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Topic: Matheology § 172
Replies: 2   Last Post: Dec 7, 2012 12:05 PM

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Scott Berg

Posts: 2,111
Registered: 12/12/04
Re: Matheology � 172
Posted: Dec 7, 2012 12:05 PM
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"WM" <> wrote in message

Matheology § 172

<snip crap>

>Regards, WM

why cut and paste gernic crap here ? Put the WHOLE ENCHALADA ON DUDE !!!



The real numbers: Stevin to Hilbert

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By the time Stevin proposed the use of decimal fractions in 1585, the
concept of a number had developed little from that of Euclid's Elements.
Details of the earlier contributions are examined in some detail in our
article: The real numbers: Pythagoras to Stevin
If we move forward almost exactly 100 years to the publication of A treatise
of Algebra by Wallis in 1684 we find that he accepts, without any great
enthusiasm, the use of Stevin's decimals. He still only considers finite
decimal expansions and realises that with these one can approximate numbers
(which for him are constructed from positive integers by addition,
subtraction, multiplication, division and taking nth roots) as closely as
one wishes. However, Wallis understood that there were proportions which did
not fall within this definition of number, such as those associated with the
area and circumference of a circle:-
... such proportion is not to be expressed in the commonly received ways of
notation: particularly that for the circles quadrature. ... Now, as for
other incommensurable quantities, though this proportion cannot be
accurately expressed in absolute numbers, yet by continued approximation it
may; so as to approach nearer to it than any difference assignable.
For Wallis there were a variety of ways that one might achieve this
approximation, so coming as close as one pleased. He considered
approximations by continued fractions, and also approximations by taking
successive square roots. This leads into the study of infinite series but
without the necessary machinery to prove that these infinite series
converged to a limit, he was never going to be able to progress much further
in studying real numbers. Real numbers became very much associated with
magnitudes. No definition was really thought necessary, and in fact the
mathematics was considered the science of magnitudes. Euler, in Complete
introduction to algebra (1771) wrote in the introduction:-
Mathematics, in general, is the science of quantity; or, the science which
investigates the means of measuring quantity.
He also defined the notion of quantity as that which can be continuously
increased or diminished and thought of length, area, volume, mass, velocity,
time, etc. to be different examples of quantity. All could be measured by
real numbers. However, Euler's mathematics itself led to a more abstract
idea of quantity, a variable x which need not necessarily take real values.
Symbolic mathematics took the notion of quantity too far, and a reassessment
of the concept of a real number became more necessary. By the beginning of
the nineteenth century a more rigorous approach to mathematics, principally
by Cauchy and Bolzano, began to provide the machinery to put the real
numbers on a firmer footing. Grabiner writes [2]:-
... though Cauchy implicitly assumed several forms of the completeness axiom
for the real numbers, he did not fully understand the nature of completeness
or the related topological properties of sets of real numbers or of points
in space. ... Cauchy did not have explicit formulations for the completeness
of the real numbers. Among the forms of the completeness property he
implicitly assumed are that a bounded monotone sequence converges to a limit
and that the Cauchy criterion is a sufficient condition for the convergence
of a series. Though Cauchy understood that a real number could be obtained
as the limit of rationals, he did not develop his insight into a definition
of real numbers or a detailed description of the properties of real numbers.
Cauchy, in Cours d'analyse (1821), did not worry too much about the
definition of the real numbers. He does say that a real number is the limit
of a sequence of rational numbers but he is assuming here that the real
numbers are known. Certainly this is not considered by Cauchy to be a
definition of a real number, rather it is simply a statement of what he
considers an "obvious" property. He says nothing about the need for the
sequence to be what we call today a Cauchy sequence and this is necessary if
one is to define convergence of a sequence without assuming the existence of
its limit. He does define the product of a rational number A and an
irrational number B as follows:-
Let b, b', b'', ... be a sequence of rationals approaching B closer and
closer. Then the product AB will be the limit of the sequence of rational
numbers Ab, Ab', Ab'', ...
Bolzano, on the other hand, showed that bounded Cauchy sequence of real
numbers had a least upper bound in 1817. He later worked out his own theory
of real numbers which he did not publish. This was a quite remarkable
achievement and it is only comparatively recently that we have understood
exactly what he did achieve. His definition of a real number was made in
terms of convergent sequences of rational numbers and is explained in [22]
where Rychlik describes it as "not quite correct". In [28] van Rootselaar
disagrees saying that "Bolzano's elaboration is quite incorrect". However in
J Berg's edition of Bolzano's Reine Zahlenlehre which was published in 1976,
Berg points out that Bolzano had discovered the difficulties himself and
