Posts Tagged ‘infinity’
Jeff Dekofsky explains Hilbert’s paradox of the Grand Hotel, a thought experiment proposed in the 1920s by German mathematician David Hilbert to illustrate some surprising properties of infinite sets, in this TED-Ed animated lecture…
As a special bonus, another amusing video (via Kottke)– an explanation of why it is that the sum of all positive integers (1 + 2 + 3 + 4 + 5 + …) = -1/12… Euler actually proved this result in 1735, but the result was only made rigorous later; and now physicists have been seeing this result actually show up in nature. (Spoiler alert: the answer turns on what one means by “sum” mathematically…)
As we pray for more fingers and toes, we might spare a thought for Harald August Bohr; he died on this date in 1951. While materially less well-known than his brother Niels, Harald was a formidable mathematician (founder of the field of almost periodic functions), a gifted athlete (an accomplished footballer who won a silver medal at the 1908 Summer Olympics as a member of Denmark’s team), an inspirational teacher (the annual award for outstanding teaching at the University of Copenhagen is called “the Harald” in his honor), and an out-spoken critic of the anti-Semitic policies that took root in the German mathematical establishment in the 1930s.
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]
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)…
How to Count to Infinity (or “Yes, Virginia, some infinities are bigger than others…”)
As we check in to Hilbert’s Hotel, we might spare a thought for Joesph Fourier; the French mathematician, physicist, Egyptologist and administrator who died on this date in 1830. Fourier introduced Jean-Francois Champollion to the Rosetta Stone, which Champollion subsequently decoded/translated. And after calculating that a body the size of earth, at earth’s distance form the sun, should be cooler than our world is, discovered what we now call “the greenhouse effect.” But Fourier is best remembered for his contributions to mathematical physics through his Théorie analytique de la chaleur (1822; The Analytical Theory of Heat), which introduced an infinite mathematical series to aid in solving conduction equations. (The technique allowed the function of any variable to be expanded into a series of sines of multiples of the variable– now known as “the fourier series.”)
True greatness is when your name is like ampere, watt, and fourier—when it’s spelled with a lower case letter.
- Richard Hamming (in a 1986 Bell Labs Colloquium)