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Файл:Integration of Natural Science and Maths in Scientific Thought and Education. The materials of Russian - German Seminar in Moscow - Cologne, 2014
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In this lecture Riemann presented the modern definition of an abstract n-dimensional
space, of manifolds, metrics, geodesics as shortest paths („generalisation of straight
lines“) , „triangles“ determind by geodesics and the notion of curvature.
As crucial conclusion it became aware that besides the traditional intuitively gained
view of space and time which is reflected by Euclidian Geometry other constellations
are possible and that a verification is not obvious.
Results like this forced an extension of the classical foundations of Mathematics in
form of an another system of axioms. Of course any such system must fulfill
requirements concerning
A) independence
B) completeness
C) consistence
of the axioms.
5. Formalisation of Mathematics
Around the middle of the 19. century there was a rise of abstraction in Mathematics.
As Jacobi put it in 1840: „One should always generalise!“ (Hersch,Davis)
Already during the 17. and 18. Century real numbers, in particular irrational numbers
were used without any hesitation. Formal justifications in form of an axiomatic
foundation were introduced around 1870 by Dedekind and Cantor either using what
is known as Dedekind`s section or by the consideration of Cauchy sequences. Both
approaches led to the notion of the continuum of real numbers. Formally the axioms
required to set up the system of real numbers consist of
I. Axioms describing the field properties
II. Axioms describing order relation implying the law of trichotomy
III. Axioms concerning completeness
Following Kronecker the integers do not need any justification, they are „given“ by
god as he put it: „Die natürlichen Zahlen hat der liebe Gott gemacht, der Rest ist
Menschenwerk.“, but already Dedekind claimed that all numbers are a human
creation:
„Die Zahlen sind freie Schöpfungen des menschlichen Geistes.“
A formal axiomatic foundation of the integer numbers has been given by Peano
similar to the set up of the real numbers.
Frege (1884) extended that approach by reducing the definition of the integers to a set
theoretic formulation. This approach considering mathematical objects as set
theoretic quantities is known as „Logicism“. This modern approach to build up
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Mathematics on a purely logical system is described in his book „Die Grundgesetze
der Arithmetik“ in 1903. Shortly before going into print Frege was confronted with
Russel`s argument known as „liars paradox“ concerning difficulties with sets
containing themselves indicating serious problems. In an addendum to his book Frege
had to admit that his sytem failed to be consistent.
First in 1880 and lateron in papers of 1895 und 1897 Georg Cantor presented his
system of a kind of „naive“ set theory. Crucial notions like cardinality were
introduced to measure various kinds of infinite sets like integer or rational as
countable sets or real numbers as power set of the integers as non-countable sets
(continua).
An important question concerning cardinality of sets is due to Hilbert in his famous
Paris lecture during the world congress of 1900 known as „continuum hypothesis“:
Is there an infinite set with cardinality in between the integers and the continuum
(0,1)?
Motivated by that kind of weakness occuring in the naive set theory A. Fraenkel and
Ernst Zermelo(1871- 1953) tried to develop an extended version of set theory.
In 1904 it was shown by Zermelo that each set can be completely ordered; for his
proof though he had to make use of the „axiom of choice“ which became part of the
„Zermelo system of axioms“ (ZF) presented in 1908; lateron once more extended by
the axiom of foundation to to system denoted by (ZF). We omit a precise formulation
since to state this new axiom more technical features are required. As Zermelo
himself once complained: „Nobody understands my principle of foundation.
(Ebinghaus p. 195))
Today we use (ZF) to describe the system of axioms without axiom of choice , the
one including the axiom of choice by (ZFC).
Roughly formulated the axiom of choice reads as:
For each set T of pairwise disjunct nonempty sets there is a set S containing precisely
one element out of each element of T.
Although the axiom of choice appears as plausible, and seems to fulfill criteria
expected of evidence it allows conclusions which sound strange, and hence are the
reason for a critical attitude with regard to this quite helpful and intuitively acceptable
axiom.
The axiom itself is equivalent to two famous theorems:
A. Wohlordnungssatz
B. Lemma of Zorn
An interesting question concerns the consistency of the new extended system.
