What We Cannot Know: Explorations at the Edge of Knowledge

by Marcus du Sautoy

Heisenberg was right when he wrote: ‘Modern physics has definitely decided in favour of Plato. In fact, the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.’ Plato’s watery icosahedron and fiery tetrahedron have been replaced by this strange new symmetrical shape SU(3).

It transpired that the patterns Mendeleev discovered in the periodic table were actually the result of these atoms being made from the more fundamental ingredients of the proton, electron and neutron. There was a feeling that the patterns in all the newly discovered particles hinted at a similar story: the existence of more fundamental building blocks at the heart of the hundreds of particles that were being detected.

At a lunch with Gell-Mann in 1963, Serber explained his idea, but when Gell-Mann challenged him to explain what electrical charge these hypothetical particles would have, Serber wasn’t sure. Gell-Mann started scribbling on a napkin and soon had the answer. The charges would be ⅔ or –⅓ of the charge on a proton. The answer seemed ridiculous. ‘That would be a funny quirk,’ Gell-Mann commented. Nowhere in physics had anything been observed that wasn’t a whole-number multiple of the charge on the electron or proton.

Given that there were three of these hypothetical new particles that could be used to build the other layers, the reference seemed perfect. The only trouble was that Joyce clearly intended the new word ‘quark’ to rhyme with ‘Mark’ not ‘kwork’. But the spelling and pronunciation Gell-Mann wanted won out.

It turns out that the blockheads were right. The up, down and strange quarks were not just a mathematical mnemonic but seemed to be a physical reality. It was discovered that these three quarks weren’t enough to cover all the new particles, and eventually we’ve found ourselves with six quarks together with their antiparticles. In addition to the three Gell-Mann named, three more appeared on the scene: the charm quark, the top quark and the bottom quark.

‘There are definitely limits in my lifetime but I’m not sure there are any other limits. In experimental physics, saying there’s no way we can do something is the perfect way to get someone to figure out how to do it. In my lifetime I’m never going to be able to measure something that decays in 10–22 seconds. I don’t think there’s any way. That’s not to say it’s provably unknowable.

You might ask: a wave of what? What is vibrating? In fact, it is a wave of information rather than physical stuff. Just as a crime wave isn’t a wave of criminals but rather information about the likelihood of a crime happening in a particular area. The wave is simply a mathematical function, and a mathematical function is like a machine or computer: you input information, and it calculates away and spits out an answer. The wave function of the electron has as input any region of space, and the output is the probability that you will find the electron in that region. Amazingly, this particle should really be thought of as a piece of evolving mathematics, not a physical thing at all. It is called a wave because the functions describing these probabilities have many of the features of classical wave functions. The peaks and troughs encode information about where the electron might be. The bigger the amplitude of the wave, the more likely you are to find the electron in that region of space.

The wave quality of light is the same as that of the electron. The wave is the mathematics that determines the probable location of the photon of light when it is detected. The wave character of light is not a wave of vibrating stuff like a water wave but a wavelike function encoding information about where you’ll find the photon of light once it is detected. Until it reaches the detector plate, like the electron, it is seemingly passing through both slits simultaneously, making its mind up about its location in space only once it is observed.

It was the code-cracking mathematician Alan Turing who first realized that continually observing an unstable particle could somehow freeze it and stop it evolving. The phenomenon became known as the quantum Zeno effect, after the Greek philosopher who believed that because instantaneous snapshots of an arrow in flight show no movement, the arrow cannot be moving at all.

My children are obsessed with the science fiction TV series Doctor Who, just as I was as a kid. The aliens we find scariest are the Weeping Angels, stone figurines like those in our local cemetery that don’t move provided you don’t take your eyes off them. But blink and they can move. The theory says that the pot of uranium on my desk is a bit like a Weeping Angel. If I keep observing the uranium, which obviously means a little more than just keeping my eyes on the pot on my desk, I can freeze it in such a way that it stops emitting radiation. Although Turing first suggested the idea as a theoretical consequence of the mathematics, it turns out that it is not just mathematical fiction. There have been some experiments in the last decade that have demonstrated the real possibility of using observation to inhibit the progress of a quantum system.

