Infinitely infinite…
The folks at Plus polled their readers for the questions they’d most like answered. The winner: “Does Infinity Exist?,” which cosmologist John D. Barrow answers…
This is a surprisingly ancient question. It was Aristotle who first introduced a clear distinction to help make sense of it. He distinguished between two varieties of infinity. One of them he called a potential infinity: this is the type of infinity that characterises an unending Universe or an unending list, for example the natural numbers 1,2,3,4,5,…, which go on forever. These are lists or expanses that have no end or boundary: you can never reach the end of all numbers by listing them, or the end of an unending universe by travelling in a spaceship. Aristotle was quite happy about these potential infinities, he recognised that they existed and they didn’t create any great scandal in his way of thinking about the Universe.
Aristotle distinguished potential infinities from what he called actual infinities. These would be something you could measure, something local, for example the density of a solid, or the brightness of a light, or the temperature of an object, becoming infinite at a particular place or time. You would be able to encounter this infinity locally in the Universe. Aristotle banned actual infinities: he said they couldn’t exist. This was bound up with his other belief, that there couldn’t be a perfect vacuum in nature. If there could, he believed you would be able to push and accelerate an object to infinite speed because it would encounter no resistance.
For several thousands of years Aristotle’s philosophy underpinned Western and Christian dogma and belief about the nature of the Universe. People continued to believe that actual infinities could not exist, in fact the only actual infinity that was supposed to exist was the divine.
But in the world of mathematics things changed towards the end of the 19th century… In mathematics, if you say something “exists”, what you mean is that it doesn’t introduce a logical contradiction given a particular set of rules. But it doesn’t mean that you can have one sitting on your desk or that there’s one running around somewhere.
And that was only the beginning; then came advances in physics and cosmology… Find out what happened, and whether infinities do in fact exist (plus, discover– finally!– the attraction of string theory) in “Do Infinities Exist?”
[TotH to 3 Quarks Daily]
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As we remind ourselves that there’s always room in Hilbert’s Hotel, we might spare a well-ordered thought for German mathematician and logician (Friedrich Ludwig) Gottlob Frege; he died on this date in 1925. Frege extended Boole’s work by inventing logical symbols, effectively founding modern symbolic logic. He worked on general questions of philosophical logic and semantics (indeed, his theory of meaning, based on distinguishing between what a linguistic term refers to and what it expresses, remains influential). But relevantly here, Frege was the first to put forward the view that mathematics is reducible to logic– thus creating the context in which mathematical infinites can “exist” (in that they do not contradict that logic)…
