by Lee Smolin
Most of us get over our inability to fly. We eventually concede that we have no influence on many of the aspects of nature. But isn’t it a bit unsettling that there are concepts existing only in our minds whose properties are as objective and immune to our will as things in nature? We invent the curves and numbers of mathematics, but once we have invented them we cannot alter them.
But even if curves and numbers resemble objects in the natural world in the stability of their properties and their resistance to our will, they are not the same as natural objects. They lack one basic property shared by every single thing in nature. Here in the real world, it is always some moment of time. Everything we know of in the world participates in the flow of time. Every observation we make of the world can be dated. Each of us, and everything we know of in nature, exists for an interval of time; before and after that interval, we and they do not exist.
Curves and other mathematical objects do not live in time. The value of pi does not come with a date before which it was different or undefined and after which it will change. If it’s true that two parallel lines never meet in the plane as defined by Euclid, it always was and always will be true. Statements about mathematical objects like curves and numbers are true in a way that doesn’t need any qualification with regard to time. Mathematical objects transcend time. But how can anything exist without existing in time?
People have been arguing about these issues for millennia, and philosophers have yet to reach agreement about them. But one proposal has been on the table ever since these questions were first debated. It holds that curves, numbers, and other mathematical objects exist just as solidly as what we see in nature—except that they are not in our world but in another realm, a realm without time. So there are not two kinds of things in our world, time-bound things and timeless things. There are, rather, two worlds: a world bound in time and a timeless world.
The idea that mathematical objects exist in a separate, timeless world is often associated with Plato. He taught that when mathematics speaks of a triangle, it is not any triangle in the world but an ideal triangle, which is just as real (and even more so) but exists in another realm, one outside time. The theorem that the angles of a triangle add up to 180 degrees is not precisely true of any real triangle in our physical world, but it is absolutely and precisely true of that ideal mathematical triangle existing in the mathematical world. So when we prove the theorem, we are gaining knowledge of something that exists outside time and demonstrating a truth that, likewise, is not bounded by present, past, or future.
If Plato is right, then simply by reasoning we human beings can transcend time and learn timeless truths about a timeless realm of existence. Some mathematicians claim to have deduced certain knowledge about the Platonic realm. This claim, if true, gives them a trace of divinity. How do they imagine they pulled this off? Is their claim credible?
When I want a dose of Platonism, I ask my friend Jim Brown for lunch. Both of us enjoy a good meal, during which he will patiently, and not for the first time, explain the case for belief in the timeless reality of the mathematical world. Jim is unusual among philosophers in coupling a razor-sharp mind with a sunny disposition. You sense that he’s happy in life, and it makes you happy to know him. He’s a good philosopher; he knows all the arguments on each side, and he has no trouble discussing those he can’t refute. But I haven’t found a way to challenge his confidence in the existence of a timeless realm of mathematical objects. I sometimes wonder if his belief in truths beyond the ken of humans contributes to his happiness at being human.
One question that Jim and other Platonists admit is hard for them to answer is how we human beings, who live bounded in time, in contact only with other things similarly bounded, can have definite knowledge of the timeless realm of mathematics. We get to the truths of mathematics by reasoning, but can we really be sure our reasoning is correct? Indeed, we cannot. Occasionally errors are discovered in the proofs published in textbooks, so it’s likely that errors remain. You can try to get out of the difficulty by asserting that mathematical objects don’t exist at all, even outside time. But what sense does it make to assert that we have reliable knowledge about a domain of nonexistent objects?
Another friend I discuss Platonism with is the English mathematical physicist Roger Penrose. He holds that the truths of the mathematical world have a reality not captured by any system of axioms. He follows the great logician Kurt Gödel in arguing that we can reason directly to truths about the mathematical realm—truths that are beyond formal axiomatic proof. Once, he said something like the following to me: “You’re certainly sure that one plus one equals two. That’s a fact about the mathematical world that you can grasp in your intuition and be sure of. So one-plus-one-equals-two is, by itself, evidence enough that reason can transcend time. How about two plus two equals four? You’re sure of that, too! Now, how about five plus five equals ten? You have no doubts, do you? So there are a very large number of facts about the timeless realm of mathematics that you’re confident you know.” Penrose believes that our minds can transcend the ever changing flow of experience and reach a timeless eternal reality behind it.
We discovered the phenomenon of gravity when we realized that our experience of falling is an encounter with a universal natural occurrence. In our attempts to comprehend this phenomenon, we discerned an amazing regularity: All objects fall along a simple curve the ancients invented called parabolas. Thus we can relate a universal phenomenon affecting time-bound things in the world with an invented concept that, in its perfection, suggests the possibility of truths—and of existence—outside time. If you’re a Platonist, like Brown and Penrose, the discovery that bodies universally fall along parabolas is no less than the perception of a relationship between our earthly time-bound world and another, timeless world of eternal truth and beauty. Galileo’s simple discovery then takes on a transcendental or religious significance: It is the discovery of a reflection of timeless divinity acting universally in our world. The falling of a body in time in our imperfect world reveals a timeless essence of perfection at nature’s heart.
This vision of transcendence to the timeless via science has drawn many into science, including myself, but now I’m sure it’s wrong. The dream of transcendence has a fatal flaw at its core, related to its claim to explain the time-bound by the timeless. Because we have no physical access to the imagined timeless world, sooner or later we’ll find ourselves just making stuff up (I’ll present you with examples of this failing in chapters to come). There’s a cheapness at the core of any claim that our universe is ultimately explained by another, more perfect world standing apart from everything we perceive. If we succumb to that claim, we render the boundary between science and mysticism porous.
Our desire for transcendence is at root a religious aspiration. The yearning to be liberated from death and from the pains and limitations of our lives is the fuel of religions and of mysticism. Does the seeking of mathematical knowledge make one a kind of priest, with special access to an extraordinary form of knowledge? Should we simply recognize mathematics for the religious activity it is? Or should we be concerned when the most rational of our thinkers, the mathematicians, speak of what they do as if it were the route to transcendence from the bounds of human life?
It is far more challenging to accept the discipline of having to explain the universe we perceive and experience only in terms of itself—to explain the real only by the real, and the time-bound only by the time-bound. But although it’s more challenging, this restricted, less romantic route will ultimately be the more successful. The prize that awaits us is to understand, finally, the meaning of time on its own terms.
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About the Author
LEE SMOLIN is a theoretical physicist who has made influential contributions to the search for a unification of physics. He is a founding faculty member of the Perimeter I
nstitute for Theoretical Physics. His previous books include The Life of the Cosmos and Three Roads to Quantum Gravity.
Footnotes
* A theory is mathematically consistent when it never gives two results that contradict each other. A related requirement is that all physical qualities the theory describes involve finite numbers.
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* The fallacious principle used here goes like this. Let us observe O and consider two explanations. Given explanation A, the probability of O is very low, but given explanation B, the probability is high. It is tempting to deduce from this that the probability is much higher for B than for A, but there is no principle of logic or probability that allows this deduction.
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* Very recently these new techniques have also been successfully applied to QCD in the real-world case of three spatial dimensions.
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