Mathematics is fundamental in all sciences and mainly in Physics, which has even had many contributions. It seems that the capacity to be applied would be the motor to be create. But this not what good mathematicians as Henry Poincarè or Hardy has said. What is the beauty in mathematics, in theoretical physics or in others which could be related subjects?
For me there are very beautiful mathematical results which sounds difficut to be applied or even against our reality, which are full of "beauty" or at least "surprise".
1.Sum of natural numbers = a negative number as - 1/12.
2. Polynomials with degree five or higher are without analytical expression of their roots.
3. Banach-Tarsky theorem
4. There cannot exit more than five regular polyhedra in three dimensions.
Dear Santanu,
Thank you very much. What is the most beautiful theorem that you know?
Dear Santanu,
Saying that every theorem has beauty doesn't help too much for understanding how this concept applies. The question is that in physics is relatively easy to find the beauty because you have something real in front of you. For instance, I can see the "beauty" of Archimedes when was taking a bath for calculating the density of material, I can understand Einstein when could reduce the gravity of Newton to a Riemann geometry or Newton finding that the laws of the planets where the same as mechanics here in the Earth. But in mathematics is more difficult for me to understand what is the explanation in mathematics except in some cases as Pythagoras theorem and mainly some theorems of geometry and topology (Gauss-Bonnet integral, Gauss' egregium theorem, Euler's theorem etc) where the "beauty" must be very different that the one in physics.
The only beauty or relationship between mathematics and theoretical physics is only Mathematical.
Perhaps the main feature of trying to find the "beauty" in mathematics is that there are conjectures or programmes which should summarize it. The most famous programme, in all that I know, is the one of Hilbert's made at the beginning of the 20th century (in 1921). The purpose was to " dispose of the foundational questions in mathematics once and for all". It was based in two main ideas:
1. Use of the axiomatic method.
2. Employing "finitary" means, one should be able to give proofs of the consistency of these axiomatic systems.
In fact, Hilbert thought that he could provide a philosophically satisfactory background of infinitery mathematics (i.e. analysis and set theory). This is obviously a very appealling idea which in some aspects remember the unification theories in physics that Einstein was working at those times. Is this the "beauty"?
Curiously, almost ten years later, Gödel's incompletness theorem and the negative solution by Church and Turing of the decisive problem showed the impossibility of the Hilbert's programme. In Physics the things are not the same and the unification theory follows because the experiment is the "king".
The usual procedure in mathematics: "axiom", "definition" and "proof", makes this science very close to "logic" or even "metalogic". And might be possible that the "dream" of a good mathematician could be to have a logic system where the mathematics were just logic deductions, without the need of the "experimental reality" and I understand that such direction is very closer to the one of an artist who only looks for the "beauty".
The most beautiful and interesting in physics and mathematics is the intuitive mind of scientists because they attack problems that do not attract the attention of others and what is the "Easy Abstain". For example :
-The discovery of gravitation by Newton, the modern binary number system of Gottfried Leibniz, etc.
I think it must have a course in primary that has for goal the development of this spirit in children
Hi Daniel
For me, the pure definition of beauty in physics, and in mathematic representations of physics, is an astounding elegant simplicity.
Causal elegance.
Chip Akins
Daniel
.Roots of Polynomial equations of degree 5 or more do exist, but just cannot be calculated in closed form analytically, except is special cases.That no. 1 cannot be is self explanatory-
What I would call beautifull in Physics and Mathematics is rather different.
You can look at the succesive explanations of Maxwells
Equations, starting from the most elementary descriptive form,
into the Standard vector analysis form, then into the tensor form,
and finally into a form used by vector análisis in Cartan form as the highest and simplest
formulation.
Then again you can see the generalization of rational mechanics, starting from Newton, into Lagrange and finally Hamilton, and then again from the principle of least action.
Im somewhat amazed that even very obvious truths can sometimes go wrong. The axiom of choice is just about obvious, but employed in unrestricted fashion takes us to
undo the moon and then créate a tenis ball with the same parts.
