Quantum Wittgenstein

I first learnt about Plato’s allegory of the cave when I was in senior high school. A mathematics and English nerd – a strange combination – I played cello and wrote short stories in my spare time. I knew a bit about philosophy and was taking a survey class in the humanities, but Plato’s theory of ideal forms arrived as a revelation: this notion that we could experience a shadow-play of a reality that was nonetheless eternal and immutable. Somewhere out there was a perfect circle; all the other circles we could see were pale copies of this single Circle, dust and ashes compared with its ethereal unity.

Chasing after this ideal as a young man, I studied mathematics. I could prove the number of primes to be infinite, and the square root of two to be irrational (a real number that cannot be made by dividing two whole numbers). These statements, I was told, were true at the beginning of time and would be true at its end, long after the last mathematician vanished from the cosmos. Yet, as I churned out proofs for my doctoral coursework, the human element of mathematics began to discomfit me. My proofs seemed more like arguments than irrefutable calculations. Each rested on self-evident axioms that, while apparently true, seemed to be based on little more than consensus among mathematicians.

These problems with mathematics turned out to be well known. The mathematician and philosopher Bertrand Russell spent much of his career trying to shore up this house built on sand. His attempt was published, with his collaborator Alfred North Whitehead, in the loftily titled Principia Mathematica (1910-13) – a dense three-volume tome, in which Russell introduces the extended proof of 1 + 1 = 2 with the witticism that ‘The above proposition is occasionally useful.’ Published at the authors’ considerable expense, their work set off a chain reaction that, by the 1930s, showed mathematics to be teetering on a precipice of inconsistency and incompleteness.

Eventually, I turned to physics, hoping to reground my Platonist aspirations in the eternal laws that governed the physical reality of the cosmos. But quantum theory exposed that, too, as a fantasy: even though we could define rules and equations for physical laws, we could not explain what they meant. Recent experiments in quantum information theory have shown that our most basic assumptions about reality, such as when something can be considered to have been observed and to have definite physical properties, are in the eye of the beholder.

Attempts to address these paradoxes date back to the dawn of quantum mechanics, when Albert Einstein and Niels Bohr debated how to interpret the baffling phenomena they’d uncovered. Yet it was only when I dived into the parallel milieu of Cambridge Philosophy, at the time of Ludwig Wittgenstein and Russell’s ascendancy, that I began to feel like my qualms about mathematics and physics might be addressed. Contemporaries of Einstein and Bohr, Wittgenstein and Russell didn’t engage with the quantum revolution directly. Yet it’s in the work of these philosophers that I began to see answers to some of our most fundamental questions about reality – answers that stem from recognising that we are not only asking the wrong questions; we are asking nonsensical ones.

The great debates about quantum physics kicked off in the 1920s. Bohr and his protégé, Werner Heisenberg, were trying to figure out how to talk about the weird behaviour of quantum particles: how they appeared to ‘know’ when they were being observed, for example, and to act as a particle when observed and a wave when not observed.

How to describe this phenomenon flummoxed theorists. Heisenberg (and later, Erwin Schrödinger) came up with equations that described particles in terms of a wavefunction, where simple numbers became entities of infinite dimensions that lived in exotic mathematical spaces. The act of observation now had a complicated description that took into account what the experimenter was doing.

But all this mathematics didn’t get at what actually happened when the properties of a particle were being measured. At that moment, all the complex infinite-dimensional mathematics suddenly compressed into individual numbers, as if the particles had been there all along. Observation after observation of photons scattering on screens revealed that no simple explanation was possible. This description was deeply unsatisfying to Einstein, because the wavefunction appeared to prevent particles from having definite attributes before they were observed. Einstein wanted the wavefunction gone, replaced with some more sensible interpretation where things retained definite properties and locations. Despite decades of haggling over the incompleteness of quantum mechanics, however, the wavefunction couldn’t be dispensed with.

