By Reuben Hersh
Collection of the main fascinating fresh writings at the philosophy of arithmetic written by means of hugely revered researchers from philosophy, arithmetic, physics, and chemistry
Interdisciplinary booklet that may be priceless in numerous fields—with a cross-disciplinary topic quarter, and contributions from researchers of assorted disciplines
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Additional info for 18 Unconventional Essays on the Nature of Mathematics
For each solution generates new problems, and depends on the solutions found for these new problems. Thus, no solution is final but is always subject to further reconsideration. 7. According to the dominant view, the method of mathematics is the axiomatic method. For “proof must begin from axioms that are not themselves proved”37. In fact, “to prove a proposition, you start from some first principles, derive some results from those axioms, then, using those axioms and results, push on to prove other results”38.
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. As a matter of fact, the reflected image does not really exist; it is illusion, nothing else. SOCRATES image? Is there nothing in common between the rock and its reflected HIPPOCRATES Well, in a certain sense, the reflected image is a faithful picture of the rock.
HIPPOCRATES I think that the sailor may be called a discoverer, and the painter an inventor. The sailor discovers an island which existed before him, only it was unknown, while the painter invents a new color which before that did not exist at all. SOCRATES Nobody could answer the question better. But tell me, the mathematician who finds a new truth, does he discover it or invent it? Is he a discoverer as the sailor or an inventor as the painter? HIPPOCRATES It seems to me that the mathematician is more like a discoverer.
18 Unconventional Essays on the Nature of Mathematics by Reuben Hersh