In 1938 Gödel proved the following result:
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If the system without the axiom of choice is consistent so the extended system
including the axiom of choice is consistent.
It was only in 1963 that Cohen was able to prove that extension by the negation of the
axiom of choice leads to a consistent system as well.
This implies for example that we are not able to decide at rresent if the continuum
hypothesis holds or not! To answer that question a new extension of the system needs
to be found (Davis-Hersch, p. 236).
6. The Crisis of Intuition
Although the axiom of choice appears as plausible and has been very useful since
many proofs could be shortened or presented more elegantly or even stronger could
not be done sofar without using it serious criticism arose as well because of some
strange conclusions. As one of the most famous the Banach-Tarski paradox is known:
Using the axiom of choice it is possible to dissect the unit sphere of the three
dimensional space into disjoint sets which can be rearranged into two new unit
spheres – a strange enlargement of volume! This paradox situation must be seen as
two aspects of volume and a different understanding of volume in mathematical resp.
physical sense. While volume in Physics is defined for example by replacement by
the same amount of water filling the volume the mathematical notion relies on
measures which are defined in an approximation procedure. Difficulties can arise
because of the existence of non-measurable sets which play a role in this setting. A
detailed and elementary presentation of the Banach-Tarski paradox can be found in
Winkler together with other examples such as the wellknown „Hilbert`s hotel“
situation: „Hilbert`s Hotel“ always has rooms available for late coming guests
although being completely booked out!
In his paper „The Crisis in Intuition“ of 1933 Hans Hahn presents further examples,
in particular from geometry illustrating weakness of intuition:
A. The classical notion of space as already mentioned as an a priori given
quantity.
B. Weierstrass` example of a continuous nowhere differentiable function (1861)
C. The Peano-curve covering a square(1890)
D. Curves consisting almost completely out of bifurcation points (Sierpinski
1915))
E. Maps of countries with an infinite number of common boundary points
(Brouwer 1910)
As conclusion Hahn puts:
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„Because intuition turned out to be deceptive in so many instances, and because
propositions that had been accounted true by intuition were repeatedly proved false
by logic, mathematicians became more and more sceptical of the validity of
intuition. They learned that it is unsafe to accept any mathematical proposition, much
less to base any mathematical discipline on intuitive convictions. Thus a demand
arose for the expulsion of intuition from mathematical reasoning, and for the
complete formalization of mathematics. That is to say, every new mathematical
concept was to be introduced through a purely logical definition; every mathematical
proof to be carried through by strictly logical means.“
Commenting contributions by Felix Klein and Hans Hahn in a lecture of 1982 Benoit
Mandelbrot, the founder of fractal geometry, arrives at a completely different point of
view:
„My reaction is very different: Fractal geometry demonstrates that Hahn was dead
wrong. Intuition is not invariable but can and must be trained to perform tasks….
Hahn draws a mistaken diagnosis, and suggests a treatment that has indeed been tried,
and has proved to be lethal.
It is both funny and pathetic to read that intuition tells man that curves have tangents.
My own recollection, together with unsystematic and limited tests, suggests the
precise contrary. Before the mathematicians`efforts, intuition must have been based
on the shapes of coastlines or of tree bark. This would lead to the conclusion that
curves have no tangent. It is the notion of tangent that has to be learned and then
made intuitive…. „
Put in other words:
Intuition is expression of some knowledge, which a hypothesis gained through
possibly unconscious experience,
7. Evolution of „schools“ in mathematical thinking
Platonists – Formalists – Constuctivists
Following Monk the community of Mathematicians seperates into three groups :
65 % Platonists (or Realists)
30 % Formalists
5 % constructivists
Platonists consider mathematical objects as really existing objects: they exist
completely indenpendently from our knowledge. They are neither invented nor
created; they just exist.
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From a formalist point of view Mathematics is seen as a compilation consisting of
axioms, definitions and theorems.
Davis und Hersh(p. 318 ff) illustrate the difference between these two groups using
the case of the continuum hypothesis which following Gödel and Cohen is neither
proven nor rejected sofar within the frame of presently known axioms. For a Platonist
that simply means that our system of axioms is still uncomplete – the continuum
hypothesis either holds or not; it is our lack of understanding that we cannot decide
yet which is true.
