SO, AT THE END of my journey to the edges of knowledge, have I found things that we can categorically say we cannot know? The things I thought we could never know, such as whether the universe is infinite, turn out not to be as unassailable as I thought. We can use mathematics to employ finite means to prove the existence of the infinite. So although we may never explore or see beyond the finite bubble that encloses our bit of the universe, we may be able, using the power of our minds alone, to discover what is beyond.
Understanding the nature of time before the Big Bang was another edge that I thought was unassailable. But chinks have opened up in that wall, too. Recent progress has provided us with ways to theorize and even perhaps detect evidence of a time before we thought it all began. And yet the question of whether time has a beginning feels like one that will remain on the scientific books for some time yet.
In contrast, the infinitely small at the heart of the die that I hold in my hand feels like something that will always remain beyond complete knowledge. Every generation imagines they’ve hit the indivisible, only for matter to fall apart into smaller pieces. How can we ever know that the current building blocks of the universe—quarks, electrons, neutrinos—aren’t as divisible as all the other particles we’ve hit as we’ve peeled the onion of reality? Quantum physics even posits a limit to how far we will be able to penetrate in our investigation of matter: beyond the Planck length, it’s simply no-go.
Whether we can ever truly understand the nature of consciousness is very much in a state of flux. Will this challenge vanish because it will turn out to be an ill-posed question? Will it be answered with a strategy similar to that used by scientists to pin down the essence of life? There was no elan vital, just a set of biological processes that means a collection of molecules has life. Or will the problem of consciousness remain something that can never be understood because we are stuck inside our own consciousnesses and can never get inside another’s?
The possibility that we cannot know because we are stuck within the system is a common theme among many of the problems I have tried to tackle. Mathematics has truths that will remain unprovable within a system. Step outside and you can know, but this creates a new system, which will have its own unprovable truths. Likewise, the idea that a quantum experiment can be repeated is an impossibility because we can never isolate the experiment from the universe it is conducted in, one that will have changed and evolved by the next time we run the experiment.
Even the mathematics of probability is something of a fantasy. What is probability? If I throw my die six hundred times, I expect to get one hundred 6s. But I just want to throw it once and know something about how it will fall. The equations of chaos theory tell us that so much of the future depends on fine-tuning the decimal places that control the input of the equations. So I can never know the present completely enough to have any real chance of knowing the future or the past.
The physical limitations of the human brain place boundaries on how much we can ever know, so there will always be things beyond knowledge. But this isn’t an absolute unknowability. It is more like light coming in from the outer reaches of the universe before we realized that the expansion of the universe was accelerating. Wait long enough and the light will arrive. Wait long enough and a computer can make its way through all the provable truths of mathematics. But what if time itself runs out while we are waiting?
The limitations of language are at the heart of many of the boundaries of knowledge, and these could evolve and change. Certainly, many philosophers identify language as a problem when it comes to the question of consciousness. Understanding quantum physics is also a problem because the only language that helps us navigate its ideas is mathematics. Try to translate the mathematics into the language of everyday experience and we create the absurdities that make quantum physics so challenging. So the unknowability of position and momentum isn’t really a genuine unknowable. Rather, it is a failure of translation from mathematics to natural language.
We are bound by the ways of thinking particular to our own moment in history. Auguste Comte thought we could never know the constituents of a star. How wrong he was! So I wonder if the safest bet is to say that we can never truly know for sure what we cannot know.
How far can we get by creating solutions to seemingly unanswerable problems? For many centuries, mathematicians looked at the equation x2 = –1 and believed it had no answer. But then a more imaginative approach was taken. We admitted imaginary numbers into the landscape of mathematics by defining i as the number whose square is –1. Why does this work? Because it doesn’t create a contradiction in the theory. We interweave this concept with the mathematics we know and we begin to know i. And most important, it gives us access to new and exciting bits of the mathematical world. Not admitting imaginary numbers into mathematics would have limited its extent and power.
What happens if I try to be creative with some of our unknowable questions? What if, for example, I were to define God as the solution to the question “Why something rather than nothing?” This concept is meant to be nothing more than the solution to that question. It doesn’t have any other properties. Even if we gain more knowledge about the answer to the question, it will just mean that we know more about this particular God.
But we have to be careful with such an approach. Just because you can write down a mathematical equation doesn’t mean it has solutions. Admitting a new concept that solves x2 = –1 was profitable because it provided access to new consistent mathematics. For the Platonist, the idea was sitting there all along, waiting to be articulated; for others, it was a creative act that enriched our mathematical world. But if I write down the equations for Fermat’s Last Theorem and try to define new numbers to solve these equations, I am going to find myself with self-contradictory statements. This is, after all, precisely how Wiles proved that the equations couldn’t be solved.
