Could Mathematics help? Mathematics may not be everyone’s cup of tea but all of us have mathematical intuitions. We understand what “three” means and what straight lines or parallel lines are. Before the 19th century, mathematics was developed mostly from intuitive understandings like these. As early as 300 BCE, the Greek mathematician Euclid laid down the foundations of geometry. He provided five “axioms”, which were common notions of space, from which all geometry could be derived. Axioms were considered self-evident statements that seemed too intuitive to be questioned. Try and see if the following axioms make sense: 1. A straight line segment may be drawn from any given point to any other. 2. A straight line may be extended to any finite length. 3. A circle may be described with any given point as its centre and any distance as its radius. 4. All right angles are congruent. 5. Given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. Geometry, as formulated by Euclid, was held in very high regard among mathematicians. To them, it represented an ideal form of the physical universe. Later, other mathematical concepts like Calculus emerged from it. The use of axioms suffered a blow in the 19th century when Russian Mathematician, Nikolai Lobachevsky ditched Euclid’s Fifth Axiom and developed “Hyperbolic Geometry”, an equally sound alternative to Euclidean Geometry. As Richard Wells writes, ‘The “disaster” to mathematics was the loss of its long-held conviction that mathematical truths were certain and that through them mankind could know the world solely by the raw power of logic and reason’. A new school of thought emerged among mathematicians called Formalism: that any mathematical system comprises its own set of “axioms” or rules which have no intrinsic truth value just like rules in chess or ludo. As Alan Weir put it, ‘Mathematics is not a body of propositions representing an abstract sector of reality’. Any statement that can be derived by reasoning with axioms is called a theorem.
This was a paradigm shift in the philosophy of maths. The question had become not one of “intuitive absolute truths” but of “truths in context”. Much like the popular belief in morality, absolute truths, which once underpinned Mathematics have been questioned to the point that the relevant question has become whether the Mathematics is useful in a given context, rather than whether it is “true”. In the 20th century, Einstein’s General Theory of Relativity proved that space is not Euclidean, contrary to our intuitions. Although we no longer have the comfort or the certainty of absolute truths, the systems we create allow us much more freedom to explore the physical and mathematical world.
The problem of choice Let’s look at an even more controversial example, the Axiom of Choice. What it essentially says is: suppose you have an infinite number of cricket teams, you can make a new team by choosing one player from each of the original teams, thus you have the power to make an infinite number of choices. It is so innate that it forms the key component of many mathematical proofs. Assuming the Axiom of Choice, however, leads to paradoxes such as the Banach-Tarski paradox which says that it is possible to cut up a ball into a finite number of pieces and reassemble them into two copies of the original. This paradox would have been enough to put off those relying on intuition to establish mathematical axioms. Even the concept of √2 was dubbed “irrational” by Greeks be-
gears that seem to run modern, popular morality is the right to liberty balanced by the axiom of reciprocity. But these gears run at their own pace and if we press the analogy too far, they may break. This is perhaps why we emphasise tolerance over moral consistency - because if we press the axioms too hard, we start noticing the cracks in our ethical gears. In mathematics, the Axiom of Choice gives some “contradictions”, but we keep it regardless, as it forms the bedrock of many essential results. One can put this poetically and say we should value choice in a society despite the apparent paradoxes that might arise from it. One such commonly-cited example is the “Paradox of Tolerance”—that tolerant societies must be intolerant to intolerance. Similarly, a free society must restrain absolute freedom to prevent basic liberties from degeneration. The key to a sound system, whether Ethical or Mathematical, is to minimise, rather than eliminate contradictions.
Back to Ethics: What have we learnt? In Ethics, although it may seem we arestraying too far from familiar notions of morality and order into primeval chaos, there is no need for alarm. Mathematicians have opted for more rigorous treatment of mathematics, but they haven’t diminished the role of intuition. They are guided by it yet resort to Formalism to make arguments more concrete. In our secular age, there might be lessons from Mathematics, and perhaps it is time for Ethics to become more rigorous in our everyday lives—to use reason to minimise contradictions. But rather than dis-
The vessels that once carried “ethical axioms” have now been shown imaginary for the secular world. cause it had an infinite sequence of digits after the decimal, without any pattern. After the transition to Formalism, mathematicians like David Hilbert wanted to axiomatize all of mathematics: to find the perfect axioms to explain everything. However, by 1931, Kurt Gödel showed that an axiomatic formal system cannot prove its own completeness or consistency via his Incompleteness Theorem. Similarly, the vessels that once carried “ethical axioms” have now been shown imaginary for the secular world. The two
missing intuition, we should acknowledge the heritage of traditional morality, and rather than base our intuition on reason, base our reason on intuition. We have to venture into the dark chaotic world of relative truths but keep the flickering lamp of intuition for context. For millennia, we confided in absolute truth, but as the world changeth, so must the Truth. Aditya Ghosh and Sea-Yun Pius Joung are both Undergraduates at Oriel College, studying Mathematics and Theology and Religion respectively.
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