A simple event – dropping a ball on the floor – is sufficient to generate a why-chain that stops not because we have reached the end of the chain, where there are no more questions to be asked, but because we are satisfied with a local explanation. Sometimes, however, we stop because a further question seems positively silly. If a child asks why 5+3=8, his dad shows him five fingers of one hand and three fingers of the other and says: this is what eight means. Why? Because when we put five objects together with three objects we call them eight objects – that’s all. At that point it looks like we have reached the end. Asking why 8, and not 9 or 15, sounds daft. Counting fingers is not just a local explanation: we are completely satisfied with it and cannot even think of any further question to ask. We do not need to repeat the finger experiment to prove that 5+3=8: all we need is one demonstration.
While proofs are ‘arguments from experience, as leave no doubt or opposition’, demonstrations are self-evident beliefs that are true on the grounds of pure reason and no empirical evidence can change. Proofs are open to Cromwell’s rule: I beseech you, in the bowels of Christ, think it possible that you may be mistaken. Should Philae send us a picture of a green rhinoceros, we would be obliged to conclude that, however fantastically unlikely it seemed, there are green rhinos on Comet 67P. But no amount of evidence can convince us that 5+3 is nothing but 8. When it comes to numbers, there are lots of questions to be asked, and some of them require a long and winding why-chain. But at the end of the chain, provided that no ring is broken, there is no other possibility: Quod Erat Demostrandum.
Q.E.D. is a thing of beauty. As such, it is in the eyes of the beholder and some people appreciate it more than other. I remember Walter, at university, a political science student who had passed all his exams except his bête noire: Maths I, and had asked me for help. Walter didn’t have any sense of Q.E.D. ‘Can’t you see? – I would tell him trying to explain some theorem – it must be true.’ ‘Why, why? – he would reply, staring at the page – why does it have to be that way? Can’t it be some other way?’ He was referring – tongue-in-cheek, but not entirely – to the art of political manoeuvring, of which his party heroes, the Christian Democrats, were refined connoisseurs. They were the ones who had coined the expression ‘Convergent Parallels‘ to denote and promote a certain degree of collaboration between them and the Communist Party, within the confines of their distinct political traditions. In Walter’s mind, such sophistication was in stark contrast with the crude rigidity of mathematical formulas. This goes a long way in explaining the muddled history of Italian politics. But, in a very different sense, Walter was right.
Any mathematical statement, be it 5+3=8 or the most convoluted theorem, is true within an axiomatic system. In fact, ‘Convergent Parallels’ is an oxymoron within the most ancient of them, Euclidean geometry. Euclid defined parallel lines as ‘straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction’ (Elements, Book I, Definition 23). He then assumed that ‘if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles’. This is known as the parallel postulate, the last of Elements‘ five axioms.
Axioms are statements assumed to be true by self-evidence or by definition, thus requiring no demonstration. Unlike the first four, however, the fifth axiom does not look as self-evident. Hence many attempts were made, over two thousand years, to demonstrate it as a theorem derived from the other axioms, until in 1868 Eugenio Beltrami showed it was impossible to do. A common line of attack in trying to demonstrate the fifth axiom was Reductio ad Absurdum, whereby a statement is shown to be true by showing that its contradiction leads to an impossible, absurd result. But when, around 1830, Nikolai Lobachevsky and János Bolyai explored what happened if they dropped the fifth axiom, they found many strange results but no contradictions. The fifth axiom can be shown to be equivalent to the Playfair’s axiom, according to which ‘in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point’. Lobachevsky and Bolyai assumed instead that more than one line never meeting the given line could be drawn through the point, and found the ensuing Non-Euclidean geometry perfectly consistent.
It was an astonishing result. Until then, geometry was Euclidean geometry. Laplace – who had died just a few years earlier – listed it alongside astronomy and mechanics as one of the supreme feats of the human mind. Spinoza had mimicked Elements in his Ethics, using definitions and axioms to demonstrate propositions – not about lines and triangles, but about such ponderous concepts as nature, God and freedom. Immanuel Kant – who had also been dead for just a few decades – would have been startled to find out that our sense of space was not a pure a priori intuition, but one of many possibilities. Starting from their alternative axiom, Lobachevsky and Bolyai gave rise to Hyperbolic geometry. A few years later, Bernhard Riemann described a new geometry founded on a different alternative to Playfair’s axiom: no line never meeting the given line can be drawn through the point. It is called Elliptic geometry, and its simplest model is a sphere, where lines are meridians, which are parallel at the equator but do meet at the poles.
