It is interesting how maths is useful for describing the physical world. But are there any branches of mathematics that are totally useless for physics? Why? Could it be that we perhaps anthropocentrically chose to follow branches of math that are interesting to us (ie. could have possible application)? To prove a point, could we invent a branch of math that is totally useless?
Could we come up with a sophisticated group theory for the game of chess? Is the reason no one has attempted that because it would be in fact utterly useless with an unexciting loss of generality?
If mathematics is of no use today it will be of use tomorrow. A good example is the geometrical concepts developed by the German mathematician Bernhard Riemann. A century later, Einstein used them to develop his general theory of relativity.
If mathematics is of no use today it will be of use tomorrow. A good example is the geometrical concepts developed by the German mathematician Bernhard Riemann. A century later, Einstein used them to develop his general theory of relativity.
One cannot simply invent new Maths, but rather discover new concepts. Since Maths relies on sets of self-consistent ideas, logic etc., it is very unlikely that any newly discovered Maths will not have some parallels with a Physical system for it not to be useful.
The virtue of useless mathematics:
Nature has published a fascinating article developing the argument that theoretical work in mathematics that has no apparent application can prove to be really useful in the future. These quotes summarise the argument:
"The mathematician develops topics that no one else can see any point in pursuing, or pushes ideas far into the abstract, well beyond where others would stop".
"There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won't have the right piece of seemingly pointless mathematics to hand".
---------------------------------
Nature 475, 166-169, 14 July 2011, 'The unplanned impact of mathematics' by Peter Rowlett
These points are then illustrated with seven examples where advances in mathematics precede, sometimes by centuries, their use in new innovations or products. One of the examples explains that the mathematics of quaternions, which were first described in the nineteenth century, turns out to be really useful in computer game programming, robotics and computer vision, and in ever-faster graphics programming.
Another such example is the concept of Hilbert space formulated in the first decade of the twentieth century and named after the German mathematician David Hilbert.
Then in the 1920s, Hermann Weyl, Paul Dirac and John von Neumann recognized that this concept was the bedrock of quantum mechanics, since the possible states of a quantum system turn out to be elements of just such a Hilbert space. Arguably, quantum mechanics is the most successful scientific theory of all time. Without it, much of our modern technology — lasers, computers, flat-screen televisions, nuclear power — would not exist.
For more of these examples refer to the article by Nature above.
What Physics? Exist very different applications.
(Human too?)
@Ana, you are quite right:
Today, what once was thought to be useless mathematics, could find applications in communications, model the population dynamics in outbreaks of viral diseases and offers prospects of reducing the risks of share-price volatility, insurance industry, help biologists understand DNA, quantum computers, making efficient conveyer belts, helping doctors to do brain scan, helping cosmologists to understand how galaxies form, helping Mobile-phone companies identify the holes in network coverage and even in our mobile phones to analyse the photos they take.
Mathematical methodologies are the prime tool to model events that we observe in nature, thereby giving us eventually a better understanding why events take place, what parameters drive and how they develop.
Although mathematic is a pure product of human thinking and cannot be derived from nature per se (just think on the imaginary unit, which is a pure imaginary thing but nonetheless is key to understand wave propagation and as such allows for RF communications, etc.) it seems that with whatever mathmatical concept humans come up with in their mind, sooner or later we will find something in nature that can be modelled best by this type of concept.
If we imagine that mathematic is nothing else than a specific language than it appears to me that the "typical" mathematical process of developing a mathmatical concept without an application is as if someone "invents" the word/concept "elevator", without yet even having the technical concept in mind - and the elevator as a technical system/concept itself is invented only but decades or centuries later.
Usually we decribe what we have built - in this exampl ehowever it is vice-versa. Is this a sort of reverse match making or simply a process of creativity? And even more along those line: Isn't it striking that mathematics allows us to calculate n-dimensional spaces, while our brain is only able to imagine but three dimensions?
So in answering the question I would tend to say that, No, I don't think that there is any mathematics for which there is absolutely no application in physics, simply because mathematics is a concept of language and creativity, therefore being bigger than natural science, which is limited by laws of nature - but then I have no "mathematical" prove for this statement...
