Can We Ever Find Absolute Truth?

A good proof has a particular kind of authority. Each step is small, each follows from the last, and when you reach the end no one can argue you out of the conclusion. It is easy to see why mathematics became, for many thinkers, the model of what knowing ought to be, and why a few of them asked a more ambitious question: could all of mathematics be placed on foundations as secure as its best proofs?

Logicomix: An Epic Search for Truth is a graphic novel about that question and the people who gave years of their lives to it. It is neither textbook nor biography, but an attempt to show what the hunger for certainty does to those who feel it most strongly.

A story told at arm’s length

The book was written by Apostolos Doxiadis, a novelist, and Christos Papadimitriou, a computer scientist, with art by Alecos Papadatos and colour by Annie Di Donna. It appeared in Greek in 2008 and in English the following year.⁶ Bertrand Russell narrates the main story, a history of the “foundational quest” in mathematics. Alongside it runs a second thread set in present-day Athens, where the creators themselves argue about what the story means.

That second thread matters. It signals that what follows is a telling, a dramatisation shaped by choices, not a transcript of events. This essay tries to keep the story the book tells apart from the history it draws on.

The dream of firm ground

By the late nineteenth century, mathematicians had made analysis rigorous, and Georg Cantor’s theory of sets had given them a language that handled infinity directly. Gottlob Frege wanted to go further. He built a formal language for logic and set out to derive arithmetic from logical principles alone, so that every step could be checked without appeal to intuition. Russell shared the ambition.

Then, in 1901, Russell found a contradiction at the heart of the enterprise.² Naïve set theory allows any property to define a set. Take the property of not being a member of oneself, and consider the set R of everything that has it. Is R a member of itself? If it is, it fails its own test for membership, so it is not. If it is not, it passes the test, so it is. In June 1902 Russell wrote to Frege with the problem. The second volume of Frege’s Grundgesetze der Arithmetik was already going to press, and Frege added a hurried appendix acknowledging that his system was in trouble.

Rebuilding, at a price

Russell’s response, developed with Alfred North Whitehead in Principia Mathematica (three volumes, 1910–1913), was to forbid the offending self-reference through a theory of types, which sorts objects and statements into levels so that a collection cannot be applied to itself.³ The known paradoxes were blocked. But the system needed further assumptions, among them an axiom of infinity and an axiom of reducibility, and many critics doubted that these were truly principles of logic. Certainty had been purchased, but the price was visible.

Here a graphic novel has an advantage over an exposition: it can set the technical labour against the personal turmoil the book weaves through it, so that a dispute about axioms is also a story about people who had staked their lives on them.

Other ways of being safe

David Hilbert proposed a different kind of security. In 1900 he had listed a proof of the consistency of arithmetic among his famous problems. In the 1920s he developed a programme to formalise mathematics and to prove, using only methods so concrete that nobody could doubt them, that no contradiction could ever arise.⁴ The foundations were to be secured by mathematics itself. In Königsberg, on 8 September 1930, he closed an address with the line later carved on his gravestone: Wir müssen wissen. Wir werden wissen. We must know. We shall know.⁵

Ludwig Wittgenstein, who came to Cambridge in 1911 to study with Russell, offered something less comforting than either project. His Tractatus Logico-Philosophicus argued that the propositions of logic say nothing about the world; they display the structure of language. For anyone hoping logic would be a vault of deep truths, this was a reframing, not a rescue.

What Gödel actually proved

In September 1930, at a conference in Königsberg, the twenty-four-year-old Kurt Gödel quietly announced a result that undercut Hilbert’s hopes; Hilbert spoke in the same city within days. Gödel’s 1931 paper established two theorems about formal systems.¹ Stated informally, but with the conditions that matter: if a formal system is consistent, has axioms that can be listed by an algorithm, and can express a modest amount of arithmetic, then there is a statement in its language that it can neither prove nor refute. The second theorem adds that such a system cannot prove its own consistency. (Gödel’s original argument assumed a slightly stronger condition than consistency, which J. Barkley Rosser removed in 1936.)

The theorems are easy to misread, so three cautions are worth stating. They do not say that mathematics is arbitrary or truth relative: a sentence one system cannot prove can be proved in a stronger one, which then has blind spots of its own. They do not apply to every theory; Presburger arithmetic, which has addition but not multiplication, is complete, as is first-order Euclidean geometry. And they do not show that mathematicians can be certain of nothing. They show that no single, fixed, mechanically checkable system can capture all the truths of arithmetic or vouch for its own soundness. That ended Hilbert’s programme in its original form, but it was a limit on one method of securing certainty, not on the pursuit of knowledge.

Logic, ambition and the human cost

What lifts Logicomix above an illustrated lecture is its suspicion that the search for certainty is never purely intellectual. The book interweaves its mathematics with its characters’ loneliness, ambition and instability, and presents the demand for total rational control as something that edges towards madness. Whether that thesis persuades is a separate question. The Berkeley philosopher Paolo Mancosu, reviewing the book in the Journal of Humanistic Mathematics, examines its historical fidelity, its logical inaccuracies and its treatment of madness and tragedy.⁷ A summary of his earlier review in the Bulletin of Symbolic Logic notes his objection to the repeated link between logic and madness and to invented meetings, such as Russell with Frege and with Cantor, while crediting the core ideas with reasonable accuracy.⁶

The link between logic and madness is best read as the authors’ thesis, not an established finding, and the creators’ own disagreement in the second thread invites exactly that reading.

What it does well, and where it stops

The book’s great strength is emotional legibility. Abstract questions acquire faces and stakes, and a reader with no mathematical training can see why a contradiction in set theory could feel like a catastrophe. The comic form also lets ideas arrive as quarrels between people rather than as definitions.

The limits are those of the genre. The mathematics is handled lightly: a reader will learn why the paradox mattered and broadly what Gödel’s theorems say, but not how they are proved. A story built around Russell also pushes rival approaches to the margins and compresses decades of untidy debate into dramatic encounters. The history was not a straight road from ambition to disaster; Gentzen’s 1936 consistency proof, and the proof theory that followed, show a programme changing shape rather than simply ending.

Returning to the question

Can we ever find absolute truth? The story Logicomix retells suggests an answer in two parts. Inside a formal system, certainty is real: a proof is a proof, and Gödel’s work does not disturb that. But the dream that one system could guarantee all mathematical truth and certify its own soundness is one that mathematics abandoned by its own standards of proof.

Certainty turns out to be local: earned system by system, never inherited from a foundation beyond question. Mathematicians went on proving theorems, but with a more exact sense of what a proof can and cannot do. That may be the most honest certainty on offer, and the book is at its best when it lets the reader feel what it cost to arrive there.