Christian Democrats were right. Parallels can converge – it depends on the geometry. I doubt my friend Walter was thinking along those lines, but his bemused protestations highlight the fact that even our hardest certainties rest upon undemonstrated assumptions. If we change the assumptions, we get different certainties.
This includes 5+3=8. Like geometry, Kant thought arithmetic contained synthetic a priori propositions. A priori, because they are independent of experience; synthetic (as opposed to analytic), because they say more than what is implied by their subject (Kant used 5+7=12 and argued that the concept of 12 is not contained in the concepts of 5, 7 and +). The ancient Greeks regarded arithmetic (from arithmos: number) as the epitome of episteme – absolute knowledge that is able to withstand any attempt at refutation. Like Euclidean geometry, arithmetic is an axiomatic system, in which a number of theorems are derived from the smallest possible set of axioms, using truth-preserving rules of inference. Given the axioms, the theorems are demonstrably true, independent of experience. But they are true within the system, i.e. relative to its syntax – the symbols, rules and principles with which the system is put together. In this sense, an axiomatic system is like a computer program, whose algorithms derive results (propositions and theorems) from initial inputs (definitions and axioms). Like a computer program, an axiomatic system is not about anything: change the inputs (e.g. the parallel postulate) and you get different results.
Axiomatic systems have two desirable properties. One is consistency: no proposition within the system can be shown to be true and false; the other is completeness: all propositions can be shown to be either true or false. In his Foundations of Geometry, published in 1899, David Hilbert showed that geometry, Euclidean as well as Non-Euclidean, is consistent and complete. But he could not say the same for arithmetic – on which geometry and most other systems are based. So, when the following year he announced his program, listing 23 unsolved mathematical problems and calling his fellow mathematicians to arms (‘This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus‘), the second problem in the list was ‘Prove that the axioms of arithmetic are consistent’.
Alas, despite Hilbert’s buoyancy, many of the problems proved hard to solve. In fact, new problems – such as Russell’s paradox, discovered the following year in set theory – kept adding to the pile. But the biggest blow to Hilbert’s program came in 1931, one hundred years after Lobachevsky and Bolyai, when Kurt Gödel demonstrated that arithmetic is incomplete. More precisely:
Gödel’s First Theorem: If an axiomatic system, capable of containing arithmetic and defined by a finite syntax, is consistent, then it is possible to construct a proposition within the system that is true, but cannot be shown to be either true or false. Hence arithmetic cannot be both consistent and complete.
Let’s call the system S and the proposition P. The theorem says that if S is consistent then: a) P is true and b) P cannot be shown to be either true or false. Let’s then set P=’S is consistent’. Hence, if S is consistent then: a) ‘S is consistent’ is true and b) ‘S is consistent’ cannot be shown to be either true or false. It follows that:
Gödel’s Second Theorem: The consistency of an axiomatic system capable of containing arithmetic and defined by a finite syntax cannot be demonstrated within the system.
So much for the ultimate goal of Hilbert’s formalist program: to demonstrate that mathematics as a whole is self-consistent. It isn’t, starting from its very base: arithmetic. Hilbert wanted to demonstrate the consistency of arithmetic from within, without recourse to external ‘intuitions’ embedded in its axioms – especially the troublesome intuition of infinity. Gödel showed that such a finitist demonstration was impossible: consistency has to come from outside the system.
Of course, arithmetic is consistent: there is no arithmetic proposition that is both true and false. But – as with the parallel postulate and its lines ‘being produced indefinitely’ – arithmetic cannot get away from infinity. In fact, Gerhard Gentzen demonstrated consistency in 1936, using transfinite induction. Once we assume the existence of the infinite set of natural numbers – whose sum, remember, equals -1/12 – arithmetic is perfectly consistent: there is no doubt whatsoever that 5+3 is nothing but 8.
Gödel – a mathematical Platonist – was firmly convinced that natural numbers exist ‘out there’, just like Kant, who viewed mathematics as synthetic a priori. Indeed, Gödel interpreted his theorems as demonstrating the very necessity of natural numbers: without them, arithmetic is not even consistent. Whether or not we share Gödel’s outlook – I do not – Gödel’s theorems show that, like geometry, arithmetic is not a self-contained corpus of absolute truths. All arithmetic propositions – including 5+3=8 – rest on undemonstrated axioms, whose truth we assume to be intuitively, and to our complete satisfaction, self-evident. Q.E.D.
Also published on Medium.