Presumably if we were to develop a special mathematics, or group theory (say), to describe the game of chess, it would be totally useless for describing the real world?
I have no answer, though I find the topic interesting. A few observations:
1) You say a group theory for chess solutions would be "utterly useless with an unexciting loss of generality". Look at all the efforts poured into Deep Blue and all the other attempts to have a computer play chess. All of them dependent upon brute force search. I posit that a group theory for chess would invite a lot of practician interest.
2) You suggest we follow research that interests us, and suggest this is based on a perceived "real world" usefulness. However, there are plenty of people who are drawn to mathematics out of a sense of beauty and aesthetics. I don't think usefulness is implied by your anthropocentricity.
@Andrew. Our sense of what we find beautiful is honed by natural selection. I suspect that what we find beautiful in maths does have utility. For example, a proof with great generality is much more beautiful than one that suffers a loss of generality. Also a short proof with great generality is much more beautiful than a long proof. That's because the utility of math is that it is compressed description. The better the compression, the more useful and also the more beautiful.
@Sandra. Of course, in a sense even our emotions are part of the real world because we are in it. However, for this discussion on physics when we talk about the "real world" we mean the world of physics as opposed to the word of human constructs.
Math is a human construct, and so is music, language, chess, poetry, economics, and our anthropocentric way of classifying data and objects.
This philosophical issue has been considered by a colleague of mine, prof. Yakir Shosani which published this work
http://physicsessays.org/doi/abs/10.4006/1.3025331
In his view, mathematics belongs to our mental world, and our mental world is part of the universe, hence necessarily mathematics describes the "real world".
What happen under the atmosphere, too? For example a stone making a parabola in the air.
This question was actually addressed many years ago in the AAPT journal "American Journal of Physics". The question was asked by Dwight Neuenschwander (Am. J. Phys. 63, 1065 (1995)) with - according to the AAPT search tool - two answers: Paul J. Dolan and Denisa S. Melichian, Am. J. Phys. 66, 11 (1998), and A. C. de la Torre and R. Zamorano, Am. J. Phys. 69, 103 (2001). Both answers to this questions were "no".
dear Derek
It can be found in our global economy crisis
some idiot guesed that world economy is dependent on stocasic processes - not so its global economy in real
thats my answer Per
I think a general answer to this question is not possible, since, as noted above, whether or not a mathematical theory/method/etc is applied in physics depends on time.
Maybe, one should discuss the question whether there is mathematics that has not been used in physics (or the sciences or engineering) _up to now_.
Since I recently have seen some mathematical work on quasi-Hadamard products of analytic functions recently, I wondered whether this type of products has been used in physics so-far.
Well, you can answer yes but only for a given amount of time. For example Sophus Lie invented Lie theory sometime in the 1800's I think and there was application till geometrical theories of mathematical physics after general relativity became THE theory in the 1920's. Indeed it was used to a large extent by the Russians to solve the physics equations because they did not have access to computers during the cold war. The Americans ignored it since they could solve such problems numerically on their computers. Then gradually the Russian research diffused into academic circles in Europe and now its a booming field for finding symmetry solutions of differential equations and for use in differential geometry in general relativity. So, yes there are many fields of mathemtaics with no current application although this term current changes all the time. Eventually somewhere an application will be found. Maybe because the human thinks in a certain way and also because the world itself asks questions of us constantly so that new approaches are needed to solve real world problems. The reasons are more difficult to answer than the original question.
las matemáticas, son solo un lenguaje. Es como preguntar ¿existen las expresiones para las que no hay absolutamente ninguna aplicación en la comunicación?
Mathematicians themselves often insist that their work has no practical effect. The British mathematician G. H. Hardy went so far as to describe his own work this way: "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." He was wrong. The Hardy-Weinberg law allows population geneticists to predict how genes are transmitted from one generation to the next, and Hardy's work on the theory of numbers found unexpected implications in the development of codes.
His famous work on integer partitions with his collaborator Ramanujan, known as the Hardy–Ramanujan asymptotic formula, has been widely applied in physics to find quantum partition functions of atomic nuclei (first used by Niels Bohr) and to derive thermodynamic functions of non-interacting Bose-Einstein systems.
