festa d ll’ dell’inquietudine d III edition 14 – 15 - 16 May 2010 Finale Ligure SV, Italian Riviera
restlessness festival 2010 restlessness & limit
The Limit in Mathematics Manfredo Montagnana
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
InquietaMente
Inquietus Celebration
Inquietus of the year
Events involving prominent personalities from the Italian and worldwide cultural, scientific and entertainment arena.
Leitmotif of the year 2010: “Restlessness Restlessness & Limit Limit” in Philosophy
Mathematics
Science & Species Sport
Economy & Technology & Organization & Resources Engineering Leadership
Life, Beyond Life, Life Life Other Worlds
Venue: Finale Ligure SV, Italy, Finalborgo (architectural ( complex omplex of S Santa Caterina) C i ) and d Fi Finalmarina l i (C ((Castelfranco C Castelfranco lf F Fortress) Fortress) Time period: 14 - 15 -16 May 2010.
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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 Th abstract b t t nature t off the th limit li it conceptt 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
Culture & Entertainment p performative event dedicated to the “Restlessness� 4
Restlessness Restlessness concerns knowledge and cultural and sentimental growing, restlessness concern not only those living characterizes anxiety states. states Restlessness surrounds and permeates lovers, who is tormented by the artistic creativity, who thirst for knowledge, knowledge who is pervaded with doubt, who is fascinated by the mystery, who is seduced by life, life those who participate in the dramas of contemporary humanity and, even more, more those who are directly afflicted afflicted. 5
Festa dell’Inquietudine 2010 Restlessness Restlessness tl F Festival ti l 2010
Limit (1) dividing line; (2) extremity t it to t which hi h th they can gett something; thi (3) term that you can not or should not be exceeded, d d [[even iin th the fi figurative ti sense]] In the III edition of the “Festa dell’Inquietudine” we work on the link:
«restlessness & limit» 6
Restlessness & Limits in … Philosophy
Sport
MATHEMATICS
Technology & Engineering
E Economy, Resources, R Environment, …
O Organizations i ti & Leadership
Science & Species
Life Life, Other Worlds, Worlds Beyond Life
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Go beyond … «We live in an age g where everything y g seems to be overcome: from sports performance to scientific knowledge, to the same "human species"». i " «For us it is obvious to think that the restlessness tl t push to h the th man to t th the limit li it and maybe beyond».
Elio Ferraris, Presidente del Circolo degli Inquieti
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“PLVS PLVS VLTRA VLTRA”,, go beyond …
«We W live li in i an age where h everything thi seems tto b 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» 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.
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They have gone beyond ‌ Of the mythology of Heracles-Hercules we like that sententious “Nec Nec plus ultra ultra" carved on the columns the same name. It came after extraordinary f feats in which h h the h hero h had h d challenged and defeated gods and monsters, and showed a llimit. But even more fascinate those who have passed that column! Ulysses, Christopher Columbus, but also Plato that "beyond" beyond places the lost Atlantis. 10
Plus Ultra We like, even, Charles V, Holy Roman Emperor (Carlos I de Espa単a )), which transforms the ban encouragement to go further, and the "Plus Ultra" becomes his motto.
Fonte: wikipedia
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Restlessness & Limit
Mathematics ď‚Ž ď‚Ž ď‚Ž
There are no borders for the human mind The limit for mathematicians The abstract nature of the limit concept Manfredo Montagnana 12
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. Genova From 1971 to 1998 he kept lessons in Analysis, Analysis Geometry, Mathematical Applications in Economy at the Turin Polytechnic where he sat in the Administrative Council and was y of Director of the Didactics Service Centre of the Faculty 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
There are no borders for the h human mind i d
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
The Limit of Mathematicians
The concept of the limit Mathematics and the limit A geometrical example of the limit Geometrical example: p further remarks Geometrical example: the method of exhaustion Archimedes and the method of exhaustion 15
The concept of the Limit
In modern mathematics the concept of the limit arises from o tthe e ttwofold o o d requirement equ e e t to spec specify y the t e nature atu e o of tthe e set of real numbers and to remove the many critiques to the Newtonian definition of the derivative.
