Group, ring and field are algebraic (abstract) structures. Some people prefere to deal with concrete objects rather than abstract objects. Therefore they sometimes seek concrete realisations (representation) of such algebraic structures either geometrical or physical. I think the best way to motivate students to study abstract structures is to use those concrete realization and then present various applications of groups , rings and fields in physics , chemistry , computer science and information theory.
Hi, in mathematics, groups and rings are the first algebraic structures so with less properties than vector fields. The theory behind groups and ring is complex. In practice, except for some problems than can be modeled by groups or rings the majority of applications (physics, applied mathematics, etc..) require et least to work in an euclidian vector space. This is perhaps why these structures are less popular (it is an assumption).
Can you explain what you mean by "not like"? We can't live without them, so why be negative about them? For one thing - they are very, very useful, and in math I would say that we should not mock something that works. :-)
Group, ring and field are algebraic (abstract) structures. Some people prefere to deal with concrete objects rather than abstract objects. Therefore they sometimes seek concrete realisations (representation) of such algebraic structures either geometrical or physical. I think the best way to motivate students to study abstract structures is to use those concrete realization and then present various applications of groups , rings and fields in physics , chemistry , computer science and information theory.
Because most of these subjects are contained in theoretical or pure mathematics ( Abstract Algebra)and the mind or brain and logic are adopted to deal with them
Is there any people of Mathematics who don't like groups, rings or fields? I belive there is non. Because concepts became familiar to anyone who starts studying Z.
Ofcouse the may exsits some abstract concepts some people of Mathematics dislike. But ofcoure that are not these
There are certainly 'people of mathematics' who dislike diffuse questions like this one.
The mathematicians have their own abilities, so they usually specialise in one style. There are analysts, geometers, algebrists etc. Presumably the geometers don't like groups and rings because they can't visualise them, while the algebrists don't like fields because they can't categorise them.
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In his influential 'Erlanger Programm' Felix Klein characterized geometries as the theory of associated group invariants.
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Galois solved the most urgent algebra problem of his time (general solvability of polynomial equations of order n) by studying properties of fields.
So my rudimentary ideas concerning the history of mathemathics suggest just the contrary of what Claude came up with.
Difffuse questions get diffuse answers!
Ulrich, I know all that. But I don't say that a geometer doesn't use groups, but that it is more at ease with visualisable structures, and so don't like or less like the former. When it comes to understand a theory or to prove a theorem, s/he naturally switch into the geometer mode even if it is algebra. Similarly for other types of mathematicians.
There are topics of mathematics that are dealt with by entirely different styles, like Riemannian, intrinsic geometry, and extrinsic one based on differential forms like Cartan. They are strictly equivalent, with theorems like Whitney's one and Gauß' Theorema Egregium making the bridge, but the used concepts, even bearing the same name like "form," are different.
It is not a matter of mathematics at all, but of mathematicians who are imperfect human beings with their strengths and their weaknesses. It is quite possible to draw boundaries between them, as research in applied psychology have shown. There are three basic aptitudes that correspond nicely with geometry (spatial,) algebra (numerical,) and analysis (verbal.) They are unified in the g factor.
I should add there are geometries that are not of Klein type, and anyway they are Lie groups. And also the so-called algebraic equations are in fact a problem in analysis: finding the zero(s) of a function. What would be wrong with Klein being an algebrist and Galois being an analyst? The progress in these disciplines would then be due to applying different ideas from the ones of the usual workers in them.
If they are not involved or not interested in or do not understand sufficiently, or do not enjoy well therefore their uninterest will probably be interpreted as dislike. I have to admit that groups are quite complex and it may not suit for every mathematician. In science almost all of us are specialising ourselves to certan extent.
I agree with what Ulrich Mutze has said. When Andrew Wiles wrote his proof for Verma last theorem ttp://math.stanford.edu/~lekheng/flt/wiles.pdf ,
he used most branchs of mathematics enhancing the tremendous integration of mathematical subjects
I fully agree with Ulrich Mutze. But I’ll go even further by advocating the idea that all significant progress in the mathematical science as a whole has come from the recognition and conceptualisation of structures : groups, rings, fields, to cite only the algebraic structures which undergraduate students must get to know …
First what does scientific progress mean ? In the eyes of the laymen, and even of many scientists in other disciplines, the image of a mathematician is that of a « problem solver », more precisely : on the lower side, a repair(wo)man with a toolbox, on the higher side a technician of a perfect science (« perfect » meaning here completed and closed). I remember a discussion I had once with a physician in an interdisciplinary commission. When I told him that I was embarked on an interesting project, but not being sure to succeed, he gave me a surprised look which clearly hinted : How can there be a mathematical problem which you could not solve using your toolbox ? He had forgotten (?) the ABC of any scientific progress : a finite toolbox can be used only for finitely many types of problems ; to attack new types, you must invent new tools.