Berg found notes by Bolzano which proposed amendments to his theory which
make it completely correct. As Bolzano's contributions were unpublished they
had little influence in the development of the theory of the real numbers.
Cauchy himself does not seem to have understood the significance of his own
"Cauchy sequence" criterion for defining the real numbers. Nor did his
immediate successors. It was Weierstrass, Heine, Méray, Cantor and Dedekind
who, after convergence and uniform convergence were better understood, were
able to give rigorous definitions of the real numbers.
Up to this time there was no proof that numbers existed that were not the
roots of polynomial equations with rational coefficients. Clearly v2 is the
root of a polynomial equation with rational coefficients, namely x2 = 2, and
it is easy to see that all roots of rational numbers arise as solutions of
such equations. A number is called transcendental if it is not the root of a
polynomial equation with rational coefficients. The word transcendental is
used as such number transcend the usual operations of arithmetic. Although
mathematicians had guessed for a long time that p and e were transcendental,
this had not been proved up to the middle of the 19th century. Liouville's
interest in transcendental numbers stemmed from reading a correspondence
between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove
that e is transcendental but he did not succeed. However his contributions
led him to prove the existence of a transcendental number in 1844 when he
constructed an infinite class of such numbers using continued fractions.
These were the first numbers to be proved transcendental. In 1851 he
published results on transcendental numbers removing the dependence on
continued fractions. In particular he gave an example of a transcendental
number, the number now named the Liouvillian number
where there is a 1 in place n! and 0 elsewhere.
One of the first people to attempt to give a rigorous definition of the real
numbers was Hamilton. Perhaps, if one thinks about it, it is logical that he
would be interested in this since his introduction of the quaternions had
shown that there were new previously unstudied number systems. In fact came
close to the idea of a Dedekind cut, as Euclid had done in the Elements, but
failed to make the idea into a definition (again Euclid had spotted the
property but never thought to use it as a definition). For a number a he
noted that there are rationals a', a", b', b", c', c", d', d", ... with
a' < a < a"
b' < a < b"
c' < a < c"
d' < a < d"
but he never thought to define a number by the sets {a', b', c', d', ... }
and {a", b", c", d", ... }. He tried another approach of defining numbers
given by some law, say x x2. Hamilton writes:-
If x undergoes a continuous and constant increase from zero, then will pass
successively through every state of positive ration b, and therefore that
every determined positive ration b has one determined square root vb which
will be commensurable or incommensurable according as b can or cannot be
expressed as the square of a fraction. When b cannot be so expressed, it is
still possible to approximate in fractions to the incommensurable square
root vb by choosing successively larger and larger positive denominators ...
One can see what Hamilton is getting at, but much here is without
justification - can a quantity undergo a continuous and constant increase.
Even if one got round this problem he is only defining numbers given by a
law. It is unclear whether he thought that all real numbers would arise in
this way.
When progress came in giving a rigorous definition of a real number, there
was a sudden flood of contributions. Dedekind worked out his theory of
Dedekind cuts in 1858 but it remained unpublished until 1872. Weierstrass
gave his own theory of real numbers in his Berlin lectures beginning in 1865
but this work was not published. The first published contribution regarding
this new approach came in 1867 from Hankel who was a student of Weierstrass.
Hankel, for the first time, suggests a total change in out point of view
regarding the concept of a real number:-
Today number is no longer an object, a substance which exists outside the
thinking subject and the objects giving rise to this substance, an
independent principle, as it was for instance for the Pythagoreans.
Therefore, the question of the existence of numbers can only refer to the
thinking subject or to those objects of thought whose relations are
represented by numbers. Strictly speaking, only that which is logically
impossible (i.e. which contradicts itself) counts as impossible for the
In his 1867 monograph Hankel addressed the question of whether there were
other "number systems" which had essentially the same rules as the real
Two years after the publication of Hankel's monograph, Méray published
Remarques sur la nature des quantités in which he considered Cauchy
sequences of rational numbers which, if they did not converge to a rational
limit, had what he called a "fictitious limit". He then considered the real
numbers to consist of the rational numbers and his fictitious limits. Three
years later Heine published a similar notion in his book Elemente der
Functionenlehre although it was done independently of Méray. It was similar
in nature with the ideas which Weierstrass had discussed in his lectures.
Heine's system has become one of the two standard ways of defining the real
numbers today. Essentially Heine looks at Cauchy sequences of rational
numbers. He defines an equivalence relation on such sequences by defining
a1 , a2 , a3 , a4 , ... and b1, b2 , b3 , b4 , ...
to be equivalent if the sequence of rational numbers a1 - b1, a2 - b2 , a3 -
