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In modern mathematics the concept of the limit arises from the twofold requirement to specify the nature of the set of real numbers and to remove the many critiques to the Newtonian definition of the …

In modern mathematics the concept of the limit arises from the twofold requirement to specify the nature of the set of real numbers and to remove the many critiques to the Newtonian definition of the derivative.

In Cauchy’s definition the limit is associated with a function’s behaviour when we approach a fixed point or when this point increases indefinitely.

A satisfactory mathematical approach to the limit concept and the computational rules appears only at the end of the XIX century.

More recently this fundamental concept was introduced in all mathematical fields, not only in the study of functions of several real variables but also in the study of general abstract spaces such as metric and topological spaces.

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- 1. festa dellÊinquietudine III edition 14 – 15 - 16 May 2010 Finale Ligure SV, Italian Riviera restlessness festival 2010 restlessness & limit The Limit in Mathematics Manfredo Montagnana
- 2. Executive Summary “Festa dell’Inquietudine” is a Culture and Entertainment performative event dedicated to the “Restlessness”. The Festa is organized on a series of events that include: Debates & meetings Exhibitions & Shows Inquietus of InquietaMente Inquietus Celebration the year Events involving prominent personalities from the Italian and worldwide cultural, scientific and entertainment arena. Leitmotif of the year 2010: “Restlessness & Limit” in Philosophy Mathematics Science & Species Sport Economy & Technology & Organization & Life, Beyond Life, Resources Engineering Leadership Other Worlds Venue: Finale Ligure SV, Italy, Finalborgo (architectural complex of ( Santa Caterina) and Finalmarina (Castelfranco Fortress) Time period: 14 - 15 -16 May 2010. 2
- 3. Contents Festa dell’Inquietudine / Restlessness Festival • Festa dell’Inquietudine 2010 / Restlessness Festival • Restlessness & Limit in … • Go beyond … The limit in Mathematics • There are no borders for the human mind • The limit for mathematicians • The abstract nature of the limit concept Places of Restlessness Festival 2010 Restlessness Festival 2010 Organization • Circolo degli Inquieti / Inquietus Cultural Club Restlessness Festival 2010 Events • Inquietus of the year Citations & Links 3
- 4. Culture & Entertainment performative event dedicated to the “Restlessness” 4
- 5. Restlessness Restlessness concerns knowledge and cultural and sentimental growing, restlessness concern not only those living characterizes anxiety states. Restlessness surrounds and permeates lovers, who is tormented by the artistic creativity, who thirst for knowledge, who is pervaded with doubt, who is fascinated by the mystery, who is seduced by life, those who participate in the dramas of contemporary humanity and, even more, those who are directly afflicted. 5
- 6. Festa dell’Inquietudine 2010 Restlessness Festival 2010 Limit (1) dividing line; (2) extremity to which they can get something; (3) term that you can not or should not be exceeded, [even in the figurative sense] In the III edition of the “Festa dell’Inquietudine” we work on the link: «restlessness & limit» 6
- 7. Restlessness & Limits in … Philosophy Sport Technology & MATHEMATICS Engineering Economy, Resources, Organizations & Environment, … Leadership Life, Other Worlds, Science & Species Beyond Life 7
- 8. Go beyond … «We live in an age where everything seems to be overcome: from sports performance to scientific knowledge, to the same "human species"». «For us it is obvious to think that the restlessness to push the man to the limit and maybe beyond». Elio Ferraris, Presidente del Circolo degli Inquieti 8
- 9. “PLVS VLTRA”, go beyond … «We live in an age where everything seems to be overcome: from sports performance to scientific knowledge, to the same "human species"». «For us, of the Restless’ Club, it is obvious to think that the restlessness to push the man to the limit and maybe beyond». “PLVS VLTRA” (Plus Ultra). In Latin means "Go beyond", exceed their limits, versus another Latin motto “NEC PLVS VLTRA” (Nec Plus Ultra), "not further", which indicates the extreme limit. 9
- 10. They have gone beyond … Of the mythology of Heracles-Hercules we like that sententious “Nec plus ultra" carved on the columns the same name. It came after extraordinary feats in which the hero had challenged and defeated gods and monsters, and showed a limit. But even more fascinate those who have passed that column! Ulysses, Christopher Columbus, but also Plato that "beyond" places the lost Atlantis. 10
- 11. Plus Ultra We like, even, Charles V, Holy Roman Emperor (Carlos I de España ), which transforms the ban encouragement to go further, and the "Plus Ultra" becomes his motto. Fonte: wikipedia 11
- 12. Restlessness & Limit Mathematics There are no borders for the human mind The limit for mathematicians The abstract nature of the limit concept Manfredo Montagnana 12
- 13. Manfredo Montagnana He has been President of the Unione Culturale Franco Antonicelli in Torino over the last ten years. As a member of the Turin City Council from 2001 till 2006 he worked in the Cultural and in the City Planning committees. Local and national leader in the CGIL School, University and Research Trade Unions. From 1961 till 1971 he taught mathematics in the Universities of Turin and Genova. From 1971 to 1998 he kept lessons in Analysis, Geometry, Mathematical Applications in Economy at the Turin Polytechnic where he sat in the Administrative Council and was Director of the Didactics Service Centre of the Faculty of Architecture. In 1969-1970 he carried on research work in the Math-Stats Department of the University of California in Berkeley. From 1940 to 1948 he lived in Australia so that English became his mother language. 13
