Lang geleden dat we nog een keer een gastschrijver hebben gehad.
Bij deze.
Ik had namelijk mijn dochter (shit! verraden! Man die gaat kwaad zijn.) in het kader van
"thuisstudie" de opdracht gegeven om een essay van een uur te schrijven over ""Over SKEPP, of wanneer wetenschap een ideologie wordt."
Alle gelijkenissen in stijl en taalgebruik zijn genetisch bepaald en dus geheel toevallig.
Hier gaan we:
Wat word (!) ik toch snel afgeleid.
Zelfs een eenvoudige zin als "wat word ik toch snel afgeleid" kan ik niet noteren zonder dat mijn brein wordt afgeleid door het opvallende woordgebruik van "worden".
Waarom schrijf ik "Wat word ik toch snel afgeleid" en niet "wat ben ik toch snel afgeleid"?
Soit.
Ik wilde het dus eigenlijk helemaal niet over Margaretha van Leuven hebben, maar over Meta.
Uiteraard begin ik altijd met het nodige opzoekingswerk, je zou wel gek zijn vandaag de dag om onvoorbereid het intellectuele debat aan te gaan.
Maar dan kom je tot de vaststelling dat "meta" kan verwijzen naar de meisjesnaam "Margaretha" en voor je het weet (!) -daar heb je het weer- zit ik dus mijn tijd (!) te verbeuzelen met de levensgeschiedenis van de fiere Margriet uit te pluizen. (Kan u zich inbeelden hoe lang het duurt voor ik zo'n zin op papier heb gezet? De lijdensweg die ik afgelegd heb nog voor ik eigenlijk begonnen ben?)
http://nl.wikipedia.org/wiki/Meta
Meta- is in het Nederlands en in veel andere Europese talen een voorvoegsel dat
betreffende het onderwerp zelf betekent.
Even vooraf een opvallend hiaat signaleren. Meta kan ook verwijzen naar een virus. Het metavirus. Dat is het meest dodelijke virus voor het brein. Het meta-virus vernietigt langzaam (!) maar zeker (!) de creativiteit waardoor het brein uiteindelijk afsterft.
Maar we dwalen dus af. Het gaat hier over het voorvoegsel Meta.
Aan de ene kant is er "onwetendheid" en aan de andere kant is er "kennis". Behalve de gekken is iedereen het daar over eens.
"Onder
informatie (van Latijn
informare: "vormgeven, vormen, instrueren") verstaat men in algemene zin alles wat kennis of bepaaldheid toevoegt en zodoende onwetendheid. onzekerheid of onbepaaldheid vermindert. In striktere zin wordt wel gesteld dat pas van informatie gesproken kan worden als die voor mensen interpreteerbaar is. Het interpreteren en integreren van deze informatie resulteert in kennis."
http://nl.wikipedia.org/wiki/Informatie
Wat zijn we toch lief en begripvol voor elkaar!
Ruis
In welke mate de informatie die overgebracht wordt aankomt bij de ontvanger is afhankelijk van verschillende factoren. Factoren die de informatieoverdracht negatief beïnvloeden worden ruis genoemd. Er bestaan verschillende soorten ruis.
Externe ruis
Onder externe ruis worden factoren verstaan die de ontvanger afleiden waardoor de informatieoverdracht wordt verstoord. Dit is bijvoorbeeld het geval wanneer je een gesprek probeert te voeren in een lawaaiige omgeving.
Interne ruis
Onder interne ruis worden factoren verstaan die gelegen zijn in de informatieontvanger of -verzender zelf. Interne ruis kan veel verschillende oorzaken hebben:
- Het gesprek vindt plaats in een taal die een of beide gesprekspartners onvoldoende beheersen
- Een of beide gesprekspartners zijn geëmotioneerd
- De informatieverzendende partij drukt zich onduidelijk uit of maakt onvoldoende onderscheid tussen hoofd- en bijzaken
- De informatieontvangende partij trekt voorbarige conclusies of interpreteert de informatie op een manier die de verzendende partij niet voorzien had
- De informatieontvangende partij ervaart de informatie als irrelevant en verwerkt deze daarom bewust of onbewust niet
- De informatieverzendende partij geeft meer informatie dan de ontvangende partij kan verwerken ("information overload").
(Als u even een pauze wil inlassen om al die informatie te verwerken kan u altijd een plaspauze nemen. In dat geval verwijs ik u graag naar
http://www.bloggen.be/excrementen/ en klikt u op de video.)
