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As even some supporters of the dominant view, like Halmos, acknowledge, mathematics “is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and he becomes convinced of their truth long before he can write down a logical proof ”56. The “deductive stage, writing the result down, and writing down its rigorous proof are relatively trivial once the real insight arrives; it is more like the draftsman’s work, not the architect’s”57.

It is often claimed that ‘plausible’ has a subjective, psychological connotation, so that it is almost equivalent to ‘rhetorically persuasive’, hence plausible arguments are of little interest in mathematics. But ‘plausible’, in the sense explained above, has nothing subjective or psychological about it. To assess whether a given hypothesis is plausible, one examines the reasons for and against it. This examination is carried out using facts which confirm the hypothesis or refute it, where these facts belong to the existing knowledge.

Thus, the method of mathematics enables us to deal with different things at the same time. HIPPOCRATES What about the following similes: If somebody looks at a city from the top of a nearby mountain, he gets a more comprehensive view than if he walks through its crooked streets; or if a general watches the movements of an enemy army from a hill, he gets a clearer picture of the situation than does the soldier in the front line who sees only those directly opposite him. SOCRATES Well, you surpass me in inventing new similes, but as I do not want to fall behind, let me also add one parable.

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