms of the metaphorical is, I think, largely due to the fact that the real mathematical reality that underpins the physical is infinitely too broad and complex for us. If that is true, then our physics may well be a work of art more brilliant than anything else we’ve done – and the true wonder is, we’ve constructed it all, not by being deliberately creative, but almost entirely by simply being physical beings and responding to physical reality around us.
When you look at a tree, the image you are seeing is very much like a combination of ideas or impressions already in our memory, and a fresh experience or idea that needs organising into analogical form. The tree is a variety of constituent parts, emitted as light into a neural signal which is transmitted from the eye to the brain, and is then aggregated into all kinds of perceptual forms, consistent with experience of other trees. From this we group together the set ‘tree’ (oak, pine, palm, etc), and form an association, immediately identifying the new image as a tree. As we’ve just said, once we zoom in on the tree we find that it is made up of, or only describable with, metaphors and analogies or numbers. So whichever way we describe the tree, we never escape this human-centred limitation. Of course, the non-physical concept of the tree can also be used in second order analogical form (Jesus’ ‘fig tree’ illustration, and Darwin’s ‘tree of life’ are two that spring to mind).
Once we begin to see how our physical representations take shape, we can start to sense how they can distract us from the mathematical reality that embeds all our physics. Our realisation of how analogies and metaphors inform our mathematical expressions is important here - they express lots of complex relations with succinct symbols. Those that think mathematics is only a human construction have, I think, mistakenly focused only on our creation of those symbolic representations while ignoring (or not realising) that what they represent is the most important part of their construction. They are like cartographers who design a map of a territory and then as soon as the map is finished they proceed to claim that the territory to which the map relates no longer exists. That the universe is a physical object is down to our human perceptions of the physical world – there is no physical picture of the universe as humans perceive it anywhere except in human minds. This can, and should, open up our minds to a more stupendous view of reality, which gets the analogy of the map, the navigation and the territory the right way around. If God and numbers are the only necessary things, then numbers (and all the concomitant mathematical laws) have their necessary existence by being inseparable from the Divine mind.
This may lead us to be able to develop the analogy with other considerations (moral, political, theological, maybe even psychological) – as long as it is understood that they only work by way of illustrative representation, not as Platonic things that we drag from outside reality into inner reality. For example, in morality the territory is goodness, justice, kindness, and the map of that territory is the moral philosophies we use to navigate the landscape. In theology, we find something a bit different; the territory is the love, grace and multi-dimensional personality of God, and the map is the way we give exhibition to those qualities (which could be what it means to be vessels that reflect Christ's light).
As humans, we don’t ever get beyond our own mental matrix, yet at the same time we don’t ever capture fully what our notions of mathematical symbols, love, grace, goodness, etc really represent – we only ever sparsely sample them. This is what it means to be human – always tapping into something far greater than ourselves – and if there is one thing in the whole of reality that we know is infinitely more complex than us, it is mathematics. Of course, part of the reason that Christianity is so illuminating yet so tough is that those tappings into love, grace, goodness, and beauty might turn out to be clusters of connected thoughts that sparsely tap into the mind’s potential that natural selection has built, but that God has not fully brought to bear.
In seeing this, we've used the necessary reality of God and mathematics to place everything else in its proper place. The reason we don't know, say, lengthy prime numbers with ease is because the numbers we’ve constructed as a map to depict the mathematical landscape outside our own forest contain complexity beyond the capacity of our endeavours. Those things awaiting discovery are the same to us as the other things awaiting discovery - only discoverable through experience - and we cannot construct something we haven't experienced, which goes right back to Hume’s insistence that experience is key, and to my initial point that the creation and the discovering are interlocked in an embrace of metaphor, analogy and symbolism.
If this is peculiar to you, I think it might be because you’re used to thinking in terms of physical reality, whereas I'm looking more towards the analogical side of our minds in order to explain the effect of our mathematical symbols being the intersecting point between the physical and the mathematical reality those symbols represent. What's stopping us uncovering the terrain of every prime number there is? It's the same thing that hinders all our discoveries of knowledge - new experiences are needed for new knowledge. In other words, we discover facts about numbers in the same way that we discover facts about nature – through new experience, and probabilistically as we increase in complexity.
One famous modelling of the prime numbers is the Riemann map, which consists of the distribution of the primes in the shape of a staircase (showing the steps as each prime is higher than the last). Then running a Gaussian curve through it Riemann composed it into a sum of simple waves, which are graphed with positive and negative discrepancies. What emerges is that the complexity of the natural numbers lies not in generating the numbers per se, but in generating true statements about those numbers. The Riemann hypothesis about prime numbers states that an infinite number of frequencies are needed to define this sequence of primes in their entirety. In other words he is telling us that it is not possible to use short-cutting compression to reduce the sequence of primes to a finite form.
This translates as; an infinite amount of data is needed to specify the primes in terms of frequencies - but this is not to be confused with the term 'infinite' which necessitates that primes will go on and on infinitely just like the natural numbers will. In actual fact, just as we can specify all the natural numbers with a very short counting algorithm, we can also define all the primes using a very short prime number generation algorithm. The infinite number of frequencies Riemann conjectures to be needed to specify the primes is not because they go on infinitely - it is because the only known prime number generation algorithm that generates primes with certainty involves the vast and lengthy task of factorising.
Here's the key to how primes serve as a good analogy for the terrain of the mystery. If God and numbers have necessary existence, then as we’ve seen, physical reality is non-necessary, because it is part of contingent reality. This means that factorising is to prime number discovery as cartography is to terrain discovery - except for one key difference. The physics is a map not the territory - it is a human-centred map constructed by minds that can only relate to the interface through physical descriptions, which is why it feels like the terrain. But, in actual fact, the territory is to physics as the prime number discovery is to factorising. Through our empirical lens of reality, mathematics is the territory, and we construct the maps. Just like in cartography, the maps are only a sparse representation of the territory in actuality. The maps are constructed through our brilliant mental effigy that is physical reality, and the symbols are part of that navigation.
And if you want to take it a step further and bring the concept of this story to a close; the universe is the created territory, and God is the cosmic mathematician! Through our theological lens of reality, love and grace are both the map and the territory, because God's genius is the map but it is also the terrain. What a beautiful story.
James Knight is a long term contributor to the Network Norwich & Norfolk website and a local government officer based in Norwich.
The views carried here are those of the author, not of Network Norwich and Norfolk, and are intended to stimulate constructive debate between website users.
We welcome your thoughts and comments, posted below, upon the ideas expressed here.