The view that emerged from this haggling came to be known as ‘the Copenhagen interpretation’ – coined by Heisenberg in 1955, and predicated on the presence of a fundamental split between the observer and the system being observed. Meanwhile, the polymath John von Neumann came up with an idealised mathematical description of what happened when you measured a particle’s wavefunction: it collapsed upon interacting with the observer. Where the rest of the wave went, or whether it was ever real in the first place, was anyone’s guess.

The Universe is a coin that’s already been flipped, heads or tails predetermined: all we’re doing is uncovering it

By the late 20th century, dozens of other interpretations had appeared under exotic names: the many-worlds theory, superdeterminism, consistent histories, the modal interpretation, superselection, Bohmian mechanics, Lindblad equations. I even invented my own: dynamic histories. While a few, like mine, proposed new theories that could come into conflict with quantum mechanics, most of them don’t. They are metaphysical, not physical.

The big question lurking behind all this is: what does the wavefunction mean? Does it represent something real or not? Most interpretations are ‘realist’ in the sense that they assume the wavefunction is a real entity and then go on to explain what it represents – but a few say it doesn’t exist at all, such as Quantum Bayesianism or QBism, as it is known. QBism owes its existence to the work of Wittgenstein’s friend and contemporary Frank Ramsey, who developed an anti-realist interpretation of probability. QBism holds that the wavefunction is purely an encoding of human uncertainty, representing a spectrum of probabilities that is updated when we make an observation. So the quantum wavefunction is not about objective reality at all, but about our future observations. QBism therefore refutes the Platonic idealism of the wavefunction and declares it to be a mere mathematical quantification of our beliefs.

Plenty of physicists have grown tired of this debate and its seemingly endless and unsatisfying arguments between realists and anti-realists. They want us to ‘Shut up and calculate!’ in the words of the physicist David Mermin: to stop trying to interpret quantum mechanics at all and get back to doing it. Philosophers, on the other hand, tend to dismiss this latter group as being philosophically ignorant. There’s a suspicion that, deep down, such physicists simply possess a metaphysics that they don’t want to admit, because they don’t want to come down on the side of an interpretation that has no scientific backing.

Yet those who follow Mermin’s injuction have a friend in one of the great philosophical minds of the 20th century – one who provides not only support for their position, but philosophical reasoning for why it is the only correct one.

Wittgenstein was a reluctant philosopher. Born in 1889 to a wealthy and powerful family in Vienna, Austria, philosophy seemed to be more of a compulsion for him than a love – a tendency to get stuck on certain questions, unable to move on without resolving them. Perhaps that’s why Wittgenstein felt the need to ‘solve’ philosophy once and for all, attacking its roots and, by doing so, tearing down all philosophical debates, including the broader quarrel between realists and anti-realists in all domains.

Wittgenstein was at once fantastically arrogant before his fellows and deeply humble before the questions he confronted. His task was no less than to discover what lay at the roots of logic. Starting out in the nascent field of aeronautical engineering in 1908, he quickly gravitated towards the philosophy of mathematics.

His German mentor Gottlob Frege sent him to the University of Cambridge to work with Russell. Of Wittgenstein, Russell wrote that: ‘An unknown German appeared … obstinate and perverse, but I think not stupid.’ Within a year, Wittgenstein had proved himself to Russell, who said: ‘I shall certainly encourage him. Perhaps he will do great things … I love him and feel he will solve the problems I am too old to solve.’

Russell’s motivations, however, were at odds with Wittgenstein’s. The grandson of an earl, Russell was raised in a noble household by his strict and devout grandmother. Finding no comfort in her religion, Russell sought it in mathematics, only to learn that the roots of the ancient discipline were rotten. He was horrified to discover that the geometer Euclid’s axioms, such as ‘two parallel lines do not intersect’, were just assumptions. Likewise, the number system is based on self-evident truths. If any were wrong, the whole thing might come tumbling down. Russell therefore dedicated his life to resolving all uncertainty in mathematics.

Russell – a product of the Victorian age – continued to look for certainty where there was none

Russell appropriated Wittgenstein’s philosophy to shore up basic logic, but Wittgenstein had other ideas. He wanted to understand what made facts true or false – not because he desired comfort from certainty, but because, well, it bothered him. Unlike Russell, Wittgenstein was devoted to the truth, no matter how ugly.

