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Working Mathematician을
위한 수학의 기초
EUREKA!
김은섭
(초록)
수학은 무한(infinite)을 연구하는 학문이다.
수학의 심연에 자리한 무한은 다른 어떤 개념보다 더욱
더 명료성이 요청된다.
Working Mathematician에게 꼭 필요한 무한의 개념을
역사 발생적 관점과 보편적 언어(Logic)를 사용하여
설명하고자 한다. 주요한 내용은 다음과 같다.
1. The limiting process.
2. Classic logic and Modern logic.
3. Logic and Set theory.
4. Intuition and Idea.
• “The true method of foreseeing the
future of mathematics is to study its
history and its actual state.” (溫故知新)
Henri Poincaré <4th International Congress of
Mathematics at Rome un 1908>
• Mathematics has gained in the course
of its history by an investigation of the
infinite.
• Spirit is freedom in the bondage of
existence, it is open to infinity.
나의 수학노정
학교수학
대학수학
수학사
고전수학
현대수학
내용
공식(Formulas)
개념(Concept)
Pythagoras(582~500 BC)
• Fundamental doctrine : the essence of
things dwells in numbers.
• Pythagoras’ theorem
Pythagoras(582~500 BC)
• The side and the diagonal of a square
are noncommensurable.
Pythagoras(582~500 BC)
• It did not seem possible to build
geometry on the basis of numbers, due
to a conflict between their notions of
number and length.
∴ geometrical quantities had to be treated
separately from numbers or, rather,
without mentioning any numbers except
rationals.
Heraclitus(535~475 BC)
• Everything changes and nothing
remains still … and … you cannot step
twice into the same stream.
Paremenides(515~460 BC)
• Reality is one, change is impossible, and
existence is timeless, uniform, necessary,
and unchanging.
Anaxagoras(510~423 BC)
• “In the small there is no smallest, but
there is always still a smaller. For what is
cannot cease to be no matter how far it
is being subdivided.”
The continuum is not composed of
discrete elements which are “separated
from one another as though chopped off
by a hatchet.”
Anaxagoras(510~423 BC)
• Space is not only infinite in the sense
that in it one nowhere reaches an end;
but at every place it is infinite if one
proceeds inward toward the small. A
point can only be identified more and
more precisely by the successive stages
of a process of division continued ad
infinitum.
Anaxagoras(510~423 BC)
This is in contrast with the state of
immobile (the calm) and completed
being in which space appears to direct
perception.
Zeno of Elea(490~430 BC)
• Zeno’s paradoxes
<The race of Achilles with the tortoise>
Conclusion : the impossibility of grasping
the continuum as a fixed being. (We
need an idealization of the continuum)
Zeno of Elea(490~430 BC)
Aristotle remarks on the solution of Zeno’s
paradox that “What is moved does not move by
counting”, or more precisely, “When you divide
the continuous line in two halves, you take the
one [dividing] point for two : you make it both
the beginning and the end. In dividing thus,
neither the line nor the motion remains
continuous … . In the continuous, there are
indeed infinitely many halves, but they are not
actually(in reality) but potentially.”
Zeno of Elea(490~430 BC)
• The First examples of a method of proof
called “reductio ad absurdum”.
Socrates(470~399 BC)
Democritus(460~370 BC)
• Formulation of an atomic theory of the
universe.
• Anaxagoras is opposed by the strictly
atomistic theory of Democritus.
Plato(429~347 BC)
• Plato’s profoundest metaphysical doctrine,
his doctrine of ideas, was clad in
mathematical garb when he expounded it
rigorous form ; it was a doctrine of ideal
numbers, through which the mind was to
apprehend the structural composition of
the world. The spatial figures and relations
investigated by geometry – half notional
category, half sense perception – were to
him the mediators between the
phenomenon and the idea.
Plato(429~347 BC)
• He refused admission to the academy to
those who were not trained in
mathematics.
Plato(429~347 BC)
• To Plato, the mathematical lawfulness and
harmony of nature appeared as a divine
mind-soul. <The twelfth book of the Law>
想起說
現象界
IDEA 世界
幾何學
Plato(429~347 BC)
• Plato, clearly conscious of his proposed
goal – the salvation of the phenomenon
through the idea, seems to have been
the first to conceive a consistent
atomism of space.