Dear Juan,
You are right the roots of polynomial equations higher than four exists, but what cannot find them as formal solutions. You need to make just substitutions or looking for tricks (this a horrible and not beautiful form to find them). Perhaps the beauty that mathematicians is related with these kind of "thoughts".
The main physical magnetudes in electrodynamics (electric field, magnetic field, currents and so on) may be represented as antisymmetric tensors and therefore they can be represented in an external algebra of differential forms (if you look in my contributions you can have a book written in this mathematical language). This is a very simple and "beautiful" algebra that is directly linked with the topology. There are closed and exact forms which directly are related with the homology classes. The Hodge dual of the electromagnetic field is one exact form (for a simply connected manifold) which allows to define the potentials.....
Physics and mathematics are absolutly related although the concept of what can be the "beauty" for these sciences is different, in all that it seems to me, Perhaps this point would be interesting for approaching at the creation of both fields of knowledge.
I know that this question is very undefined and even within mathematics it is very different to work in statistics than in algebraic topology, for instance.
theoremstrivialisationtheorem: applying similarity in 2 right-angled triangles=>pythagoras`theorem, using twice pythagoras`theorem, gives pythagoras` generalized theorem (cosine theorem), applying cosine theorem two times => stewart`s theorem; using stewart`s theorem twice, I can create my own theorem (MOT), as well as, when applying 2xMOT I can find MOT2-Theorem, then MOT3, and so on: MOTi theorems. Well, one can postulate that this method simply shows the trivialization of compiling theorems. Apart from the question , I would suggest that this (previous) text is a theorem! The theorem named: TTT, alias tritium, alias theoremstrivialisationtheorem. that`s Beauty too :))
Paul,
Let me tell you that this issue is more serious that you can think. I'm a theoretical physicist and the form to choice one subject or another is very close to something as what could be called "beauty". And I suppose that this procedure is more important in mathematicians, I'm afraid that the answery is yes.
I think that it is a difficult question to define because there are two contradictions in its answer:
1. People who is speciallist in mathematics or theoretical physics are accustomed to solve problems and no to think on these psicological aspects. Thus it is easy to employ the word "beauty" as a form to "interesting", "worthy" or so on. Avoiding to enter in details.
2. Really there is some appealling interest which finally gives something stetic and close to what could be named "beauty".
dear Daniel, I am someone who tries to find the Errors made by scientists when using maths (unfairly, improperly, wrongfully). hereby first are ment famous scientists: they are the more abusively... apropos: could You mention one single Demonstration given by eintein himself to his own equations?
Imho, "beauty" has a common nature, no matter where it is. In aesthetics there is such a (so far not a formalized) category as "harmony" - a concept that denotes orderliness of diversity, integrity, coherence of parts and balance of their tension.
Ultimately, our physical brain makes sensation, something beautiful or ugly.
https://www.researchgate.net/publication/270705947_Human_Physiological_Benefits_of_Viewing_Nature_EEG_Responses_to_Exact_and_Statistical_Fractal_Patterns
The research data only about visual content, of course.
"They found that people overwhelmingly preferred images with a low to mid-range D (between 1.3 and 1.5.) To find out if that dimension induced a particular mental state, they used EEG to measure people’s brain waves while viewing geometric fractal images. They discovered that in that same dimensional magic zone, the subjects’ frontal lobes easily produced the feel-good alpha brainwaves of a wakefully relaxed state. This occurred even when people looked at the images for only one minute."
This is another reason for thinking about symmetry, invariants and universality.
The universality is near 4/3 = 1.(3) ; c = 1.327...
The aesthetic sense of beauty is about the state of the interconnections of our brains, i.e. some physical state of interrelations in physical structures.
Here's https://arxiv.org/abs/1102.3764 another good reason to reflect on the beauty.
NB: It is obvious that in the case of each person this sensation is slightly different, because implies tensions of relationships in personal specific state (knowledge) and specific position in relation to the environment also. In other words, everyone discovers beauty in what it encounters, whether it be a prehistoric man or a 21st century resident.
