This chapter considers the mental pictures of thought and images in relation to algebraic symbols. According to Ludwig Wittgenstein, “We make to ourselves pictures of facts.” For Wittgenstein, the picture is a model of what we take to be real. The geneticist Francis Galton claimed that his thoughts almost never suggested words, and when those rare moments did suggest words, they were nonsense words like “the notes of a song might accompany thought.” As for words, the French mathematician Jacques Hadamard suggested that words are neither followed by thoughts, nor thoughts by words. He goes on to say that this is also the case when he is thinking about algebraic symbols. More revealing is Hadamard's presentation of his mental pictures of the steps in a proof that there are an unlimited number of prime numbers. The chapter also discusses thought without verbal language and in relation to proofs.
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