Hardy preferred his work to be considered pure mathematics, perhaps because of his detestation of war and the military uses to which mathematics had been applied. He made several statements similar to that in his Apology:
"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."
Though Hardy wanted his maths to be "pure" and devoid of any application, much of his work has found applications in other branches of science.
There is always the chance that what we are doing will eventually be applied on some way or the other. My position is that, specially in developing countries, we cannot afford the luxury of spending lots of research budget on doing research just "for the sake of it", or with the vague justification that there will eventually be a connection to reality that no one is seeing yet. I'm not advocating against Pure Math research here, but I feel that every country should have Scientific Program, and of course, pure abstract Math should have its place there. I think the world needs more socially engaged mathematicians, which ask themselves these questions, and try to figure out ways to use mathematics to make a positive impact on society. With that said, I do believe that there is a lot of development which is very unlikely to be ever applied, at least in a way that justifies the public investment being made on it. Of course, history is full of surprises, like the ones mentioned in some of the posts above.
The Usefulness of Useless Knowledge, October 1939, by the American educator Abraham Flexner:
“The real enemy is the man who tries to mold the human spirit so that it will not dare to spread its wings.”
"[Hertz and Maxwell] had done their work without thought of use and that throughout the whole history of science most of the really great discoveries which had ultimately proved to be beneficial to mankind had been made by men and women who were driven not by the desire to be useful but merely the desire to satisfy their curiosity."
"Curiosity, which may or may not eventuate in something useful, is probably the outstanding characteristic of modern thinking. It is not new. It goes back to Galileo, Bacon, and to Sir Isaac Newton, and it must be absolutely unhampered. Institutions of learning should be devoted to the cultivation of curiosity and the less they are deflected by considerations of immediacy of application, the more likely they are to contribute not only to human welfare but to the equally important satisfaction of intellectual interest which may indeed be said to have become the ruling passion of intellectual life in modern times."
http://www.brainpickings.org/index.php/2012/07/27/the-usefulness-of-useless-knowledge/
Seemingly-obscure research can often have unexpected applications. The Nobel-prize-winning cancer researcher, Paul Nurse , discovered a gene for cell division after watching yeast for no particular reason. The discovery, which has important implications in understanding how cancer cells grow, earned him a knighthood and a Nobel Prize.
After all, how can we expect science to advance without a few brave people jumping feet first into the glorious waters of investigation just out of sheer curiosity.
There is no physics, applied or otherwise, without concomitant mathematics -- that mathematics preceding, a priori, in 'our' linear time and thinking (even scientific thinking without Einstein's 'jump of intuition,' or imagination, more important than or a priori all knowledge) all physics; all science was once philosophy and all science to be is philosophy now. Frankly, the mathematics we have to date is faulty, without ever yet a noncircular foundation, which if it were built on a noncircular foundation (something tangible in 'reality'), would once and for all introduce absolute certainty into all equations and physical representations, or physics, creating much more reliable applications, aligned precisely with a reality Einstein equates as bounded infinity. I mean why not, why can't we know? Since we know something is faulty, how do we know that, by what yardstick, by what utility, what constitutes what works? Once that mathematical foundation (grounded in reality) is to be articulated in science and precision applications derived, it will (again in our linear thinking) be called physics!
In turn, and most unfortunately, this is why as soon as philosophy offers something concrete, it's called science. Case in point, even in Newton's time, so Newton's very Laws themselves, physics was referred to as Natural Philosophy. I have noticed physics recently referred to as mathematical physics, which per above I find redundant and, as for that matter, mathematical may be placed in front of every term. In reality, without violating Godel's Incompleteness, it is a Venn Diagram with Mathematics the Universal set and all else certain subsets. And in this, there is no loss of generality or in my opinion excitement at all.
The answer to this question depends on what is understood by physics.
If the human brain working way is considered under the scope of physics, then any mathematical construction being carried out by any human brain belongs to physics, and can be included in its scope.
From this viewpoint if there is some mathematical construction with no application in physics, must be performed by some intelligent being who does not belong to our Universe.
Extremely abstract math has a way of finding unforeseen applications in physics later on:
Amongst many such instances, Fields Laureate Connes's work on non-commutative differential geometry was found to have applications in string theory (Witten applied some of the results to develop the Lagrangians for superstrings).