Article Human Physiological Benefits of Viewing Nature: EEG Response...
Dear Christian,
I remember when I was a teenage reading two books of Henri Poincare (translated in spanish) about how he solved and created mathematics. It was wonderful to see that it was not a question of making strange calculations, but to have the same kind of ideas (also related with the "beauty") that I could have had playing chess. Only looking at the chessboard, without entering in details of the difference of material or the position, I could see what was "beautiful" or not. For instance, one fantastic and with great "beauty" in his game was Capablanca who really discovered the positional form of playing. He winned the world championship in 1921 against Emanuel Lasker, who was longest time champion of chess in the history (during 27 years) of this game. He was also mathematician (doctor in mathematics contributing to the Noether's theorem so important in Physics) besides chess player and his main characteristic was not to play with great precision but using the psicology of his opponent looking for the mistake. Both these players have a clear idea of what was the "beauty" within the game.
It is clear that "combination of simplicity, clarity, logical rigour and completeness" can form good ingredients for cooking the beauty, but I think that there is more. For me it same kind of symmetry as we can have in the music of Mozart or in the decoration of certain walls painting with geometrical figures.
Dear Vasily,
Your psicological point of view is quite important, but I think that it must be based (also) in something external to us. It is curious that pure sciences as mathematics or theoretical physics "coincides" with the basic "anarchic" form of creation in arts.
This is the conjecture of Poincarè:
Every simply connected, closed 3-dimensional manifold is homeomorphic to a 3-sphere.
Which was created by Poincarè almost one century before Perelman solved it in 2003. Is this a kind of beaty that the mathematicians have for creating their science?
One yet another interesting point sounds in the topic, which should be noted in the light of the cognitive process. The effect of the unexpected result, which dear Daniel speaks about.
Obviously, this works only with new information for an individual. The real surprise is experiencing with new data. Apparently, this is another component, which later forms harmony: two different emotions, a sharp peak and steady pacification.
The sharp peak corresponds to the moment of understanding: the so-called "afflatus", the so-called "enlightenment", the so-called "unexpected reward", or the "eureka" effect. There is a trigger in the brain for it, which allows not to abandon the "strange" result at the time of cognitive dissonance, but to undertake to revise the old relatively harmonious structure of knowledge.
I will not clutter up with a detailed list of studies on monkeys etc., just a reference to an essay based on person's experience, who had the opportunity to study the features of the process quite extensively in clinical settings.
https://aeon.co/essays/the-dopamine-switch-between-atheist-believer-and-fanatic
Do not pay attention that it is mainly about religion. The principle is general, understanding of something is based on the current belief system of a person. Unexpected results affect your expectations, accordingly, beliefs. In other words, you have to change your axioms, or at least add new ones (obviously, when according to the legend Archimedes climbed into the bath, the world did not collapse).
That is revision of beliefs leads to revealing of new invariants, since we have to get along with old and new data.
Beauty have links with what you like.
For me,
- simplicity - of a formula that explain and structure, which solve a lot and force us to get rid of less simple equations or arbitraries
- some kind of symetry or correspondance with what we already know
- soundness of prediction
- fluidity, easyness, usefullness, powerness, those kind of considerations
From the standpoint of the theoretical-physics:
Beauty=Symmetry
I think so...
Dear Denys.
I'm quite agree with these ingredients of a theory in Physics, But perhaps we need to go deeper for understanding what we mean for beauty in mathematics or theoretical physics.
1. Soundness prediction. That is fundamental in Physics (not perhaps within the concept of beauty) but not so important in mathematics. The concept of prediction, at least, has another form to be understood.
2. some kind of symetry or correspondance with what we already know
Symmetry and correspondance are two very different concepts, but it is clear that if we want to relate them with what is already known, then they are difficult to consider them as a part of the beauty of the present theory. For instance, Newton summarize the knowledge of many people in his gravitation equations and the interesting is to see how these aspects were "parts" of his theory, but using the fact that the past was incomplete or no so "symmetric". The coexistence of the past, present and future are difficult to present as a whole. The past appears as something which was explained better with the new.
