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  • Mathematics In Europe - Axioms
    we might never agree on what an integer is We would be like children who quickly figure out that to every answer to a why question there is another such question It was the field of geometry that for the first time in the history of mathematics was treated axiomatically Over 2000 years ago Euclid laid an axiomatic foundation in his Elements that has become the model for many other sciences and in particular for the other branches of mathematics Thus somewhat simplified we agree on what is meant by point line and so on and which constructions are possible with straightedge and compass End of discussion Now let us begin the actual work Today axiomatic systems are grounded in set theory Set theory is well suited as an axiomatic foundation because few axioms are needed to describe it To begin one need know only that a set is a collection of certain objects assembled into a new independent object This is nothing particularly special for we do the same in our everyday lives string instruments is the collective name for violin viola violoncello and contrabass European capitals is the set comprising Athens Berlin and so on When a theory is described as objects of the theory are sets M with such and such properties one can thereby avoid discussions over questions that properly belong to set theory An example In group theory we begin thus A group is a set G together with a composition law with the following properties 1 The composition law is associative 2 There is an element e of G called the identity such that e x x e x for every x in G 3 For every x in G there corresponds an element y of G such that x y y x e

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/137-axioms (2013-11-18)
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  • Mathematics In Europe - Proofs
    law x y z for three variables In using it twice we were able to extend the distributive law to four variables 2 Proof by Contradiction Another method of proof is called proof by contradiction This method is frequently used to disprove some assertion that is to prove that a certain claim is false The process goes like this One assumes that the assertion is true infers other assertions from this assumption and eventually reaches some contradiction One may then conclude that the original assumption was false Example We wish to show that the square root of 2 is not a rational number For a proof by contradiction we assume the contrary that the square root of 2 is rational It therefore can be written as a fraction p q where p and q are relatively prime that is the fraction has been reduced to lowest terms But by squaring we then have that 2 p² q² or equivalently 2q² p² But then p² is an even number which implies that p is even as well But if p is even then by the relative primality of p and q it follows that q must be odd If we now divide the equation 2q² p² by 2 we see that q² is an even number since p² contains a factor of 4 and so q must also be even But we concluded above that q must be odd and we have thus achieved the desired contradiction We may therefore conclude that 2 is an irrational number 3 Proof by Induction Many conjectures are of the following form a certain property holds for all natural numbers greater than or equal to some natural number k To verify this property for each of the infinitely many natural numbers involved would be of

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/138-proofs (2013-11-18)
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  • Mathematics In Europe - Vectors, dimension
    4 104 5 We express the addition by writing 3 2 4 2 0 0 1 2 0 3 4 104 5 2 0 4 2 3 4 104 5 where we point out the subtle fact that we use the same plus sign for addition of vectors as we use for addition of numbers Another parallel that vectors have with numbers is that there is a distinguished vector namely 0 0 0 0 0 that when added to any vector leaves that vector unchanged This corresponds of course to the number 0 Moreover every vector has an additive inverse whose components are the individual additive inverses Thus for example the additive inverse of 2 4 0 8 3 is 2 4 0 8 3 since these two vectors sum to the zero vector There is also multiplication of a vector by a number which is carried out term by term For example 7 2 5 3 2 1 31 14 35 21 14 7 217 There are also ways of forming the product of two vectors but we shall not go into that here Addition of vectors and multiplication of a number by a vector inherit from the analogous operations on numbers the well known associative and commutative properties Any space possessing all these properties is called a vector space Therefore R n is a vector space Dimension Why do we call R n an n dimensional space In particular why does R 5 have dimension 5 The reason is this To specify an element of the space you need to provide exactly n real numbers Fewer will not suffice and more would be too many This concept is completely general through all of mathematics Any mathematical object that requires say exactly 27 numbers to specify each of its members is said to have dimension 27 and we speak of a 27 dimensional space We confess that this is a great simplification Two examples 1 To specify a point on a curve a spiral for example we must specify a starting point called the origin and after that we require a single number to represent any point on the spiral for example by saying how many centimetres it is from the origin We specify the direction with a plus or minus sign For example you arrive at point 5 3 by moving 5 3 centimetres to the right from the origin and to the point 6 1 by moving 6 1 centimetres to the left We may conclude that curves are one dimensional objects 2 For our second example let us consider the surface of a sphere such as the Earth s surface It is well known that a point on such a surface can be specified by giving its latitude and longitude Therefore the surface of sphere is two dimensional Similarly a plane is two dimensional and points are usually described by specifying its x and y coordinates As an aside we note that these coordinates are called

