A Mathematical Proof of Pluralism? (With apologies to Wittgenstein and Godel)

As the airwaves and Internet fill with loud voices proclaiming certain truths, it’s worth taking a quiet moment to remember two men who proved, mathematically, that there was no such thing as absolute and complete truth.  Kurt Gödel (pictured) and Ludwig Wittgenstein are rightly renowned in the esoteric worlds of logic and philosophy.  But although their intent was never overtly political, their work has a deep political significance.  In a way, these two logicians proved the logical necessity of pluralism.

 

Over the holidays, I read again “Logicomix” the wonderful graphic novel about Bertrand Russell’s life-long quest to understand the nature of truth.  This led me to the rather more daunting but equally satisfying biography of Russell by Ray Monk.  Russell laid the groundwork for what was to follow, in his epic search for absolute truth.  But Russell’s is in some ways a tragic story, for his attempt to prove the foundations of mathematics – what can be certainly known –  was ultimately a failure.  

 

A brilliant Austrian mathematician proved that Russell’s attempt to find certainty at the base of mathematics was doomed.  There was no such thing as complete knowledge; there are no absolute truths.  But in demonstrating the incompleteness of mathematics – or any formal system – Kurt Gödel had also, perhaps unintentionally, suggested something vital and fundamental about politics and the world.

 

As a student of mathematics, Russell had doubted his professors who told him that it should simply be accepted that 2 +2 = 4.  These were self-evident truths – axioms.  But reliance on such axioms troubled Russell: if mathematics relied upon self-evident statements, provable only by reference to themselves, its foundations were shaky indeed. Everyone knew that 2 + 2 = 4, but why? This simple axiom had never been proven.  And if this was the case, the very basis and foundation of mathematics remained unproven.

 

Along with his friend Alfred Whitehead, Bertrand Russell devoted himself to an attempt to prove the basis of mathematics, why 2 +2 = 4. The attempt was to take the two friends over ten years, and resulted in the Principia Mathematica, a prodigious masterpiece of logical exposition.  But though highly complicated and voluminous (I cannot pretend to have read it or understand it), Russell sensed at its conclusion that it wasn’t enough.  He drove Whitehead over and over again to find further proofs to those they had already demonstrated.  Under one foundation, there had to be another, but where would it end?

 

Russell himself believed that there was an ultimate end, a final proof that underpinned all others, and thus rendered solid – at last – the very foundations of mathematics: a complete and proven system.  This was the holy grail of mathematics – the paradigm-defining challenge first proposed in 1920 by David Hilbert, a German mathematician.  

 

In the 1930’s, as dark clouds gathered over Europe, a young Austrian mathematician read the Principia.  Kurt Gödel was the only person Russell believed who fully read and understood his masterwork.  But Gödel’s conclusion was that Russell and Whitehead’s proof was insufficient. 

 

Gödel’s critique of the Principia was devastating, and saw the first appearance of his famous incompleteness theorem.  Gödel showed not only that Russell’s attempt was inadequate, but also that the very quest for solid foundations, for completeness and thus certainty itself, was futile.  There was no holy grail.

 

Gödel’s incompleteness theorems state – and professional logicians must forgive my pathetic simplifications – that no formal system, such as mathematics, is complete.  There will always be true statements that cannot be proved within that system.  A complete logical system is thus – oddly – a logical impossibility.  This sounds perverse, but also in its way obvious, and often such is the nature of the most brilliant observations.  

 

The implications of Gödel’s insight are profound, not only in proving the futility of the quest for a complete mathematical system, but also, to take his analysis a stage further, in demonstrating the impossibility of absolute truths – about anything.  For it follows from Gödel’s conclusion that if no formal system, such as mathematics, can be complete, how can anything else be? No absolute truth can be shown to be true.  There is always something else.

 

Gödel’s intent, it seems, was not political though logicians like him and Russell knew that they were wrestling with some of the fundamental questions of human knowledge, questions that lie at the heart of all questions, including economics, politics and how we run the world.  But his incompleteness theorems nonetheless carry considerable political implications.  If there is no absolute complete and provable truth, we must contend with multiple claims for truth.  We must contend with – and accept – pluralism, thus demonstrating too the necessity of tolerance.  

 

Pluralism is therefore shown not only as a desirable thing in itself – which is after all merely an axiomatic claim – but as a logical and now proven inevitability.  The fascists and jihadists – and indeed some of our own supposedly pluralist politicians – who claim absolute certainty about the world can only be wrong.

 

Ludwig Wittgenstein, another brilliant Austrian logician, and Jewish, came to a similar conclusion, although by a different  route.  Wittgenstein was Russell’s student, and he too saw in the Principia a fundamental question – but one inadequately answered.  In his Tractatus Logico-Philosophicus, the only book published in his lifetime, Wittgenstein concluded that whatever the logical basis of any formal system, logic itself – like reason – could not be proven.  It could only be shown. 

 

In Tractatus, Wittgenstein famously concluded that what could not be described or talked about with a logical language, should not be spoken of at all – since there was no logical way to do so.  Beyond any logical system, there was inevitably something else. 

 

Wittgenstein, a deeply religious man, believed that the most important things in life belonged in this realm of what could not be logically described.  The profound implications of this insight – nothing less than a demarcation between the rational and irrational –  are immense: more of this in a later post. But in a way similar but different to Gödel, Wittgenstein had demonstrated the same truth of what we could know – or say – of truth itself.

 

Both Wittgenstein and Gödel were driven from their homeland by the absolutism of the Nazis.  In their different ways they had shown the absurdity of absolute claims, such as those proffered in Hitler’s grotesque philosophy.  Gödel feared the Nazis’ assault on the pursuit of rationalism, including the murder – by a Nazi sympathizer – of his academic colleague, the logical positivist Moritz Schlick.  After Germany’s Anschluss with Austria, he also feared conscription.  Wittgenstein spent the war working as a porter in a South London hospital, and was never to return to live in Austria.  It was continental Europe’s loss, but humanity’s gain that they were able both to find refuge, Wittgenstein in England and Gödel at Princeton in the United States – where Albert Einstein witnessed his application for citizenship. 

 

To these two refugees from totalitarianism, who were also the most incisive critics of total theories of anything, and to Russell, Hilbert and others who prepared the way, we owe immeasurable gratitude – and the duty of recalling, every now and then, the truth de profundis that they articulated – that there is no truth, but truths, as Albert Camus once put it in The Myth of Sisyphus.  Our search for truth is never-ending; anyone who claims to possess it complete and absolute, secure from all challenge, is always wrong.  This truth – the truth that there is no single truth – is perhaps the most important and deep-seated defence against the absolutists, fascists and all those – who seem troublingly many – who aggressively proclaim certainty. 

 

This proof of what cannot be proven should not be taken as an endorsement of anything-goes moral relativism, where every claim is equally valid.  For although Wittgenstein showed that we cannot prove reason, we still have it, and must apply it, as Wittgenstein did to all parts of his life (as Ray Monk vividly describes in his outstanding biography).  Not all claims are of equal merit. Every one must be tested rigorously to the limits – but there are limits, as Gödel and Wittgenstein showed us too: there is always something else.

 

In discovering that there is no bedrock of mathematics and indeed of logic itself, and that therefore nothing is completely certain, these heroic and often troubled figures instead showed us something of equal if not greater importance – the necessity of pluralism, of tolerance, and thus the very bedrock of that uncertain venture, a worthwhile civilization.

 

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