3. I strongly agree also with the first point:
simplicity - of a formula that explain and structure, which solve a lot and force us to get rid of less simple equations or arbitraries
But I have some difficulties too for gluing it with what I understand by beauty. Perhaps the theory in theoretical physics which closer to this conditions is the Schrödinger quantum mechanics. And that theory is not the one that, personally, could say that it has "beauty" or ,at least, so beautiful as others in Physics.
Dear Yusef,
You apply too seriously the concept of simplicity within the concept of "beauty" in mathematics or theoretical physics. The equation
beauty = symmetry
is very appealing but it needs explanation to understand it. Clearly the symmetry is one important aspect of the beauty, but I think that the beauty in this scientific context is more than that, and even the symmetry is of very different kinds. Let me put you one example of music:
Bach (very symmetric), Mozart (symmetric), Beethoven (less symmetric)
Have they different concepts of beauty? Apply that to
Newton mechanics (?), Einstein's special relativity (??) and Maxwell electrodynamics (???)
Dear Daniel, I completely agree that there are really invariant things that are not dependent on anything specifically, psychology and physiology is only a manifestation of self-organization, that is, the manifestation of these invariant processes. This is the essence of the "anthropic principle". The nature of mathematics, as a means of communication and prediction of processes and as a phenomenon, lies in the same, really objective, invariants (imho).
Dear Yusef, I'm afraid that symmetry is an unattainable phenomenon in nature, otherwise, the very philosophical concept of existence is in question.
By the way, about the music, with this logic, the best musical work should be something that sounds (including) equally good, regardless of the direction of the time.
It's a funny fact that Lewis Carroll was entertaining playing the music box in the opposite direction. It's one thing to play linear program in the opposite direction in existing physics, quite another - reversed pink noise (including the beats of the heart, for example). This is an unreal world beyond the bounds of irreversibility.
There is dissonant and psychedelic things, which, for example, I very like in music, along with sharp changes and a huge dynamic range (of tempo, spectrum and instruments set). All these are also components of aesthetics. Beauty is not a white noise and not a simplest order.
Beauty lies in simplicity. Simple face of a child is beautiful whereas a manipulative person always wants to hide his true self in the fear of seeming ugly. Similarly beauty of science in general is in it's ability to describe diverse natural phenomenon in terms of simple sets mathematical expressions, such as the Maxwell's equations of Electrodynamics.
However, beauty is always seen or measured in contrast with ugliness. It is not an absolute concept at all; it is quite a relative one. Both beauty and ugliness were always there and will always remain. There is nothing to complain against.
Dear Biswajoy,
Simplicity as the only characteristic of the beauty in mathematics or in theoretical physics doesn't seems to be the key word. At least in what would be necessary for defining clearly the meaning of such concept in general.Because, although it is easy to agree that it would be fantastic to have a theory that could summarize in a simple equation all the "knowledge", which seems quite impossible; in any case, this would not to have a real knowledge of everything and perhaps of the "beauty·" behind so appealling purpose. The monochromatic spectrum is not the one that we usually like in our common life which fortunatally is provided by different colors..
Perhaps one of the persons who was thinking more about the beauty in general (not in mathematics and theoretical physics as it is our issue) was Leonardo da Vinci, who takes from mathematics the old Pythagoras' (and Euclid) concepts of the harmonic proportions (the golden ratios). For instance, he represented a man's body in a pentagon using the mathematical golden ratio of the proportions in his famous De Divina Propotione and in Vitruvian Man. Without any kind of doubts many architets, painters and sculptors has accepted some mathematical criteria of this kind.
In any case, this is not the main ingredient of our discussion devoted to pure thoughts in mathematics and theoretical physics, obviously, but surely is related with these kind of proportions, harmonics or something even more intellectually sophisticated.
Dear Daniel,
I'm less enthusiastic with the soundness of predictions in physics. In many cases are the predictions only approaches or simply wrong.