b3 , a4 - b4 , ... converges to 0. Heine then introduced arithmetic
operations on his sequences and an order relation. Particular care is needed
to handle division since sequences with a non-zero limit might still have
terms equal to 0.
Cantor also published his version of the real numbers in 1872 which followed
a similar method to that of Heine. His numbers were Cauchy sequences of
rational numbers and he used the term "determinate limit". It was clear to
Hankel (see the quote above) that the new ideas of number had suddenly
totally changed a concept which had been motivated by measurement and
quantity. Similarly Cantor realised that if he wants the line to represent
the real numbers then he has to introduce an axiom to recover the connection
between the way the real numbers are now being defined and the old concept
of measurement. He writes about a distance of a point from the origin on the
If this distance has a rational relation to the unit of measure, then it is
expressed by a rational quantity in the domain of rational numbers;
otherwise, if the point is one known through a construction, it is always
possible to give a sequence of rationals a1 , a2 , a3 , ..., an , ... which
has the properties indicated and relates to the distance in question in such
a way that the points on the straight line to which the distances a1 , a2 ,
a3 , ..., an , ... are assigned approach in infinity the point to be
determined with increasing n. ... In order to complete the connection
presented in this section of the domains of the quantities defined [his
determinate limits] with the geometry of the straight line, one must add an
axiom which simple says that every numerical quantity also has a determined
point on the straight line whose coordinate is equal to that quantity,
indeed, equal in the sense in which this is explained in this section.
As we mentioned above, Dedekind had worked out his idea of Dedekind cuts in
1858. When he realised that others like Heine and Cantor were about to
publish their versions of a rigorous definition of the real numbers he
decided that he too should publish his ideas. This resulted in yet another
1872 publication giving a definition of the real numbers. Dedekind
considered all decompositions of the rational numbers into two sets A1 , A2
so that a1 < a2 for all a1 in A1 and a2 in A2. He called (A1, A2) a cut. If
the rational a is either the maximum element of A1 or the minimum element of
A2 then Dedekind said the cut was produced by a. However not all cuts were
produced by a rational. He wrote:-
In every case in which a cut (A1, A2) is given that is not produced by a
rational number, we create a new number, an irrational number a, which we
consider to be completely defined by this cut; we will say that the number a
corresponds to this cut or that it produces the cut.
He defined the usual arithmetic operations and ordering and showed that the
usual laws apply.
Another definition, similar in style to that of Heine and Cantor, appeared
in a book by Thomae in 1880. Thomae had been a colleague of Heine and Cantor
around the time they had been writing up their ideas. He claimed that the
real numbers defined in this way had a right to exist because:-
... the rules of combination abstracted from calculations with integers may
be applied to them without contradiction.
Frege, however, attacked these ideas of Thomae. He wanted to develop a
theory of real numbers based on a purely logical base and attacked the
philosophy behind the constructions which had been published. Thomae added
further explanation to his idea of "formal arithmetic" in the second edition
of his text which appeared in 1898:-
The formal conception of numbers requires of itself more modest limitations
than does the logical conception. It does not ask, what are and what shall
the numbers be, but it asks, what does one require of numbers in arithmetic.
Frege was still unhappy with the constructions of Weierstrass, Heine,
Cantor, Thomae and Dedekind. How did one know. he asked, that the
constructions led to systems which would not produced contradictions? He
wrote in 1903:-
This task has never been approached seriously, let alone been solved.
Frege, however, never completed his own version of a logical framework. His
hopes were shattered when he learnt of Russell's paradox. Hilbert had taken
a totally different approach to defining the real numbers in 1900. He
defined the real numbers to be a system with eighteen axioms. Sixteen of
these axioms define what today we call an ordered field, while the other two
were the Archimedean axiom and the completeness axiom. The Archimedean axiom
stated that given positive numbers a and b then it is possible to add a to
itself a finite number of times so that the sum exceed b. The completeness
property says that one cannot extend the system and maintain the validity of
all the other axioms. This was totally new since all other methods built the
real numbers from the known rational numbers. Hilbert's numbers were
unconnected with any known system. It was impossible to say whether a given
mathematical object was a real number. Most seriously, there was no proof
that any such system actually existed. If it did it was still subject to the
same questions concerning its consistency as Frege had pointed out.
By the beginning of the 20th century, then, the concept of a real number had
moved away completely from the concept of a number which had existed from
the most ancient times to the beginning of the 19th century, namely its
connection with measurement and quantity.

References (28 books/articles)
Article by: J J O'Connor and E F Robertson

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