Children have little trouble understanding zero. When we teach them to count to ten on their fingers, two closed fists mean zero fingers. Easy, and not that interesting. In their attempt to size up the world, children are much more interested in upper limits. As every parent knows, they bombard us with measurement questions: What is the strongest animal? The fastest car? The best footballer? And as soon as they get into numbers and figure out that they go far beyond ten, to millions, billions and gazillions, comes the fateful question: what is the biggest number? To which the standard answer – there is no biggest number: take any number, if you add one to it you get a bigger number – is at first puzzling and rather upsetting. Until they get a name for it: infinity.
But this goes only some way to appease them. Infinity sounds like the biggest number, but it is a weird one. What is infinity plus one? Infinity. And infinity plus infinity? Still infinity. You know – I told them to assuage their perplexity – infinity is not really a number: it is a concept. ‘What’s a consett?’ Well, it’s an idea that we create in our mind to talk about things. ‘Hmm’ – I could hear their brain whirring.
No problem with zero, puzzled by infinity. Interestingly, it was the other way around with the ancient Greeks, who had trouble with ‘nothing’, but were quite comfortable with the unlimited – a-peiron. Anaximander saw it as the principle of all things. Euclid used it to define parallel lines and demonstrated the infinity of prime numbers: ‘Prime numbers are more than any assigned multitude of prime numbers. (Elements, Book IX, Proposition 20). ‘More than any assigned magnitude’ is the concept of infinity that we teach our children. Aristotle called it potential infinity:
The infinite, then, exists in no other way, but in this way it does exist, potentially and by reduction. (Physics, Book III, Part 6).
According to Aristotle, there is no such thing as actual infinity. To demonstrate it, he contrasted arithmetic infinity with physical infinity, and infinity by addition with infinity by division:
It is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude, but that there should be thought to be such an infinite in the direction of division. For the matter and the infinite are contained inside what contains them, while it is the form which contains. It is natural too to suppose that in number there is a limit in the direction of the minimum, and that in the other direction every assigned number is surpassed. In magnitude, on the contrary, every assigned magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude. The reason is that what is one is indivisible whatever it may be, e.g. a man is one man, not many. Number on the other hand is a plurality of ‘ones’ and a certain quantity of them. Hence number must stop at the indivisible: for ‘two’ and ‘three’ are merely derivative terms, and so with each of the other numbers. But in the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time.
With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens. (Physics, Book III, Part 7).
According to Aristotle, in physics there are no infinitely large things, but there are infinitely small things. In arithmetic, however, it is the other way around: numbers are infinitely large, but not infinitely small. To make the point, he used Zeno’s Dichotomy argument: any magnitude can be bisected a potentially infinite number of times. Hence numbers are potentially infinite: ‘in the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite’. On the other hand, there is no number smaller than one: ‘one is indivisible whatever it may be, e.g. a man is one man, not many’…’Hence number must stop at the indivisible’.
For the ancient Greeks there were only natural numbers: ‘A unit is that by virtue of which each of the things that exist is called one’. ‘A number is a multitude composed of units’. (Elements, Book VII, Definitions 1 and 2). So, strictly speaking, one was not even a number (let alone zero) and there was definitely no number smaller than one. A fraction was not seen as a number per se, but as the ratio of two numbers: ‘A ratio is a sort of relation in respect of size between two magnitudes of the same kind’. (Elements, Book V, Definition 3). On the other hand, there was no largest number: the number of times a magnitude can be bisected is infinite. Hence, magnitudes can be infinitely small.
Aristotle was wrong on both counts. In mathematics, there are lots of real numbers smaller than one – in fact, an infinity of them – whereas in the physical world nothing is smaller than a Planck length. On the side of the large, however, he was right on both counts: numbers are infinite, but there is no such thing as an infinite physical magnitude.
Whether mathematical infinity is merely potential or fully actual has been and still is the subject of a heated debate. The prince of mathematicians, Carl Gauss, agreed with Aristotle:
So first of all I protest against the use of an infinite magnitude as something completed, which is never allowed in mathematics. The infinite is only a way of speaking, in which one is really talking in terms of limits, which certain ratios may approach as close as one wishes, while others may be allowed to increase without restriction. (Letter to H. C. Schumacher, no. 396, 12 July 1831).
On the other side, Georg Cantor was adamant about actual infinity, and regarded its staunch defence as a mission from God. If there is an actual infinity of natural numbers, infinity can be treated as a number. But then, since Cantor’s set theory implies that there is an infinity of infinities, our childish quest to get our arms around the biggest number is thrown into even deeper despair. Numbers are infinite – or even infinitely infinite.
At the same time, however, Aristotle made clear that there are no infinite physical magnitudes. When we call something ‘infinite’, what we usually mean is that it is really, really big. Otherwise, if we purposely intend to say that it is actually infinite, we don’t understand what we are talking about. Clearly, no thing can be infinite: the difference between the most ginormously big thing and infinity is, well, infinite. Take the 1080 atoms in the observable universe. That’s a lot of atoms, but it is still a finite quantity. Call it U1. How big is U1 compared to U2=U180? As big as an atom relative to the observable universe. So is U2 compared to U3=U280. And that’s just a miserable 3: how about U1Million, U1Billion, U1Gazillion? They are all next to nothing compared to actual infinity.
Actual physical infinity is not an awe-inspiring immensity that we are too small to comprehend. It is an ill-considered, meaningless and unusable concept. There is no such thing as actual physical infinity. Nor is there potential infinity: ‘For the size which it can potentially be, it can also actually be’. In the physical world, potential infinity – we can call it indefiniteness – coincides with actual infinity: nothing can be bigger than the universe, otherwise it would itself be the universe.
Once properly rearranged, Aristotle’s crucial distinction between the mathematical and the physical world should not be forgotten. In mathematics, there is zero and there is infinity, and we can speak and think of both. There are infinitely small numbers and infinitely large numbers. Zero is itself a number and infinity can be treated as a number. But neither zero nor infinity are something: there is no such thing as nothing and no such thing as infinity. There are no infinitely small things and no infinitely large things. In the physical world, zero and infinity are just useful signs: zero indicates that something is absent and infinity indicates that something is indefinitely big. There are, however, three fundamental differences between them:
1. We can observe zero: it is what we call negative evidence. There is nothing – zero things – on the table. But we cannot observe infinity: there is no infinity of things, on the table or anywhere else.
2. Something becomes nothing after a finite number of bisections. Zero is the magnitude of nothing. But something cannot become infinite: no finite number of operations can turn something into infinity. No thing has infinite magnitude.
3. We cannot observe the absence of everything – Nall – but we can imagine it. While Nall may be impossible, it is not senseless. But we can neither observe nor imagine the infinity of everything: it is an impossible and senseless concept.
Infinity is a sublime, breathtaking, but much abused word. We should never forget it is a consett – perfect to express a parent’s love for his children, but inapplicable to the size of any thing.
