# Mathematics is the art of giving the same name to different things

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1 1 154 NAW 5/13 nr. 3 september 2012 Mathematics is the art of giving the same name to different things Ferdinand Verhulst Ferdinand Verhulst Mathematical Institute Utrecht University P.O. Box TA Utrecht An interview with Henri Poincaré Mathematics is the art of giving the same name to different things It looked like a daunting and perhaps even impossible enterprise to interview the famous professor Henri Poincaré. However, it turned out to be possible on the condition that professor Poincaré was permitted to formulate the questions himself. We were happy to accept this condition. The interview is elaborated by Ferdinand Verhulst. [We start with some questions about the foundations of mathematical thinking.] Question: What is the nature of mathematical reasoning? Is it really deductive as is usually Henri Poincaré receives visitors in his office at home believed? [1, essay Sur la nature de la raisonnement mathématique ] Answer: A deep analysis shows us that it is not, that it uses to a certain measure inductive reasoning and in this way it is fruitful. Opening an arbitrary book on mathematics, we find the author announcing that he wants to generalize a known proposition. So, the mathematical method proceeds from the particular to the general and how is it then that we can call it deductive? [Thinking about the foundations of mathematics can be interesting, but a question for the mathematician at work is whether exploring the foundations actually improves our mathematical reasoning.] Question: Have we achieved absolute rigour [in mathematics]? In each stage of the evolution, our predecessors believed to have achieved this. If they were wrong, are we not also wrong like them? [2, essay L intuition et la logique en mathématiques ] Answer: We believe that in our reasoning, we don t need intuition. The philosophers tell us that this is an illusion. Pure logic will lead us only to tautologies. This cannot create anything new, from logic alone no science can emerge. For the other thing we need, we have no other word than intuition. If one wants to take the trouble to be rigorous in todays analysis, there are only syllogisms or an appeal to the intuition of numbers [induction], the on-

3 3 NAW 5/13 nr. 3 september 2012 Mathematics is the art of giving the same name to different things Ferdinand Verhulst Illustratie: Ryu Tajiri 156 3

4 4 Ferdinand Verhulst Mathematics is the art of giving the same name to different things NAW 5/13 nr. 3 september numbers. This topic was important in your time but it is still important.] Question: What is exactly this continuum, this subject of mathematical reasoning? [1, essay La grandeur mathématique et l expérience ] Answer: The continuum is nothing else than a set of elements, sequentially arranged, certainly infinite, but separated from each other. Our definition is however not complete. [...] One can ask whether the concept of mathematical continuum has not simply been derived from experience. One is forced to conclude that this idea has been created completely by the human spirit, but that experience has induced it. Question: Is the creative power of the mind regarding the mathematical continuum exhausted? [1, essay La grandeur mathématique et l expérience ] Answer: No, the work of Du Bois-Reymond shows this in a remarkable way. One knows that mathematicians distinguish between infinitesimal small [quantities] of different orders and that those of second order are not only infinitesimal small in an absolute sense, but also with respect to those of first order. It is not difficult to imagine infinitesimal small quantities of fractional or even irrational order and in this way we find again an ordering of the mathematical continuum. [The styles and methods of proofs in mathematics have evolved since your time. With the enormous growth of both pure and applied mathematics there is now an abundance of mathematical styles, there are even computerassisted proofs. Does it make sense to distinguish between beautiful and ugly mathematics, between elegant and graceless reasoning?] Question: Mathematicians attach a great importance to the elegance of their methods and of their results, and this is not mere dilettantism. What is it that gives us the feeling of elegance in a solution or proof? [3, essay L avenir des mathématiques ] Answer: It is the harmony of the different parts, their symmetry, and their happy adjustment; it is, in a word, all that introduces order, all that gives them unity, that enables us to obtain a clear comprehension of the whole as well as of the parts. Elegance may result from the feeling of surprise caused by the unlooked-for occurrence of objects not habitually associated. In this, again, it is fruitful, since it discloses thus relations that were until then unrecognized. Mathematics is the art of giving the same names to different things. [In your time, you were one of the last universal scientists, but then, scientists started to specialize in their research. Is this a danger or a necessity, or both?] Question: How should one view specialization? [3, essay L avenir des mathématiques ] Answer: As science develops, it becomes relatively more difficult to grasp it in its entirety. Then an attempt is made to cut it in pieces and to be satisfied with one of these pieces in one word, to specialize. Too great a movement in this direction would constitute a serious obstacle to the progress of science. As I have said, it is by unexpected concurrences between its different parts that it can make progress. Too much specialization would prohibit these concurrences. [You wrote many textbooks, but why are some people afraid of mathematics? To prove and invent new mathematics, one needs intuition, but to understand school mathematics one needs only some natural logic.] Question: How is it that there are so many minds that are incapable of understanding

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