The interesting point for me – one that is often missed – is that up to the point of observation, quantum physics is totally deterministic. There is no question of what the nature of the wave equation is that describes the electron as it passes through the slits. When he came up with his theory in 1926, Schrödinger formulated the differential equation that provides this completely deterministic prediction for the evolution of the wave function. Schrödinger’s wave equation is as deterministic in some sense as Newton’s equations of motion. The probabilistic character and uncertainty occurs when I observe the particle and try to extract classical information.

That said, there are a number of ways in which people have tried to overcome this apparently in-built uncertainty. One is the hypothesis that, at the point of observation, reality splits into a superposition of realities. In each reality the photon or the electron is located in a different position so that the wave in some sense doesn’t collapse but remains, describing the evolution of all these different realities. It’s just that, as conscious beings, we are now stuck in one of the realities and are unable to access the other realities in which the photon or electron ended up somewhere else on the photographic plate. This is a fascinating attempt to make sense of the physics. It is called the ‘many worlds’ interpretation and it was proposed in 1957 by American physicist Hugh Everett.

The trouble for us is that as part of this wave function, we are denied access to other parts of it. It traps us inside, confined to one branch of reality, and it may be an in-built feature of our conscious experience that we can never experience these other worlds.

Perhaps it’s just greedy to believe that I can know more than I can measure. Hawking has certainly expressed such a view: I don’t demand that a theory correspond to reality because I don’t know what it is. Reality is not a quality you can test with litmus paper. All I’m concerned with is that the theory should predict the results of measurements.

If I pin down the location of the electron to within the radius of an atom, its speed could change by as much as 1000 kilometres per second in any direction. It’s like trying to fit a strange quantum carpet: every time I pin down the position end of the carpet, the momentum end pops out; try to pin the momentum end down and the position end no longer fits.

Following Heisenberg’s original 1927 paper detailing this strange inverse relationship between knowledge of position and knowledge of momentum, Earle Kennard and later Howard Robertson mathematically deduced the trade-off in knowledge.

But can we trust the maths? This theoretically derived behaviour predicted by the mathematics of Heisenberg’s uncertainty principle has been confirmed in experiment. In a paper published in 1969, American physicist Clifford Shull describes the results of firing neutrons at slits of decreasing width. The increased knowledge of the location of the neutrons given by a narrower slit resulted, as theory predicted, in a greater spread of possible values for the momentum. And when the neutrons arrived at the detector plate, they were spread in a distribution whose standard deviation corresponded exactly to that predicted by the equation of Heisenberg’s uncertainty principle.

If I try to measure precisely the coordinates of one of the electrons inside my dice, then, as the error in the coordinates goes down, this has to be paid for by an associated uncertainty in the momentum and hence energy. Heisenberg’s equations tell me the mathematical relationship between the trade-off. But there is an extra twist. Since energy and mass are related by Einstein’s equation E = mc2, if the energy is high enough it can lead to the spontaneous creation of new particles. The trouble is that if I am trying to pin down the location of a particle and by doing that I create lots more particles, it significantly muddies any attempt to investigate the location of the original particle. The scale at which this complication starts kicking in is called the Compton wavelength of a particle. For an electron it is about 4 × 10–13 metres.

Things get even more uncertain as I zoom in closer on my particle. There is a point at which the energy uncertainty is so large that the corresponding mass becomes so great that it causes a black hole to materialize. As I shall explore in the Fifth Edge, a black hole by its very nature appears to trap any information within a certain radius of the centre of the hole and prevent it being released. This means that the uncertainty principle implies an in-built limit to how far I can probe nature. Beyond a certain scale I seem to be denied access to what’s going on. That scale is very small. It is of the order of 1.616 × 10–35 metres, a distance known as the Planck length. This is ridiculously tiny. If I scale up the full stop at the end of this sentence to be the size of the observable universe, then something the size of the Planck length would scale up to look like the full stop before I enlarged it.