- 14. There are no borders for the human mind There exists a deep contradiction between the perception of real space and time as bounded entities, on the one hand, and our mind’s refusal to accept the idea that “nothing else” exists on the other side of any spatial or temporal border, on the other hand. (what was there before the big bang? what is there at the end of our more or less known universe?). The long and tiring transition from a “bounded” number of things to the concept of an infinite set of numbers (Bolzano, Weierstrass) begins with this attempt to understand what we mean with the word “infinity”. It was even more difficult to accept the existence of different numerical infinities (numerable, continuous) and to understand what distinguishes one infinity from the other; to the point that few yet understand how the set of rational numbers (fractions) can contain as many elements as the set of positive integers. 14
- 15. The Limit of Mathematicians The concept of the limit Mathematics and the limit A geometrical example of the limit Geometrical example: further remarks Geometrical example: the method of exhaustion Archimedes and the method of exhaustion 15
- 16. The concept of the Limit In modern mathematics the concept of the limit arises from the twofold requirement to specify the nature of the set of real numbers and to remove the many critiques to the Newtonian definition of the derivative. In Cauchy’s definition the limit is associated with a function’s behaviour when we approach a fixed point or when this point increases indefinitely. A satisfactory mathematical approach to the limit concept and the computational rules appears only at the end of the XIX century. More recently this fundamental concept was introduced in all mathematical fields, not only in the study of functions of several real variables but also in the study of general abstract spaces such as metric and topological spaces. 16
- 17. The Mathematicians of the Limit Gottfried Wilhelm von Leibniz (1646 – 1716), German philosopher, mathematician, scientist. Sir Isaac Newton (1643 – 1727), English physicist and mathematician "one of the greatest minds of all time”. Bernard Placidus Johann Nepomuk Bolzano (1781 – 1848) Bohemian mathematician, philosopher, logician. Augustin-Louis Cauchy (1789 – 1857), French mathematician and engineer. Karl Theodor Wilhelm Weierstrass (1815 – 1897), German mathematician, "father of modern analysis”. Ludwig Wittgenstein (1889 – 1951), Austrian philosopher and logician. Source: Wikipedia 17
- 18. A geometrical example of the limit Consider a polygon inscribed in a circle … When the number of sides increases the polygon looks more and more like the circle. If we refer to the polygon as an n-gon, where n is the number of its sides, we can suggest some mathematical remarks … 18
- 19. Geometrical example: further remarks As n increase the n-gon gets like the circle. When n tends to infinity the n-gon approaches the circle. The n-gon’s limit, when n tends to infinity, is the circle! circle “The n-gon never identifies with the circle but it gets so near that in practice it can be considered as a circle”. 19
- 20. Geometrical example: the method of exhaustion Consider a circle and all the inscribed n-gons. As the number of sides increases the n-gons exhaust the portion of plain occupied by the circle. The area An of each n-gon is easily computed as the sum of the areas of all the triangles in which it may be divided. When n increases indefinitely the areas An approach what we shall call the area of the circle. Mathematicians say that, when n tends to infinity, the areas An tend towards the area A of the circle and they write lim An = A n→ ∞ 20
- 21. Archimedes and the method of exhaustion About 2300 years ago Archimedes (287-212 a.C.) used this idea: by computing the areas of the first n- gons, he obtained an excellent approximation for the area of the circle. In this way he found the first two decimals of the number π = 3,14159265358979 . . . The method of exhaustion that Archimedes described in The Method represents the basis for the concept of the integral developed by Newton and Leibniz in the XVII century. 21
- 22. The abstract nature of the limit concept Abstract spaces Painting the derivative Infinite and infinitesimal Source: Calculus has practical applications, such as understanding the true meaning of the infinitesimals. (Image concept by Dr. Lachowska, MIT) 22
- 23. Abstract Spaces The abstract nature of Cauchy’s definition of the limit gains new value only when it is extended to abstract spaces and anyway it doesn’t seem to overcome the doubts regarding the definition of the derivative. Infact Newton’s and Leibniz’s approach to differential calculus was opposed by other scholars and among them by Karl Marx. 23
- 24. Definition of the derivative Actually the definition of the derivative given by Newton presents an obvious inconsistency. If we consider the ratio (mean velocity) between the increase ∆s of the quantity s (distance covered) and the corresponding increase ∆t of the variable t (time taken), it has sense only if the denominator ∆t is different from zero. On the other hand, simple algebraic computations show that the ratio can always be transformed so that we can put ∆t = 0 and so get the “derivative” (instantaneous velocity) of the quantity s. In other words, we accept a posteriori an operation which a priori had been ruled out. 24