In Cauchy Cauchy’s 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 M tl thi this ffundamental d t l conceptt was iintroduced t d d iin 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 spaces. 16
The Mathematicians of the Limit Gottfried Wilhelm von Leibniz (1646 – 1716), 1716) German philosopher, mathematician, scientist. Sir Isaac Newton (1643 – 1727), English physicist and mathematician "one one of the greatest minds of all time 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 p p and logician. g Source: Wikipedia
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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 n mber of its sides number sides, we e can ssuggest ggest some mathematical remarks … 18
Geometrical example: f h remarks further k
As n increase the n n-gon gon gets like the circle. When n tends to infinity the n-gon approaches the circle. Th n-gon’s The ’ li limit, it when h n ttends d tto iinfinity, fi it iis th 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 circle”. 19
Geometrical example: th method the th d off exhaustion h ti
Consider a circle and all the inscribed n n-gons 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 n-gon 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, th areas An the A tend t d ttowards d th the area A off th the circle i l and they write lim An = A n→ ∞ 20
Archimedes and the method of exhaustion About 2300 years ago Archimedes (287 212 a.C.) (287-212 C ) used d thi this id idea: b by computing the areas of the first ngons, he obtained an excellent approximation i ti ffor the th area off the th 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 p the basis for the concept p of the integral developed by Newton and Leibniz in the XVII century.
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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 Dr. Lachowska Lachowska, MIT) 22
Abstract Spaces
The abstract nature of Cauchy’s definition of the limit gains i new value l only l when h it iis extended t d d tto abstract b t t spaces and anyway it doesn’t seem to overcome the doubts regarding g g 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.
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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. O the On th other th hand, h d simple i l algebraic l b i computations t ti show that the ratio can always be transformed so that we can p put ∆t = 0 and so g 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 out. 24
Painting g the derivative … The first figure gives us the value of the derivative at each point: it is the slope of the tangent line to the function’s 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 Newton s definition definition, that Cauchy made rigorous by introducing the limit of the ratio ∆s/ ∆t.
Grafico di f(x)=1/x
Grafico di f’(x)=-1/x2
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 g to a certain algorithm. g In our case,, the given function is 1/x and we associate its “derivative function” - 1/x² 25
Infinite & Infinitesimal
The concepts of “i fi it i l = point” “infinitesimal i t” and d off “infinite = beyond any bound” suggest a similarity with the identification between the infinitely small and the infinitely bi that big th t appears iin H Hebrew b mystic literature. This remark induces to build a bridge between mathematics, logic and philosophy (already existing since a long time time, e e.g. g Wittgenstein’s work). 26
Where will all this? … At Finale Ligure, “locus finalis”
Finalborgo
Finalmarina «We like to think
that, h for f three h days, the Pillars of g there will Knowledge mark the location of boundary». Finale Ligure, Savona, Italy 27
Finalborgo_Architectural Complex off S Santa t C Caterina t i
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 p with semi-circular in between medieval walls and still well ppreserved,, interspersed 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
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. Genoa After a long and suffered history 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
Restlessness estlessness Festival 2010 organization i ti Promotional Committee:
Comune di Finale Ligure
Fondazione A. De Mari di Risparmio Ri i di S Savona
Provincia di Savona
Cassa
Planning and organization: Circolo degli Inquieti di Savona 30
Circolo degli Inquieti I Inquietus i t C Cultural lt l Club Cl b
Member profile: Temperament emotional and imaginative, and at the same time self-critical. Ill suited for conformity to rigid rule. Cultural C lt l ttraveller ll always l available il bl tto lleave ffor unusuall destinations. Develop and sustain a lifelong desire for knowledge. Maintain a Socratic ignorance ignorance. Know and develop yourself yourself. Be pervaded by doubts. Aim at understanding others and their differences. Be aware of well-known well known and knowable matters matters. Perceive magic and mystery. Embark on new adventures and initiatives.
Club motto: “The The more I understand, understand the more I do not know” know , philosopher Tommaso Campanella. 31
Restlessness estlessness 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 off artistic ti ti creativity. ti it InquietaMente: Inquieta Mente: Innovative projects dedicated to yyoung gp people, p , 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 restless. 32
Inquietus of the year “The Year” 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1998 1997 1996
Edition Celebration Inquietus of the year XIII 2010 ? XII 2009 Don Luigi Ciotti XI 2008 Milly & Massimo Moratti X 2007 Raffaella Carrà IX 2006 Règis Debray VIII 2005 Costa Gavras VII 2004 Oliviero Toscani VI 2003 Barbara Spinelli V 2002 Antonio Ricci IV 2001 Gino Paoli III 1999 Francesco Biamonte II 1998 Gad Lerner I 1997 Carmen Llera Moravia 33
Citations & Link
The logo of the “Circolo degli Inquieti” was designed by Ugo Nespolo www.nespolo.com 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 Ardesia, Pietra di Liguria Liguria. “Slate 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 g medieval churches and their black and white Riviera,, the region's 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” life , www.portale-ardesia.com www portale ardesia com 34
See you at the Restlessness estlessness Festival F ti l 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
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