Then how does invention work ? Whatever the problem, in no way could there exist a general in chief commanding an army and directing his troops to here and there. The reality is rather a mob of searchers with different interests, different formations, different backgrounds… This mob does advance without battle order, but there comes a moment when some map road map emerges and some geographical map must be drawn. In an abstract science as maths, this moment of « coagulation » consists, as I said, in the recognition and conceptualisation of a structure. « A mathematician, like a painter or a poet, is a maker of patterns » (G. H. Hardy). This is not a question of « like » or not as on Facebook. Abstraction is not for fancy, it is a unifying process, and the pattern which emerges from this process is not only a tool, but also a user guide (to understand past theorems and discover new ones). To come back to the examples given by Ulrich, there is no doubt that the theory of groups has pervaded all the domains of maths, and more recently, modular forms have introduced a new paradigm in the theory of numbers, what M. Eichler nicknamed « the fith operation of arithmetic » (the 4 previous ones coming from the structure of rings).
One last remark about the word invention , which is commonly distinguished from discovery . A recurrent question about mathematical theorems is whether they are discovered or invented. Putting it in a Platonician way : do the mathematical structures on which we work pre-exist in Plato’s « world of ideas», or are they manufactured by us in the « wold of men » ? As a personal experiment I have sometimes taken advantage of the « 3-rd half » of a dissertation defence to conduct an informal poll on the question. The result : a large majority of algebraists (resp. analysts) are «platonicians» (resp. are not) . No comment, since my polls had no statistical pretention . /.
Who do not like groups, rings and fields are non-social and do not want to marry and make a family, ha ha ha ha ...
I am very thankful all of you. U sharing ur ideas to solve my question. I hope that u will reply in future of my problem.
The biggest progress of mathematics have been to subject it to formal logics. There are sets, functions, and other abstract (i.e. invented or imaginary) relations, then everything else is discovering interesting particular cases. A group (or other structure) is nothing else than a particular case of a function from the Cartesian product of a set with itself, and into itself. Sets and functions are themselves interesting particular cases of abstract relations, the other important one being the equivalence relation. The groups in geometry are nothing else than an interesting equivalence relation called congruence or symmetry. Fields in algebra are functions etc. etc. Of course mathematics, as an incarnation of formal logics (a set of conventional relations) about abstract relations is unified. But anyway, there are different manner to make mathematics. A group is not necessarily a purely algebraic concept from this very unification, but can be considered in geometric of analytic light. These ways though are not unified, since they can't be reduced to abstract relations but are different cognitive processes. Visualisation is different from language, and different from calculation. They process data in different and not interchangeable forms. There are relation between images, words, and figures which can be thought mathematically, but what they refer to can't be unified.
So I think there is a misunderstanding about Platonism. Platon spoke about pedagogy and concepts as opposed to immediate perception. The ideas are then related to cognitive process, not to the sifting of abstract relations. For example in the realm of ideas, a colour is represented by a wave length, which can be measured by comparison with a standard, that is, something physical. It makes no useful sense to oppose immediate perception and abstract (invented, imaginary) relations.
The groups should have been called "symmetry" or something like that, since what makes this particular case interesting is that it has the properties of a symmetry relation. Now symmetry is manifestly of geometrical nature. Even though in Euclidean geometry, groups aren't formalised, they are underlying it. Like M. Jourdain, superposing geometrical figures the geometers were unknowingly using groups. These have been useful only for generalising geometry, especially because of the unprovable Euclide postulate. But in spite of many attempts, it has never been possible to define a straight line otherwise than implicitly. That makes a definitive distinction between mathematics and geometrical ideas.
