Infinity is back. Or rather, it never (never, never …) went away. While mathematicians have a good sense of infinity as a concept, cosmologists and physicists find it much more difficult to make sense of infinity in nature, writes Peter Cameron.
Each of us faces a moment, often quite early in our lives, when we realize that a loved one, once a permanent presence in our life, was not infinite, but has left us, and that one day we too will have to leave this place.
This experience, probably as much as the experience of looking at the stars and wondering how far they go, shapes our visions of infinity. And we urgently want answers to our questions. It has been like this since the time, two and a half millennia ago, when Malunkyaputta addressed his doubts to the Buddha and asked for answers: among them he wanted to know if the world is finite or infinite, and if it is eternal or not.
Recently we listened to the words of John Donne who promised us that eternity is done
“no noise or silence, but the same music;
without end or beginning, but an equal eternity “.
Hard to imagine, and surely the same music would soon become intolerable!
There are many approaches to infinity across the two pillars of science and religion, but I will limit my attention here to the views of mathematicians and physicists.
Aristotle was one of the most influential Greek philosophers. He believed that we could consider “potential infinity” (we can count objects without knowing how many others are coming) but that “completed infinity” was taboo. For mathematicians, infinity was off-limits for two millennia after Aristotle’s ban. Galileo tried to tackle the problem, noting that an infinite set could be matched to a part of himself, but eventually he withdrew. He was left to Cantor in the nineteenth century to show us the way of thinking about infinity, which is now accepted by most mathematicians. There are infinite counting numbers; any number you write is a negligible step along the road to infinity. So Cantor’s idea was to imagine having a package containing all these numbers; put a label on it that says “Natural numbers” and treat the package as a single entity. If you want to study the individual numbers, you can open the package and take them out to look at them. You can now take any collection of these packs and group them to form another single entity. Thus was born the theory of sets. Cantor studied ways to measure these sets, and set theory is the most common basis for mathematics today, although other bases have been proposed.
If you flip a coin 100 times, it’s not impossible (just very unlikely) that it falls tails every time. But if you can imagine flipping a coin infinitely often, then the chance of not getting heads and tails as often is zero
One of Cantor’s discoveries is that there is no greater infinite set – given any set you can always find a larger one. The smallest infinite set is the set of natural numbers. What comes next is an enigma that cannot be solved at the moment. It could be real (decimal) numbers, or maybe not. Our current foundations aren’t solid enough, and building larger telescopes won’t help with this question. Perhaps in the future we will adopt new foundations for mathematics that will solve the question. But for now, since math is a mental construction, we can decide whether the universe we’re playing in satisfies the “continuum hypothesis” or not.
These questions keep set theorists awake at night; but most mathematicians work at the bottom of this dizzying hierarchy, with small infinitives. For example, Euclid proved that prime numbers “go on forever”. (Aristotle would say, “Whichever prime I find, I can find a bigger one”; Cantor would simply say, “The set of primes is infinite.” Mathematicians (including this year’s James Maynard Oxford Fields Medal) seem to be approaching the conjecture, not yet proven, states that there are infinite. But these are the infinitives of natural numbers, the smallest infinity.
Physics alone cannot answer the big questions
By Sabine Hossenfelder
While Kronecker (a fierce opponent of Cantor’s ideas) thought in the nineteenth century that “God created natural numbers; the rest is the work of man”, we can now construct natural numbers using the tools of set theory, starting from nothing. (more precisely the empty set).
Mathematicians know, however, that there is a huge gap between the finite and the infinite. If you flip a coin 100 times, it’s not impossible (just very unlikely) that it falls tails every time. But if you can imagine flipping a coin infinitely often, then the chance of not getting heads and tails as often is zero. Of course, you could never actually perform this experiment; but mathematics is a conceptual science and we are happy to accept this claim on the basis of rigorous proof.
Infinity in physics and cosmology has not been resolved so satisfactorily. The two great theories of 20th century physics, general relativity (the theory of the very large) and quantum mechanics (the theory of the very small) have resisted attempts to combine them. The one thing most physicists can agree on is that the universe was born at a finite time (about 13.7 billion years) – large, but not infinite.
They deny that the infinitely small can exist in the universe, but prescribe a minimum possible scale, essentially the so-called Planck scale
The James Webb Space Telescope has just begun to show us unprecedented details in the universe. In addition to nearby objects, it sees the farthest objects ever observed. Since light travels at a finite speed, these are also the oldest objects observed, having formed near the beginning of the Universe. The finite speed of light also places limits on what we can see; if an object is so far away that its light could not reach us if it has traveled the entire age of the universe, then we are not aware of its existence. So Malunkyaputta’s question as to whether the universe is finite or infinite is moot. But is it eternal or not? This is a real question, and it is undecided so far.
Attempts have been made to reconcile relativity and quantum theory. Those currently most promising adopt a very radical attitude towards infinity. They deny that the infinitely small can exist in the universe, but prescribe a minimum possible scale, essentially the so-called Planck scale.
The infinite riddle
With David Malone, Laura Mersini-Houghton, Peter Cameron, Julian Barbour
Such a solution would put an end to Zeno’s paradox. Zeno denied the possibility of motion, since to pass from A to B one must first go to a point C in the middle of B, and before that to a point D in the middle from A to C, and so on ad infinitum. If space is not infinitely divisible, then this infinite regress cannot occur. (This solution had already been grasped by Democritus and the first Greek atomists.)
Of course, this leaves us with a conceptual problem similar to that raised by the possibility that college is over. If so, the obvious question is “If the universe has an advantage, what is beyond it?” In the case of Planck’s length, the question would be “Given the length, however small, why can’t I just take half of it?”
Perhaps because conditioned by Zeno’s paradox, we tend to think that the points of a line are, like real numbers, infinitely divisible: between any two we can find another. But the current thinking is that the universe is not built that way.
Time, however, remains a problem
More important to physics, the atomism hypothesis also gets rid of another troublesome infinity event in physics. Black holes in general relativity are points in spacetime where the density of matter becomes infinite and the laws of physics fail. These have been a thorn in the flesh of cosmologists ever since their existence was first predicted, as by definition we cannot understand what goes on there. If space is discrete, we cannot put together infinite things that are infinitely close, and the paradox is avoided. We can still have an extremely high density; the recently observed and photographed black hole in the center of our galaxy is (according to this theory) just a point of such high density that light cannot escape, but it does not challenge our ability to understand it.
Time, however, remains a problem; current theories cannot decide the ultimate fate of the universe. Does it end in thermal death, a cold dark universe where nothing happens? Does the mysterious “dark energy” get so strong it tears apart the universe? Or does the Big Bang expansion go the other way around, so that the universe ends up in a Big Crunch?
The Big Bang did not happen
By Eric J Lerner
Physicists in the 19th century developing the science of thermodynamics observed that, over time, a complicated system such as the universe becomes more disordered. (Let’s say its entropy increases.) Recently it has been suggested that this is reversed; it is the growing disorder of the universe that somehow makes time pass. This is part of a movement in which the traditional units of space, time, matter and energy are being replaced by information as the fundamental currency of the universe. But these are the first days for such theories.
None of this interests us individually. The sun will expand and engulf the earth long before the universe reaches its end. But we have an insatiable curiosity to know the answer to Malunkyaputta’s question. As mathematician (and optimist) David Hilbert said: “We have to know; we will know.)
Apostolos Doxiadis, Logicomix, Bloomsbury, 2009.
Carlo Rovelli, Reality is not what it seems, Riverside Books, 2017.