Your complimentary articles
You’ve read one of your four complimentary articles for this month.
You can read four articles free per month. To have complete access to the thousands of philosophy articles on this site, please
Logicomix by Apostolos Doxiadis et al
Grant Bartley scrutinizes an epic graphic biography of Bertrand Russell.
“Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair.” Russell, Prologue to Autobiography.
Does Gödel’s Incompleteness Theorem prove the non-existence of God? Consider: if God is omniscient, then by definition he must be capable of knowing everything. But Gödel’s theorem says that there are some things that cannot be known (ie proved) from within a given system of ideas, such as mathematics, but only seen as true from outside it. Therefore, someone might argue, Gödel has proved omniscience, that is, knowing all ideas together, to be impossible, since he’s demonstrated that there would be some things knowable about this set of ideas only from outside this omniscient state. This is a contradiction. So an omniscient God does not exist.
Although it’s superficially seductive, I don’t think this argument is valid. Omniscience would surely include knowledge of those things known only in a Gödelian way. We might suppose for example that in some weird infinite way, God’s knowledge loops back in on itself to include even its own knowledge about itself. After all, Infinity is a counterintuitive land (to put it very finitely), and many ships of philosophy, logic and mathematics have been lured onto its rocks.
Logic, infinity, madness and God (or not) are depicted as driving the mind and life of Bertrand Russell, in the intriguing, intelligent and humane saga in comic form, Logicomix: An Epic Search For Truth by authors Apostolis Doxiadis and Christos H. Papadimitriou, artist Alecos Papadotos, and colourist Annie Di Donna. Logicomix is “a tragedy with logicians as heroes!” (p.98), in that it is the story of the men, their ideals, and their utter disappointment. Like Gödel’s proof, the story is concerned with showing the incoherence built into a puritanical search for logical certainty, especially as pursued through a frail humanity such as the third Earl Russell thoroughly demonstrated. It uses the history of philosophical logic in the early twentieth century to demonstrate that despite the passionate endeavours of logicians to seek the basis of reason, pure reason can never claim to be the ultimate ruler of human life.
The plot involves Russell coming to desire a logic as incontrovertible as the truths of maths, on which he hopes to unassailably base knowledge of philosophical truths. Only one of the postmodern ironies of this biographic, is how it interprets Russell’s quest as showing that even the search for ultimate logic is ultimately illogical. Like Gödel’s proof, it shows that logic can only be understood from a perspective outside of logic. Thus the narrative entwines the discoveries of the logicians with revelations of the logicians as emotionally complex beings, Russell especially. There is more than pure logic in the quest for truth: the fundamental truths of human life, of desire, love and disappointment are vital too. ‘If only Russell had integrated this in his thinking!’ the saga seems to be saying. Not absolutely coincidentally, the story is also an examination of the link between logic and madness, or as I might say, it focuses on the moments where the extremes of thinking meet. Logic and emotion – reason and passion – provide the central threads of the book, and the link between them is reflected in its denouement, which demonstrates that truth must know the heart as well as the head. This itself demonstrates what I think is the deeper theme of the book: that there are many different ways to understand, and they’re not necessarily in conflict, but may be complimentary, and indeed necessary, so that one can only possess truth when one has a full hand of ways of knowing, and balances one against another.
The book is the demonstration of different perspectives in several senses. First, it is a story within a story. It starts with Apostolis introducing the other creators of the book and its themes. I like to think the subtext here is that in some Gödelesque way, Russell’s life can only be understood by looking in from the outside. In a similar vein, we have postmodern-style references by Apostolis to the reader, who has a further perspective on what’s going on.
What Do You Call A Man In A Brown Paper Suit?
The story proper begins on 4th September 1939, the day the UK declared war on Germany. Russell is scheduled to give a talk at an American university on ‘The Role of Logic in Human Affairs’. He is pressed by demonstrators to say that nothing is more irrational than war. (Russell was a famous pacifist, having gone to prison for five months for peace campaigning during WWI.) He proceeds to explain the reasons, or you could say the logic, for his pacifism; starting with an analysis of the nature of logic itself. He does this by telling the life story of “one of its most ardent fans”: himself.
Thus a story within a story within a story begins. Young Bertie Russell arrives at the home of his grandfather, a former British Prime Minister. In Dickensian tradition, Bertie is told by his stern grandmother that he must follow rules according to definitions. Wait a minute … As Russell later comments: “Logic is all about rules. In fact, it begins with definitions and continues with rules” (p.36).