Often these results are trotted out to knock down any attempt to claim that there is a mechanism at work that determines when my pot of uranium spits out an alpha particle. But really these results should be interpreted only as the conditions that a hidden machine must satisfy. There could be a hidden mechanism – it’s just going to be very weird. As Bell, who was responsible for proving that such hidden machines must span the universe, said: ‘What is proved by impossibility proofs is a lack of imagination.’

The current majority interpretation in physics is that we are mistaken to believe that the particles inside my uranium can be said to have simultaneously a precise momentum and position. This interpretation turns epistemology into ontology. Our inability to know is actually an expression of the true nature of things. As Heisenberg put it: ‘The atoms or elementary particles themselves are not real; they form a world of potentialities or possibilities rather than one of things or facts.’

According to Heisenberg’s uncertainty principle, it follows from the fact that space exists that you will get particles appearing from nothing. You don’t have any need for a creator.

I have always been intrigued by the question: does infinity physically exist? My attempts to create infinity by cutting the dice up into infinitely small pieces ran aground when I hit indivisible quarks. It even seems that I can’t divide space infinitely often, since space may be quantized. So my quest to know if infinity exists will turn in a different direction: I shall look out, not in. What happens if I keep going in a straight line? Do I go on forever? It is a question that I think everyone who looks up into space must contemplate at some point in their lives.

Cicero writes of ancient Greek astronomers making models of this celestial globe with stars marked on them, early forerunners of my cut-out universe. Sadly, none of the Greek models have survived, so I popped into one of my favourite museums in Oxford, the Museum of the History of Science, to see some others that have. There was a beautiful globe about half a metre high dating from the early sixteenth century.

A Cepheid star twinkles, and in 1912 American astronomer Henrietta Leavitt discovered how to use these twinkling stars to navigate the universe. She was employed at the time not as an astronomer but as a ‘computer’ at the Harvard College Observatory, extracting data from the photographic plates for 30 cents an hour. Women weren’t allowed to operate the telescopes. She’d been assigned the task of analysing stars that grew brighter and dimmer over a period of time. Curious to know if there was any pattern to the pulse of these stars, Leavitt focussed on a batch of stars that were located in the Small Magellanic Cloud and were therefore believed to be at similar distances from the Earth. When she plotted luminosity against the period of pulsation, she discovered a very clear pattern. The time it takes a Cepheid star to pulsate is directly correlated with its luminosity: the longer the period of pulsation, the brighter the star is shining. So to know how bright a Cepheid star really is, all you need to measure is the period of pulsation, something that is much easier to do than measuring missing light frequencies. These stars were perfect for measuring distances.

The first such region to come under investigation was a small cloud identified by the Persian astronomer al-Sufi in the tenth century. It is bright enough to be detected by the naked eye and became known as the Andromeda nebula. The suggestion that this and other clouds might actually be galaxies in their own right was first voiced in 1750 by English astronomer Thomas Wright. After reading about Wright’s ideas, Immanuel Kant romantically referred to them as ‘island universes’.

Leavitt’s use of Cepheid stars as a way of navigating space transformed our picture of the universe so much that the Swedish mathematician Gösta Mittag-Leffler wanted to nominate her for the Nobel Prize in 1924. He was devastated to discover that Leavitt had died four years earlier from cancer and was therefore ineligible for the award.

But mathematicians have a third alternative, which posits a universe which has no boundary but is nonetheless finite. In this universe you travel out into space until, rather than carrying on to infinity, eventually you find yourself heading back to your starting location, just like a terrestrial explorer rounding the Earth.

But how can I wrap up a three-dimensional universe so that it has finite volume but no edges? This is the power of mathematics, which allows us to embed our three-dimensional universe in a higher-dimensional space and wrap it up like I did with the game of Asteroids. Although I can’t physically picture the wrapping up, the language of mathematics gives me the equations to describe and more importantly explore the properties of these finite three-dimensional universes.

Although mathematics provides us with candidates for a finite yet unbounded universe, how can we ever know whether our universe is finite and what shape it might be if it is? Do we have to wait for an astro-Magellan to circumnavigate the universe? Given the scale of the known universe, human exploration seems a rather hopeless way of proving whether the universe is finite. But there are explorers out there that have been navigating the universe for billions of years and can provide us with some insight into whether it is finite or not: photons of light.

