There are four types of mathematical proofs, by definition, by contradiction, by contrapositive and by induction. The second – proof by contradiction – works on the principle that if you start with an assumption and use it to logically deduce a contradiction (a paradox with something that’s true) then you’re assumption is false. This assumption that’s proved to be false may be one of your logical steps, but it has to be something, falsehood cannot be created from truth alone.
And so philosophically, contradictions and paradoxes are challenges to assumptions. In the same way that two even numbers can never add together to produce an odd number, truth plus truth can never equal something that’s false. When we come across a paradox, it must be because there is something false in whatever it was that led us to the paradox. This doesn’t mean that nothing is paradoxical or counter intuitive, only that there is no such thing as a true paradox or contradiction. This can be demonstrated in philosophy, science and theology.
One of the most famous philosophical ‘paradoxes’ is of Achilles and the tortoise, this is sometimes phased in terms of an arrow and a target but the philosophy is the same. If Achilles can run ten times faster than the tortoise and he gives the tortoise a 100 meter head start, then how far will the tortoise get before Achilles catches him? In the time it takes Achilles to run the 100 meter starting difference, the tortoise will have gone ten meters. In the time it takes Achilles to run that ten meters the tortoise will have gone one meter. In the time it takes Achilles to run that one meter the tortoise will have gone a tenth of a meter and so on. This is an infinite series where Achilles asymptotically approaches the tortoise but never catches it, even though he’s running ten times faster – a philosophical paradox.
But this isn’t actually a paradox, it’s a proof by contradiction that space and time are quantised. The underlying assumption to the philosophy is that space and time (if you use seconds instead of meters) can infinitely divided i.e. you can always divide any given amount of space (or time) into ten. There has to be a length, probably the width of an electron, (and a time, probably the time it takes the speed of light to travel the width of an electron,) that cannot be further divided. Once Achilles is an electrons distance from the tortoise, he catches him in the next time step.
The classic scientific paradox is the wave-particle duality of light. In Young’s double-slit experiment, he fired a single photon (particle of light) at two slits and observed two slits of light on the other side. The single particle went through both slits at the same time – a scientific paradox. But this is actually a proof by contradiction that light has properties of waves as well as particles. The experiment disproves the assumption that photons behave in the same way as protons and electrons, which isn’t that surprising given that they have mass where as photons don’t.
In theology, one of the biggest ‘paradoxes’ is in God’s sovereignty and human responsibility. If God is sovereign over everything that happens, then how can people be held responsible for their actions? But again, this is no paradox, only an exposure of the false dichotomy between God’s sovereignty and our responsibility. Left to our own devices we all choose the path of selfishness, but God uses our bad choices for his good purpose. Like the dual authorship of the Bible, God’s sovereignty and our responsibility is the dual authorship of history. The two are not pulling in different directions but running parallel. Your choices and actions can be with or without God, but never instead of God.
Other theological concepts like the trinity and Jesus’ nature may appear to be paradoxical, but God’s word doesn’t contain contradictions. When a perceived contradiction arises, it is our assumptions that are challenged, not God. This gives rise to the principle hermeneutic; that the Bible must be read without drawing dichotomies with itself. This principle for understanding God’s special revelation – the Bible – is also used for our understanding of his general revelation in science. A scientific hypothesis is made in conjunction with other scientific laws, not in opposition to them. So too, theological conclusions must be made in coherency with all of God’s word. Furthermore, if the God who created the universe has revealed himself in the Bible, then dichotomies can’t be drawn between God’s special revelation and his general revelation. Our assumptions about how they are true may be challenged, but our challenge is to hold them both as true.