Allemaal goed en wel, maar om van "onwetendheid" naar "kennis" te kunnen switchen moeten we op een bepaald moment die ruis toch kunnen detecteren en uitschakelen (dat heb ik uit de cursus "evidence based HRM" die ik gevolgd heb).
Gelukkig is er dan het meta-niveau.
"Informatie kan op inhoudsniveau en betrekkingsniveau gezonden worden. Het inhoudsniveau betreft de concrete inhoudelijke informatie. Betrekkingsniveau is informatie op meta-niveau: bijvoorbeeld informatie over hoe een boodschap moet worden opgevat en hoe de verhoudingen zijn tussen de betrokkenen in een relatie."
Meta-informatie is informatie over de informatie zelf.
Dat kan ik begrijpen.
Informatie over hoe de verhoudingen zijn tussen de betrokkenen in een relatie.
Dat kan ik begrijpen.
Informatie die van een hoger niveau komt is andere informatie dan informatie die van een lager niveau komt.
Meta-niveau informatie is hetzelfde als meta-informatie.
Meta-niveau is geen substantief, het is een bijvoeglijk naamwoord.
Meta-niveau informatie is een bepaald soort informatie.
Net zoals de informatie is de meta-informatie altijd afkomstig van een informatieverzendende partij.
Nochtans heb ik de indruk dat meta-niveau dikwijls als een substantief gebruikt wordt, meta-niveau is ongemerkt een eigen leven gaan leiden in onze maatschappij. Meta-niveau is door een substantief te worden ook een subject geworden. Met dank aan jullie vriend Aristoteles. (Door Aristoteles kwam het begrip subject in gebruik in de zin van
substantie, dat is de wezenskern van het ding, datgene wat blijft bestaan onder wisselende omstandigheden.
http://nl.wikipedia.org/wiki/Subject_(filosofie)
Meta-niveau kennis is geen kennis van de kennis meer, het is niet langer een bepaald soort kennis, het is kennis die afkomstig is van de informatieverzendende partij "het meta-niveau".
Er is onbepaaldheid en er is bepaaldheid.
De weg tussen die beide polen is de weg van het meta-niveau.
Het meta-niveau is een weg.
Het is zoals de weg vragen.
Als je met de wagen vertraagt naast een voetganger en je draait het raampje naar omlaag en je vraagt "Kan u me zeggen waar ik café terminus kan vinden", dan vraagt u niet naar de locatie van café terminus, dan vraagt u de weg naar café terminus.
Over café terminus gesproken, ik heb er dorst van gekregen. Laten we iets gaan drinken.
Ik heb nog les gekregen van een prof die ooit met zijn studenten op café ging (het was een zeer kleine faculteit).
"Wat je schrijft is helemaal niet belangrijk. Of denk je nu echt dat er iets is tussen hemel en aarde wat nog niet beschreven is? Hoe je erover schrijft is belangrijk".
Dat zei hij terwijl hij achteloos met een lepeltje in zijn koffie roerde.
Meer geleerd op café dan in de les. (Weet u wie de beschermheilige van de horeca is?)
Wat je schrijft is hoe je schrijft.
Café terminus is de weg naar café terminus.
Het is zo ver gekomen dat men de weg vraagt naar het meta-niveau.
Kan u me zeggen waar ik het meta-niveau kan vinden?
Is het de wetenschap?
Bestaat er zoiets als wetenschap van de wetenschap?
Filosofiewetenschap bestaat ja, maar filosofie is dat een wetenschap?
Is het de wiskunde?
Misschien is dat wel zo.
In de meeste talen is het woord voor wiskunde afgeleid van het Griekse woord μάθημα (
máthèma), dat wetenschap, kennis of leren betekent.
Wiskunde is kennis.
http://pauli.uni-muenster.de/~munsteg/arnold.html
On teaching mathematics
by V.I. Arnold
This is an extended text of the
address at the discussion on teaching of mathematics in Palais de Découverte in
Paris on 7 March 1997.
Mathematics is a part of physics. Physics is an experimental science, a part
of natural science. Mathematics is the part of physics where experiments are
cheap.
The Jacobi identity (which forces the heights of a triangle to cross at one
point) is an experimental fact in the same way as that the Earth is round (that
is, homeomorphic to a ball). But it can be discovered with less expense.