But there is another problem about really knowing the curvature of the universe. Much of our exploration of space depends on an assumption: that where we sit in the universe isn’t particularly special. It’s called the Copernican principle. Once we thought we were at the centre of it all. But Copernicus put paid to that. So we now believe that what the universe looks like around us is pretty much what it looks like everywhere. That is certainly what the evidence tells us. But it need not be the case. It’s possible that the bit of the universe we see is rather special.

The same thing happens to light. As a star speeds away from us, its light shifts towards the longer red wavelength. If it moves towards us, the light shifts towards the shorter blue wavelength. Having already discovered that our galaxy was not special but just one of many, Edwin Hubble turned his attention in 1929 to analysing the light from these galaxies to see how they were moving relative to our own. To his surprise, the light from distant stars in the galaxies he observed were all shifted towards the red. Nothing seemed to be coming towards us.

As you head out into space, you are heading back in time. Since there were no stars before 13.8 billion years ago, this means there is a sphere around us beyond which there is nothing to see. The wonderful thing is that we are back with the model of the universe proposed by the ancient Greeks. There is a huge sphere with the Earth at its centre, and photons from beyond that sphere have not had time to reach us yet. That sphere is getting larger as time passes, and the question of how much space is contained inside this expanding horizon will turn out to have an unexpected answer.

You might think we should be able to see light going back to the first moment after the Big Bang. But, tracing back the state of the universe, we believe there is a moment when no light could travel through space because space was opaque. Photons just found themselves buffeted between one particle and the next. It took 378,000 years following the Big Bang before the density of particles dropped sufficiently for the first photons to start their uninterrupted journey through space.

Those first photons that we see today in the microwave background radiation were only 42 million light years away from the Earth when they started their journey. Today, the distance between that starting point in space and the Earth has stretched to an estimated 45.7 billion light years. This is the edge of the visible universe, the visible cosmic horizon. But light isn’t everything.

As time goes on, this cosmic horizon is growing, allowing us to see further and further into space. At least that’s what we thought. However, a discovery in 1998 revealed the alarming fact that, rather than extending further into space, our cosmic horizons are actually contracting. Although the cosmic horizon is growing at a constant rate, the underlying fabric of space itself isn’t just expanding, this expansion seems to be accelerating. As it does so, it is pushing things we can see out beyond our horizon with devastating implications for what future generations can ever know.

It seems that about 7 billion years ago something dramatic happened. Up to this point, the expansion appeared to be slowing down, as one would expect as the gravitational force of the matter in the universe exerted a braking effect. But at this point, halfway through the current life span of the universe, the expansion rate changed character and started to increase, accelerating as if something had suddenly put its foot on the pedal. The fuel driving the acceleration is what scientists call dark energy. It seems that in the first half of the universe’s existence, the density of matter was enough to assert a slowing gravitational pull, but as the universe expanded this density decreased to such a point that the underlying dark energy was strong enough to take over. Dark energy isn’t thought to be something whose density decreases with expansion. It is a property of space itself.

With the cosmic microwave background radiation redshifted to such an extent that it can no longer be detected and galaxies having disappeared from view, it is amazing to think that cosmologists in the future may have no evidence to suggest that we live in an expanding universe. Future civilizations will perhaps return to the model of the universe held by the ancient world: our local galaxy surrounded by the void – everything contained in the paper icosahedron I made to navigate space.

The rapid inflation didn’t last for long: some 10–36 seconds according to the current model. That’s a billionth of a billionth of a billionth of a billionth of a second. Yet in that time the inflaton is thought to have expanded space by a factor of 1078. It’s as if there was a build-up in pressure which suddenly got released, and once the release had happened, the universe settled down into the more sedate expansion we have picked up since.

I was keen to talk to a cosmologist who had accepted the prize. Professor John Barrow, based in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, received the Templeton prize in 2006. His response to my email got straight to the heart of why cosmology in particular has implications for my concept of God as those things we cannot know: ‘Most of the fundamental questions in cosmology are unanswerable. In fact, I will talk about some them in a lecture I’m giving this Saturday.’

