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- Mathematics In Europe - Historia/Filosofía

2008 pdf 68 kB Bourguignon Cartan NL70 2008 pdf 45 kB Brüning Euler NL66 2007 pdf kB Baas Selberg NL65 2007 pdf 291207 kB Geiges Huygens NL65 2007 pdf 265 kB Schindler Gödel NL62 2006 pdf 66 kB Reiter Boltzmann NL61 2006 pdf 161 kB Lützen On the Mutual Influence of Mathematics and Physic NL60 2006 pdf 138 kB Samuel Cartan NL53 2004 pdf 263 kB Fokking van Kampen NL52 2004 pdf 1130 kB Rédei von Neumann NL51 2004 pdf 178 kB Gray Legendre NL45 2002 pdf 78 kB Mann Burnside NL 44 2002 pdf 78 kB Bekken Abel NL43 2002 pdf 124 kB Barner Fermat NL42 2001 pdf 293 kB Fauvel Napier NL39 2000 pdf 155 kB Gray Kovalevskaja andBeltrami NL35 2000 pdf 143 kB Barrow Green Celestial mechanics NL34 1999 pdf 193 kB Gray Klein and Krull NL32 1999 pdf 252 kB Philosophy of mathematics The question of the firmness of the foundations of mathematics has various answers depending on one s philosophical perspective The issue was discussed with particular intensity at the beginning of the twentieth century Although it is a question of fundamental importance it is not the focus of interest of most working mathematicians engaged in research or practical applications Here you can find an overview of the most important philosophical viewpoints Introduction Overview of the Philosophy of Mathematics logicism formalism intuitionism these articles are translations of contributions that were written originally for www mathematik de The following articles appeared originally in the Newsletter NL of the European Mathematical Society EMS They are republished here with permission of the EMS and the authors Corfield Nominalism and Realism NL75 2010 pdf 1395 kB Artstein Platonism NL75 2010 pdf 1362 kB Davies Platonism NL72 2009 pdf 108 kB Gardner On the Philosophy of Mathematics NL72 2009 pdf

Original URL path: http://mathematics-in-europe.eu/es/informacion/historia-filosofia (2013-11-18)

Open archived version from archive - Mathematics In Europe - Historia/Filosofi

kB Bourguignon Cartan NL70 2008 pdf 45 kB Brüning Euler NL66 2007 pdf kB Baas Selberg NL65 2007 pdf 291207 kB Geiges Huygens NL65 2007 pdf 265 kB Schindler Gödel NL62 2006 pdf 66 kB Reiter Boltzmann NL61 2006 pdf 161 kB Lützen On the Mutual Influence of Mathematics and Physic NL60 2006 pdf 138 kB Samuel Cartan NL53 2004 pdf 263 kB Fokking van Kampen NL52 2004 pdf 1130 kB Rédei von Neumann NL51 2004 pdf 178 kB Gray Legendre NL45 2002 pdf 78 kB Mann Burnside NL 44 2002 pdf 78 kB Bekken Abel NL43 2002 pdf 124 kB Barner Fermat NL42 2001 pdf 293 kB Fauvel Napier NL39 2000 pdf 155 kB Gray Kovalevskaja andBeltrami NL35 2000 pdf 143 kB Barrow Green Celestial mechanics NL34 1999 pdf 193 kB Gray Klein and Krull NL32 1999 pdf 252 kB Philosophy of mathematics The question of the firmness of the foundations of mathematics has various answers depending on one s philosophical perspective The issue was discussed with particular intensity at the beginning of the twentieth century Although it is a question of fundamental importance it is not the focus of interest of most working mathematicians engaged in research or practical applications Here you can find an overview of the most important philosophical viewpoints Introduction Overview of the Philosophy of Mathematics logicism formalism intuitionism these articles are translations of contributions that were written originally for www mathematik de The following articles appeared originally in the Newsletter NL of the European Mathematical Society EMS They are republished here with permission of the EMS and the authors Corfield Nominalism and Realism NL75 2010 pdf 1395 kB Artstein Platonism NL75 2010 pdf 1362 kB Davies Platonism NL72 2009 pdf 108 kB Gardner On the Philosophy of Mathematics NL72 2009 pdf 108 kB Davis

Original URL path: http://mathematics-in-europe.eu/sv/information/historia-filosofi (2013-11-18)