For the Formalist that interpretation does not make sense because our system of
numbers does not exist on its own – it has been created by us using axioms:
„Formalists and Platonists are at opposite sides on the question of existence and
reality; but they have no quarrel with each other on what principles of reasoning
should be permissble in mathematical practice.“
Constructivists have a completely different point of view: they only accept in
mathematics what can be constructed in a finite amount of steps. They do not care
about the continuum hypothesis since the real numbers or other infinite quatities are
not available at all.
The situation can be illustrated by a wellknown example due to Brouwer who is one
of the key representatives of that school.
Intuitively the law of trichotomy applies to the real numbers in the following sense: a
real number is either positive, negative or equal to zero.
Brouwer suggests to construct a number for which it is not possible to decide which
of these cases holds. He starts with the irrational number Pi; for this number one can
compute as many digits as wanted but it is also known that is never possible to know
all digits. It might occur that at some point the decimal representation of Pi contains
a segment consisting of hundred times the number zero. This property either holds or
not but this is not true for the constructivist as the decimal representation of an
irrational number is never known completely. (In case it should occur for a sequence
already computed one can change the requirement to 1000 times the digit zero.)
Starting with that setting Brouwer constructs another number Pi^ by the following
procedure: if the sequence of 100 zeros starts after an odd number of decimal digits
all following digits are substituted by zero; if it starts after an even number the
sequence is continued by 1. Then it holds that Pi = Pi^ if no such sequence of a
hundred (or 1000) zeros exists in the decimal representation of Pi. Since it is
unknown which case holds one cannot decide if the difference Pi – Pi^ is less, equal
or larger than zero.
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Of course, in the opinion of the Platonist one of these statements holds although it
remains open which one but for the constructivist none is fitting.
The construtivist`s point of view is rigorous but ignores a substantial part of
Mathematics which is simply based on existence and not on construction by finitely
many steps.
Davis und Hersh (p. 321) comment the situation ironically in the following way:
„The typical working mathematian is a Platonist on weekdays and a formalist on
Sundays. That is, when he is doing Mathematics he is convinced that he is dealing
with an objective reality whose properties he is attempting to determine. But then,
when challenged to give a philosophical account of this reality, he finds it easiest to
pretend that he does not believe in it after all.“
8. The philosophical approach
Following the argumentation sofar intuition can be seen as understanding or insight
based on plausibility or apparent evidence. Statements gained on such a base need not
be true; rather they can be seen as hypotheses.
Since every theory derived by deduction must be based on hypotheses (axioms) its
truth depends on the truth of the underlying hypotheses. Modern philosopy deals with
the question if such a theory can be seen as „knowledge“.
Discussing artificial examples philosophers such as Gettier have started to state more
precisely the notion of knowledge leading to characterizations such as „justified true
belief“. Sometimes various interpretations arise which are frequently due to
linguistic differences. In addition ethnical related differences seem to play a role (s.
Deutsch).
The situation can be illustrated by the wellknown „ 10 coins case“ due to Gettier:
„Suppose that Smith and Jones have applied for a certain job. And suppose that Smith
has strong evidence for the following conjunctive proposition:
d. Jones is the man who will get the job, and Jones has ten coins in his pocket.
Smith´s evidence for (d) might be that the president of the company assured him that
Jones would in the end be selected, and that he, Smith, had counted the coins in
Jones´s pocket 10 min ago. Proposition (d) entails:
e. The man who will get the job has ten coins in his pocket.
Let us suppose that Smith sees the entailment from (d) to (e), and accepts (e) on the
grounds (d), for which he has strong evidence. In this case, Smith is clearly justified
in believing that (e) is true.
But imagine, further, that unknown to Smith, he himself, not Jones, will get the job.