I remind the prediction of renowned scientists that superconductivity would never happen at temperatures exceeding 30 K. Hardly spoken, Paul Chu find a material with 90 K transition temperature.
Symmetrie is a criterion of beauty also at faces of peoples you know. And you can trace the symmetrie in physics finally to the duality of two elementary particles: Electron and positron (see the link below).
I'm not a follower of the depictions of the reality with Powerpoint presentations or pictures created by theories resp. mathematical derivations. People do mostly enormous efforts to see the world how it has to be in their opinion even if the evidence shows that they are in error (the last sentence according to Robert B. Laughlin).
Thesis The Reason of a realistic View to Particles and Atomic Nuclei
Dear Hans,
Frankly, I don't understand what is the beauty that you refers. Obviously, one of the most beautiful achievements in Physics is the one of the Standard Model where three of the four interactions appear unified. There are six leptons (electron, muon, tau, electron neutrino, muon neutrino, tau neutrino) and six quarks (up, down, charm, strange, top, bottom) with the gauge bosons ( photon, W+, W-, Z, eight gluons) which locally follow the symmetries U(1)x SU(2)X SU(3) associated to the Yang-Mills action.
Please, what do you put your paper ? What are the symmetries that you consider?
Thinking in terms of simplicity and symmetries as the ingredients of the beauty in mathematics or theoretical physics, is not always a form to get realistic theories. One could think that in the same form that the motion equations are obtained as one extremal of a simple action, we could also find simple and beautiful theories to get much deeper theories. And that is true, for instance, the grand unified theories go further than the Standard Model and use the fact that
SU(3) x SU(2) x U(1)
is a subgroup of SU(5) with 24 infinitesimal generators which lead to interesting conservation laws and even it is possible to go further using a special orthogonal group SO(10) ( 45 generators) which contains SU(5) as subgroup.
For a mathematician, perhaps, this is very beautiful theory (I suppose) ,but for a physicist it is full of new predictions as proton decay, mass of the neutrinos or electric dipolar moments which needs to be found. But for the moment not was found a form to find one agreement experiment-theory. Could we follow to think that these theories contain "beauty"?
The idea of beauty of symmetries builds on a philosophical view of the world. Beauty may be defined otherwise, but for living entites, symmetry is a reassuring data.
Dear Paul,
Do you realize that what I am asking is exactly what you understand for beauty of mathematics? What is that? Why nature cannot be so beautiful?
Dear Denys,
I find that your answer leaves too many degrees of freedom to be interpreted. Please, could you try to define a little bit more the concepts for seeing what are you really understanding by beauty.
I have written my last post on symmetries to show that simplicity and symmetries can be joined in a theory, but this doesn't mean that this is a good theory. Sometimes seems that somepeople identifies those concepts and it is not a logic conclusion. What can be is a good form of creating a theory.
It was only a precision of a precedent post.
Some equations may be exact but doesn't lyes on anything we appreciate, so are less «beatiful».
Beauty have links with what you like.
For me,
- simplicity - of a formula that explain and structure, which solve a lot and force us to get rid of less simple equations or arbitraries
- some kind of symetry or correspondance with what we already know
- soundness of prediction
- fluidity, easyness, usefullness, powerness, those kind of considerations
What is "beauty" in mathematics and theoretical physics?. Available from: https://www.researchgate.net/post/What_is_beauty_in_mathematics_and_theoretical_physics/3 [accessed Jun 21, 2017].
Dear Denys,
By definition all the equations are exact, even if they have not a defined solution. That is not an special feature of these objects.
"Beauty have links with what you like"
This sentence seems to be tautological without more extension or explanation.
The other are aspects that you have said previously but also no proper of mathematics or theoretical physics, because they could be for architecture, music or other arts.
My question tries to find what was the beauty that Poincaré or Einstein could say when they used such a word for defining their work.
Yes It is, like any postulate as the first stone of any theory.
What Poincaré or Einstein defines as «beauty», is necessary in relation with what they value. Applied to mathematics, i still find that a good demonstration blowing my mind in new possibilities is beauty, especially if not always, when it find simplicity.