Open archived version from archive - Mathematics In Europe - Chance as Composer

the same building block more than once Nonetheless there remains the immense number 759 499 669 166 482 of compositions Therefore after one has finished throwing the dice one may be fairly certain that the work created has never before been heard Chance plays a much more important role in contemporary music In the music of Xenakis for example chance determines not only which notes are played and in what sequence but also the waveform that is used to produce the sound Perhaps Xenakis s music evokes great enthusiasm in a relatively small number of listeners However it is interesting to ask what role chance plays in classical music What moved Schubert in measure six of a C major waltz to modulate suddenly to E major Why did Mozart decide to set the alla turca finale of his A major sonata in A minor Is it a question of a genius who takes inspired dictation from another world or did perhaps the random firings of certain neurons in the brain play a role We are not yet able to see so deeply into the brain But surprises are not out of the question for in the last few decades we have achieved the insight that random influences can have quite a productive and stabilizing influence Mozart from a Computer Anyone who takes the trouble to play through a large number of Mozart s dice compositions will realize that after a while one has the feeling that one has heard all of this before even if the notes have never been played in this particular arrangement The reason is that our brains are capable of recognizing musical structures What harmonies were used and in what order What is the rhythmic structure What intervals are preferred If these aspects are the same in two musical compositions they appear very similar to us One can make use of this fact to program a computer after a careful analysis of their works to compose music that sounds like Mozart or Bach One has simply to filter out the important aspects of the musical structure and with these parameters create something new With what probability say in a C major piece does the note B appear after the sequence G followed by C With what probability was it an E Then the computer will do the same if a G and a C have appeared in sequence then a B or an E will follow with the prescribed probability To the average listener the result sounds something like Mozart or something like Bach To be sure there are no new ideas and one wouldn t go so far as to call the music inspired Building on this approach the composer Orm Finnendahl developed a method of creating hybrid compositions One begins with the analysis of two compositions A and B and initiates the hybrid composition using the parameters of A That is all the harmonies rhythms and note patterns are based on those of composition

Original URL path: http://mathematics-in-europe.eu/tr/44-information/math-and/752-chance-as-composer (2013-11-18)

Open archived version from archive - Mathematics In Europe - Mathematics that you can hear (Fourier analysis)

easier to distinguish the deep voices of men from one another than the higher female voices That is because men s voices have a large number of overtones in the audible range which gives the ear many chances to make a differentiation A Black Box There are other mathematical results that you can verify with your ears at least qualitatively Imagine a black box in which one can input signals which are then somehow processed in the box s internal mechanism and then output Electronics hobbyists can imagine some wildly complex circuit into which an electric signal is introduced at some point and measured at some other point see Figure 2 Figure 2 The input and output of a black box This black box should have the following properties It should be linear That is if the strength of the input signal is doubled the output should be doubled as well and if one inputs a signal that is the resultant of two partial signals then the output is the same as what would result from the combination of the outputs of the two partial signals It should be time invariant That is if a wave in input and the output is logged the output for a given input should be the same today as it was yesterday For the electronics hobbyist this means that transistors may not be used they are not linear and no settings can be changed during the experiment One should limit oneself to resistors inductors and capacitors and the currents and voltages that arise should not be too great Although such black boxes describe a rather general situation they all have one property in common Sine waves the building blocks of Fourier analysis pass through such a black box essentially unchanged They can be weakened or pushed out of phase but that is all that can be done with them The audible consequence is this a filter for acoustic signals high pass low pass band pass and so on that can be described as a black box with the properties described above does not change the character of sine waves If you whistle into such a filter which gives a good approximation to a sine tone a whistle of the same frequency should come out the other end On the other hand a sung tone can have its character changed completely for example it could be much duller or much shriller A Recipe for Periodic Waves Fourier s Formula Periodic oscillations are combined according to Fourier s theory in the form of sine waves What is the exact recipe That is in what proportions do the various sine functions appear Suppose we have a function f whose graph is shown in Figure 3 Figure 3 A periodic function f There is a number p the period such that the function evaluated at x p is always the same as at the point x Therefore it suffices to know the values of the function on an

Original URL path: http://mathematics-in-europe.eu/tr/44-information/math-and/746-mathematics-that-you-can-hear-fourier-analysis (2013-11-18)

Open archived version from archive - Mathematics In Europe - On Semitones and Twelfth Roots

the same as in the old one Out of this difficulty was born the idea of dividing the octave democratically into twelve equal parts From one semitone to the next the frequency increases by the twelfth root of two thus by a factor of 1 059463094 It was over three hundred years ago that the equal tempered scale was developed and in his Well Tempered Clavier Johann Sebastian Bach 1685 1750 demonstrated by presenting a collection of pieces preludes and fugues written in each of the twenty four major and minor keys that one could play in any key without having to retune the instrument This development by no means exhausted the possibilities of relating mathematics to music In the twentieth century many composers used a variety of mathematical relationships in their compositions from the method of tuning to the large scale compositional form For example the composer Iannis Xenakis 1922 2001 used probabilistic methods game theory and group theory as organizing principles in his compositions However no matter how high a value is placed on mathematics it will never be possible to understand our enjoyment of a Schubert sonata or our favorite pop song in terms of a mathematical formula Pythagorean versus Chromatic Why did the twelfth root of two pop up in our discussion of equal temperament Suppose that the octave is to be divided into n parts where n is any positive integer A guitar builder would then have to supply n frets up to the middle of the fingerboard where the last fret would be exactly in the middle to sound the octave See Figure If all the musical intervals are to be of the same size then the frequency relationship between the note on the first fret and that on the open string must be