In the middle of the twentieth century it was attempted to divide physics and
mathematics. The consequences turned out to be catastrophic. Whole generations
of mathematicians grew up without knowing half of their science and, of course,
in total ignorance of any other sciences. They first began teaching their ugly
scholastic pseudo-mathematics to their students, then to schoolchildren
(forgetting Hardy's warning that ugly mathematics has no permanent place under
the Sun).
Since scholastic mathematics that is cut off from physics is fit neither for
teaching nor for application in any other science, the result was the universal
hate towards mathematicians - both on the part of the poor schoolchildren (some
of whom in the meantime became ministers) and of the users.
The ugly building, built by undereducated mathematicians who were exhausted
by their inferiority complex and who were unable to make themselves familiar
with physics, reminds one of the rigorous axiomatic theory of odd numbers.
Obviously, it is possible to create such a theory and make pupils admire the
perfection and internal consistency of the resulting structure (in which, for
example, the sum of an odd number of terms and the product of any number of
factors are defined). From this sectarian point of view, even numbers could
either be declared a heresy or, with passage of time, be introduced into the
theory supplemented with a few "ideal" objects (in order to comply with the
needs of physics and the real world).
Unfortunately, it was an ugly twisted construction of mathematics like the
one above which predominated in the teaching of mathematics for decades. Having
originated in France, this pervertedness quickly spread to teaching of
foundations of mathematics, first to university students, then to school pupils
of all lines (first in France, then in other countries, including Russia).
To the question "what is 2 + 3" a French primary school pupil replied: "3 +
2, since addition is commutative". He did not know what the sum was equal to and
could not even understand what he was asked about!
Another French pupil (quite rational, in my opinion) defined mathematics as
follows: "there is a square, but that still has to be proved".
Judging by my teaching experience in France, the university students' idea of
mathematics (even of those taught mathematics at the École Normale Supérieure -
I feel sorry most of all for these obviously intelligent but deformed kids) is
as poor as that of this pupil.
For example, these students have never seen a paraboloid and a question on
the form of the surface given by the equation xy = z2 puts the
mathematicians studying at ENS into a stupor. Drawing a curve given by
parametric equations (like x = t3 - 3t, y = t4 -
2t2) on a plane is a totally impossible problem for students (and,
probably, even for most French professors of mathematics).
Beginning with l'Hospital's first textbook on calculus ("calculus for
understanding of curved lines") and roughly until Goursat's textbook, the
ability to solve such problems was considered to be (along with the knowledge of
the times table) a necessary part of the craft of every mathematician.
Mentally challenged zealots of "abstract mathematics" threw all the geometry
(through which connection with physics and reality most often takes place in
mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard
were recently dumped by the student library of the Universities Paris 6 and 7
(Jussieu) as obsolete and, therefore, harmful (they were only rescued by my
intervention).
ENS students who have sat through courses on differential and algebraic
geometry (read by respected mathematicians) turned out be acquainted neither
with the Riemann surface of an elliptic curve y2 = x3 + ax
+ b nor, in fact, with the topological classification of surfaces (not even
mentioning elliptic integrals of first kind and the group property of an
elliptic curve, that is, the Euler-Abel addition theorem). They were only taught
Hodge structures and Jacobi varieties!
How could this happen in France, which gave the world Lagrange and Laplace,
Cauchy and Poincaré, Leray and Thom? It seems to me that a reasonable
explanation was given by I.G. Petrovskii, who taught me in 1966: genuine
mathematicians do not gang up, but the weak need gangs in order to survive. They
can unite on various grounds (it could be super-abstractness, anti-Semitism or
"applied and industrial" problems), but the essence is always a solution of the
social problem - survival in conditions of more literate surroundings.
By the way, I shall remind you of a warning of L. Pasteur: there never have
been and never will be any "applied sciences", there are only applications of
sciences (quite useful ones!).
In those times I was treating Petrovskii's words with some doubt, but now I
am being more and more convinced of how right he was. A considerable part of the
super-abstract activity comes down simply to industrialising shameless grabbing
of discoveries from discoverers and then systematically assigning them to
epigons-generalizers. Similarly to the fact that America does not carry
Columbus's name, mathematical results are almost never called by the names of
their discoverers.