Then there is a mysterious howling in the night. No-one will answer Bertie’s questions about the ghost he clearly heard, denying the evident truth, which sows the seeds of Russell’s epistemological belligerence – his cynicism about claims to knowledge. His grandfather, like the Serpent, points out forbidden books in his library ( “Philosophers, in the top shelves to my right, are definitely no-nos”). The turning point in Russell’s education is shown to be his introduction to the ‘magic’ of Euclid’s geometry, including axioms which were true apparently of logical necessity: “Geometry showed me the only way to reality: Reason,” as Russell says. “Proof thus became my Royal Road to Truth.” (p.57). Science is similarly extolled as the only way of explaining the physical world. Thus the diamond core of Russell’s intellect is established.
In an episode reminiscent of the life of the Buddha, on the day young Bertie first encounters death, in the form of the tomb of his parents, he also encounters the bitter ravages of life, in the form of an amputee Crimean war veteran. When he discovers the howler, his mad uncle, the shock pushes Russell towards madness himself – with only the hope of pure reason as glimpsed through mathematics to save him. Russell is therefore horrified to learn that some of Euclid’s ideas are mere axioms – reasonable-looking assumptions which nevertheless have not been proved! The security of certainty is still elusive.
So to Cambridge University, where Russell has a Romantic Awakening – to the hope of pure reason! This may seem a paradox, but it reveals the heart of the book, that human truth includes the unrationalized truths of experience. After gaining a First in Mathematics, Russell migrates to Philosophy, where he isn’t impressed with the philosophers’ stew of highly contingent arguments. The only logical step for Russell is to seek out those who seek a formal logic, an algebra, for reason, so that philosophy can emulate maths in its technical precision. In a logical twist, Russell wants to use a formal philosophical logic to provide demonstrated foundations for the axioms of mathematics.
Thus Bertrand Russell becomes: a Logician.
After a short interlude for romance and marriage, Russell heads off to the continent to see Gottlob Frege, for whom the task of logic is not to calculate, but to model reality. Then to Georg Cantor, whom Russell aptly calls “The man who ate of the tree of knowledge of the infinite.” To corral his herds of wild infinities, Cantor developed Set Theory, which was to give Russell a Trotsky-sized headache for his life’s project of discovering secure foundations for knowledge. Ominously, Russell finds Cantor in an asylum. Yet Russell still has enough faith in logic to work obsessively on the foundations of mathematics, using set theory as his basis – until he discovers a paradox. This riddle says: “Does the set of all sets which do not contain themselves contain itself?” and Russell thinks it destroys any hope of a foundation for thought. This wasn’t what he was hoping to be remembered for.
The paradox’s implications are illustrated by the town where every man is required to shave, and all and only those who don’t shave themselves are obliged by law to be shaved by the barber. Who will shave the barber? If he doesn’t shave himself, then he must; and if he does shave himself, he must not. What’s a barber to do?
Tricky, huh? (One solution: employ an extra barber.) Actually, as Russell was traumatically aware, logical paradoxes such as this and the liar paradox of Euboulides ( ‘This statement is false’) challenge the possibility of knowledge. Russell says the set paradox “subverts the notion of ‘set’ as a collection defined by a common property … and with it, logic!” (p.168). Thus the rarified world of the logicians is put into an ‘uproar’, even prompting Frege to add this Addendum to his Foundations of Arithmetic, Vol 2: “The collapse of one of my laws, to which Mr Russell’s paradox leads, seems to undermine not only the foundations of my Arithmetic, but the only possible foundations of Arithmetic as such.” (p.171). If you can’t find the foundations of Arithmetic, you can’t find the foundations of anything; we’re hurled into eternal ignorance; and our knowledge is “turtles all the way down.”
I think the truth about self-referential paradoxes is that they are technically meaningless – there is nothing that they’re saying, no concept to which they are referring, despite appearances to the contrary. The problem is a problem of the appearance of meaning. Suspiciously, they only occur when the language refers to nothing but its own use. As long as there is an external referent to which the language can be seen to refer, there is no logical paradox. Russell’s suggested solution to his Frankenstein’s monster is metalinguistic: he says that self-reference generates distinct levels of meaning, which may be compared non-paradoxically. His adaptation of this idea to Set Theory through the theory of ‘types’ leads him and Alfred Whitehead to work together on the Principia Mathematica (1910). Except that ‘types’ doesn’t work (they’re like logical epicycles, I think), and no other possible foundation is evident. And for a moment Russell thought he’d proved 1+1=2 in only 362 pages! It took them ten years for the three volumes trying to fix logic and maths – yet still they could not provide it with foundations.
