his wife died in 1993. he says part of her is still thinking

part i

Parma, 1524. Francesco Mazzola is twenty-one and about to leave for Rome to try his luck with the new pope. He wants something small to show what he can do. According to Vasari, he goes to a turner, has a wooden ball made and sawn in half, and paints on the dome. The thing he copies is what a barber's convex mirror shows him, a cheap glass already old-fashioned by then. The beams of the ceiling bend. The window pulls into a bow. The panel is 24.4 centimetres across, about the size of the mirror, and along the right edge he even paints a sliver of the gilded frame.

The hand in front is his left. The glass flipped it, and because a convex mirror swells whatever is nearest, the hand comes out bigger than his head. Vasari says he enlarged it a little on purpose. His face sits small and calm in the middle, in a fur-collared coat, a boy's face. The painting went to Clement VII that summer with two other small pictures. It did not get him the Vatican commissions he came for. It went on to Pietro Aretino, then Alessandro Vittoria, then Rudolf II in Prague, and in 1777 it entered the Habsburg collection in Vienna, filed under Correggio. It hangs in the Kunsthistorisches Museum now, under his nickname.

Vienna, 1931. Kurt Gödel is twenty-four. His paper has just come out in the Monatshefte für Mathematik und Physik, under a title that takes a breath to say: On Formally Undecidable Propositions of Principia Mathematica and Related Systems. The trick inside it is small enough to hold in one hand. He gives every symbol, every formula, every proof in arithmetic its own number. Once statements are numbers, arithmetic can talk about statements, and so it can talk about itself.

Then he builds one particular number. Read as arithmetic, it says that no number has a certain property. Decoded, the property means being a proof of this very sentence. The sentence says: I cannot be proved. If the system could prove it, the system would be proving something false. So it can't. Which makes the sentence true. A barber's mirror, in symbols: the room in the glass, the glass in the room.

Douglas Hofstadter gave the shape a name in 1979, in Gödel, Escher, Bach. A strange loop is what you get when you move up through the levels of a hierarchy, level after level, and arrive back where you started. His everyday example is a video camera pointed at its own monitor. Point it at an empty room and nothing is there; turn it on the screen and a corridor of screens opens up, a shape that lives nowhere but in the feedback.

Hofstadter's claim, made at length in 2007 in I Am a Strange Loop, is that the I is that shape. A brain is a mass of symbols. Some of those symbols stand for the brain itself. Over years, the symbol for the self gets fed its own output until it is the loudest thing in the room, and the brain calls that the person. He calls it a hallucination perceived by a hallucination. He does not mean it as an insult.

strange loop, i've seen the cycle repeat

Look at the Parmigianino again. The hand is what he does and the face is who he is, and the hand is bigger.

part ii

Madrid, 1656. Velázquez paints a room in the Alcázar, the old quarters of the dead prince Baltasar Carlos that Philip IV gave him for a studio. He paints himself inside the picture, brush raised, looking straight out. The canvas he is working on faces away from us. It is enormous, about the size of Las Meninas itself, which is 318 by 276 centimetres. In the mirror on the back wall, small and bright, the king and queen. They are standing where you are.

The Infanta Margarita is five. Her two maids of honour bend to her. The dwarf Nicolás Pertusato nudges a sleeping mastiff with his foot. In the lit doorway at the back, José Nieto, the queen's chamberlain, has paused on the stairs. Everyone in the room has stopped for whoever just walked in, and whoever just walked in is you, or the royal couple, or both. Foucault spent the first chapter of The Order of Things on the fact that the painter looks at you only because you happen to be standing in his subject's place.

Trento, 1993. Douglas Hofstadter is three months into a sabbatical year in northern Italy with his wife Carol and their two children, Danny, five, and Monica, two. Carol dies, very suddenly, of a brain tumour. She was forty-two. He flies home a single parent and keeps speaking Italian to the children because she would have.

Years later, in the book, he writes about looking at a photograph of Carol and feeling that he was behind her eyes. He tried, for a while, to look at Monica for her, with her way of looking. His conclusion, worked out slowly and with a scientist's care, is that a person is a pattern, and that a pattern can run on more than one piece of hardware. Eight years of marriage had copied a rough, low-resolution Carol into his skull, and that copy was still running, slower and blurrier than the original but running. The reviewers were polite and uneasy about it.

Back to the painting. The red cross on Velázquez's chest is the cross of the Order of Santiago. He did not have it in 1656. He was knighted in 1659 and died in August 1660. Palomino, writing half a century later, says the king ordered the cross painted onto the finished picture after the painter's death, and adds that some say his Majesty painted it himself.

So the man in the picture was given his honour by a hand that was not his, after he was gone, and the picture went on changing without him. The painter is gone. The painting is still being finished.

The painting: Parmigianino, Self-portrait in a Convex Mirror (c. 1524), Kunsthistorisches Museum, Vienna. Public domain. Part II of the video uses Diego Velázquez, Las Meninas (1656), Museo del Prado, Madrid. Public domain.

Sources