J. Bellisard's work applied the same work towards explaining the Hall effect. Connes himself later used his work to describe space time as a non commutative space rather than a set of points ....
There are many such examples .....
Mathematics which looks useless may find applications in future. Integral calculus when looked as inverse process of differentiation as mathematicians often take it was hardly of any use to physicists or engineers. But when appreciated as area summation process brought in its significance. Similarly complex algebra and analysis when used to exploit residue theorem blossomed into applications. So though mathematicians may not necessarily realize the significance of abstract mathematics but physicists often bring its glory out. It happened in the past and it is likely to happen in the future. Moral of the story is no effort goes waste!
One would be a fool to make such a claim. If pressed I would shoot at mathematics of extremely large cardinal or ordinal numbers. But I would not be the least surprised to find out I was wrong. What's the famous phrase? "The unreasonable effectiveness of mathematics"
Almost any branch of mathematics may be very useful in the mathematical physics.
Practically any part of mathematics may become (soon or later) very fruitful in the language of any science. "The unreasonable effectiveness of mathematics" is quite reasonable because of the unity of mind.
However no “non-mathematical” science is a part of mathematics. No science is a part of its own language.
Since the 70s of last century, when we pass by the movement of modern mathematics, in Brazil, in the curricula of school, work with complex numbers. Often we psra our students the idea that the imaginary unit "i", only serves to extract a negative root and not provide our students the opportunity to realize the impotância the set of complex numbers in all its applications. Lose yourself in this way the opportunity to disclose to beginning students that, although there is further study, we know that today, in contemporary physics, the application of the set of complex numbers is so great that it is even possible to think of a authentic "complejificación de la física", as quoted by the author Federico Rubio y Galy on "The Role of Mathematics in the Rise of Science." So credit to have little influence mathematics teachers in training bases for the continuity of research on mathematical applications in physics to multiply and thrive in other parts of mathematics.
So far all of mathematics can be used in physics. Although, I can say that some of these concepts are not used in physics as the same way as mathematics. For example the concept of the Dirac delta is used in the physics partially. That is, mathematicians does not conceptualized this as a function, but physicist does it because for us, it works!! to describe the physical phenomenon. Things like this happen with mathematics. But to say that a branch of mathematics cannot be used in physics, I do not think.
Also agree with those who contend that right now "is not found a use in physics" but this may be due to many factors. For example, a definition can be erroneous, incomplete or partial. This could cause science and mathematics not agree on its use.
Please google these words "ALL YOU WANTED TO KNOW ABOUT MATHEMATICS,BUT WERE AFRAID TO ASK" Louis Lyons....and also the worlds greatest ever mathematician "Ramanujan's Death Bed Problem" u know ...its being used to model the formation of black holes..so sir its all our head .how to apply is to be known....after a certain stage mathematics speaks better science and engineering than theory.....
also my dear friends who say they know some use less mathematics....please make a list of all those concepts you claim to be of no application...and then google it
" APPLICATION of the XYZ function u say is of no use".....then comeback here to this post and tell that....yeah! this one is a useless function.....!
Mathematics id the language in which laws of nature bis written.
It may apparently appear that some mathematical concept is too abstract
to conceptualize. But, really speaking it has its own logic.
I have given a number of examples earlier where mathematics of the past thought to be useless, even sometimes by its authors, only to appear years later to find a useful applications in the world we live in.
In the words of Mario Livio:
"The list of such 'anticipations' by mathematicians of the needs of various disciplines of later generations just goes on and on .One of the most fascinating examples of the mysterious and unexpected interplay between mathematics and the real (physical) world is provided by the story of knot theory-the mathematical study of knots. A mathematical knot resembles an ordinary knot in a string, with the string's ends spliced together. That is, a mathematical knot is a closed curve with no loose ends. Oddly, the main impetus for the development of mathematical knot theory came from an incorrect model for the atom that was developed in the ninteenth century. Once that model was abandoned-only two decades after its conception-knot theory continued to evolve as a relatively obsecure branch of pure mathematics. Amazingly, this abstract endevour suddenly found extensive modern applications in topics ranging from the molecular structure of DNA to string theory-the attempt to unify the subatomic world with gravity."
