Is everything a thing?

A word that tries to hold everything

Start with a game. Name everything in the room you are sitting in: the chair, the lamp, the dust on the sill. Now add the building, the street, the city, the planet, the sun and the other stars, the galaxies. Keep going until nothing is left out. The word for the result is the universe, and the definition is easy: everything that exists, taken together, with nothing outside it and nothing left over. The Greeks had a word for it too, kosmos, which you will meet in Lesson 3.

Now ask a question that sounds like a riddle. Is the total itself one more thing? The books on a shelf are not themselves another book on the shelf. A crowd is not one more person in the crowd. But when we say “everything,” we talk as if we could point at it, give it a size, a shape, an age, and ask what it is made of. Can we? Is “the universe” a thing at all, in the way that a tree is a thing?

This lesson follows that question through two of the famous episodes of Western thought. In 1781 the German philosopher Immanuel Kant argued that if we treat the universe as a thing, our reasoning contradicts itself. In 1901–02 the English philosopher Bertrand Russell found the same kind of tangle in the foundations of mathematics. The two stories are about a century apart, and they teach one lesson.

One more note on what kind of lesson this is. This course takes a philosophical position, and it is better to say so now than to hide it. The position is that thinking about “the universe” begins in earnest at the moment the universe stops being a comfortable word and becomes a problem: a thing you cannot quite say anything about without trouble. In this reading, philosophy is first of all a hard training for the mind, a workout in catching your own assumptions, and the paradox of the whole is the first exercise. That is a thesis of the course, and it is not the only way to define philosophy. Where the history comes in, it has been checked against the sources; where the argument comes in, we say so.

Kant’s first antinomy

Immanuel Kant (1724–1804) spent his whole life in Königsberg, in what was then East Prussia. In 1781 he published the Critique of Pure Reason, a book about what human reason can and cannot know. One of its most famous chapters shows that when reason tries to think about the world as a whole, it can prove two opposite statements with apparently equal force. Kant called such a pair an antinomy, a clash between two laws, and found four of them. The first is about the size of the world.

• Thesis: the world has a beginning in time, and is also limited in space.
• Antithesis: the world has no beginning and no limit; it is infinite in both time and space.

Each side comes with a proof, and each is a proof by contradiction: assume the opposite, and show that it cannot be. Take time first. For the thesis, suppose the world had no beginning. Then, by now, an infinite series of moments has gone by. But an infinite series is exactly the one that can never be completed by adding one moment after another, so the series up to this moment could never have been finished. The past must have a first moment. For the antithesis, suppose the world did begin. Then before it there was an empty time. But in an empty time no moment differs from any other, so nothing could make something start at one moment rather than another. The world could never have got going.

The proofs for space run the same way. If the world is limited, it sits in an empty space beyond, and so stands in a relation to nothing. If the world is an infinite whole already given in space, then to take it all in we would have to run through an endless series of steps, and that is impossible.

Notice what the first proof leans on. Kant defines the infinite as that which can never be completed by adding one step after another. That is his own stipulation. Mathematicians since Georg Cantor in the 1880s have worked with infinities as completed wholes, and for them an infinite series is not a contradiction.

The idea behind the antithesis is old. In the fourth century BCE the philosopher Archytas of Tarentum is reported to have asked what happens if you reach the outermost edge of the heavens and stretch out a hand or a staff. If you can, there is space beyond the edge, so the edge was not the edge, and you may ask the question again from the new place. If you cannot, something stops you, and that something is outside too. The Roman poet Lucretius put the argument into verse, with a flying spear: wherever you place the edge, what at last becomes of the spear?

Chinese readers will know the same argument in two forms. In the Liezi, a collection of Daoist stories compiled around 300 CE, the founder of the Shang dynasty asks the sage Xia Ge whether the universe has an edge. Xia Ge first says he does not know. Pressed, he says that beyond the limitless there is no further limitless, so he knows the universe is without limit, though he cannot know that it has one. Then he adds a line that goes to the heart of this lesson: heaven and earth are also a thing (天地亦物也), perhaps one thing inside a larger heaven and earth. In the 1300s the Ming statesman Liu Ji (刘基, 1311–1375) put the dilemma in a few short lines: 天有极乎？极之外又何物也？天无极乎？凡有形必有极。 “Does heaven have an edge? Then what is outside the edge? Does it have none? Everything that has a form has an edge.” Two centuries later the scholar Yang Shen (1488–1559) quoted the lines with approval. Liu Ji’s own point was modest: the sage cannot know the answer, and honesty means saying “I do not know,” not “I will not speak of it.”