In order to avoid being misquoted, I have to note that my own achievements
were for some unknown reason never expropriated in this way, although it always
happened to both my teachers (Kolmogorov, Petrovskii, Pontryagin, Rokhlin) and
my pupils. Prof. M. Berry once formulated the following two principles:
The Arnold Principle. If a notion bears a personal name, then this
name is not the name of the discoverer.
The Berry Principle. The Arnold Principle is applicable to itself.
Let's return, however, to teaching of mathematics in France.
When I was a first-year student at the Faculty of Mechanics and Mathematics
of the Moscow State University, the lectures on calculus were read by the
set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old
classical calculus course of French type in the Goursat version. He told us that
integrals of rational functions along an algebraic curve can be taken if the
corresponding Riemann surface is a sphere and, generally speaking, cannot be
taken if its genus is higher, and that for the sphericity it is enough to have a
sufficiently large number of double points on the curve of a given degree (which
forces the curve to be unicursal: it is possible to draw its real points on the
projective plane with one stroke of a pen).
These facts capture the imagination so much that (even given without any
proofs) they give a better and more correct idea of modern mathematics than
whole volumes of the Bourbaki treatise. Indeed, here we find out about the
existence of a wonderful connection between things which seem to be completely
different: on the one hand, the existence of an explicit expression for the
integrals and the topology of the corresponding Riemann surface and, on the
other hand, between the number of double points and genus of the corresponding
Riemann surface, which also exhibits itself in the real domain as the
unicursality.
Jacobi noted, as mathematics' most fascinating property, that in it one and
the same function controls both the presentations of a whole number as a sum of
four squares and the real movement of a pendulum.
These discoveries of connections between heterogeneous mathematical objects
can be compared with the discovery of the connection between electricity and
magnetism in physics or with the discovery of the similarity between the east
coast of America and the west coast of Africa in geology.
The emotional significance of such discoveries for teaching is difficult to
overestimate. It is they who teach us to search and find such wonderful
phenomena of harmony of the Universe.
The de-geometrisation of mathematical education and the divorce from physics
sever these ties. For example, not only students but also modern
algebro-geometers on the whole do not know about the Jacobi fact mentioned here:
an elliptic integral of first kind expresses the time of motion along an
elliptic phase curve in the corresponding Hamiltonian system.
Rephrasing the famous words on the electron and atom, it can be said that a
hypocycloid is as inexhaustible as an ideal in a polynomial ring. But teaching
ideals to students who have never seen a hypocycloid is as ridiculous as
teaching addition of fractions to children who have never cut (at least
mentally) a cake or an apple into equal parts. No wonder that the children will
prefer to add a numerator to a numerator and a denominator to a denominator.
From my French friends I heard that the tendency towards super-abstract
generalizations is their traditional national trait. I do not entirely disagree
that this might be a question of a hereditary disease, but I would like to
underline the fact that I borrowed the cake-and-apple example from Poincaré.
The scheme of construction of a mathematical theory is exactly the same as
that in any other natural science. First we consider some objects and make some
observations in special cases. Then we try and find the limits of application of
our observations, look for counter-examples which would prevent unjustified
extension of our observations onto a too wide range of events (example: the
number of partitions of consecutive odd numbers 1, 3, 5, 7, 9 into an odd number
of natural summands gives the sequence 1, 2, 4, 8, 16, but then comes 29).
As a result we formulate the empirical discovery that we made (for example,
the Fermat conjecture or Poincaré conjecture) as clearly as possible. After this
there comes the difficult period of checking as to how reliable are the
conclusions .
At this point a special technique has been developed in mathematics. This
technique, when applied to the real world, is sometimes useful, but can
sometimes also lead to self-deception. This technique is called modelling. When
constructing a model, the following idealisation is made: certain facts which
are only known with a certain degree of probability or with a certain degree of
accuracy, are considered to be "absolutely" correct and are accepted as
"axioms". The sense of this "absoluteness" lies precisely in the fact that we
allow ourselves to use these "facts" according to the rules of formal logic, in
the process declaring as "theorems" all that we can derive from them.
It is obvious that in any real-life activity it is impossible to wholly rely
on such deductions. The reason is at least that the parameters of the studied
phenomena are never known absolutely exactly and a small change in parameters
(for example, the initial conditions of a process) can totally change the
result. Say, for this reason a reliable long-term weather forecast is impossible
and will remain impossible, no matter how much we develop computers and devices
which record initial conditions.
