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.

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.
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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
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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.
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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
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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
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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