What Kant concluded

Kant did not conclude that the world is finite, or infinite. He concluded that both sides make the same mistake. Each assumes that “the world” is something that exists as a finished whole, with a size, in the way that a table does, and that reason can reach out and describe it. But we never meet the world as a whole. Everything we ever meet is a part of experience: this tree, that star. The total of all the things that appear to us is not itself something that appears. In Kant’s language, the world as a whole is an idea of reason, a goal that thinking aims at, and not an object of experience. Asking how big it is, is asking about something that is not there to be measured, and that is why the answers contradict.

Kant gave an everyday picture of the mistake. Of any body, you might say it must either smell sweet or smell foul. But a body can have no smell at all, and then both statements are false. In the same way, “the world is finite or infinite” leaves out the third case: that the world is not a given thing with a size. This course takes from Kant a simple way to say it: the universe is not one more thing among things. It puts it as a slogan, “the universe is not a thing,” and draws a large conclusion from it: that the universe belongs not to science but to philosophy. Kant himself drew a narrower line: science describes what can appear in experience, and the totality of what appears cannot.

Russell’s set

The second story is about numbers. By the 1870s mathematicians had a new tool, the set: a collection of things, treated as one thing. The set of the days of the week, the set of whole numbers, the set of all circles. In the 1880s and 1890s the German logician Gottlob Frege (1848–1925), working in Jena, set out to show that all of arithmetic follows from pure logic, with sets as the building blocks. His rule for making sets looked innocent: any property you can state defines a set, the set of all things with that property.

In 1901 the young Bertrand Russell asked a simple question about this rule. Some sets are members of themselves, and most are not. The set of all spoons is not a spoon, so it is not a member of itself. The set of all things that are not spoons is not a spoon either, so it is a member of itself. Call a set ordinary if it is not a member of itself. Now use Frege’s rule to form the set R of all ordinary sets. Is R ordinary? If it is, it belongs to the set of all ordinary sets, so it is a member of itself, so it is not ordinary. If it is not ordinary, it is a member of itself, so it belongs among the ordinary sets, so it is ordinary. Either answer contradicts itself.

On 16 June 1902 Russell wrote to Frege to tell him. Frege’s second volume was already in the hands of the printer. He answered on 22 June, honestly and briefly: the discovery had surprised him beyond words, he wrote, and left him thunderstruck, because it had rocked the ground on which he meant to build arithmetic. He added a hurried appendix to the volume, which appeared in 1903, admitting that hardly anything worse can befall a scientific writer than to see one of the foundations of his edifice shaken after the work is done. His attempt at a repair failed, and he never fully recovered from the blow. Russell, who admired the grace of Frege’s reply, added an appendix on types to his own book. (The German logician Ernst Zermelo said he had found the same contradiction independently, a little earlier, and had told David Hilbert.)

The cure took a few years. Russell himself proposed a “theory of types” in 1908, which sorts sets into levels so that a set can only contain things from a lower level. The same year Zermelo published a list of axioms for set theory, later extended by Abraham Fraenkel and others. The central change is that you can no longer form a set from any property whatever. You may only separate out of a set you already have the members that have the property. With that rule, there is no set of all sets, and no set R. (A few exotic set theories, such as Willard Quine’s “New Foundations” of 1937, do allow a universal set, and pay for it elsewhere; in the standard ones there is none.) Mathematicians still talk about “the collection of all sets” when they need to, and they call it a class, not a set, to remind themselves it cannot be handled like one.

Chinese textbooks often tell this episode as the third of three “crises” in the history of mathematics, after the discovery of irrational numbers and the argument over infinitesimals in calculus. Historians of mathematics are more guarded: they speak of a single “foundational crisis” of the early 1900s, and the stories of the earlier crises are partly legend. The practical result was a more careful foundation, and most mathematicians went on with their work.

What the two paradoxes share

Look at the shape of the two arguments. Each starts by gathering all of something into a single whole: all the things there are, in Kant’s case, and all the ordinary sets, in Russell’s. Each then treats the whole as one more of the things gathered: the world as an object with a size, the set R as a set among sets. And each trips on exactly that step.

That is the course’s first conclusion, put in its own words: the total of all things is not a thing of the same kind as the things. This is also what the course means by saying that “the universe is not a thing.” It can sound like a verdict that the universe does not exist, which is not the claim. The claim is that “the universe” is not a name for one more object on the list, and that if you use it as though it were, contradictions follow.