Russell’s psyche is depicted as if constantly trying to inform itself of experiences or truths beyond logic, but he’s only able to see over his logical horizons at the epiphanic moments of the heights or depths of his intellectual achievements, where extremes of feeling can no longer be channelled intellectually. For instance, the publication of the imperfect Principia pushes Russell to an emotional quake, as he declares his love to Whitehead’s wife, Evelyn. After the (unsurprising) failure of Russell’s marriage, Ludwig Wittgenstein appears as Russell’s obsessive young student, and is sketched as emphasizing the lack of proof even of objective reality – further undermining Russell’s logical security. Then Evelyn apparently nearly dies, leading Russell to discover the need to act reasonably in ‘human affairs’, just as WWI is breaking out. Even this reasonable intention is almost instantly undermined, as at the declaration of war with Germany, in an Orwellian episode a peace rally in Trafalgar Square turns into a jingoistic mob. Russell momentarily finds himself swept along by the emotionalism, until he regains his composure, soon becoming a ‘militant pacifist’ in the face of a culture of passionate patriotism.
In the Tractatus Logico-Philosophicus (1921), Wittgenstein developed his logical atomism and picture theory of language. Here, logic is the form of language, or, as I might paraphrase, logic provides the structure through which linguistic meaning is possible – the Kantian implication being that you’d need to get outside of thought to assess logic. “Logic… you can only show,” Wittgenstein says: logic itself cannot be proved, since, like unavoidable axioms of thought, logic is the presupposition of rational thinking. Russell’s reported response to the Tractatus is: “There it was: for twenty years I had sweated to justify the existence of a machine for manufacturing tautologies.”
Waiting For Gödel
So to the ‘solution’ of the tragedy, as Aristotle would have put it.
Bertrand and his new wife Dora have a son. Russell abandons his Logical Grailquest, opting for Education as a worthy project, running an experimental school, which turns i nto an anti-authoritarian mess. Thus Russell’s biggest adventure into pure human affairs is a failure. Then Gödel sets out to test the assumption that logical propositions must be provable, and eventually proves instead that not all truths are logically provable. Thus “There will always be unanswerable questions” (p.286). “It’s all over,” Von Neumann says to Russell, acknowledging the end of a long dream of philosophy. Gödel has given humanity a mathematical proof that there will never be perfectly logical, that is, perfectly rationally-justifiable knowledge. Thus at the end of his lecture on logic in human affairs, Russell can only say we might have to rely instead on “Responsibility, Justice, even a sense of Good vs Evil… my story… tells you that applying formulas is not good enough” (p.297). He can only conclude that it’s up to every individual to make up their own minds.
The book ends self-referentially, with its creators watching a performance of Aeschylus’ tragedy Oresteia. Orestes takes revenge for the murders of various family members by killing his mother. The goddess Athena, representing the spirit of reason as good judgement, asks the citizens of Athens to decide upon his guilt, and his life. The vote is equal. Athena has the deciding vote. She acquits him; and thus the force of reason breaks the imprudent cycle of revenge. Yet instead of banishing them, Athena invites the Furies, representing the primal passion for vengeance, to remain to influence the city’s future decisions. As Hume would agree, reason needs its motives. “To achieve wisdom, you must… allow for a lot that’s usually left out as un-wise,” Apostolos comments on p.311. As Athena proclaims, “Rejoice, rejoice, rejoice, you happy citizens who love true wisdom!”
Gödel knew that the experience of truth has many aspects, not all of them demonstrable purely by reason (if any). This higher principle is demonstrated by the logicians’ story too: that knowledge must incorporate truths which reason sometimes cannot even express. These include the feelings of uncertainty that let us know we’re not omniscient. Indeed, it appears to be an essential truth that we can never have a perfected perception of reality. To be human is to be uncertain.
© Grant Bartley 2010
Grant is Assistant Editor of Philosophy Now. His book The Metarevolution is available as a free download from philosophynow.org (go to About us). You can also order it in physical form from Amazon for only £8.99, or $17.99.
• Logicomix: An Epic Search For Truth by Apostolis Doxiadis, Christos H. Papadimitriou, Alecos Papadotos and Annie Di Donna, Bloomsbury 2009, 348 pages, £16.99, ISBN: 978-0747597209.