New from the The Institute of Art and Ideas. Paradoxes of self-reference are found in mathematics, literature and philosophy from the Greeks to Derrida. Can we ever solve them? And do we need to? Literary critic Patricia Waugh, mathematician Peter Cameron and philosopher Hilary Lawson and tackle the problem here:
Calculators are an often used example in the philosophy of mind. Sometimes they’re used analogously, to show how computational algorithms can be implemented in a variety of mediums (say, the very different circuitries of the calculator and the human brain). Other times, they’re used metaphorically, as objects that we can attribute intentional states: the calculator ‘knows’ how to add and ‘believes’ that 2+2=4. But how appropriate are comparisons between calculators and humans? Is it a matter of implementing the same (or nearly the same) algorithm? Or is the comparison a mere metaphor? Stanislas Dehaene is the champion of the surprising view that neither of these (caricatured) approaches can be right: calculation is neither a matter of merely attributing intentional states, nor do humans and calculators implement algorithms in the same way.
(Apologies if this topic seems old hat to any – if you are a person already familiar with Dehaene, ‘cultural re-mapping’, number sensing, and the like, the payoff to re-reading this extremely cool and interesting stuff about human mathematical capabilities, is some very exciting and interesting new advances in brain localization and machine-learning)
Dehaene’s view is that our mathematical abilities result from the mixture of two evolved mechanisms, and, importantly, a sprinkling of language. The first of these evolved mechanisms is a capacity to distinguish a certain amount of discrete quantities, or numerosity: the ability to tell apart one, two, three, and maybe four and five. Then, there is the capacity to distinguish differences in quantity: that six is bigger than one, or that twenty is less than sixty. Both of these abilities can be found in animals, and, yes, human children. And it’s easy to understand why such mechanisms might persist over time*: as an organism, it is very handy to have a capacity to determine between alternatives; whether option (a) was better than (b) because more nutrients, or less competition, or what have you. Continue reading “Machine Math?”
Philosophers do like a bit of Lewis Carroll. When Humpty Dumpty exclaimed to Alice, “There’s glory for you!” and meant “there’s a nice knock down argument for you!”, Donald Davidson took it as an illustration of how intention can override convention in determining what one said. When the Tortoise said to Achilles to use logic to force him to accept Z, given that If A and B then Z, Barry Stroud and Robert Brandom (among many others) took this to indicate something important about meaning and inference. And there have been various occasions when the Jabberwocky has been wheeled out to illustrate some point about sense or nonsense.
Last month, DPhil student Melanie Bayley Continue reading “Why is a raven like a writing desk?”