Dear C.-P. Massé,
I don't quite agree with your assertion that "symmetry is manifestly of geometrical nature", which could be interpreted as "geometry is more natural than Galois theory". On the contrary, one could say that the discovery (or invention) of Galois symmetry, because it is more deeply hidden, reveals the pre-existence of the idea (in the platonist sense) of symmetry. The italics everywhere are warnings that here we are on the borderline of an endless metaphysical (not to say tautological) discussion on the notion of reality. Speaking of platonist concepts, let me stress that they seem to me historically dated, corresponding to a certain - primitive - stage of scientific knowledge. Nowadays one could as well lean on the discoveries in neuro-sciences to assert that Plato's ideal world is just the world of our brain, which is the indispensable intermediary in our perception of reality. Then, no wonder when we recognize patterns / structures in our description of this reality since it is the same organ which organizes (regurgitates) what is has perceived (ingurgitated). Surely metaphysicists would howl at such a materialist interpretation. But to stay with the platonist imagery, I would say that we discover mountains and streams, but that we buld (invent) bridges and roads to come across.
Back to mathematics. One example is better than one hundred speeches, as the French say, so let me give an example which I think is the perfect illustration of what we are discussing about. The starting point is the famous Bloch-Kato conjectures on the "special values" of the Zeta function (or more generally, of the L-functions), which generalize Dedekind's "analytical class number formula" and reveal a deeply mysterious relationship between the transcendental world (the above functions) and the algebraic world (the ideal class group and the higher K-groups). See the 2nd part of the appended text if you have some leisure. These conjectures have been proved recently, at least for abelian number fields, thanks to an original - not to say platonist - idea of Grothendieck, the so called search for "motives". Remember Plato's "apologue of the cavern": we live in a cave, and the physical reality which we perceive consists in shadows cast on the walls by the sun in our backs; to really understand the true reality, we must turn around and face the archetype which projects these shadows. Grothendieck applied this philosophical concept to algebraic/arithmetic geometry : around a given variety are floating a host of dissimilar cohomologies (Betti, de Rahm, étale...), which become isomorphic when passing to an algebraic closure, but such a passage destroys all the arithmetical properties we are interested in. Following Plato, Grothendieck suggested to look not at the shadows but at the archetype, the conjectural "motivic cohomology". A harebrained idea, it may seem, but which has occupied people for decades, until the final success of Voevodsky (Fields medalist in 2002). Vv. immediately used his new weapon to bring down two other conjectures (by Bloch-Kato and Lichtenbaum-Quillen) giving precise relationship beween K-theory and Galois/étale cohomology, and that was how the conjecture on special values was settled. So, as a mathemathician are you a Platonist or not ?
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Thong, when you plot the graph of an algebraic polynom, the symmetries are obvious and of geometrical nature (or after some simple geometrical transformation.) Indeed, I doubt you can find a symmetry (group) that can't be translated into geometrical language.
In mathematics, many notations and approaches are used, so that it seem very complex while there are connections between them that reduces it to something manageable. That is only a wealth of mathematicians and of traditions, not of ideas in the Platonic sense.
I can still go farther. Mathematics is not something isolated. In fact, it is only one of the forms of the cognitive functions. They work just like computer science, physics, or ordinary language. The environment is modelled by objects having a state and a behaviour. The brain is only able to extract useful information from a much too complex environment. That is done by recognising patterns, be it a prey, a tool, an abstract idea. As I said, in mathematics the patterns are particular cases of relations. Mathematics is unable to manage the general case, it study only very simple and easy cases. In this, they aren't different, and in modern language, what is closest to Platonic idea is a pattern. I don't think at all that ideas are independent and floating in a "beyond" were they can be perceived by the mind. I rather think that all our idea are false, but they do a not so bad job for our survival.
@Claude Pierre Massé
Before going on, I feel we need to define more precisely the main terms of our discussion, namely "geometry" and "symmetry". According to Wikipedia:
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space .
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance : the property that something does not change under a set of transformations.
Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure (…) In abstract algebra, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.
Even if these definitions are not entirely satisfying, let us take them as a starting basis.
1° You say : « When you plot the graph of an algebraic polynomial, the symmetries are obvious and of geometrical nature (or after some simple geometrical transformation) ». This could be true in certain simple cases, and "on the surface", but certainly not in general, and in the "depth" of the structure. To stick to your example, take e.g. a cubic polynomial. The geometrical symmetry of the graph certainly catches the eye, but it reveals nothing of the hidden permutation group of the roots: that this group is A_3 or S_3 , and that the discriminant of the polynomial determines what case occurs. Even further, take the example of a plane projective smooth cubic - called an elliptic curve (not an ellipse !) - which is endowed with a natural commutative group law, revealed probably by the study of complex torii by people like Weierstrass. This addition law has been widely popularized by drawing intersections on the graph of the fuction, but such a do-it-yourself job reveals nothing of the vertiginous depth of a theory which has led - among other achievements - to Wiles’ proof of FLT. We can even take examples in analysis : what do the graphs of a polynomial and of its derivative reveal about the parenthood of the functions ? Nothing in general. Conclusion : instead of focusing on a supposed geometrical essence of symmetry, why wouldn’t we apply Occam’s « razor principle » and adopt the working hypothesis that the symmetry lies in the structure of the mathematical object itself, not in the shadows - geometic, algebraic, analytic - cast by it ?
