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The Divine mind, morality and mathematics

MathematicalWorldRegular columnist James Knight ponders the relationship between God, morality and mathematics, and considers how human minds frame this in order to understand reality.

In Plato’s famous dialogue we are given the classic philosophical conundrum called Euthyphro’s dilemma:

Are things good because God commands them, or does God command them because they are good?

Philosophers, theists and atheists have debated this for centuries, mostly reaching a stalemate.  I don’t think the issue is very complex to solve.  If we mean the Christian God who is thought to be omnipotent, omniscient, and the all-loving creator of everything in nature then it stands to reason that God does not command good things because they are good, because that would be to suggest that goodness and morality have an existence independent of God.  

I don’t think Christians should subscribe to this idea because it suggests some kind of standard that exists outside of God Himself – something to which even God must adhere, and that is very dodgy ground, particularly given that morality and goodness requires a mind in order to attain a place in reality. 

So we may say that things are good because God commands them, but unless we want to fall for the counter-claim that this is arbitrary, we must say that the goodness that emerges because God commands it is, in fact, due to God being the epitome of goodness, love and grace. Hence, there’s no real problem if we say that goodness, love and grace are inherent properties of the Divine mind, and are thus inextricably attached to Divinity. 

But even having got that far, the work is still ahead of us, because I think the real issue at hand is not to do with morality; it is to do with God and mathematics.  I say this because unlike morality, mathematics is something that has a reality independent of human minds.  In other words, morality makes no sense without the interaction of minds, but mathematics contains properties (numbers) that are true without any interaction of minds – in fact, 2 + 2 = 4 is true even without a universe.  

Physics is how our minds engage with reality, and physical reality is only a tiny fraction of mathematical reality.  So in physics a model is an approximation to outside reality, whereas in mathematics, a model is in reference to the primary reality itself.  A mathematical theory is a particular approximation or tapping into that mathematical reality as a whole, it is just that a fraction of mathematical reality appears to us in the form of a physical universe. A theory can be taken to be a set of axiomatic forms that describe the real world, and we just happen to live in a world that we describe using the part of mathematical reality that gives us physical theories. 

This is one of the reasons why so many people mistakenly think mathematics is merely a model of physical reality created by humans. Studies of physical reality show quite clearly that our mathematical symbolism is meagre in comparison to the grandness of mathematical reality in its entirety (think of factorising large prime numbers as a good example).  The mathematical maps we use for deciphering physics are very much part of the symbols we create - and I think this is what is being confused with mathematics as a whole.  

Our symbolic constructs are useful because they enable us to tap into a system of pattern deciphering concomitant with the physical world. I think the only reason why anyone would confuse physical reality with the ultimate reality is because we have evolved to see the physical as primacy. This is an error on our part, I fancy, because, being physical beings in a physical universe we cannot easily escape the first person physical perspective and conceive of other realities beyond the physical.  So I think a much more interesting Euthyphroian dilemma would be:

Does mathematics exist because God thinks mathematically, or does God think mathematically because mathematics is a primary tenet of reality?

Just as with goodness, if mathematics is an inherent property of the Divine mind then it is not in the least bit suspect to suppose that mathematics and logic are inextricably attached to Divinity.  Although morality and mathematics differ in their respective realities, the Christian seems to me to be compelled to think of God’s personality as containing properties or dimensions that define what mind is.  That is to say, I’m not sure it can make any sense to think of any kind of Divine mind that is bereft of mathematical foundations, and the foundation of goodness to which all morality is subordinate.  Perhaps the German mathematician Leopold Kronecker was onto something when he said:

"God made the integers, all the rest is the work of man".

Knowledge and Faith
The nature of knowledge from within our worldview is interesting. A table exists, but 4 men observe a different table from different angles. Justice exists, but 4 men have different ideas about it. Mathematics exists, but it goes beyond the physical universe. 

The question 'Does everything have a cause?' is a question derived from our understanding of the physical, but it is metaphysical in the breadth of its scope. Quine was sceptical of metaphysics, because he thought science gives us all the genuine knowledge we need - but still the human mind ponders - almost lives to ponder - big questions that are clearly beyond the reach of science. Even concepts like love, purpose, justice and morality have aspects that science is ill-equipped to tackle. 

A more respectable idea about metaphysics would be the notion that it describes the nature of our thoughts, ideas and impressions of the world. For me the subject is enriching not simply when we apply it to those big questions, but when we consider it in terms of the different kinds of reality with which we interface; that is, physical objects, concepts that leads us into the abstract, and mathematics, which seems to be the most primary thing we tap into (albeit sparsely).
 Once we get into the realms of knowledge and discussions about the different epistemological problems and about the different lenses through which we contemplate things that exist (physical nature, concepts, mathematics) things open up into centuries of debate from philosophers of all cultures and generations. Taken to its logical limit this becomes wholly open-ended - and that is the nature of philosophers - we are always on a journey because journeys are more realistic than destinations. 

Throughout this lengthy and ambiguous process we can question and express doubt about how things can be said to exist, whether other minds are like one's own, whether everything needs a cause, the issue of meta-mathematics, the controversy over absolute morality, whether our perceptions accurately reflect the outside world, and more general subset tenets of philosophy concerning what we can justifiably call knowledge (see Gettier).

So in my opinion philosophy of knowledge isn’t most enriching when we debate these issues to death (although it is good to do that for other reasons) - it is most enriching when we look to compress the vast intractable data into some succinct chapters or sentences that give us a reliable set of manifolds through which to make probability-based claims about the reality in which we live. The reason people have a tendency to debate it to death is probably because it is mentally challenging looking for those all important short-cutting methods. 

There is always going to be the logical hiatus that creates a gap between our impressions of reality, and what those impressions represent in terms of physical nature, concepts and mathematics, because humans only have access to the self's own representations. As far as humans go our representations 'are' the basis for understanding reality, and our job is to construct the best chapters or sentences within that framework, where one must get the balance right between premature closure and promiscuous open-endedness.  Somewhere in between we find that there is an invitation to have faith in God, and that the Cosmic Mathematician who gave us the wonders of the universe and the breadth of mathematics itself, humbled Himself by living and dying to bring us salvation.

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. 
You can also contact the author direct at at j.knight423@btinternet.com

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