In exactly the same way a small change in axioms (of which we cannot be
completely sure) is capable, generally speaking, of leading to completely
different conclusions than those that are obtained from theorems which have been
deduced from the accepted axioms. The longer and fancier is the chain of
deductions ("proofs"), the less reliable is the final result.
Complex models are rarely useful (unless for those writing their
dissertations).
The mathematical technique of modelling consists of ignoring this trouble and
speaking about your deductive model in such a way as if it coincided with
reality. The fact that this path, which is obviously incorrect from the point of
view of natural science, often leads to useful results in physics is called "the
inconceivable effectiveness of mathematics in natural sciences" (or "the Wigner
principle").
Here we can add a remark by I.M. Gel'fand: there exists yet another
phenomenon which is comparable in its inconceivability with the inconceivable
effectiveness of mathematics in physics noted by Wigner - this is the equally
inconceivable ineffectiveness of mathematics in biology.
"The subtle poison of mathematical education" (in F. Klein's words) for a
physicist consists precisely in that the absolutised model separates from the
reality and is no longer compared with it. Here is a simple example: mathematics
teaches us that the solution of the Malthus equation dx/dt = x is uniquely
defined by the initial conditions (that is that the corresponding integral
curves in the (t,x)-plane do not intersect each other). This conclusion of the
mathematical model bears little relevance to the reality. A computer experiment
shows that all these integral curves have common points on the negative
t-semi-axis. Indeed, say, curves with the initial conditions x(0) = 0 and x(0) =
1 practically intersect at t = -10 and at t = -100 you cannot fit in an atom
between them. Properties of the space at such small distances are not described
at all by Euclidean geometry. Application of the uniqueness theorem in this
situation obviously exceeds the accuracy of the model. This has to be respected
in practical application of the model, otherwise one might find oneself faced
with serious troubles.
I would like to note, however, that the same uniqueness theorem explains why
the closing stage of mooring of a ship to the quay is carried out manually: on
steering, if the velocity of approach would have been defined as a smooth
(linear) function of the distance, the process of mooring would have required an
infinitely long period of time. An alternative is an impact with the quay (which
is damped by suitable non-ideally elastic bodies). By the way, this problem had
to be seriously confronted on landing the first descending apparata on the Moon
and Mars and also on docking with space stations - here the uniqueness theorem
is working against us.
Unfortunately, neither such examples, nor discussing the danger of
fetishising theorems are to be met in modern mathematical textbooks, even in the
better ones. I even got the impression that scholastic mathematicians (who have
little knowledge of physics) believe in the principal difference of the
axiomatic mathematics from modelling which is common in natural science and
which always requires the subsequent control of deductions by an experiment.
Not even mentioning the relative character of initial axioms, one cannot
forget about the inevitability of logical mistakes in long arguments (say, in
the form of a computer breakdown caused by cosmic rays or quantum oscillations).
Every working mathematician knows that if one does not control oneself (best of
all by examples), then after some ten pages half of all the signs in formulae
will be wrong and twos will find their way from denominators into numerators.
The technology of combatting such errors is the same external control by
experiments or observations as in any experimental science and it should be
taught from the very beginning to all juniors in schools.
Attempts to create "pure" deductive-axiomatic mathematics have led to the
rejection of the scheme used in physics (observation - model - investigation of
the model - conclusions - testing by observations) and its substitution by the
scheme: definition - theorem - proof. It is impossible to understand an
unmotivated definition but this does not stop the criminal
algebraists-axiomatisators. For example, they would readily define the product
of natural numbers by means of the long multiplication rule. With this the
commutativity of multiplication becomes difficult to prove but it is still
possible to deduce it as a theorem from the axioms. It is then possible to force
poor students to learn this theorem and its proof (with the aim of raising the
standing of both the science and the persons teaching it). It is obvious that
such definitions and such proofs can only harm the teaching and practical work.
It is only possible to understand the commutativity of multiplication by
counting and re-counting soldiers by ranks and files or by calculating the area
of a rectangle in the two ways. Any attempt to do without this interference by
physics and reality into mathematics is sectarianism and isolationism which
destroy the image of mathematics as a useful human activity in the eyes of all
sensible people.
I shall open a few more such secrets (in the interest of poor students).
The determinant of a matrix is an (oriented) volume of the
parallelepiped whose edges are its columns. If the students are told this secret
(which is carefully hidden in the purified algebraic education), then the whole
theory of determinants becomes a clear chapter of the theory of poly-linear
forms. If determinants are defined otherwise, then any sensible person will
forever hate all the determinants, Jacobians and the implicit function theorem.