Wittgenstein’s life was no less unusual than his thoughts. He worked with Russell intensely from 1911-13, retreating to an isolated hut in rural Norway for months at a time in order to work out his ideas. In 1913, he returned to Austria, only to be swept up in the chaos of the First World War.

It was a time of massive upheaval at all levels of European society. Empires were in decline, and the old monarchical order was ebbing away. Women’s suffrage was in full swing, with the vote in Britain and the United States arriving after the war. Science and mathematics likewise were throwing off the shackles of 19th-century classicism. Einstein’s theory of relativity, both special and general, banished Isaac Newton’s concept of universal time and space, while Heisenberg’s uncertainty principle destroyed the certainty of measurement some years later.

Russell, meanwhile – a product of the Victorian age – continued to look for certainty where there was none. It fell to young Wittgenstein, picking up the zeitgeist, to seek to resolve the realism debate once and for all, even if that meant destroying it.

The war years were not easy on Wittgenstein. Poor health exempted him from conscription, but he volunteered for service and eventually to go to the Front. His reasons were complex, but from his letters it seems he was seeking something that he felt he could not find in intellectual pursuits. Writing from the Eastern Front, he expressed the hope that ‘the nearness of death’ would bring about a spiritual transformation in him. Wracked by loneliness and spiritual longing, he contemplated suicide, only to be saved by faith. While before the war he’d rivalled Russell in his distaste for religion, a chance discovery of Leo Tolstoy’s The Gospel in Brief (1902) in a bookshop caused him to become a devout Christian. His faith would influence his later work, and vice versa. Captured by the Italians in 1918, he spent months in a prisoner-of-war camp.

It was during the war that he formed much of his ideas for his first great work, the Tractatus Logico-Philosophicus (1922) – a book that applied modern logic to metaphysics via language to relate facts about reality to reality itself. He called it the theory of meaning.

In the Tractatus, Wittgenstein developed a philosophy that was deeply embedded in the world – not in idealised realms of thought like the rational idealist Russell, but in how we talk about the world. Rather than coming up with a theory about how words and facts represent reality, which is crucial to both realists and anti-realists, he determined that representation is irrelevant. No one needs to say what facts and objects represent. They are simply there, embedded in our picture of reality. To say what they represent is actually nonsense, absurd. As in the visual realm, a second picture is not necessary to explain what a first picture means; if it were otherwise, we’d fall prey to infinite recursion.

Wittgenstein was serious in that he believed we could not talk about things that are not in the world

What Wittgenstein understood is that you can’t use words to explain representation, because words are representations themselves. It would be like trying to travel outside the Universe to show somebody what the Universe is – a feat that’s both impossible and unnecessary. A sentence shows what it means by its own sense. Thus, if I say ‘Jenny has an apple,’ I do not have to explain how the words ‘Jenny’ and ‘apple’ represent physical objects in the world; nor do I have to explain what ‘has’ means. We mutually understand that, if Jenny is right there and she has an orange in her hand, the proposition is false. It shows its sense. There is nothing more to say about it, as long as we both understand the rules of the language.

Thus, Wittgenstein, even in his early work, suggests that the realist versus anti-realist debate is meaningless because both sides are trying to say things that are only showable. From this early Wittgensteinian perspective, a mathematical equation – in fact, any equation, including the ones governing quantum mechanics – is like a photograph of reality. Like photographs, we do not need anyone to interpret its meaning as realist or anti-realist. We do not need a Copenhagen or a many-worlds to indicate the sense of the equation to us, because it is already as apparent as it is ever going to be. To ask what the wavefunction represents is like asking what Michelangelo’s statue of David or Van Gogh’s painting The Starry Night represents: any explanation beyond the mere facts is insufficient and subjective.

You might find this explanation unsatisfactory. Yet Wittgenstein was serious in that he believed we could not talk about things that are not in the world. While we might talk about quantum mechanics in terms of particles, measurements and calculations, any philosophical attributes that ascribe significance to what we can observe (such as ‘real’ or ‘unreal’) are nonsense. We must be silent on ascribing additional meaning to the wavefunction.