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/139-vectors-dimension (2013-11-18)
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  • Mathematics In Europe - Finite vs. infinite
    numbers c Finite sets are simple in the sense that one can at least in principle solve all problems regarding them simply by considering each element in turn To prove statements about all the natural numbers one can use induction Even for the most hypercritical philosopher finite sets and constructions that involve only a finite number of steps pose no difficulties Infinite sets Georg Cantor was the first to investigate the infinite systematically in the context of sets He first defined two sets M and N to have the same size if the elements of the two sets can be placed in one to one correspondence Simplified somewhat this means that one can use the elements of M to enumerate the elements of N For example one can use the natural numbers 1 2 3 to enumerate the squares 1 2 2 2 3 2 Thus the set of natural numbers and the set of squares are of the same size The precise definition is somewhat more demanding of the same size means that there is a mapping f M N with the following two properties 1 For every x y in M if x is not equal to y then f x is not equal to f y 2 For every z in N there is an x in M such that f x z Then Cantor explained what it should mean that M has at most as many i e as many or fewer elements as N There must exist a mapping with property 1 above For simplicity when this condition is satisfied we shall say that M is less than or equal to N The following facts should be noted a If we consider only finite sets then things are exactly as you would expect The sets 1 2 3 are of the same size while 4 5 is less than or equal to 6 7 8 etc b In the domain of infinite sets a number of remarkable phenomena occur For example a proper subset of a set can have the same number of elements as the entire set We saw this above in the case of the natural numbers and the squares c The following are well known results due to Cantor more on this can be found at countable vs uncountable The set of rational numbers and the set of natural numbers have the same number of elements naively one would expect there to be many more rational numbers than natural numbers It can be proved that there are more real numbers than natural numbers Put another way no matter how you try to order the real numbers so that there is a first a second a third and so on you will never be able to include them all d There is no set that is bigger than any other set No matter how big M is one can find a set N such that M is less than or equal to N

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/141-finite-vs-infinite (2013-11-18)
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  • Mathematics In Europe - "Existence" in mathematics
    1 The formalistic point of view For formalists the problem disappears as soon as one scratches the surface there exists is simply shorthand for If one postulates axioms A1 A2 which includes existence axioms then one may conclude the existence of additional objects with other properties For example There exists a real number x such that x x 119 is short for From the axioms of the real numbers that is from the field axioms the order axioms and the axiom of completeness one may conclude using allowable methods of inference that there exists a real number x such that x x 119 For a mathematician this is the most secure philosophical foundation and he or she need have no fear of further debate on the subject 2 The Platonic point of view A Platonist imagines that the objects of mathematics exist somewhere somehow The word Platonic comes from the philosophy of Plato who argued that all ideas have a reality in the ideal realm For the Platonist the Pythagorean theorem has always been true as has the fact that there are infinitely many prime numbers and so on Such facts have gradually been discovered by humankind This point of view is widespread among mathematicians the great majority of whom view their subject in this light That such is the case likely results in part from the fact that after long involvement in a subdiscipline of mathematics its objects take on an aura of reality as real say as the Leaning Tower of Pisa and while much is known about these objects there is also much that one has may not yet have observed concretely Caution One is perhaps better off keeping this point of view to oneself Otherwise one might encounter provocative questions Where then precisely are the circles