I remember when I was studying very basic geometry in (12 years old) that our teacher of mathematics came with only five polyhedra made in beautiful wood. He asked us if we could think to have more. For me it was very difficult to believe that this was the only possibility and when I got a proof of this (that the greeks had) I was really full of what perhaps is called this kind of beauty. But that was, surely, very fast of understanding the Euler invariants and the algebraic topology surrounding such result! Is this just pure mathematics? That is so real as the most experimental physical phenomenology, isn't?
Beauty is in the eye of the beholder but it seems more applicable to the arts or say a rose. I have heard "elegant" used for proofs since they were remarkable. I guess it depends on your first exposure to the word beauty and in what context you use it for most of you life. A cultural bias I guess.
G.H. Hardy wrote in his autobiographical book A mathematician's apology:
"beauty is the first test; there is no permanent place in the world for ugly mathematics"
Henri Poincaré wrote too:
“would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility.”
And a mathematical physicist as Hermann Weyl declared,
“My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.“
This was fully reflected in his own career, when he first attempted to unify electromagnetism with gravitation.
What makes the theory of general relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical "beauty". This is a quality which cannot be defined easily, any more than beauty in art can be well defined in all that I know, but which people who research on mathematics, or mathematical physics, usually have no difficulty in appreciating. The theory of general relativity of Einstein introduced mathematical "beauty" to an unprecedented extent into the description of Nature: the curvature of the spacetime (pure Riemann geometry) is equivalent to the enery-momentum tensor (pure physics). Perhaps these kind of results are a good definition of beauty in mathematical physics.
when using maths to analyse physics many physical beauties become nonsense(s).
Dear Paul,
Please, could you put one example? Thanks.
I don't know any beautiful physics without having a mathematical component.
dear Daniel, it`s a great pleasure to expose more than one example: as mathematicians saw that physicists operated an integral of a product of 2 functions, where one so-called function not even was a function (as it is defined), they analysed how it could be that the result still can be correct? so they created distributions theory! for details please google: generalized functions, Laurent Schwartz, Heavyside, test functions, functionals,set functions (postilica and harvaneanu), dirac`s delta function... and put them together. if you are interested I can show more examples.
When you wrote:
when using maths to analyse physics many physical beauties become nonsense(s).
I thought that said that the physicists didn't employ mathematics for presenting their theories. By the way, the distribution theory was in fact used by physicists as Dirac or Heaviside before a pure mathematical formalism and they are also developed by Richard Feynman for developping its concept of path integral.
If you want to follow putting examples, that would be very interesting and also saying what is the beauty behind them. Thanks.
distribution theory was developed as mathematicians saw that physicists operated impermissible. You are right that physicists used distributions first, but they did not accurately know what they do.
Dear Paul,
There are many different forms to understand it, the delta as a distribution was introduced quite well by Dirac (now keeps its name) and Feynman introduced with them the transition amplitude (or correlated function in a more classical language of fields). For me these achivements are the most beautiful parts of this mathematics, but perhaps I'm wrong.
Please, could you say where is the main beauty that you see in this mathematical formalism? It would be very worthy if you could find a pure mathematical achievement instead of the mathematical physics that I have shown.
dear Daniel, one of 2 functions in the product to be integrated is a test function (infinite derivable, etc. in other words with properties) which permits to tranfer derivative from one function to the other: i.e. integral(uv`) = - integral(u`v). I personally use distributions theory to analyse RT and QT, in other words to find the results of unifying QT with RT. one of the results of unifying the theories (that`s no joke: I know that physicists try to unify this theories since 100 years) is: natural constants are not constant indeed! (see my work published at congress of mathematicians and here on RG: unifying RT and QT without using strings)
Dear Shailaj,
Without mathematics is impossible to fully understand physics, although the inverse is not truth: there are many results in mathematics without an application in Physics.
