1+1 & Mathematical Faith

We all know that one plus one equals two, but can we say what this really means? Suppose I start by reaching for the obvious: “If I have one thing and I add another thing, then I’ll have two things.” Straightforward as it may seem, on a moment’s reflection this interpretation is not always true. If I put one splash of milk into my coffee and then add another, I don’t see two splashes of milk, just milky coffee. One crowd added to another crowd yields one bigger crowd. One tonne of uranium and another tonne of uranium doesn’t always end up as two tonnes of uranium. When considering things, one plus one only equals two when we know what sort of things to count.  

In the case of speeds, it’s rather curious. On board a train crawling out of a station at 10 km per hour, if I hurry forwards down the aisle at 10 km per hour, I will of course be travelling at 20 km per hour relative to the platform. If a hypersonic aircraft travelling at 10 km per second could fire a projectile forwards at 10 km per second, the resulting speed of the projectile would be 20 km per second… except not quite! According to special relativity theory, an observer on the ground would measure the projectile’s speed to be just fractionally less than this. (1) It’s not intuitive, but apparently (and Einstein’s theory is very well attested) this is how God’s creation of time and space works: there is an absolute speed that we call c (as in the famous E = mc²), and everything that moves does so with reference not to absolute stationarity, but to c. There is no Archimedean reference point within the creation!

At least for mathematical physicists, then, speeds can’t just be added up any more than physical things can. What about distances? We all know how they can’t always be added up. If I walk a mile in one direction and then turn and walk a mile in another, I’ll certainly be less than two miles away from where I started. This is quite intuitive, I think, and the Cosine Rule turns out useful for finding the exact value of such a displacement. This too is a facet of God’s law-ordered creation.

Istine Rodseth Swart’s insightful piece in the last issue of The Big Picture pointed out that numerical conventions, like measuring time in 12 or 24-hour cycles or the binary code of computers, are ubiquitous and essential to our thinking. And at a deeper level, mathematics has its essential axioms that cannot be proven (there’s more to say about that another time!) so that, as Swart says, a kind of faith is also present. Above, I’ve tried to show that simple arithmetic – and by extension, more advanced mathematics too – has a rather subtle meaning that’s independent of the physical, kinetic or spatial aspects of any particular scenario. (2) 

All this raises some far-reaching questions when we reflect on faith commitments. For example, if you believed that all reality were fundamentally physical, how would you interpret or justify the truths of mathematics, like 1+1=2? Philosophers of maths, unlike their colleagues in the sciences, have tended to steer clear of materialist worldviews, but more often they have embraced a Platonism where numbers are independent absolute entities of some kind – which rescues maths but raises problems for its application to the rest of reality. How would all those entities that we know we can count up – from planets and cedars to people and nations, not to mention distances, weights and prices – be submitted to a platonic One, Two or the Square Root of Three? Similar questions arise with perhaps the most prominent contemporary faith – I mean, philosophy! – among mathematicians: that numbers and their relationships are merely systems of conventions that humans have created to play with. Why, then, is there only one overall game in town?

Such problems, thankfully, are diminished for those who believe that reality is a richly-ordered creation resting, as a unified cosmos, on the law-word of its creator. If the word that became flesh is also the one in whom all things hold together, we can rejoice in the remarkable discoveries of how numbers and their relationships seem to be woven through the fabric of space, time and material things – and how the physical aspect of the cosmos fosters living things, which in turn harmonises with the many layers of human culture. Kuyper’s worldview, and the Reformational tradition flowing from it, offer a vista in which God’s grace permits and undergirds all kinds of scholarship, from maths to theology, advanced by unbelievers as well as believers, even while the revelation of Jesus Christ calls us to turn from idolatry – including any misplaced faith in numbers or equations.