Of completeness and little men

Ignacio Escañuela Romana

And suddenly, my friend Gödel’s incompleteness theorems. I imagine that on the day he came to them he looked at the trees outside his window, breathed heavily, and set off in search of the clear and distinct perception that must wander through the university garden. I think he prayed to all the gods of logic that day, and sighed for the Leibnizian dream of all-encompassing knowledge. No doubt he laughed with glee at the benefits of intuition, so neglected and reviled in mathematics. In intimate alliance with Plato, they both toasted the eternal forms, more real than any thought of the fundamentals, so human at bottom.

And since then, Gödel’s ghost has haunted us, reminding us that we will not succeed in establishing absolutely the propositions of a certain type, which will inevitably resemble each other, without asking whether they are appropriate. From that moment on, when we went for a walk in the countryside, a little man would come up to us and say with a smile: «I lie». Since then, the phrase «I am wrong» has tended to run through the texts. And at such moments, out of helplessness, you want to take your whole library and set it on fire.

In short, we would like to have the algorithm: something that would make us less human by assuring us beyond any doubt of the truth of what we think. We would introduce into every discussion the procedure of methods which, like a judge, would evaluate all the opinions put forward. Suddenly, Gödel smiles at us and warns us: «When you receive the book of books in the Borgian library, you will not know what the book of books is. You may sense it, but you may not. Look for something that includes its self-reference, take a deep breath, decide if this book is lying to you.

It must be because of all this that I sometimes have nightmares at night in which Feynmann’s equation of everything appears: only to be erased when I wake up, and the fog of reason prevents me from returning to that first intuition. Dream or reality? «I am a dream,» I say to myself. «True or false?», I add.