And, also unknown to Smith, he himself has ten coins in his pocket. Proposition (e) is
then true, though proposition (d), from which Smith inferred (e), is false. In our
example, then, all oft he following are true: (i) (e) is true, (ii) Smith believes that (e)
16
is true, and (iii) Smith is justified that in believing that (e) is true. But it is equally
clear that Smith does not know that (e) is true; for (e) is true in virtue of the number
of coins in Smith´s pocket, while Smith does not know how many coins are in
Smith´s pocket and bases his belief in (e) on a count of the coins in Jones`s pocket,
whom he falsely believes to be the man who will get the job.“
It is common in mathematics to define notions by names which can be chosen
arbitrarily although usually suggestive; the precise definition is given by
characteristic properties. In that sense they are just abbreviations.There is a famous
illustration due to Hilbert:
„man muss jederzeit anstelle von „Punkte“, „Geraden“ und „Ebenen“ auch „Tische“,
„Stühle“, „Bierseidel“ sagen können.“ (see Behnke et al. P. 22))
Any mathematical theory depends on the validity of the underlying axioms. Usually
mathematical statements are built up by the scheme: Hypothesis – Claim-Proof. The
weaker the hypothesis, the stronger the theory appears. In practical applications there
is the difficulty to verify the assumptions in particular if simplified models must be
used. For example a typical situation is given by assumptions concerning smoothness
required in proofs (see a famous statement in KAM-theory requiring 350 –times
differentiabilty for technical reasons).
Although such restrictive assumptions might not be verified in practice such results
are useful; it might be worthwhile to investigate if they are due to technical
requirements or due to fundamental reason. Nonsmooth analysis is a rather new
subject which is arisen out of such inquiries. Similar dilemmas occur in any theory
built up to model specific situations. Based on comparision between theoretic
predictions and reality the model can be refined step by step by adjusting the
aaumptions.
9. Conclusion
Intuition is needed in situations when just incomplete knowledge is available but
there always remains the challenge to test if the intuitive approach has led to the
correct path. Sometimes it may last several centuries as in the case of Euclidian
Geometry to find out about more sophisticated concepts. Nevertheless, it is important
to support the development of a „good“ intuitive understanding, and this is an
important task in scientific education. The study of historic cases of treacherous as
well as of correct intuitions are a good part of such a training to form correct
intuitive faculties.
10. References:
( 1) Behnke -Bachmann -Fladt -Süß: Grundzüge der Mathematik I
17
( 2) Cantor, G.: Beiträge zur Begründung der transfiniten Mengenlehre, Math. Annalen 46, 481-512
(1895) und Math.Annalen 49, 207-2246, (1897)
( 3) Cohen, P. J.: Set Theory and the Continuum Hypothesis. New York: W.A. Benjamin 1966
( 4) Davis, P. J., Hersh, R. : The Mathematical Experience, Boston: Birkhäuser 1981
( 5) Deutsch, M.: Intuitions, Counter Examples and Experimental Philosophy, Rev. Phil. Psych.,
2010 (Doi10.1007/s13164-010-0033-0)
( 6) Dummett,M.: Elements of Intuitionism. Oxford: The Clarendon Press 1977
( 7) Ebinghaus,H.-D.: Ernst Zermelo, Berlin: Springer 2007
( 8) Euklid: Elemente
( 9) Feferman, S.: Mathematical Intuition vs. Mathematical Monsters, Synthese 125, 317-332
(2000) (http://math.stanford.edu/~feferman/papers/intuition.pdf)
( 10) Fraenkel, A. A.: The recent controversies About the Foundations of Mathematics. Scripta
Mathematica 13, 17-36 (1947)
( 11) Frege, G.: Die Grundlagen der Arithmetik, Breslau: Koebner 1884
( 12) Gödel, K.: What is Cantor`s Continuum Problem? In: P. Bernacerraf and H. Putnam (eds.):
Philosophy of Mathematics pp. 258-273. Englewood Cliffs:Prentice Hall 1964
(13) Hahn, H.: The Crisis of Intuition, in: J. R. Newman (ed.) The World of Mathematics, 19561976, Simon and Schuster, New York 1956
(http://www-gap.dcs.st-and.ac.uk/~history/Extras/Hahn_crisis_in_intuition.html)
( 14) Helmholtz, H. v., Über den Ursprung und die Bedeutung der geometrischen Axiome, in: Raum
und Kraft- Kapitel 10 (http://gutenberg.spiegel.de/buch/raum-und-kraft-6606/10)
(15) Heuser, H.: Gewöhnliche Differentialgleichungen, Stuttgart, B. G. Teubner 1991
(16 )Hilbert, D.