-------------------------------------------------------------
Is God a Mathematician, By Mario Livio, 2009.
A circular string needs a miracle to make a knot...
(Sorry, this inspire me to write)...
So, Mr.Hussein Hamzeh you have indirectly , rightly pointed out the meaning of research.....Keeping aside the necessity , Half of the research is driven by thirst for new discoveries and the excitement to innovate.....So the point is not to stop....but to chose wat is relevant and to decide where to stop...a good point surely came out of your comment.... :)
Derek Abbot> But are there any branches of mathematics that are totally useless for physics? Why?
It seems for me that there are (It seems for me that such the branches are so deep into mathematics that people which are far away from pure mathematics may not even know about them. :-) ) For instance, metamathematics (because it is a tool for the interior of the mathematics, not for the world) or cardinal arithmetics (because the world is too small for it. :-) It seems for me that physics does not depend on the axiom of the existance of a measurable cardinal. :-) ).
(to surya narayana Agnimitra: I googled on the applications of the above notions. :-) )
Physics is interested only in interpretations, not in formal systems. If we regard mathematics (like Pythagoreans), as the language of science (but, as a mathematician, I disagree with this claim :-) ), as the language in which the laws of nature are written, then we apriori should work with interpretations, not with (the theory of) formal systems. So, like a mathematician-Platonist, we should choose a mathematics which we regard as the mathematics of the world.
Luis Ateca Torres> For example the concept of the Dirac delta is used in the physics partially. That is, mathematicians does not conceptualized
We did. Delta function is well known in functional analysis as an element of a completion of a certain space of functions (if I remember it right).
Alex Ravsky> "Luis Ateca Torres> For example the concept of the Dirac delta is used in the physics partially. That is, mathematicians does not conceptualized
We did. Delta function is well known in functional analysis as an element of a completion of a certain space of functions (if I remember it right)."
There is a well-developed branch of analysis that extends the usual Reimannian and Lebesguean approach to calculus by extending the notion of what a function is. It is called Distribution theory. The dirac delta function is quite an ordinary object in this realm, and well-developed, rigorous theory supports differentiation and integration of such. One carries on with differential equations involving such objects quite naturally.
In general if something CAN be formalized, there is mathematics that does so (what I mean is that the very act of formalization constitutes the development of such mathematics). If something CANNOT be formalized then it is inconsistent and simply bad/invalid science. So that math is applicable to all matters of science (physics or otherwise) is tautological. It is not even controversial. Rather it is a misunderstanding of mathematics to think otherwise.
I think that mathematics in not completely formal (especially, from the point of view of working mathematicians) (see, for instance, Imre Lakatos's introduction to his classical book "Proofs and Refutations: The Logic of Mathematical Discovery" (http://file3.webfile.ru/6374030/lakatos.djvu?filename=lakatos.djvu). Dirac was mentioned there too. :-) ).
Measure theory perhaps! Indeed, most of the people of applied sciences might never have heard about measure theory. In fact, without even knowing that a branch of mathematics is known as measure theory, all sorts of researches in physics are perhaps possible.
It seems that measure theory is used in some physical models which are abstract dynamical systems and in an abstract space corresponding to such a model the invariance of the measure is equivalent to something like the conservation of energy law in theoretical mechanics or to uncompressness of a liquid in hydrodynamics.
Fauth: Well, if the Lebesgue integral is used in quantum mechanics, then of course my answer has to be wrong! However, I wanted to say something a bit different. Measure theory is a branch of mathematics, which is avoided like plague even by most of the mathematics fraternity. Therefore, I thought, perhaps the applied people will not touch it at all!
Ravsky: I did not know that. Do you mean, measure theory is used in chaos theory?
I expect, yes. If I remember it right, in general, when the evolution transformation of a system preserves the measure on the space of states, and this space has a finite measure then the behavior of the evolving system tends to chaos. For instance, see
http://en.wikipedia.org/wiki/Baker's_map
Ravsky: Thank you for your answer. I in fact did not know this. In that case, we are back to the original question.
I think, there is no branch of mathematics that does not have any application in Physics.