One of the components of Mathematics is that it has more degrees of freedom to stablish its results because they depend of the pure logic or mind (this is a great clue for my question of the beauty in Mathematics and no in Physics). For instance, the Cantor's transfinite numbers were never used in Physics in all that I know.
The beauty of theoretical physics is that Maths is it’s language. The beauty of mathematics is in its remarkable success of describing the natural world.
it is therefore not surprising that most research mathematicians and theoretical physicists pepper their description of important research work with terms like “unexpected,” “elegance,” “simplicity” and “beauty.”
Let me make it easy for you:
Can you imagine a bride without a wedding dress?
Dear Issam,
Can you imagine a bride without a wedding dress?
Perhaps this question is not the best if I understand you well. There are surely many people that not only can imagine the bride with another dress different of the wedding, but without a dress at all.
The maths has its own field of work as, for instance, can be a Cantor's transfinite number or a Hilbert functional space, both are objects which doesn't exist out of our minds and which don't need to have a physical reallity for being used.
Mathematics is much more than a language or clever tool for other sciences. Plato worked with the same five polyhedra in three dimensions as we do: tetrahedron, cube, octahedron, dodecahedron and icosahedron. I'm sure that they are going to be less discussed than the six quarks (with the three leptons) that we have nowadays as the fundamentals of the matter which surrounds us.
https://arxiv.org/abs/math-ph/0303071
It is said that Mathematics remains beauty until and unless it is not applied
Dear Daniel
your question asks what’s the beauty in mathematics and theoretical physics ?
I answered and ended my answerer by asking the question :
Can you imagine a bride without a wedding dress?
In other word both are beautiful and both complement each other.
people, who as you describe, can imagine a bride without a wedding dress or without a dress at all are not imagining beauty! they are imagining other things that have nothing to do with your question but to do with perhaps their own other issues!
Many branches of mathematics when formulated, its authors never dreamed of any useful applications in explaining the natural world. One of the most “beautiful “ example is the Riemannin geometry formulated by Bernhard Riemann in 1854 and published 12 years later in 1868. 100 years later used by Einstein to formulate his celebrated theory of General Relativity .
Any abstract mathematics that you may think has no useful applications nowadays will have in the future.
Both mathematics and theoretical physics are beautiful and both complement each other.
can you imagine theoretical physics without mathematics ?
I hope you can now see why I asked the question: Can you imagine a bride without a wedding dress? and I hope I explained to you what the real meaning of “beauty“ in science as related specifically to mathematics and theoretical physics.
Mathematics is beautiful as it is a structure of logic and reason to validate truth of reality.
by using mathematics one can merely confirm/counter theories! that`s the great thing what maths does!
Dear Issam,
I think that I'm understanding you but we are not in the same path with respect to this question. Mathematics is much more than a language and the dependence of mathematics and theoretical physics is quite subtle. In principle theoretical physics depends of mathematics as the history has shown: Newton contributed to create differential calculus, Dirac to the theory of distributions and so on. But the mathematics don't need the theoretical physics and it is not a wedding dress. In fact you contradict this idea when you said that the Riemann was introduced as a generalization of the Euclidean geometry without taking into account General Relativity. That is true!
Thus, in spite that it seems that mathematics is not only a pure logic structure independent of the reality, I don't know how it is possible that people as Poincarè could say that he was guided by the "beauty" of the mathematics for chosen the creation of one theorem. This is very difficult to understand because for me it is not easy to see this kind of beauty and only when I understand the whole results can grasp same emotional feelings which could be of these kind of things. But I haven't a clear form to say what is that.
Dear Dejenie,
Mathematics is beautiful as it is a structure of logic and reason to validate truth of reality
Sorry but I don`t understand you. What is truth of reality? Why logic is beautiful?
Dear Paul,
Mathematics introduce new objects a derivatives, integrals, distributions, curvatures, tensors, homotopies, algebras, etc...which everyday are used for calculating things. They don't validate theories, at least in Physics where the experiment does it.
dear Daniel, you`re maybe right, but I do analize physics by using maths (for instance: distributions theory). I really found out what the results of unification RT and QT look like!
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