2° You say : « Indeed, I doubt you can find a symmetry (group) that can't be translated into geometrical language ». Yes, I can. Consider for instance the Lorentz group, which is the symmetry group of the quadratic form x^2 + y^2 + z^2 - t^2 and which plays a prominent role in the theory of Special Relativity. Unless you postulate that all physics are geometry, the only geometrical object I can see here is the « light cone » x^2 + y^2 + z^2 - t^2 = 0, but this is just an illustration, nothing more than a drawing. And if you accept the definition proposed by Wiki., I could find a dozen other examples without even thinking hard. At this point, perhaps we should make a small distinguo between « structures » and what you call « patterns » (Hardy also, in his « Apology of a Mathematician »). I think that the word « structure » is more rigid than « pattern », in the sense that it already implies the existence of underlying relations/laws behind the regularity of a « pattern ». Here is a developped example. It has been known from Euclide (at least) that there exists an infinite number of primes, but the question was whether they are randomly distributed. Actually the answer is « No », there are patterns in the distribution of primes (see Dirichlet, Hadamard &La Vallée-Poussin, Green & Tao…), and at the time being, even if we don’t know if there will ever be symmetry groups, striking symmetries appear in many deep results. A classical example is Gauss’ famous « quadratic reciprocity law » : if p and q are two distinct odd primes, then (p/q)(q/p) = (-1)^a, where 4a = (p-1)(q-1) and (*/*) denotes the Legendre symbol. Even without knowing the definition of a Legendre symbol, anybody can appreciate the striking symmetry of this formula. And this is number theory, not geometry in the slightest way. Gauss himself was so aware of the importance of his law that he gave half a dozen different proofs of it during his career (none of them geometrical). Then the usual process of research/discovery that I described previously went on its way, until 150 years later, it culminated in the « splendor » (in Hasse’s own words) of the so called « class field theory » (CFT), which describes all the abelian extensions of a number field K in terms of parameters contained in K itself. And the story still goes on, with the search for a non abelian CFT, the famous « Langlands program », of which Wiles’ work can be considered as the first milestone.
3° You say : « The environment is modelled by objects having a state and a behaviour. The brain is only able to extract useful information from a much too complex environment. That is done by recognising patterns, […], in mathematics the patterns are particular cases of relations [which are], in modern language, what is closest to Platonic idea. I don't think at all that ideas are independent and floating in a "beyond" were they can be perceived by the mind ». You don’t believe in independent « ideas » floating in a beyond, but you postulate the existence of « objects » floating … where ? This is the type of tautological discussion on « reality » which I don’t want to get stuck in. Apart from this objection, I think we agree on the role of our brain in the perception of « reality ». As I said, platonist concepts should be nowadays taken only as metaphores, and the « beyond » you alluded to is inside our brain. Let me illustrate this materialistic-neuro-physiologist axiom by one of the two dozens intriguing clinical cases reported in Oliver Sacks’ famous book « The Man who mistook his Wife for a Hat » (1985) (*). Around the mid 60’s, Sacks had the opportunity to study in a public hospital the strange faculties of two autistic , gravely retarded twins. At the age of 26, the twins were not even able to lace their shoes, but if you told them a date, any date in an interval of 40.000 years in the past or the future, they could give you instantly the corresponding day of the week. Let us not be mistaken : they were not prodigy calculators like the ones we can see at times in a circus or a theater, they were hardly able to add/substract (and with errors), and to multiply/divide was beyond their ability, so their computing gift was a mystery. Sacks had the opportunity to witness a few intriguing events. One day, the content of a box of matches fell accidently on the ground, and one of the twins immediately said 111, and the other repeated three times 37, 37, 37. Another day, Sacks caught the two in the middle of the funny (funny for them) game of exchanging numbers : one issued a 6 digit number, the other answered by another 6 digit number, and so on. They were, in Sacks’ words, like two experts in wine tasting a rare bottle. He noted a few numbers and, at home, had the idea to consult a table of primes. Bingo ! The twins had been exchanging 6 digit prime numbers ! Coming back to the hospital with his table, Sacks invited himself into their game. At the first opportunity, he launched an 8 digit prime, of course taken from his table. For the first time, the twins seemed to notice his presence, and for the first time, they hesitated for a few minutes, and then one of them came up with a 9 digit prime, the other answered with a similar number, and the cheating Sacks followed with the largest prime in his table, with 10 digits. After an unusually long silence, one twin issued a 12 digit number, the other called again, and when Sacks left them, they were exchanging 20 digit numbers, which were probably primes, but Sacks had no possibility to check. I let the reader conclude by himself ./.