(17)Kant, I.: Kritik der reinen Vernunft, Stuttgart, Ph. Reclam 1966
(18)Kripke, S.: Naming and Necessity, Harvard University 1980
(19) Lehn, M.: Welche Geometrie gilt? Vortragsmanuskript Mainz 2005
(20) Mandelbrot, B.: A Crisis of Intuition as viewed by Felix Klein and Hans Hahn and its
resolution by fractal geometry, in: The Fractal Geometry of Nature 1982
(http://math.yale.edu/mandelbrot/web_pdfs/WKleinCrisisOf Intuition.pdf)
(21) Monk, J.D.: On the Foundations of Set Theory, American Math. Monthly 77, 703-711, 1970
(22) O`Shea, D.: Poincaré`s Vermutung: Frankfurt, Fischer 2007
(23) Risse, Th.: Zur Rolle der Intuition in der Mathematik-Ausbildung, Global J. of Eng. Edu. 9, 15, 2006
(24) Winkler, R.: Wie macht man 2 aus 1? Das Paradoxon von Banach-Tarski, Didaktik-Hefte der
Österr. Math. Gesellschaft, Schriftenreihe zur Didaktij der Math. An Höheren Schulen 33, 166-196,
2001
(http://dmg.tuwien.ac.at/winkler/pub/)
(25) Zermelo, E.: Untersuchungen über die Grundlagen der Mengenlehre. I., Math, Annalen 65,
261-281, 1908
(26) http://fr.wikipedia.org/wiki/intuition
(27)http://de.wikipedia.org/wiki/Die Grundlagen der Arithmetik
(28)http://www.iep.utm.edu/dummett/print
(29)http://www.philosophie-woerterbuch.de/online-woerterbuch/
(30)http://de.wikipedia.org/wiki/Intuition
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THE EDUCATION IN THE RUSSIAN FEDERATION
Natalia S. Purysheva, Alexandra Zhelyabovskaya, Nika Rybalka
Moscow State Pedagogical University
Key words: education, the Russian Federation, principle of education, federal
state educational standard, clip-thinking, universal studying activity, academic results
Education in the Russian Federation is provided mostly by the state. The
Ministry of Education and Science is the only federal body regulating all education
levels in all types of institutions. In 2008 the state expenditures for education
amounted to 4.1% of GDP. [1] The quality of education is ensured through federal
state educational standards, licensing and state accreditation provided by the Federal
Service for Supervision in Education and Science. Regulation of education in the
constituent parts of the Russian Federation and municipalities is also carried out by
the local authorities. [2]
The basic principles of education in Russia postulated by the Law on Education
of 2012 are as follow:
- The priority of education, its humanistic and secular nature, its accessibility
and adaptability to pupils and students’ development and training;
- Education pluralism and freedom, education management democratic and
public nature and educational institutions autonomy;
- The priority of human values, human health and life and human rights and
freedom;
- Civic education, respect and love for the environment, homeland and family;
- The unity of educational space within the multinational state along with
protection of ethnocultural peculiarities and traditions of different nations;
- Encouragement of the Russian Federation education system and
education systems of other countries on a mutually eligible and beneficial basis [3 pp.
10-12; 4].
The types, levels and programs of the Russian education system are presented
on picture 1.
From 5 to 9 March, 1990 the first World Education Conference on Education
for All took place in Jomtien, Thailand. Since that remarkable event Russia has
experienced considerable politic and socio-economic changes which influenced the
reformation of education system as much as the application of commitments of the
EFA World Declaration. [5] Globalization, new social demands of the Russian
society, the necessity for application of the latest achievements of scientific thought
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Pic. 1. The Russian Education System [6]
in education, new educational aims proclaimed among the commitments of the
Conference in Jomtien and the World Education Forum in Dakar, Senegal (26-28
April 2000) and approved by Russia resulted in the two series of state educational
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