The trouble with most religions is that the God that is served has so many properties that have nothing to do with the definition. It’s as if we are working backward, focusing on the strange properties conjured up over the generations without really understanding the original definition. We come across this bastardized picture early on as kids, and then when we ask the question “Why something rather than nothing?” it doesn’t really work as a solution. But we’ve been shown the wrong thing.
Being an atheist means, for me, that I reject the classical solutions that religion seems to offer for these unknowns. But maybe I shouldn’t throw everything out. There are things that will always remain unknown, so perhaps God does exist. The traditional argument against the God of the gaps is that we should strive to know God, to have a personal relationship with it. And this God, defined as the transcendental or unknown, precludes by its definition the possibility of knowing it.
The trouble with this definition of God is that it doesn’t really get you much further. While defining a number whose square is –1 resulted in a rich array of consequences, defining something as the solution to “Why something rather than nothing?” doesn’t give rise to anything new. You need to make up properties for this thing that don’t follow from its definition. As Karen Armstrong put it, this high God is too high.
There are several responses to identifying the unknowable. One is to leave it at that; if it is unknowable, then it can’t be known. But there is also the temptation to make a choice and live your life according to that choice. Perhaps the logically most consistent response is to be open to many solutions and allow all the solutions to run parallel until new ideas collapse the possibilities.
I wonder, though, whether, as I come to the end of my exploration, I have changed my mind about declaring myself an atheist. With my definition of a God as that which we cannot know, to declare myself an atheist would mean that I believe there is nothing we cannot know. I don’t believe that anymore. In some sense I think I have proved that this God does exist. The challenge now is to explore what quality this God has.
My statement about being an atheist is really just a response to the rather impoverished version of God offered by most religions and cultures. I reject the existence of a supernatural intelligence that intervenes in the evolution of the universe and in our lives. This is a rejection of the God that people assign strange properties to—such as compassion, wisdom, love—that make no sense when it comes to the idea that I am exploring.
Such a position and definition will probably not satisfy either side of the divide. The militant atheists don’t want to admit anything named God into the debate, while those who believe in a God will criticize the concept of God as the unknown as impotent and missing the point. So how do we engage with this God of the gaps?
Perhaps the important lesson is to maintain a schizophrenic state of mind. A multi-mind-set. On the one hand, as humans we must recognize that we cannot know it all. There are provable limits to knowledge. Such a state of humility is intellectually important, or we will live in a state of delusion and hubris. Yet the other lesson is that we cannot always know what will forever transcend our understanding. This is why it is essential for a scientist not to give in too early. To believe that we can find answers. To believe that perhaps we can know it all.
My journey through science has thrown up a number of challenging questions. But there is also the basic epistemological question of whether we can actually know anything at all. More than two thousand years ago Socrates declared, “True knowledge exists in knowing you know nothing.” An acknowledgment of your ignorance is the only true statement of knowledge.
There have been volumes of philosophy that tackle the theory of knowledge and try to pin down what we can know, to define what we mean by knowledge. Plato proposed that knowledge should be defined as “justified true belief,” but Bertrand Russell and then the American philosopher Edmund Gettier in the 1960s questioned whether this truly captures its meaning.
The classic example proposed by Bertrand Russell tells the story of a woman who looks at a clock that says two o’clock. She believes therefore that it is two o’clock. She seems justified in her belief because the clock says so. And it does indeed happen to be two o’clock. Yet actually the clock stopped twelve hours ago, and it’s just a coincidence that she happened to look at the clock exactly twelve hours later.
Gettier created similar scenarios to challenge “justified true belief.” You are looking at a field and see what you believe is a cow. You infer that there is a cow in the field. Actually, there is a cow in the field, so the inference is true, but it can’t be seen because it is in a dip in the field. You are in possession of a true statement. It was based on a justifiable belief, and the thing you were looking at certainly looked exactly like a cow. But the fact that the statement you made is true does not imply knowledge.
We can imagine a situation in which we have come up with a true statement about the universe. But the justification for the statement is completely incorrect, even if it led us to make a true statement. Surely that doesn’t constitute knowledge. I have often cooked up proofs of true mathematical statements that turn out to have a logical flaw in them (which I hope to spot before I send them to a journal for publication). But my false proof can’t really justify my knowledge that the mathematical statement is true.