(*) There have been at least 2 movies inspired by Sacks’ book, « The Awakenings » with Robert De Niro & Robin Williams, and « Rain Man », with Tom Cruise & Dustin Hoffman.
Undoubtedly some quite successful applied mathematicians, analysts, geometers and combinatorists have taken one or two courses in abstract algebra and hated it, perhaps just managing to pass the course. It requires a different skill set. It's a pity when someone like this becomes Dean or Chair and seeks to exclude algebra.
Thong, patterns are not structures, it is a much more general term, applying to every cognitive and perceptual function. The brain is a machine to recognise patterns, because this is important for our survival, and mathematics is not an exception. Beyond the various structures and theories, we found that mathematics are organised around patterns, they don't study anything haphazardly. (Some of them have a name, like cohomology.) These patterns are very often the sames as in physics and other sciences, including financial and management ones. That is because, just like for perceptual functions, it is the more efficient way to extract useful information from a very complex environment, and in mathematics the environment is the huge wealth of general relations.
Groups, rings and fields are structures of mathematics which are in sequence of complexities.These are foundational structures of mathematics and in some way or another they are related or one grows upon the other. A true matematician recognizes and loves the vitality of each structure, since it is a part in the galaxy of mathematics. To know about one is a natural imperative to know about the other, and one is embedded in the other as a substructure. A dislike of one is a dislike of mathematics per se.
Thong, I take on the example of the Galois group of the roots of a polynomial. In its formulation, it is baffling. But it is the way the mathematicians express things, they often conceal the insight that got them to the result. Let's take the polynomial in its fully factorised form. The Galois group is but a consequence of the commutativity of the multiplication among the monomials. Then it is a matter of whether the product or the sum of two roots is rational (and real) or not. If not, the corresponding permutation is removed from the group.
(x-a)(x-b) = (x-b)(x-a) = x^2 + (a+b)x + ab = x^2 + (b+a)x + ba.
Now the multiplication and its commutativity has an obvious geometrical interpretation, it is a (hyper)volume. The permutation group is a subgroup of the (proper and improper) rotation group.
The Lorentz group is still easier. For simplicity, let's take its two dimensional version. A Lorentz transformation is a squeezing along a diagonal that preserves the area of the surface elements: x^2-t^2 = (x+t)(x-t) which is the formula of a surface in the so called light-cone coordinates. It is also sort of a rotation, since rotations are essentially volume preserving transformations (together with the translations, that is, rigid motions.) It can be distance and angle for ordinary rotations and inversions, or neither for pseudo rotations.
@ Claude Pierre Massé
Come on ! I can't believe your geometrical "idée fixe" leads you to issue such - excuse the word - rather absurd affirmations.
1) About Galois, you say : " The Galois group is but a consequence of the commutativity of the multiplication among the monomials (...) Now the multiplication and its commutativity has an obvious geometrical interpretation, it is a (hyper)volume. The permutation group is a subgroup of the (proper and improper) rotation group ". I don't even insist on your opening sentence, which would make such giants as Fermat, Lagrange, Legendre, Gauss... turn in their graves. Don't you think they knew that ? From what I understood of your 2nd degree example, I guess that you are talking of the so called symmetric functions of the roots of a polynomial, which appear hundreds of times in the works of those who came before Galois. The revolutionary new idea in Galois theory is the systematic use of groups acting on roots of polynomials (instead of the roots by themselves) to study and classify algebraic extensions of fields. I think everybody agrees on that, so our discussion must be centered on the "naturality" of the notion of group. It is generally accepted that it originates from the 4 operations which we (more or less painfully) learned to master in elementary school. But just think of it : these 4 operations actually involve 2 distinct laws, + and x, and what we learned to master in school was indeed the manipulation of the structures of rings and fields. No wonder we blundered ! The group structure is simpler but more fundamental (in the sense of "nearer to the foundations"), that is why I said in one of my previous posts that the "symmetries" in Galois theory are deeper buried than those in geometry. And if we recall that a group law can be non commutative, then, in terms of exploration/discovery, the notion is not just on the other side of the hill, but on the other side of the mountain.