What is a group? Algebraists teach that this is supposedly a set with
two operations that satisfy a load of easily-forgettable axioms. This definition
provokes a natural protest: why would any sensible person need such pairs of
operations? "Oh, curse this maths" - concludes the student (who, possibly,
becomes the Minister for Science in the future).
We get a totally different situation if we start off not with the group but
with the concept of a transformation (a one-to-one mapping of a set onto itself)
as it was historically. A collection of transformations of a set is called a
group if along with any two transformations it contains the result of their
consecutive application and an inverse transformation along with every
transformation.
This is all the definition there is. The so-called "axioms" are in fact just
(obvious) properties of groups of transformations. What axiomatisators
call "abstract groups" are just groups of transformations of various sets
considered up to isomorphisms (which are one-to-one mappings preserving the
operations). As Cayley proved, there are no "more abstract" groups in the world.
So why do the algebraists keep on tormenting students with the abstract
definition?
By the way, in the 1960s I taught group theory to Moscow
schoolchildren. Avoiding all the axiomatics and staying as close as
possible to physics, in half a year I got to the Abel theorem on the
unsolvability of a general equation of degree five in radicals (having on the
way taught the pupils complex numbers, Riemann surfaces, fundamental groups and
monodromy groups of algebraic functions). This course was later published by one
of the audience, V. Alekseev, as the book The Abel theorem in problems.
What is a smooth manifold? In a recent American book I read that
Poincaré was not acquainted with this (introduced by himself) notion and that
the "modern" definition was only given by Veblen in the late 1920s: a manifold
is a topological space which satisfies a long series of axioms.
For what sins must students try and find their way through all these twists
and turns? Actually, in Poincaré's Analysis Situs there is an absolutely
clear definition of a smooth manifold which is much more useful than the
"abstract" one.
A smooth k-dimensional submanifold of the Euclidean space
RN is its subset which in a neighbourhood of its every point
is a graph of a smooth mapping of Rk into R(N -
k) (where Rk and R(N - k) are
coordinate subspaces). This is a straightforward generalization of most common
smooth curves on the plane (say, of the circle x2 + y2 =
1) or curves and surfaces in the three-dimensional space.
Between smooth manifolds smooth mappings are naturally defined.
Diffeomorphisms are mappings which are smooth, together with their inverses.
An "abstract" smooth manifold is a smooth submanifold of a Euclidean space
considered up to a diffeomorphism. There are no "more abstract"
finite-dimensional smooth manifolds in the world (Whitney's theorem). Why do we
keep on tormenting students with the abstract definition? Would it not be better
to prove them the theorem about the explicit classification of closed
two-dimensional manifolds (surfaces)?
It is this wonderful theorem (which states, for example, that any compact
connected oriented surface is a sphere with a number of handles) that gives a
correct impression of what modern mathematics is and not the super-abstract
generalizations of naive submanifolds of a Euclidean space which in fact do not
give anything new and are presented as achievements by the axiomatisators.
The theorem of classification of surfaces is a top-class mathematical
achievement, comparable with the discovery of America or X-rays. This is a
genuine discovery of mathematical natural science and it is even difficult to
say whether the fact itself is more attributable to physics or to mathematics.
In its significance for both the applications and the development of correct
Weltanschauung it by far surpasses such "achievements" of mathematics as the
proof of Fermat's last theorem or the proof of the fact that any sufficiently
large whole number can be represented as a sum of three prime numbers.
For the sake of publicity modern mathematicians sometimes present such
sporting achievements as the last word in their science. Understandably this not
only does not contribute to the society's appreciation of mathematics but, on
the contrary, causes a healthy distrust of the necessity of wasting energy on
(rock-climbing-type) exercises with these exotic questions needed and wanted by
no one.
The theorem of classification of surfaces should have been included in high
school mathematics courses (probably, without the proof) but for some reason is
not included even in university mathematics courses (from which in France, by
the way, all the geometry has been banished over the last few decades).
