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Science and Hypothesis 

Jules Henri Poincaré (1854–1912) was one of France’s greatest mathematicians and theoretical physicists, and a philosopher of science. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. In his research on the threebody problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is considered to be one of the founders of the field of topology. Poincaré introduced the modern principle of relativity and was the first to present the Lorentz transformations in their modern symmetrical form. He discovered the remaining relativistic velocity transformations and recorded them in a letter to Lorentz in 1905. Thus he obtained perfect invariance of all of Maxwell’s equations, the final step in the formulation of the theory of special relativity. First Page:it will be true for a = 1.1 Now, it is true for a = 1, and therefore is true for a = 2, a = 3, and so on. This is what is meant by saying that the proof is demonstrated “by recurrence.” (2) I say that a b = b a: 1For ( 1) 1 = (1 ) 1 = 1 ( 1).—[Tr.] science and hypothesis 10 The theorem has just been shown to hold good for b = 1, and it may be verified analytically that if it is true for b = , it will be true for b = 1. The proposition is thus established by recurrence. definition of multiplication. We shall define multiplication by the equalities (1) a 1 = a; (2) a b = a (b 1) a: Both of these include an infinite number of definitions; having defined a1, it enables us to define in succession a 2, a 3, and so on. properties of multiplication. Distributive.—I say that (a b) c = (a c) (b c): We can verify analytically that the theorem is true for c = 1; then if it is true for c = , it will be true for c = 1. The proposition is then proved by recurrence. nature of mathematical reasoning... Continue reading book >> 
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