Eudoxus(408~355 BC)
• The late Pythagorean description of the
quadrivium ;
arithmetic : numbers at rest
music
: numbers in motion
geometry : magnitudes at rest
astronomy : magnitudes in motion
Eudoxus(408~355 BC)
•
But Eudoxus eliminated the dualism of number
and magnitude. His idea was this : rather than
say what the ratio of two magnitudes is, it
suffices to define a notion of two such (possibly
nonexistent) ratios being equal, and this he did
by a subtle quantification over all the
Pythagorean numbers. Despite the fact that it
was in the Elements for all to read, this
construction of the real number system was not
understood - not even by Galilei. It was just last
century that the notion was re-invented by
Richard Dedekind. I know of no parallel to this in
the history of human thought.
Eudoxus(408~355 BC)
Eudoxus defined a notion of two things
being equal in order to construct the
things themselves. Here was a triumph of
formalism, a victory of syntax over
semantics!
Eudoxus(408~355 BC)
1. In Place of the untenable
commensurability he sets down the
axiom : if a and b are any two segments,
then a can always be added to itself so
often that the sum na exceeds b. This
means that all segments are of a
comparable order of magnitude, or that
there exists neither an actually infinitely
small nor an actually infinitely large in
the continuum.
Eudoxus(408~355 BC)
2. And what is it that characterizes the
individual segment ratio? Eudoxus replies :
two segments ratios, a:b and a’:b’, are
equal to each other if, for arbitrary natural
numbers m and n, the fulfillment of the
condition in the first line below invariably
entails the validity of the corresponding
condition in the second line:
(Ⅰ)
na > mb
na’ > mb’
(Ⅱ)
na = mb
(Ⅲ)
na’ = mb’
na < mb
na’ < mb’
Aristotle(384~322 BC)
• Aristotle’s doctrine ;
the infinite is forever being on the way and
therefore exists only “potentially” not
“actually”.
• The logic of Aristotle – the greatest
logician before Gődel – is based on the
theorem of the excluded third(排中律) and
essentially a logic of the finite. It is
inadequate for mathematics (It was already
inadequate for the mathematics of his day).
Euclid(about 300 BC)
• Euclid’s Elements.
Archimedes(287~212 BC)
• Eudoxus – Archimedean Axiom.
• General method of exhaustion.
e.g., the area of circle is that it is equal to
the area of a triangle whose height is equal
to it radius and whose base is equal to its
circumference :
<Summary up to now>
Antiquity has bequeathed to us two
important contributions to the problem of
the continuum :
(1) a far-reaching analysis of the mathematical
question of how to fix a single position in
the continuum, [The pure geometry of the
Greeks, in elevating itself above the
inexactitude of the sense date, applies the
idea of existence (not only to the natural
numbers but also) to points in space.],
<Summary up to now>
(2) the discovery of the philosophical
paradoxes which have their origin in the
intuitively manifest nature of the
continuum.
F. Vieta(1540~1603)
put letters
Geometrical
Problem
calculations
Algebraic
Problem
Solution
F. Vieta(1540~1603)
Example. (Trisection of an angle). The
famous classical problem “Datum
angulum in tres partes aequales
secare” becomes, with the help of
and of some simple calculations, the
algebraic equation
(see Viète 1593, Opera, p.290)
This perfection of this idea led to Descartes’s
“Geometry”.
René Descartes(1596~1650)
<Constructing the product and quotient of lengths (segments)>
The aggregation of segments is a semifield.
度量衡統一
Pierre de Fermat(1601~1665)
• The method of infinite descent.
Blaise Pascal(1623~1662)
• Mathematical induction(ascent form).
Isaac Newton(1642~1726)
• Newton immediately gives a general
definition of number. We recall that in
antiquity number denoted a collection of
units (i.e., natural numbers), and that
rations of numbers (rational numbers) and
ratios of like quantities(real numbers) were
not regarded as numbers. Claudius Ptolemy
(2nd century AD) and Arab mathematicians
did identify ratios with numbers, but in
16th- and 17th-century Europe the Euclidean
tradition was still very strong. Newton was
the first to break with it openly.
Isaac Newton(1642~1726)
He wrote :
By a ‘number’ we understand not so much a
multitude of units as the abstract ratio of any
quantity to another quantity which is
considered to be unity. It is threefold : integral,
fractional, and surd. An integer is measured by
unity, a fraction by a submultiple part of unity,
while a surd is incommensurable with unity.
Isaac Newton(1642~1726)
With characteristic brevity, Newton
goes on to define negative numbers :
Quantities are either positive, that is, greater
than zero, or negative, that is, less than zero,
… in geometry, if a line drawn with
advancing motion in some direction be
considered as positive, then its negative is
one drawn retreating in the opposite
direction.
To denote a negative quantity … the sign – is
usually prefixed, to a positive one the sign +.