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/142-existence-in-mathematics (2013-11-18)
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  • Mathematics In Europe - Extreme values
    that mathematicians have given much thought to such problems Here is some further information on the topic 1 Existence It is unclear a priori that there in fact exists an x for which f x is maximal For example there is certainly no longest route from Istanbul to Seville since given any route one can take a detour by adding a loop thereby making the route longer One has to think through such problems carefully Usually one may simply make use of known results An example Let a b be a closed interval of the real numbers and f a b R a continuous function Then there is guaranteed to be an x such that f x is a maximum Advanced students will know that this result depends on the notion of compactness 2 Uniqueness In real world problems there is almost never a single value of x that produces a largest value of f x Such is generally the case for instance in the example above In the extreme case that f is constant every value of x produces a maximum How does one find such an x Case 1 If the set M is finite and not too large at most several million elements perhaps a billion at most then one can use a computer to calculate all the values of f x and then choose the x value that gives the maximum Unfortunately even with finite sets which are too large this technique is infeasible Case 2 The set M is an interval a b and f is differentiable This case has many applications and it is studied in first year calculus Here is a method for finding the x having the maximal f x Determine all x such that f x 0 that is all x such that the graph of the function has a horizontal tangent Let us call these x 1 x 2 x n Now calculate f a f b f x 1 f x 2 f x n and see which of these is the largest there may of course be more than one such x In this way we have been able to reduce an extreme value problem on an infinite set M namely a b to the complete solution of the equation f x 0 and then carrying out a finite number generally not very large of test computations Case 3 Now suppose M is a subset of n dimensional Euclidean space and f is again differentiable We now proceed similarly to what we did in Case 2 except that now we must solve the vector equation grad f x 0 to go into more detail here would take us too far afield Case 4 If we may not assume differentiability and we have only some sort of black box for the function rule which on input x produces the output f x one may have to rely on probabilistic methods Consider for instance a traveller who is lost in

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/143-extreme-values (2013-11-18)
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  • Mathematics In Europe - What means "equality" in mathematics?
    something is measured by a single real number since such numbers can always be compared That is how the results of many sporting events are determined Things become more complicated when several measurements are combined It is clear that a student who receives grades of B and C is better than a student earning a D in all subjects But what if the second student is the school s superstar in some special subject for example the winner of the national mathematical Olympiad The moral here is that in many cases it is not at all clear how an order in the sense of ranking is to be defined and what such an order should look like And now back to mathematics Here too there are many situations in which one would like to compare objects sets functions and so on There are two important points to be noted here 1 Mathematicians have approached the topic of order axiomatically They speak about a partially ordered set when that set has been equipped with a partial order namely a relation that is reflexive transitive and antisymmetric More details on this can be found below This may all sound a bit intimidating but it is simply the axiomatic formulation of what any reasonable definition of size must require In this regard transitivity means simply that if I regard A as greater than B and at the same time regard B as greater than C then I must regard A as greater than C In this way such varied problems as comparing numbers comparing functions and comparing sets can be handled in a uniform way 2 Comparison is often made by assigning a number to every object under consideration and then considering the numbers themselves An example from real life Something like this is done in the decathlon For each of the ten events points are awarded and the winner is the one who earns the most points If it were required that the winner be the one who had been the victor in each event then there would practically never be a winner We should emphasize that in applications the choice of such an assignment one speaks of an objective function is almost never made by a mathematician It makes a huge difference whether in evaluating an industrial product one seeks to optimize longevity profit utility or some other characteristic The mathematics required to solve the optimization is independent of such an assignment We have pointed out these obvious facts because many people seem to have the impression that the application of mathematics to a problem immediately makes everything value free and objective On the contrary the important decisions those involving subjective values frequently have nothing to do with mathematics and no amount of mathematics brought in later can eliminate that subjectivity Relations In dealing with some aspect of mathematics or of real life one often wishes to assert that one object stands in a particular relation to another object in real

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/145-what-means-equality-in-mathematics (2013-11-18)
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  • Mathematics In Europe - Equality in Mathematics
    a role than equivalence with respect to some aspect For jotting a quick note a napkin can be as good as a pad of paper Or with respect to the goal of getting to the opera this evening a compact car a limousine and a taxi are all equivalent However as soon as cost estimated time of travel or prestige comes in the door any idea of equivalence flies out the window The situation is similar even in elementary mathematics If you want to explain to your child the meaning of five you can as well produce five apples as five children The apples and the children with respect to fiveness are entirely equivalent The complete truth is somewhat more complicated If one wishes to explain what the abstraction five really means one generally begins nowadays with equivalent with respect to number Then fiveness is the entirety of all objects that are equivalent in respect to the set of the fingers on one hand This principle permeates all of mathematics In geometry two triangles are equal if one can be superimposed on the other by a translation rotation and possible flip And in probability theory it makes absolutely no difference

    Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/292-equality-in-mathematics (2013-11-18)
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