Wittgenstein’s exploration about what we can and cannot talk about in philosophy, however, would evolve over the next several decades, and lead to a rejection of even those philosophical concepts such as the picture theory upon which he built the Tractatus.

Having written the Tractatus, Wittgenstein believed that he had ‘solved’ philosophy. In his strange, haughty humility, he left the discipline in the 1920s, and worked various jobs as a gardener, teacher and architect.

This interregnum came to an end, however, when Wittgenstein was exposed to logical empiricism. This was a movement arising from a group of philosophers known as the Vienna Circle. They emphasised empirical knowledge and the theory of ‘logical positivism’, meaning that we can only ascribe meaning to what can be measured or observed. A strict logical positivist is unconcerned with explanations or interpretations; rather, they believe that understanding the world is built upon measurement and its prediction.

The Tractatus was a foundational pillar of the Vienna Circle, and this galvanised Wittgenstein to continue his work. He decided to return to Cambridge in 1929, but moved away from the philosophy of mathematics and logic, and towards ordinary language and psychology.

He rejects the theory of meaning entirely while making one of his most powerful contributions to it

Wittgenstein eventually collected his ideas in a book called Philosophical Investigations – probably one of the strangest books of philosophy ever published (and, perhaps for this reason, only released after Wittgenstein’s death from cancer in 1951). Rather than being organised as a sequence of topics or propositions, the Investigations is a stream-of-consciousness series of points, arguments and statements. That was in fact in keeping with its own philosophy, which is that philosophy itself can discover nothing. It is simply a form of therapy that can quickly become a disease of the intellect. Its only job is to remind us of that which we already know.

From this later Wittgensteinian position, all the varying quantum interpretations would be the result of diseased minds, and ultimately self-destructive. That’s because all philosophy is actually a debate over mere grammar. If we take seriously these metaphysical debates, he argues, we are not only wrong, but ill.

In his Philosophical Investigations, Wittgenstein rejects the theory of meaning entirely while making one of his most powerful contributions to it. All language, he says, gains its definitions from how it is used in specific cases. All language is a game like chess or poker – we learn the rules by playing, not theorising or defining. So the very notion of a universal definition is an artifice, a bit of subterfuge. One cannot talk about what words really mean; one can only use them. This applies as much to mathematics as it does to ordinary words.

Wittgenstein wants to show us that we need to stop trying to interpret language. Take the example of a road sign pointing to a village. We see the road sign and instantly understand its meaning. While there is an element of symbolic decoding involved, there is no deeper interpretive step, he says. In other words, we do not need to figure out how the sign represents reality, either in the ideal world of Plato or some subjective concept of reality in our heads. The sign could contain almost any kind of symbols, colour coding or numbers, as long as the action that people take upon seeing it is the correct one. The sign ‘shows’ us where the village is, because that is how signs of that kind are used. That is its true meaning.

The late Wittgenstein entirely rejects his own picture theory of reality. Pictures are nice and satisfying, but usage is what actually matters. The wavefunction, on this reading, isn’t like a picture of reality at all. All that matters is that physicists now have the ability to do calculations, which lead to predictions that can be verified by measurements. The point is not the measurements themselves, however – as a logical positivist might claim – but how the physicists behave. Do they calculate in a way that leads to more and better physics? Language and mathematics are a means of controlling and modifying collective human action so that work gets done.

This is language as culture rather than language as picture. And culture includes ritual. Like all ritualistic communities, physics contains its rules, interpretations, specialised vocabulary, a community of adherents who are admitted to the arcane arts, levels of indoctrination, and gatekeepers. While some societies relate ritual to the appeasement of gods and spirits, in science they serve to therapeutically appease our philosophical needs. Competition between interpretations is not unlike competition between clan gods, trying to achieve cultural dominance.