Isaac Newton(1642~1726)
Then Newton formulates rules of operation
with relative numbers. We quote his
multiplication rule : “A product is positive
if both factors are positive or both
negative and it is negative otherwise.”
He provides no “justifications” for these
rules.
Isaac Newton(1642~1726)
It is tempting to think that power series
“complete” the rational functions in the
same way that the real numbers complete
the rational numbers, and indeed Newton
was extremely impressed by analogy :
I am amazed that it has occurred to no one … to fit the
doctrine recently established for decimal numbers in similar
fashion to variables, especially since the way is then open
to more striking consequences. For since this doctrine in
species has the same relationship to Algebra that the
doctrine in decimal numbers has to common Arithmetic, its
operations of Addition, Subtraction, Multiplication, Division
and Root extraction may be easily learnt from the latter’s.
Newton(1671), p. 35.
Isaac Newton(1642~1726)
Cutting a line segment into equal parts.
• He constructed the number line.
Gottfried Wilhelm Leibniz
(1646~1716)
• Symbolic Logic
It is obvious that if we could find characters or
signs suited for expressing all our thoughts as
clearly and as exactly as arithmetic expresses
numbers or geometry expresses line, we could
do in all matters, insofar as they are subject to
reasoning, all that we can do in arithmetic
and geometry.
- Gottfried Wilhelm Leibniz
Preface to the General Science, 1677.
<The limiting process>
• The Parabola
. If
then increases to
increases by
,
.
∴
.
Therefore, the slope of the line connecting
with
is equal to
.
If
tends to zero, this slope will approach that
of the tangent to the parabola.
<The limiting process>
Leibniz imagines that
and
become “infinitely
small” and denotes then
by
and
. Then we
neglect the term
,
which is “infinitely smaller”
than
, and obtain
Tangent to parabola
or
.
<The limiting process>
• From Cauchy to Weierstrass.
<The limiting process>
Symbolic form :
<The limiting process>
(Example)
Pigeonhole Principle
• If n pigeons fly into few than n
pigeonholes, then one hole has more
than one pigeon.
• The pigeonhole Principle occurred in
Dirichlet’s Vorlesungen … , edited and
published by Dedekind in 1863.
대수학 – 논리학 비교
Algebra
Logic
constant
variables
numbers
(e.g., rational numbers)
propositions
constant operations
+, -, ×, ÷
∼, ∨, ∧, →
variables
x, y, z, …
x, y, z, …
operations
+, -, ×, ÷
∼, ∨, ∧, →
extensions
polynomials or
polynomial functions
propositional
functions
방정식과 항등식 만들기
(수의 생성)
명제 만들기
(명제 생성)
대수학 – 논리학 비교
• George Boole(1815~1864)
• Gottlob Frege(1848~1925)
Logic
Set Theory
Intentional
Mathematics
內包
Extensional
Mathematics
外延
Richard Dedekind(1831~1916)
• In 1888, he published Was sind und sollen
die Zahlen?
• He was the first to define infinite
set(complete infinite), with the definition
being a set for which there is a one-to-one
correspondence with a proper subset. This
is just the negation of the Pigeonhole
Principle. Dedekind in effect had inverted a
negative aspect of finite cardinality into a
positive existence definition of the infinite.
George Cantor(1845~1918)
Set theory was born on that day in
December 1873 when Cantor established
that the continuum is not countable :
There is no one-to-one correspondence
between the natural numbers
w={0,1,2,3,…} and the real numbers ℝ.
Transfinite logic
• Suppose several pieces of chalk are lying in
front of me ; then the statement ‘all these
pieces of chalk are white’ is merely an
abbreviation of the statement ‘this piece is
white & that piece is white & …’ (where
each piece is being pointed at in turn).
Similarly ‘there is a red one among them’ is
an abbreviation of ‘this is red ∨ that is red
∨ … ‘ But only for a finite set, whose
elements can be exhibited individually, is
such an interpretation feasible.
Transfinite logic
In the case of infinite sets, the meaning
of ‘all’ and ‘some’ involves a profound
problem which touches upon the core of
mathematics, the very secret of the
infinite. The situation here may be
compared to the transition from finite to
infinite sums; the meaning of the latter is
tied to special conditions of convergence,
and one may not deal with them in
every respect as with finite sums.
Transfinite logic
• Lagrange’s theorem
Transfinite logic
•
Transfinite logic
Conclusion :
We can only represent the completed
infinite in the symbol. Yet up to now it
has been only in mathematics and physics,
as far as we can see ; that the symbolic –
theoretical construction has taken on such
solidity that it is compelling for everyone
whose mind opens itself to these science.