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History of mathematics and mathematical research. Some considerations

We think that a reflection on the relationship between the history of mathematics and the research into mathematics is opportune in order to understand the same nature of mathematics. The importance and the significance of a mathematical result are dependent on its historical position, on previous

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History of mathematics and mathematical research. Some considerations

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    1. History of mathematics and mathematical research. Some considerations Paolo Freguglia Dept. of Pure and Applied Mathematics University of L’Aquila, Italy

    2. We think that a reflection on the relationship between the history of mathematics and the research into mathematics is opportune in order to understand the same nature of mathematics. The importance and the significance of a mathematical result are dependent on its historical position, on previous researches and on the theoretical development of some ideas.

    3. The work of a historian of mathematics lies in a philological analysis of various sources, that is of texts and documents which contain (preceding) theories, theorems, problems and mathematical concepts. Sometimes by reading a historical text, it is possible to find some interesting mathematical results. Of course, it is possible to have expertise both on themes of current research and historical results. However, when this synergy is not possible in one same person, then collaborations can be achieved. Vieri and I have carried out an assignment and we would like to propose some remarks about the ideas of space and time. We are also planning an essay about the rationalization of the infinity concept.

    4. An example where from an analysis of a historical text it is possible to find some interesting mathematical results. A historical analysis of Vičte’s Zeteticorum libri quinque (1593) and a consequent opportune (from historical point view) comparison with Pierre de Fermat’s Remarks on Diophantus (1644 - 54) has brought forth an interesting result about the elementary solutions of indeterminate third degree equations.

    5. During the period between the second half of the XVIth century and the first half of the XVIIth century, in the Western Latin culture we find a considerable interest for Diophantus's work. This interest has led to a lot of writings where we can find a partial or complete translation, a commentary and an enrichment of Diophantus problems. In particular, we have the following works: - the third book of Bombelli's Algebra, (1572) - Rerum arithmeticarum libri VI by G.Xylander, (1575) - the Appendix to the second book of S.Stevin's L'Arithmétique (1585, ed. A.Girard 1634) - Zeteticorum libri quinque by F.Vičte, (1593) - Diophanti Arithmeticorum Libri Sex, by Bachet de Meziriac, C.G., (1621) - Remarks on Diophantus by Pierre de Fermat (1644 - 54)

    6. Vičte’s third degree indeterminate problems. The Zetetici IV, 18, 19, 20, which respectively correspond to Bachet’s Quaestiones I, II, III, concern indeterminate third degree problems. In Diophantus’s V book we find the Quaestio, 16 where a porism concerning the equation X 3 + Y 3 = B 3 – D 3 is mentioned. Vičte examines two other cases (Zet. IV, 19 and IV, 20). Hence, he tries to go mathematically beyond the Diophantus scheme, even if he only gives one solution for each equation, relative to these zetetici.

    7. Zeteticum IV, 18 (Bachet, Quaestio I): Datis duobus cubis, invenire numero duos alios cubos, quorum summa aequalis sit differentiae datorum [Given two cubes (cubic numbers), find numerically two other cubes the sum of which is equal to the difference between the two given cubes]. Symbolically, we must solve the following indeterminate equation: X 3 + Y 3 = B 3 – D 3 where B, D ? Q+ and the condition: B3 > 2D3 is assigned. The solution is:

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