Evolutionary cultural anthropology backs up this view, having demonstrated that language is deeply connected to ritual and religion. Likewise, the vocabulary, grammar and procedures of science are themselves ritualistic, with each subdiscipline having its own mores and norms. These are necessary because it is impossible for scientists to evaluate new research purely based on factual merits; it often takes years to validate a new theory or experimental result. The role of ritual also makes perfect sense from an evolutionary perspective. Humans have spent hundreds of thousands of years navigating a hostile planet by encoding information crucial for survival into ritual, which can then be transmitted across generations. When we invented the scientific method only a few hundred years ago, we had to graft it onto that part of our nature in order to pass it down the generations, hijacking an ancient and effective cultural mechanism for a new purpose.

The activity of science, rather than its interpretations, defines what words and symbols mean

Hence, quantum interpretation is not really an investigation into reality, and it tells us nothing new about the world. Rather, it is a grammatical investigation or, in anthropological terms, a cultural one. It is a competition between differing philosophical therapies, satisfying different emotional-cultural needs.

Crucially, it is in the activity of science, whether via experiment or calculation, that all its useful information it generates exists. Wittgenstein explains, for example, how the act of determining the length of an object is not a case for learning theories and definitions, but an activity:

What ‘determining the length’ means is not learned by learning what length and determining are; the meaning of the word ‘length’ is learnt by learning, among other things, what it is to determine length.

This description indicates that to learn what quantum physics means is to learn to calculate it – and vice versa.

Wittgenstein suggests that even mathematics is potentially a shared language and activity. He asks:

what would this mean: ‘Even though everybody believed that twice two was five it would still be four’?—For what would it be like for everybody to believe that?—Well, I could imagine, for instance, that people had a different calculus, or a technique which we should not call ‘calculating’. But would it be wrong?

He suggests that ‘odd’ would be a better word for it, but we would have no common frame of reference to call it wrong. He goes on to suggest that mathematics is very much an activity, like a game, and we all know the same rules that form a system. Hence, we all come to the same conclusions and never argue about what is proved. Yet, some alien species could come up with different rules for their mathematical game that are no less valid because they are following their rules.

If Wittgenstein were alive today, he might have couched his arguments in the vocabulary of cultural anthropology. For this shared grammar and these language games, in his view, form part of much larger ritualistic mechanisms that connect human activity with human knowledge, as deeply as DNA connects to human biology. It is also a perfect example of how evolution works by using pre-existing mechanisms to generate new behaviours.

The conclusion from all of this is that interpretation and representation in language and mathematics are little different than the supernatural explanations of ancient religions. Trying to resolve the debate between Bohr and Einstein is like trying to answer the Zen kōan about whether the tree falling in the forest makes a sound if no one can hear it. One cannot say definitely yes or no, because all human language must connect to human activity. And all human language and activity are ritual, signifying meaning by their interconnectedness. To ask what the wavefunction means without specifying an activity – and experiment – to extract that meaning is, therefore, as sensible as asking about the sound of the falling tree. It is nonsense.

I have come to think of the world not as filled with sharply defined truths but as a place of myriad possibilities

As a scientist and mathematician, Wittgenstein has challenged my own tendency to seek out interpretations of phenomena that have no scientific value – and to see such explanations as nothing more than narratives. He taught that all that philosophy can do is remind us of what is evidently true. It’s evidently true that the wavefunction has a multiverse interpretation, but one must assume the multiverse first, since it cannot be measured. So the interpretation is a tautology, not a discovery.

I have humbled myself to the fact that we can’t justify clinging to one interpretation of reality over another. In place of my early enthusiastic Platonism, I have come to think of the world not as one filled with sharply defined truths, but rather as a place containing myriad possibilities – each of which, like the possibilities within the wavefunction itself, can be simultaneously true. Likewise, mathematics and its surrounding language don’t represent reality so much as serve as a trusty tool for helping people to navigate the world. They are of human origin and for human purposes.

To shut up and calculate, then, recognises that there are limits to our pathways for understanding. Our only option as scientists is to look, predict and test. This might not be as glamorous an offering as the interpretations we can construct in our minds, but it is the royal road to real knowledge.

Timothy Andersen

Source: AEON