It may seem strange to include a post on mathematics on a web site about atheism and spirituality. But questions about the nature of mathematics inform our view of the natural world and our place in it. There are many component domains of mathematics, ranging through number theory, arithmetic, geometry, algebra, algorithms, set theory, so called ‘pure’ and ‘applied’ mathematics. These may each have different characteristics.
There are three aspects of the nature of mathematics which are important for philosophy. The first is the process of abstraction. Mathematics is able to start well calibrated to the real physical world, to then perform a myriad of logical operations in virtual abstract space, and then to return to earth, to physical reality, correctly calibrated. For example, numerical arithmetic allows us to represent a real number of real objects as 1,2,3,4 etc, much as an abacus represents them as beads. Euclidean geometry represents real objects such as triangles and circles, and derives truths about them, such as formulae for their area and circumference, which are then found to apply in reality. We can generalise from the exact values of 1,2,3,4 by writing the algebraic ‘x’ to represent any value a variable might take.
These are abstractions which are easy to follow. So far they apply to positive, finite and up to 3 dimensional objects. It’s much more difficult to follow when mathematics enters negative numbers, infinity, or 4+ dimensions. Take a negative number. We can write -1, but there is no physical counterpart of -1. Minus one of anything does not exist in physicality. Nevertheless, using equations including a term of -1 works. The same is even more true when we move into infinite mathematics using the ∞ operator, or into complex numbers involving i = √-1, or into non-Euclidean n-dimensional geometry. We can move from reality into abstract virtual space, perform a few pirouettes there, and then return to reality, finding our conclusions are correct. Real and virtual, physical and abstract relate exactly to each other. Plato is right that abstractions are valid. Aristotle and J S Mill were wrong to insist on total empiricism. There is some parallel here to the way the human physical body generates abstract virtual consciousness and personality which operate in harmony. Physicalism is not the whole story.
Invented or discovered?
Whether mathematics is something we invent or something we discover is a crucial question. Philosopher of mathematics Stewart Shapiro, tackles the question in his paper ‘The Objectivity of Mathematics’ (Synthese 156). Using criteria earlier proposed by Crispin Wright, Shapiro tests mathematics for objectivity. He concludes that the likelihood is that mathematics is discovered. One of his main arguments for this conclusion is his ‘epistemic constraint’. For example, we take it that there is an infinite number of prime numbers. But we cannot know them all. How could we not know what we had invented? Therefore they are objective, they are discovered, not invented. He also reasons that there is one unique correct conclusion to a mathematical problem, suggesting again that mathematics is discovered, is mind-independent, is objective.
So what? The implication is that the natural world includes more than physicality. It includes logic. It includes mathematics, as number systems, as formulae. These are metaphysical elements of nature – the metaphysical domain exists.
Consistency and paradox
Nearly all mathematical systems justify themselves on the basis of being consistent. But why is consistency a valid criterion? Philosopher Graham Priest explores the validity of paradox, the opposite of consistency. Much human behaviour is logically inconsistent. Do we live in a consistent world? And if so, how come? Once again consistency is a metaphysical quality. Nature includes the metaphysical. We are therefore justified in invoking metaphysical concepts in our understanding of humanity. This includes the metaphysics of intellectual ideas, of emotional feelings, and of spiritual virtue. They are valid naturally, and are not excluded by atheism.