By Reuben Hersh

This startling new number of essays edited via Reuben Hersh comprises frank evidence and critiques from prime mathematicians, philosophers, sociologists, cognitive scientists, or even an anthropologist. each one essay presents a demanding and thought-provoking examine fresh advances within the philosophy of arithmetic, demonstrating the chances of pondering clean, sticking on the subject of genuine perform, and fearlessly letting cross of ordinary shibboleths.

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Moreover, the idea that mathematics is a body of absolutely certain and hence irrefutable truths, overlooks the fact that we cannot be sure of the current proofs of our theorems. For, by Gödel’s incompleteness theorems and related results, we cannot be sure of the hypotheses on which they are based. 68 69 70 71 72 Feferman 1998, p. 77. Jaffe 1997, p. 135. Rota 1997, p. 142. Jaffe 1997, pp. 139-140. Maddy 1997, p. 163. “Introduction” to Filosofia e matematica 31 As regards certainty, mathematics has no privilege and is as risky as any other human creation.

HIPPOCRATES What is it? SOCRATES There is another reason why the study of the mathematical image of the world may be of use. If mathematicians discover some property of the circle, this at once gives us some information about any object of circular shape. 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 Do you see that rock on the other side of the river, there where the river broadens out and forms a lake? HIPPOCRATES SOCRATES I see it. And do you see the image of the rock reflected in the water? HIPPOCRATES Certainly I do. SOCRATES Then tell me, what is the difference between the rock and its reflection? HIPPOCRATES The rock is a solid piece of hard matter. It is made warm by the sun. If you touched it, you would feel that it is rough. The reflected image cannot be touched; if I put my hand on it, I would touch only the cool water.