Word: formalities
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Dates: during 1990-1999
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...mathematics included a brilliant young mathematician, Alan Turing, who was giving his own course that term on the same topic. Turing too had been excited by the promise of mathematical logic and, like Wittgenstein, had come to see that it had limitations. But in the course of Turing's formal proof that the dream of turning all mathematics into logic was strictly impossible, he had invented a purely conceptual device--now known as a Universal Turing Machine--that provided the logical basis for the digital computer. And whereas Wittgenstein's dream of a universal ideal language for expressing all meanings...
Happily, in those days before tape recorders, some of Wittgenstein's disciples took verbatim notes, so we can catch a rare glimpse of two great minds addressing a central problem from opposite points of view: the problem of contradiction in a formal system. For Turing, the problem is a practical one: if you design a bridge using a system that contains a contradiction, "the bridge may fall down." For Wittgenstein, the problem was about the social context in which human beings can be said to "follow the rules" of a mathematical system. What Turing saw, and Wittgenstein...
Although every teacher in training memorizes Piaget's four stages of childhood development (sensorimotor, preoperational, concrete operational, formal operational), the better part of Piaget's work is less well known, perhaps because schools of education regard it as "too deep" for teachers. Piaget never thought of himself as a child psychologist. His real interest was epistemology--the theory of knowledge--which, like physics, was considered a branch of philosophy until Piaget came along and made it a science...
...theorem, it is crucial to understand how mathematics was perceived at the time. After many centuries of being a typically sloppy human mishmash in which vague intuitions and precise logic coexisted on equal terms, mathematics at the end of the 19th century was finally being shaped up. So-called formal systems were devised (the prime example being Russell and Whitehead's Principia Mathematica) in which theorems, following strict rules of inference, sprout from axioms like limbs from a tree. This process of theorem sprouting had to start somewhere, and that is where the axioms came in: they were the primordial...
...symbols in which statements in formal systems were written generally included, for the sake of clarity, standard numerals, plus signs, parentheses and so forth, but they were not a necessary feature; statements could equally well be built out of icons representing plums, bananas, apples and oranges, or any utterly arbitrary set of chicken scratches, as long as a given chicken scratch always turned up in the proper places and only in such proper places. Mathematical statements in such systems were, it then became apparent, merely precisely structured patterns made up of arbitrary symbols...