- 25. Painting the derivative … The first figure gives us the value of the derivative at each Grafico di point: it is the slope of the tangent line to the function’s f(x)=1/x graph, where the tangent line in a point is defined as “the limit position” of all straight lines passing through that point. Here we have the derivative according to Newton’s definition, that Cauchy made rigorous by Grafico di introducing the limit of the ratio f’(x)=-1/x2 ∆s/ ∆t. In the figure below (following Marx’s approach) the derivative is an “operator”, i.e. a mathematical instrument that associates to any given function another function according to a certain algorithm. In our case, the given function is 1/x and we associate its “derivative function” - 1/x² 25
- 26. Infinite & Infinitesimal The concepts of “infinitesimal = point” and of “infinite = beyond any bound” suggest a similarity with the identification between the infinitely small and the infinitely big that appears in Hebrew mystic literature. This remark induces to build a bridge between mathematics, logic and philosophy (already existing since a long time, e.g. Wittgenstein’s work). 26
- 27. Where will all this? … At Finale Ligure, “locus finalis” Finalborgo Finalmarina «We like to think that, for three days, the Pillars of Knowledge there will mark the location of boundary». Finale Ligure, Savona, Italy 27
- 28. Finalborgo_Architectural Complex of Santa Caterina The place name Final Borgo derives from Burgum Finarii, a border town (ad fines, at the border) at the time of the Romans and administrative centre of the marquisate of the Del Carretto family between the 14th and 16th centuries. Closed in between medieval walls and still well preserved, interspersed with semi-circular towers and interrupted only by the gates, Borgo di Finale immediately offers the visitor a feeling of protection and welcome (Source: www.borghitalia.it). 28
- 29. Finalmarina_Castelfranco Fortresss www.scalo.org/images/finaliu.jpg Castelfranco Fortress is placed on the height of Gottaro, the promontory that divides the Sciusa Valley from the Pora's one. The fortress date back to the XIVth century and it was built by the Republic of Genoa. After a long and suffered history, the fortress, under the domain of the Regno di Sardegna, became a jail at first, and then, it became an infirmary. Nowadays, it's just a cultural and tourist destination. 29
- 30. Restlessness Festival 2010 organization Promotional Committee: Comune di Finale Ligure Fondazione A. De Mari - Cassa di Risparmio di Savona Provincia di Savona Planning and organization: Circolo degli Inquieti di Savona 30
- 31. Circolo degli Inquieti Inquietus Cultural Club Member profile: Temperament emotional and imaginative, and at the same time self-critical. Ill suited for conformity to rigid rule. Cultural traveller always available to leave for unusual destinations. Develop and sustain a lifelong desire for knowledge. Maintain a Socratic ignorance. Know and develop yourself. Be pervaded by doubts. Aim at understanding others and their differences. Be aware of well-known and knowable matters. Perceive magic and mystery. Embark on new adventures and initiatives. Club motto: “The more I understand, the more I do not know”, philosopher Tommaso Campanella. 31
- 32. Restlessness Festival 2010 Events Debates & meetings: Promotion of restlessness as a condition of being human and a synonym of knowledge and cultural growth. Exhibitions & Shows: Proposition differing aspects of artistic creativity. InquietaMente: Innovative projects dedicated to young people, work and businesses. Inquietus Celebration (IV edition): "Celebration" of restless personalities who have distinguished themselves for their high intellectual and emotional vitality in specific areas of human activity. Inquietus of the Year (XIII edition): Celebration of personality that has stood out for being restless. 32
- 33. Inquietus of the year “The Year” Edition Celebration Inquietus of the year 2009 XIII 2010 ? 2008 XII 2009 Don Luigi Ciotti 2007 XI 2008 Milly & Massimo Moratti 2006 X 2007 Raffaella Carrà 2005 IX 2006 Règis Debray 2004 VIII 2005 Costa Gavras 2003 VII 2004 Oliviero Toscani 2002 VI 2003 Barbara Spinelli 2001 V 2002 Antonio Ricci 2000 IV 2001 Gino Paoli 1998 III 1999 Francesco Biamonte 1997 II 1998 Gad Lerner 1996 I 1997 Carmen Llera Moravia 33
- 34. Citations & Link The logo of the “Circolo degli Inquieti” was designed by Ugo Nespolo www.nespolo.com Logo of the “Festa dell’Inquietudine” by Oliviero Toscani & La Sterpaia www.lasterpaia.it “Inquietudine e Limite” logo by Marco Prato www.manolab.it Pictures by Emilio Rescigno www.emiliorescigno.it Presentation background: Ardesia, Pietra di Liguria. “Slate in Liguria: One of the most striking features of Liguria is the extent to which slate is used: the dappled grey roofs, the resorts along the Riviera, the region's medieval churches and their black and white striped facades, the homes of the aristocracy with their grand slate stairways, overdoor decorations, … wherever you look this fascinating stone has left its mark on the region's history and everyday life”, www.portale-ardesia.com 34
- 35. See you at the Restlessness Festival 2010 … The unique atmosphere of Finale Ligure, historic Finalborgo, fascinating Varigotti and the Italian Western Riviera, the curiosity of the events offered at the Festa dell’Inquietudine and the flavors of the cuisine and fine wine from Liguria make the three days of Restlessness celebration really unforgettable. www.festainquietudine.it 35

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