I don’t know whether the Riemann hypothesis is true or false. However, a few people have come up with what they believe are proofs of the truth of this hypothesis, and they have pages and pages of equations to back up their belief. Most of the time a fault is found. Once shown to the proposer, that justified belief vanishes. But what if that false proof convinces everyone? Suppose the fault is actually quite subtle. We cannot say we know that the Riemann hypothesis is true despite our justified true belief. Surely the justification has to be true to lead to justified true belief.
Some ancient astronomers proposed that the Earth goes around the sun, but their justification of this fact was not correct. The Indian philosopher Yajnavalkya justified this belief in a heliocentric solar system in the ninth century BC by proposing, “The sun strings these worlds—the Earth, the planets, the atmosphere—to himself on a thread.” Can we say that he knew that the Earth goes around the sun?
I think that I side with my colleague in New College, Timothy Williamson, who asserts in his book Knowledge and Its Limits that knowledge should be regarded as something fundamental, not something that can be defined in terms of other things. It seems like we all seem to know what “to know” means. It is one of only a hundred or so phrases that have a comparable translation in every language on Earth, which is not the case for as basic a word as eat.
It was also from Williamson that I learned about a fantastic piece of logical trickery called the paradox of unknowability, which proves that, unless you know it all, there will always be truths that are by their nature unknowable. This paradox is attributed to American logician Frederic Fitch, who published it in a paper in 1963. Fitch admitted that the original source of the argument was actually a comment made in 1945 by an anonymous referee about a paper Fitch had submitted for publication but had never made it into print. For many years, the referee responsible for this logical gem remained a mystery. But subsequent detective work has tracked down the handwritten copy of the report, and an analysis of the handwriting reveals the author to be the famous American logician Alonzo Church, who made major contributions to the understanding of Gödel’s incompleteness theorem.
Church’s argument has a whiff of the self-referential strategy that Gödel employed, but this time there is no mathematics involved, just pure logic. And while Gödel proves that there are mathematical truths that can never be proved within a particular consistent axiomatic system of mathematics, Church goes one further, promising a truth that will never be known by any means.
Suppose there are true statements that I do not know are true. In fact, there are lots of such statements. For example, my house is full of dice, not just the casino die that I have on my table. There are dice in our Monopoly set, our box of ludo, dice that have gone missing down the side of our sofa, dice that are buried in the chaos of my kids’ rooms. I do not know whether my house contains an odd or even number of dice. This by itself is, of course, not an unknowable statement, because I could make a systematic search of my house to determine the answer. But it is certainly something that, at this point in time, I do not know the answer to.
Now hold on tight—this is the bit that leaves my brain in a spin every time I read it. Let p be the true statement between the two options: “there are an even number of dice in my house” and “there are an odd number of dice in my house.” I don’t know which one is true, but one of them must be. The existence of an unknowable truth is squeezed out of the existence of this unknown truth. The following statement is an unknowable truth: “p is true but unknown.” It is certainly true. Why is this unknowable? Because to know this means I know that p is true and unknown, but that’s a contradiction because p can’t be unknown and known simultaneously. So the statement “p is true and unknown” is itself an unknowable statement. It’s not that p itself is unknowable. As I said, I can go and find all the dice in my house and know whether it is an odd or even number. It is the meta-statement “p is true and unknown” that is unknowable. The proof works, provided something exists that is true but unknown. The only way out of this is if I already know it all. The only way that all truths are knowable is if all truths are known.
Although it has become known as a paradox, there is, as Williamson points out, no paradox involved. It is simply a proof that there are unknowable truths. After our long journey to the outer limits of science, it turns out that a clever logical riff produces the answer I was looking for.
Many philosophers of knowledge question how much we can ever really know about anything. The eighteenth-century Scottish philosopher David Hume identified one of the fundamental problems we’ve had with many of the questions I’ve been tackling—that of being stuck inside the system. If we are going to apply scientific methods to establish that we actually know something, we get into a loop because we are using scientific and logical arguments to prove that these methods are sound. It is impossible to assume an outside position. Wittgenstein summed this up colorfully: “You cannot shit higher than your arse.”
What about mathematics? Surely there we have certain knowledge. Doesn’t proof give us complete certainty that, for example, there are infinitely many prime numbers? Yet even mathematical proofs have to be processed by the human brain to check whether they are correct. What if we are all convinced of the truth of an argument that nonetheless has a subtle hole in it? Of course, one of the things that we take advantage of is the fact that any fatal hole should eventually reveal itself. But doesn’t this imply that mathematics, like science, is subject to an evolutionary process? The mathematical philosopher Imre Lakatos believed so. He developed a philosophy of mathematics that was modeled on Popper’s view of science as something that could only be falsified, not proved true. For Lakatos, it can never be known whether a proof might hide a subtle flaw.