The return of mathematical teaching at all levels from the scholastic chatter
to presenting the important domain of natural science is an espessially hot
problem for France. I was astonished that all the best and most important in
methodical approach mathematical books are almost unknown to students here (and,
seems to me, have not been translated into French). Among these are Numbers
and figures by Rademacher and Töplitz, Geometry and the imagination
by Hilbert and Cohn-Vossen, What is mathematics? by Courant and Robbins,
How to solve it and Mathematics and plausible reasoning by Polya,
Development of mathematics in the 19th century by F. Klein.
I remember well what a strong impression the calculus course by Hermite
(which does exist in a Russian translation!) made on me in my school years.
Riemann surfaces appeared in it, I think, in one of the first lectures (all
the analysis was, of course, complex, as it should be). Asymptotics of integrals
were investigated by means of path deformations on Riemann surfaces under the
motion of branching points (nowadays, we would have called this the
Picard-Lefschetz theory; Picard, by the way, was Hermite's son-in-law -
mathematical abilities are often transferred by sons-in-law: the dynasty
Hadamard - P. Levy - L. Schwarz - U. Frisch is yet another famous example in the
Paris Academy of Sciences).
The "obsolete" course by Hermite of one hundred years ago (probably, now
thrown away from student libraries of French universities) was much more modern
than those most boring calculus textbooks with which students are nowadays
tormented.
If mathematicians do not come to their senses, then the consumers who
preserved a need in a modern, in the best meaning of the word, mathematical
theory as well as the immunity (characteristic of any sensible person) to the
useless axiomatic chatter will in the end turn down the services of the
undereducated scholastics in both the schools and the universities.
A teacher of mathematics, who has not got to grips with at least some of the
volumes of the course by Landau and Lifshitz, will then become a relict like the
one nowadays who does not know the difference between an open and a closed set.
V.I. Arnold
Translated by A.V. GORYUNOV
Men vraagt niet meer naar de weg van het meta-niveau.
Iedereen heeft een gps.
Men is vergeten hoe de krasse knar langs de kant van de weg zijn klak van zijn hoofd nam en nadenkend over zijn kalende kop wreef om dan behoedzaam te antwoorden. (it's a prose poem!)
De weg naar het meta-niveau is altijd rechtdoor.
Altijd rechtdoor tot je bij een groene deur komt.
En dan is het terug rechtdoor tot een rode deur.
Die stroeft een beetje.
Ja, en dan is het eigenlijk terug rechtdoor tot je bij een zwarte deur komt.
Dat is een hele zware deur, die gaat echt wel moeilijk open.
Wacht hé, efkes denken.
Ja, best is van dan gewoon rechtdoor te gaan.
Rechtdoor tot aan een gele deur.
En daar moet ge het nog maar een keer vragen.
"
Indien een en ander anders is geschied dan ik het heb geschreven, moet dat veeleer verweten worden aan hen door wie deze feiten werden verteld."
middeleeuws hagiograaf, genaamd Caesarius, tekende de legende van Margaretha van Leuven voor de eerste keer op in 1222 ter illustratie van de deugd
eenvoud in zijn werk
Dialogus miraculorum
http://nl.wikipedia.org/wiki/Margaretha_van_Leuven
Is dat niet prachtig?
Is het geen tijd (!) om terug te keren naar de eenvoud?
Complex models are rarely useful (unless for those writing their dissertations).
Is het geen tijd (!) om de "duistere middeleeuwen" te opwaarderen?
Is het geen tijd (!) om wat de prof in zijn vrije tijd zegt "leerstof" te maken, om een leerstoel "cafépraat" op te richten.
Nog een tip voor onderweg: vergeet je niet te vaccineren.
De term vaccineren komt van het Latijnse woord
vaccinia, koepokken. De term vaccineren werd namelijk oorspronkelijk gebruikt voor de aan het einde van de 18e eeuw door Edward Jenner ontwikkelde methode om mensen met de koepokken te besmetten, waardoor ze ook weerstand kregen tegen de voor mensen gevaarlijke 'gewone' pokken.
"Over SKEPP, of wanneer wetenschap een ideologie wordt."
Dat is de titel van een essay.
Wetenschap en ideologie worden bij aanvang als twee aparte entiteiten naar voor geschoven.
En het blijven ook twee aparte entiteiten.
Behalve in bepaalde (!) gevallen.
Wie kan daar nu een uur over schrijven pap?
Zelfstudie is de slaap van de rede. ;-)
Yours truly,
Isabel Korstjens.
Ik denk dat ik haar een nieuwe dekbedovertrek van Winnie-the-pooh cadeau ga doen.