Original URL path: http://mathematics-in-europe.eu/tr/44-information/math-and/738-on-semitones-and-twelfth-roots (2013-11-18)

Open archived version from archive - Mathematics In Europe - M.C. Escher and Infinity

allow one to model infinity on what in the usual geometry is a finite portion of the plane It is thus that Escher s famous snake and fish motifs arose Finally we should mention Escher s impossible pictures such as the staircase that while finite spirals forever upward Each detail looks reasonable enough but the viewer is unable to put together all the local impressions to form a three dimensional image that conforms with reality Even if we used the terms local and global in their mathematical sense there would still be an inexplicable remainder that belongs to the psychology of perception In looking at Escher s pictures it becomes clear that the eye generally works as a quiet and unobtrusive censor that sends only preprocessed messages to our consciousness Do It Yourself Escher Would you like to make your own Escher print Perhaps you require a space filling motif for some wallpaper or gift wrap There is no limit to the possibilities that await the enterprising hobbyist though it must be said that it has long been known that there are only twenty eight essentially different construction methods all of which were actually used in Escher s images What follows is a relatively simple pattern to get you started In tiling jargon it is basic type CCC To understand it you will have to know what a C line is It is a curve that joins two points A and B such that the midpoint of the line segment from A to B lies on the curve and such that rotating the curve 180 degrees about the midpoint yields the exact same curve Some examples are shown in Figure 1 Figure 1 1 Three C lines Now you can let your imagination take flight Take a clean sheet of

Original URL path: http://mathematics-in-europe.eu/tr/44-information/math-and/751-m-c-escher-and-infinity (2013-11-18)

Open archived version from archive - Mathematics In Europe - The Leipzig Town Hall and the Sunflower

approximation Fibonacci numbers pop up in nature for example in the arrangement of the seeds of a sunflower And if you have a tape measure handy you can search for the golden ratio on your own person The relationship distance from elbow to fingertips divided by distance from elbow to wrist is just one of many possible examples However such results lead one quickly into the realm of speculation Perhaps the ratio of the number of bad characters in Grimm s fairy tales to the number of good characters is equal to the golden ratio Continued Fractions The golden ratio is a remarkable number for yet another reason This time it comes about in a type of approximation In dealing with numbers that cannot be represented as fractions it is convenient to replace such a number with a fraction that has a relatively small numerator and denominator and that represents a good approximation to the original number For example the fraction 22 7 3 14285 represents a good approximation to the circle number π Pi whose value to five decimal places is 3 14159 This approximation was known to the Egyptians 2500 years ago and such an approximation suffices for many everyday applications The best rational approximations to a number are obtained by continued fractions these are fractions that arise from a rather complicated process This is how it works A continued fraction is written as a finite sequence of natural numbers enclosed in square brackets The notation is interpreted as follows a 0 a 0 a 0 a 1 a 0 1 a 1 a 0 a 1 a 2 a 0 1 a 1 a 2 a 0 a 1 a 2 a 3 a 0 1 a 1 a 2 a 3 If that seems too abstract here are a couple of concrete examples 3 9 3 1 9 28 9 2 3 5 7 2 1 3 1 5 1 7 266 115 If one approximates a number by the best possible continued fraction then the larger the numbers in the continued fraction the better the approximation in general That is because all the numbers after the first appear in the denominator of the fraction that the continued fraction represents and the larger the numbers the faster the denominator grows therefore more decimal places of the number being approximated will be accurately represented Returning to the golden section this number has the remarkable property that among all irrational numbers it is the one that is least well approximated by a continued fraction in the sense that the numbers in its representation are as small as possible and thus it takes a relatively large number of terms to approximate the number to a given accuracy Indeed the best continued fractions for the golden section are 1 1 1 1 1 1 1 1 1 1 and so on This fact plays an important role in KAM theory One can conclude from this theory that a vibrating system

Original URL path: http://mathematics-in-europe.eu/tr/44-information/math-and/745-the-leipzig-town-hall-and-the-sunflower (2013-11-18)

Open archived version from archive - Mathematics In Europe - Matematik i Europa

matematik Landskab Forskning Historisk Filosofisk Musik kunst Matematik i Europa PR aktiviteter for matematik Konkurrencer Hjælp til Matematik Matematik som erhverv Andet materiale Mission Velkommen EMS Sponsor Sprogene Kontakt Juridisk ansvarlige Søg The European Mathematical Society Our Sponsor Munich RE Mathematics in Europe Detaljer Kategori math in europe Mathematicians are being educated in all the countries of Europe The significance for the development of the subject and the intensity with

Original URL path: http://mathematics-in-europe.eu/da/information/matematik-i-europa (2013-11-18)

Open archived version from archive