Once we agree that groups must intervene in the symmetries we are discussing about, the question shifts to the "naturality" of geometry vs Galois (or anything else). When you say that multiplication and its commutativity has an obvious geometrical interpretation, it is a (hyper)volume, I hope you don't maintain that this is more natural than learning to multiply in elementary school ! If you don't, then why invoke this interpretation, which has no more relevance than : "A camel and a peanut shell share a common property, they both have bumps". Given n objects, say n pebbles, what you claim is that it is more natural, because more geometrical, to relate them by rotations rather than by permutations. When children play shell game ("bonneteau"), they simply permute the pebbles. What you ask from them is to draw a circle (center, radius), place the pebbles on it (according to what repartition ?), rotate (angles depending on the repartition), etc. This is soundly absurd, and methodologically all the more absurd. Methodologically, to speak of rotations, you add an irrelevant metric structure to a problem which is combinatorial in essence. Finally and mathematically, your assertion that the permutation group is a subgroup of the (proper and improper) rotation group is completely false. If you distribute your n objects regularly on your circle, your rotation group Cn will be cyclic of order n, whereas the symmetric group Sn has order n! , and it is Sn which contains subgroups isomorphic to Cn (= n-cycles, or circular permutations, the name speaks by itself). You could say you allow rotations of any angle (in which case you introduce an even more irrelevant "unnatural" notion, the real numbers), but you still remain wrong, because Sn is not commutative for n >2.
2) About Lorentz, I regret to say that I can't make any valuable comment on arguments which are not written in a precise mathematical language. In particular, you seem seem to argue by analogy and - excuse the word - hand waving when you speak of sort of a rotation, or rotations are essentially volume preserving transformations. When I try to translate this into mathematical language, I can't but disagree. To speak of volumes or surfaces or angles, you need a metric. The Minkowski space-time M, i.e. the vector space R4 equipped with the quadratic form t2 - x2 - y2 - z2 (note the slight change of notation, in coherence with (1,3) below) is not a metric space since it admits isotropic vectors (= the "light cone"). The Lorentz group L we speak about is the group of linear automorphisms of M which preserve the above quadratic form. Denoted O(1,3) in the usual termino-logy of quadratic forms, L is a Lie group of dimension 6 which admits a subgroup isomorphic to the usual rotation group O(3) (this is natural, since the Minkowski space-time contains the "usual" space). Historically the Lorentz transformations were determined by trials and errors (from 1892 to 1904) to account for the "contraction of lengths" in the famous Michelson-Morley experiment (1887). The math. definition and determination of the Lorentz group as described above are due to Poincaré (1905) and Minkowski (1907).
To be honest I should add that there is a connection with geometry which I forgot about in my previous post, more precisely an isomorphism between the Lorentz group and the complex projective special linear group PSL(2, C), which is in turn isomorphic to the symmetry group of conformal geometry on the Riemann sphere (a conformal transformation is one which preserves angles). But, as I said before, there is no reason to interpret this far fetching parenthood as a sign of a supposed geometrical essence of symmetry, rather than a manifestation of the profound unity of mathematics ./.
EDIT : At the beginning of the last §, PSL(2, C) is not isomorphic to the whole Lorentz group, but to the so called "restricted" Lorentz subgroup consisting of transformations which preserve orientation, direction of time, and have determinant +1. Sorry for the misprint.
"Naturality." No, there is no "naturality" in mathematics. I knew that you would answer something like this. There are ways of thinking that are more or less usual to such or such mathematician, and that's all. Everything that is geometrical seems more "natural" to me, because I have a "natural" ability to handle geometrical data and to see in space. I see no point in discussing tastes and colours here.
I see no point in discussing tastes and colours here. So do I, and since you don't give new math. arguments, we should stop here.
Your shouting false proves only one thing, you haven't been able to see it geometrically. You introduce a circle, which shows that you aren't able to see in more than two dimensions. The permutation group of n objects Sn is of course represented in a n dimensional space.
