Documenta Mathematica Optimization Stories - Mathematics

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18 Eberhard Knobloch only those who had suffiently penetrated in the mysteries of his differential calculus. This he had predicted for the brother of Johann Bernoulli and the Marquis de l’Hospital, for Huygens if he were alive, for Hudde if he had not given up such pursuits, for Newton if he would take the trouble. The words were carelessly written because their obvious meaning was that Newton was indebted to the differential calculus for his solution. Even if Leibniz did not want to make such a claim, and this is certain in 1697, his words could be interpreted in such a way. There was indeed a reader who chose this interpretation: the French emigrant Nicolas Fatio de Duillier, one of Newton’s closest followers. Fatio was deeply offended at not having been mentioned by Leibniz among those authors who could master the brachistochrone problem. In 1699 he published a lengthy analysis of the brachistochrone. Therein he praised his own mathematical originality and sharply accused Leibniz of being only the second inventor of the calculus. Fatio’s publication was the beginning of the calculus squabble. But this is another story. References [1] H.H.Goldstine, Introduction, in: DieStreitschriftenvonJacobundJohann Bernoulli, Variationsrechnung, bearbeitetundkommentiertvonH.H.Goldstine mit historischen Anmerkungen von P. Radelet-de Grave, Birkhäuser, Basel-Boston-Berlin 1991, pp. 1–113. [2] E. Knobloch, Le calcul leibnizien dans la correspondance entre Leibniz et Jean Bernoulli. in: G. Abel, H.-J. Engfer, C. Hubig (eds.), Neuzeitliches Denken, Festschrift für Hans Poser zum 65. Geburtstag, W. de Gruyter, Berlin-New York 2002, pp. 173–193. [3] Eberhard Knobloch, Galilei und Leibniz, Wehrhahn, Hannover 2012. [4] Jeanne Peiffer, Le problème de la brachystochrone à travers les relations de Jean I. Bernoulli avec l’Hospital et Varignon, in: H.-J. Hess, F. Nagel (eds.), Der Ausbau des Calculus durch Leibniz und die Brüder Bernoulli. Steiner, Wiesbaden 1989, pp. 59–81 (= Studia Leibnitiana Sonderheft 17). Eberhard Knobloch Berlin-Brandenburg Academy of Sciences and Humanities Technische Universität Berlin H 72 Straße des 17. Juni 135 10623 Berlin eberhard.knobloch@tu-berlin.de Documenta Mathematica · Extra Volume ISMP (2012) 15–18
Documenta Math. 19 Leibniz and the Infinite Eberhard Knobloch 2010 Mathematics Subject Classification: 01A45, 28-03 Keywords and Phrases: Leibniz, infinite, infinitely small, mathematical rigour, integration theory TheGermanuniversal genius Gottfried Wilhelm Leibnizwas bornin Leipzigon the 21st of June according to the Julian calendar (on the 1st of July according to the Gregorian calendar) 1646. From 1661 he studied at the universities of Leipzig and Jena. On February 22, 1667 he became Doctor of Laws at the university of Nürnberg-Altdorf. He declined the professorship that was offered to him at this university. For a short time he accepted a position at the court of appeal of Mainz. From 1672 to 1676 he spent four years in Paris where he invented his differential and integral calculus in autumn 1675. From1676uptotheendofhislifeheearnedhislivingaslibrarianatthecourt of the duke, then elector, of Hannover. In 1700 he was appointed president of the newly founded Electoral Academy of Sciences of Berlin. He contributed to nearly all scientific disciplines and left the incredibly huge amount of about 200 000 sheets of paper. Less than one half of them have been published up to now. In Paris he became one of the best mathematicians of his time within a few years. He was especially interested in the infinite. But what did he mean by this notion? His comments on Galileo’s Discorsi give the answer. Therein Galileo had demonstrated that there is a one-to-one correspondence between the set of the natural numbers and the set of the square numbers. Hence in his eyes the Euclidean axiom “The whole is greater than a part” was invalidated in the sense that it could not be applied there: infinite sets cannot be compared with each other with regard to their size. Leibniz contradicted him. For him it was impossible that this axiom failed. This only seemed to be the case because Galileo had presupposed the existence of actually infinite sets. For him the universal validity of rules was more important than the existence of objects, in this case of actually infinite numbers or actually infinite sets. Hence Leibniz did not admit actual infinity in mathematics. “Infinite” meant “larger than any given quantity”. He used the mode of possibility in order to characterize the mathematical infinite: it is always possible to find a quantity that is larger than any given quantity. Documenta Mathematica · Extra Volume ISMP (2012) 19–23
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Documenta Math. 107 A Brief History
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A Brief History 109 first-generatio
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A Brief History 113 in practice. Ka
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A Brief History 119 [10] Crowder, H
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A Brief History 121 Robert E. Bixby
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Documenta Math. 123 Discrete Optimi
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Documenta Math. 127 The Origins of
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Documenta Math. 155 On the History
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Transportation and Maximum Flow Pro
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Acknowledgement The Ongoing Story o
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Optimisation and Utility Functions
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