His book Proofs and Refutations presents a fascinating discussion among students exploring the proof of Euler’s theorem on the relationship between the number of vertices, edges, and faces of a three-dimensional polyhedron. It mirrors the history of the evolution of this theorem: E = V + F – 2. At first, the students think they’ve got a proof. Then a student proposes a shape with a hole in the middle. The formula doesn’t work on this shape. Nor does the proof. One interpretation is that the proof works for the shapes that it was intended to work on. But now a new proof and theorem are introduced pertaining to a new formula that in addition to vertices, edges, and faces, includes the number of holes in the shape. The story reveals a much more evolutionary approach to mathematical knowledge than many mathematicians will dare admit to, more akin to the process of scientific investigation. So how effective is either in discovering the truth?
One of the reasons for believing that science is producing true knowledge is its success rate. Science is so successful in its description and prediction of the way things appear to be that we feel like it must be getting close to a reality that most of us believe does exist. The fact that science works so well at making predictions and explaining phenomena is perhaps the best measure that we are close to the truth. If the map that you are using consistently gets you to your destination, that’s a good sign that the map is an accurate representation of reality.
Science has mapped the universe pretty well. Space probes land on distant planets thanks to our discoveries about the nature of gravity. Gene therapies help to tackle previously untreatable conditions thanks to our discoveries about the biology of the cell. We find our way with a GPS by exploiting our discoveries about time and space. When the scientific map doesn’t work, we are ready to redraw its contours to find a description that successfully helps us navigate our environment. It is survival of the fittest theory: continued success at making predictions and controlling our environment means the theory survives. Science may not really represent reality, but nothing comes close as an alternative.
Ever since Kant, we have had to wrestle with the unknowability of “things in themselves.” The limits of human perception, highlighted by how easily our senses can be tricked, raise questions about how much our brain can really know about reality. Isn’t everything being viewed through the spectacles we wear to look at our universe?
One of the key problems with our attempt to know the world is that we use our senses to gain knowledge of the world around us, and analytic argument to extend that knowledge. We come up with stories that match all the information that we gather with our senses. The inventions of the telescope, the microscope, and the fMRI scanner have extended how much we can perceive with our senses.
Yet what if there are things in the universe that our senses can’t detect? We have more senses than many people realize; in addition to sight, hearing, taste, touch, and smell, we also have a sense called proprioception that gives us an awareness of how our body is located in space. There are also senses that give us information about the inner state of our body. The fluid in the inner ear tells us how our body’s position is changing in relation to gravity. But are there physical phenomena that we miss because we don’t have the sensory tools to interact with them?
Consider an organism without an eye or neurons to detect light. If it has no way of accessing electromagnetic waves, how could it ever come up with a theory of electromagnetism? We have done very well to combine our sense of sight, which can see part of the electromagnetic spectrum, with mathematical analysis to deduce other parts of the spectrum. And then we developed tools that can detect these waves and convert them into things that we can interpret. But could we have got going without being able to access, via our sense of sight, some bit of the spectrum?
It is possible that the limitations of our senses also limit the mathematics we can know. Despite mathematics being all in the mind, there is a school of thought that suggests that because our intelligence is ultimately an embodied intelligence, the knowledge we can obtain about mathematics is restricted to that which can be embodied. It is certainly true that if you look at the mathematics we do know, it often has its origins in descriptions of the physical world. Take imaginary numbers—you may question how they are embodied. And yet they emerge from the act of measuring lengths in geometric shapes. The act of understanding the diagonal across the face of my cube-shaped die led the Babylonians to consider the square root of 2. And from here we begin the journey that leads to the idea of the square root of –1.
There are proponents of artificial intelligence who assert that if we are to create an intelligence that matches ours, it must be physically embodied. In other words, a brain that lives exclusively in computer hard drives cannot generate intelligence like ours without physically interacting with the world through a body. It is a challenging hypothesis. Could there really be parts of the mathematical world that are off limits to me because they don’t originate in physically embodied concepts?
And yet there is the deep philosophical issue of the extent to which our senses allow us to know anything for sure. We’ve already seen how our senses can be fooled into believing things that turn out to be tricks of the mind. How, for example, can we be sure that the universe as we apprehend it isn’t a simulation? As we saw in the Sixth Edge, we can make someone believe they are in another person’s body. So how can we be sure that we aren’t just brains in a jar being fed artificial sensory information by a computer and that the whole world around us isn’t just a trick?