The multiplication isn't introduced this way in elementary school, and that's the very reason why most pupils can't understand it and are branded: "bad in maths." That's awful. I gave private lessons, and I have noticed that most pupils are able to do mathematics if they are presented in the appropriated way for them. There is a quotation attributed to Einstein: "Everybody is a genius. If a fish is judged by its ability to climb a tree, it will think all its life that it is stupid."
I can't give new math arguments if you aren't willing to receive them. That's not the first time I experience the psycho-rigidity of the mathematicians. Other people are also able to reason in a correct way, even if it isn't yours.
Considering your polemical tone, I'll answer only to the "mathematical" part of your last post. You say : Your shouting false proves only one thing, you haven't been able to see it geometrically. You introduce a circle, which shows that you aren't able to see in more than two dimensions. The permutation group of n objects Sn is of course represented in a n dimensional space . I wonder why on earth n given objects must be imperatively be studied in an n-dimensional space (poor pupils in elementary school !), but I accept this diktat, and I'm still shouting false.
Since Sn contains S3 for n > 2 , let us examine first the special (and central) case of S3 , the group of permutations of 3 points living in R3 according to your wish. Here R3 is considered indifferently as a vector space or an affine space with a fixed origin. Suppose that S3 is contained in the group of rotations of R3. Speaking of rotations, we assume implicitly that the vector space R3 is equipped with its usual structure of an euclidean space. Then the group of isometries of R3 (= linear automorphisms preserving the euclidean inner product) is usually denoted O(R3), and the subgroup of rotations O+(R3) (= isometries of determinant +1). If O+(R3) contains a subgroup H isomorphic to our S3 , then necessarily H stabilizes the plane P (of dimension 2) determined by our 3 points, and because isometries preserve orthogonality, H will also stabilize the subspace orthogonal to P , which is a line D (of dimension 1). So for all s in H, we can consider the restrictions sP and sD of s to P and D respectively, and obviously sD = IdD (i.e. D is the axis of the rotation) and sP is a plane rotation in the usual sense. In other words, S3 is isomorphic to a subgroup O+(P) = O+(R2), which is commutative : contradiction.
This argument carries over almost without change to arbitrary n > 2. Pick up 3 points and construct the plane P as before, stabilized by a subgroup H of O+(Rn) isomorphic to S3 . As before, H stabilizes also the orthogonal of P, which is a subspace Q of dimension n-2. This means that H is the direct product of its restrictions HP and HQ to P and Q , with HP = H = S3 : same contradiction as before ./.
Almost all mathematicians I know who have studied and understood groups, rings, or fields, have ended up loving such algebraic structures. Ah, I can state the same after including semigroups, modules, and algebras in your list of algebraic objects. All of them are simply beautiful.
We have here an example of somebody who don't like geometry. In general, the mathematicians are used to complicate thing as much as they can. And when someone point to an angle of sight that simplifies them, they merely say it is false. Mathematics is less science than poetry. Some argue that odd lines or hugged rhymes are false, while actually they only don't like them.
Perhaps groups, rings and fields are too simple, but in applied mathematics, we can't afford endless discussions about regularity hypotheses or uniform convergence. These structures, apart perhaps from rings, are of much use in physics.
Perhaps some structures aren't liked because of the way they are taught. I take the example of the complex numbers. One usually insists on the purely abstract algebraic aspect. But abstraction is always from something. A better way could be to start from symmetries of the plane that preserve angles and a fixed point, subset of two-dimensional matrices, sinusoidal functions of the same frequency, and to investigate what composition properties they have in common. Then explain that they all can be treated in a abstract way only using the same algebraic properties. The algebraic magic is then plain to see, and that avoids consistent and fruitless admonishment of not wondering what the imaginary unit really is, since it can represent different things with no obvious relation between them.
Dear Claude, I highly disagree with the statement that mathematicians like to complicate thing and I even more that they state that something is false for the sake of being simple. Indeed, mathematicians love to find simple solutions to difficult (not complicated) problems. For example, in the sixties John Thompson and Walter Feit proved the following statement "all non-abelian finite simple groups have even order". As you can see this is a simple, clear, and concise problem. But it was also a cornerstone in the classification of finite simple groups (one of the most important and systematic project in mathematics history), which took a century of hard work and the labor of many brilliant mathematicians. The proof given by Feit and Thompson was published in the Pacific Journal of Mathamatics and took a whole issue (about 250 pages long). But it didn't take 250 pages long because they wanted to make it complicated; it was so long because they were answering a really hard problem. I am pretty sure that if someone else had answered the same important question using only a simple ten-page argument, the mathematics community would have rather preferred such a (non-existent) simpler proof. One of the main task of mathematicians is trying to make theories, theorems, and proofs as simple as possible. This is why they try to create common and succinct systems of axioms, reason for which areas of mathematics such as category theory has been created. To finish, let me restate that a mathematician never stipulates that something is false just for the sake of being too complicated, ugly, or unappealing; even a math college student knows that mathematicians only declare a statement false after having found a counterexample.