Looking back at the Edges I’ve visited so far, nothing seems quite as unanswerable as the question of whether the universe is infinite. Chaos theory told me that the future is unknowable, but I can just wait until it becomes the present and then I’ll know. As I slice my dice, I might actually hit a point at which space is quantized, so it’s possible that there are only finitely many steps inside my dice before I hit the indivisible. It’s true that it might be almost impossible to navigate this descending staircase, but it isn’t a challenge that is a priori unsolvable. And Heisenberg’s uncertainty principle does not so much challenge us to give an answer but rather to consider whether the questions we are asking are well posed. It’s not that we can’t know position and momentum simultaneously; the revelation is that it is not a question that it makes sense to ask.

But then I had a revelation. Perhaps the question of an infinite universe is not as unknowable as one might think. Might there not be more indirect ways that would lead us to the conclusion that the universe is infinite? The answer might lie in my own field of expertise. Mathematics has been a very powerful telescope through which to look at the universe. What if the current laws of physics lead to a mathematical contradiction under the assumption that the universe is finite? That would force us to conclude that the universe must be infinite, or that our laws of physics are at fault. This is after all how we discovered irrational numbers, numbers whose decimal expansions continue infinitely and never repeat themselves.

For example, 600 million years ago the Earth rotated once on its axis every 22 hours and took 400 days to orbit the Sun. But the tides of the seas have the strange effect of transferring energy from the rotation of the Earth to the Moon, which results in the Earth’s rotation slowing down and the Moon gradually moving away from us. Similar effects are causing the Earth and Sun to drift apart, changing the time it takes to complete an orbit.

The force of gravity is an illusion. There is no force. Objects are just free-falling through the geometry of space-time, and what we observe is the curvature of this space. But if massive bodies can distort the shape of space, they can also have an effect on time.

But in 1930 the Indian physicist Subrahmanyan Chandrasekhar realized that there is a problem with this. Stuck on a boat sailing from India to do his doctoral studies in Cambridge, Chandrasekhar recognized that special relativity puts a speed limit on how fast these particles can move. So if the mass of the star is great enough, gravity will win out over this speed limit and the star will continue to collapse, creating a region of space of increasingly high density. His calculations made on board ship revealed that any star that was more than 1.4 times the mass of our Sun would suffer such a fate. The supernovae responsible for creating elements like gold and uranium are the result of these cataclysmic collapses.

Popularized by inventor and futurist Ray Kurzweil in his book The Singularity Is Near, the singularity is due to hit humanity in 2045. At this point, Kurzweil believes, humans will be able to create artificial intelligence that exceeds our own. This moment will be accompanied by a breakdown in our ability to predict what life after such a singularity will be like.

Hawking now believes that the information that plunges into a black hole is actually encoded on the surface of the event horizon enclosing the black hole and is then imparted back to the particles that are emitted. Weirdly, this surface is two-dimensional yet seems to encode information about the three-dimensional space inside. It has led to the idea of the holographic universe: the whole of our three-dimensional universe is actually just the projection of information contained on a two-dimensional surface. Although Hawking conceded the bet, Thorne has stood his ground. He still believes that the information is lost. Roger Penrose, like Thorne, believes that Hawking conceded the bet too quickly.

I have learnt not to be surprised by sentences formed in natural language that give rise to paradoxes like the one captured by the circular logic of the two sentences on my cracker card. Just because you can form meaningful sentences doesn’t mean there is a way to assign truth values to the sentences that makes sense.

It was the Indians in the seventh century AD who successfully developed a theory of negative numbers. In particular, Brahmagupta deduced some of the important mathematical properties of these numbers: for example, that ‘a debt multiplied by a debt is a fortune’, or minus times minus is plus. (Interestingly, this isn’t a rule, it’s a consequence of the axioms of mathematics. It is a fun challenge to prove why this must be so.) It took Europeans till the fifteenth century to be convinced that there were numbers that could solve these sorts of equations. Indeed, the use of negative numbers was even banned in thirteenth-century Florence.

In his paper ‘Computational Capacity of the Universe’, Seth Lloyd calculates that since the Big Bang the universe cannot have performed more than 10120 operations on a maximum of 1090 bits.