My response to this attempt to undermine everything we know is to counter that this book has tried to explore how we can know anything about that simulation. Kant believed that the way things really are will always remain hidden from our view. All we can ever know is the appearance of things. I think most scientists spend some time reading about this debate concerning ontology and epistemology, and listening to philosophers who question whether science is really telling us how it is. And then they get back to the science, telling themselves that if we can never know what reality is really like, then let us at least try to say what the reality we apprehend through our senses is like. After all, that is the one that affects us.
So perhaps the best we can hope for is that science gives us verisimilitudinous knowledge of the universe; that is, it gives us a narrative that appears to describe reality. We believe that a theory that makes our experience of the world intelligible is one that is close to the true nature of the world, even if philosophers tell us we’ll never know. As Niels Bohr said, “It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.”
But what of the things that we cannot know? If something is beyond scientific investigation, if it is unknowable, perhaps some other discipline will have a better grip on the unknowable? Here is Martin Rees wrestling with the “something rather than nothing” question: “The preeminent mystery is why anything exists at all. What breathes life into the equations, and actualized them in a real cosmos? Such questions lie beyond science, however: they are the province of philosophers and theologians.”
Perhaps that is to give in too easily, but it is certainly true that science flourishes when we share the unknowable with other disciplines. If the unknowable has an impact on how we lead our lives, then it is worth having ways to probe the consequences of choosing an answer to an unknowable. Music, poetry, stories, and art are powerful tools for exploring the implications of the unknowable.
Take the question of whether the universe is infinite. There are interesting consequences if you believe that space goes on forever. The fact that across the universe there may be infinitely many copies of you reading this book might have a profound effect on the way you lead your life, even if you will never know whether it’s true.
Chaos theory implies that not only my casino die but also humans are in some ways part of the unknowable. Although we are physical systems, no amount of data will help us completely predict human behavior. The humanities are the best language we have for understanding as much as we can about what it is to be human.
Studies into consciousness suggest boundaries beyond which we cannot go. Our internal worlds are potentially unknowable to others. But isn’t that one of the reasons we write and read novels? It is the most effective way to give others access to that internal world.
What we cannot know creates the space for myth, for stories, for imagination, as much as for science. We may not know, but that doesn’t stop us from creating stories, and these stories are crucial in providing the material for what one day might be known. Without stories, we wouldn’t have any science at all.
Wittgenstein concluded his Tractatus Logico-Philosophicus with the famous line: “Whereof one cannot speak, thereof one must be silent.” I think that is defeatist, as did Wittgenstein in later life. A better denouement would be: “Whereof we cannot know, there our imaginations can play.” After all, it’s by telling stories that we began our journey to know what we know.
That journey has always been driven by what we do not know. As Maxwell declared, “Thoroughly conscious ignorance is the prelude to every real advance in science.” I certainly think that’s true when it comes to mathematics. I need to believe that there is a solution, and that I can find it if I’m going to have any chance of maintaining my faith as I venture into the unknown. Being aware that we don’t know is crucial to making progress. Stephen Hawking appreciates the danger of believing we know it all: “The greatest enemy of knowledge is not ignorance but the illusion of knowledge.”
For me, the conjectures of mathematics, the things we haven’t proved, are its lifeblood. It is the things I do not know that drive me to continue my mathematical quest. I want to know if the Riemann hypothesis is true, and whether the PORC conjecture to which I have dedicated the last few decades of my research is false. As Jacob Bronowski put it, “Human knowledge is personal and responsible, an unending adventure at the edge of uncertainty.”
The importance of the unattained destination is illustrated by the strange reaction many mathematicians have when a great theorem is finally proved. Just as there is a sense of sadness when you finish a great novel, the closure of a mathematical quest can have its own sense of melancholy. I think we were enjoying the challenge of Fermat’s equations so much that there was a sense of depression mixed with the elation that greeted Andrew Wiles’s solution of this 350-year-old enigma.
It is important to recognize that we must live with uncertainty, with the unknown, the unknowable. Even if we eventually manage to produce a theory that describes the way the universe works, we will never know that there isn’t another chapter in the story, waiting for us to discover it. As much as we may crave certainty, to do science we must always be prepared to move on from the stories we tell now. But that’s why science is alive and will never ossify.
So maybe it is important that I embrace the uncertainty of my casino die as it rattles around in my hand. And once it falls from my palm, perhaps not knowing how it will land will drive me to keep looking.