@Felix Gotti
And don't forget Wiles' proof of Fermat's LT (more accurately, of the Shimura-Taniyama-Weil conjecture): if added to the preliminary (resp. complementary) work of Ribet (resp. Taylor-Wiles), it amounts to about 150 pages in a math. journal, i.e. without recalling known results (known only to the experts!). To have an idea of the total amount of knowledge needed to thoroughly explain the demonstration to, say, a Phd student, just think of the book by Morgan & Gang Tian, who took 489 pages to completely expose Perelman's (originally a bit sketchy) proof of the Poincaré conjecture. As you say, " it didn't take XXX pages long because they wanted to make it complicated; it was so long because they were answering a really hard problem ". I would add : this is the natural course in the development of science (more generally of knowledge), because the frontiers between the known and the unknown are exponentially expanding. And this renders all the more hopeless (or unconscious) the search for "elementary" solutions to problems which are, in spite of possibly elementary statements, not elementary in essence.
And let me also make a distinction between "complication" and "complexity".
Dear Felix, you have here someone who shouted "false" because the simple geometrical picture escaped her. I'm not used to speak in the air. That's not the first time I experienced that, because I have a very strong intuition that help me simply and concisely understand even difficult matter. Anyway, if a proof take many pages, that's not necessarily because it is complicated, it is because it is proven for the first time, and the work of shortening and simplification have not yet been done, nor did care the author to do it, because like you he think the longer the better. Often, other more simple proofs are found afterwards, and it is way more difficult to find an elegant and terse proof than blackening paper.
Dear Thong, I am glad you brought up Fermat's Last Theorem and Poincare Conjecture. I just mentioned the example of Feit-Thompson result because the initial question was oriented towards algebraic structures, but I pretty much agree on the fact that Mathematics is permeated of many exciting simple-to-state conjectures (now theorems) whose proofs had ended up being very involved and challenging. Names such as Andrew Wiles, Terence Tao, and Harald Helfgott (just to mention a few) will remain in math history forever for providing brilliant proofs to some those beautiful and intrinsically challenging questions any college student can easily understand.
The Weyl group of the classical group U(n) is the group of permutations of n elements Sn. U(n) being the group of rotations of Cn that preserve the inner product, the group of permutations is clearly of geometrical nature.
That's a complicated way to state a very simple idea. Take Rn, then there are elements of the group of rotation O(n) that permute the n coordinates of any point. Those elements make up the group Sn which is then a subgroup of O(n).
The proof is very simple too. Any permutation can be written as the composition of exchanges of two elements. Similarly, any element of O(n) can be written as the composition of reflections with respect to linear subspaces of n-1 dimensions. Take for them diagonal subspaces that exchange two coordinates of any points, e.g. x-y=0 for exchanging x and y, and the subgroup is constructed explicitly. Its elements are represented by matrices whose every line and column has one 1, and 0 elsewhere. They are manifestly orthogonal and of determinant +-1.
The well known Frobenius theorem (about the integrability of a distribution in a manifold) is perhaps one of these simple to state questions. Perhaps a many pages proof has been churned out by this brillant mathematician with an extensive knowledge and convoluted mind. But what impress me more is the short, but brief proof that have been given since, and that have much more usefulness when making use of this theorem, than it did in its original version.
Article A Short Proof of the Frobenius Theorem
And honestly, I don't see how examples where the problems are already complicated could make a point about the habit of the mathematicians to complicate things, especially when they are simple. They are not counter examples in the logical meaning, while mine is.
Better, Rn-1 is enough to represent Sn. For those who are limited in two dimensions, they can see this with the triangle group. Any two vertices of an equilateral triangle whose center it at the origin can be exchanged by an element of O(2), namely a reflection with respect to one of its heights. S3 is then represented in R2. Similarly, in R3 there is the tetrahedron group ~ S4, and so on.
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