Concerning the original question, a few posts above I proposed metamathematics and cardinal arithmetics as such the branches.
cardinality is a topic that at least is touched in Penrose, "The road to reality". Thus, it is at least not irrelevant for physics. what is the cardinality of strange attractors and of fractals, by the way? Simply that of the real numbers?
> what is the cardinality of strange attractors and of fractals, by the way? Simply that of the real numbers?
I denote this cardinality as C and the cardinality of the set of integer numbers as W. Both these cardinals are very small. Moreover, it is independent of ZFC that any uncountable set has the cardinality at least C. Hence these cardinals plays very small role in cardinal arithmetics.
Even one of the simplest fractals, Cantor set, has the cardinality C. Any uncountable separable complete metric space contains a homeomorphic copy of Cantor set and hence has the cardinality C. In particular, each uncountable closed subset of a finitely dimensional Euclidean space and each uncountable metric compact have the cardinality C.
>Moreover, it is independent of ZFC that any uncountable set has the cardinality at least C.
Does that mean that C=aleph_1 has been proven?
It means the opposite: “C=aleph_1” cannot be proved in ZFC and “Caleph_1” cannot be proved in ZFC too. (Of course, provided ZFC is consistent)
How about the study of problems without solution? This may be another philosophical inquiry, but the classification of differential equations which have no solution (eg local non-solvability of PD Ops) may be considered non-physical, unless one considers such results to be apriori refutation of certain physical theories.
How about the theory behind Normal Numbers (NOT normally distributed RVs)
I think, not just the Transportation Problem, things like Linear Programming, Integer Programming, Stochastic Programming, Dynamic Programming, Assignment Problem etc. are possibly not used in Physics.
Correct. The word 'discovery' is used for something that was already there; one uncovers it. Whereas the word 'invention' is used for something that was not there earlier, and one creates it. Natural laws are already there; one can only discover them, one can not create a natural law, and therefore one can not invent a natural law. Hence physics discusses discoveries only, not inventions. On the other hand, using some laws of physics, when one creates a device, the telephone for example, that is an invention, because it was not there earlier before it was invented. Now the telephone is not something of physics, it is a matter of technology. For something of technology, we might need things like operations research, but not for something in physics.
Linear programming is actually used in physics to calculate the exact ground states of three-dimensional Ising spin glasses. Problem of spin glasses is of great interest both in solid state physics and in statistical physics.
Linear programming is also used in plasma physics to calculate Optimal stellarator design.
Sijab: Is that so? Thank you for this information. It is really difficult to name a branch of mathematics that is not used in physics. What about the transportation problem, integer programming etc? Are they also used in physics? Theory of queues for example? Is it used?
I just mentioned two applications of linear programming in physics that I happened to know. I would not be surprised if there were others. I believe dynamical programming is used in medical physics but can't be sure. I will provide references in my next post.
Your other branches of mathematics, transportation problem, integer programming and queues theory, I have no idea. But you got me so interested I will research them and find out. I will not be surprised to find applications for them in physics. I have a feeling all these have applications in statistical physics, mathematical physics and computational physics but don't hold me on this! I will let you know, if somebody else doe not answer before me.
In fact, what has not been used till this day may actually be used in future. Perhaps, number theory too will be applied in physics some day!
Highly recommended (I think I mentioned this before) : "Modern Mathematics in the Light of the Fields medal" by Michael Monastyrsky .
Lists some applications of abstract math that were absolutely not foreseen when the math was first developed.
"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. ... it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation."
— Paul A. M. Dirac
Introduction to a paper on magnetic monopoles, 'Quantised singularities in the electromagnetic field', Proceedings of the Royal Society of Lonndon (1931), A, 133 60. In Helge Kragh, Dirac: a Scientific Biography (1990), 208.
1-There's a website called "Number Theory and Physics Archive" that contains a vast collection of links to works in this interface.
http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm
2-Matilde Marcolli has a nice paper entitled "Number Theory in Physics" explaining the several places in Physics where Number Theory shows up.
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.140.1998
Hard to say for sure, but I would guess that some areas of set theory, e.g., large cardinals, have no physical applications.
Number theory does show up in physics, but so far things like Galois theory do not show up.
Andrew
Google it and you will be surprised. Below is just one reference:
Applications of Galois theory in physics
http://freepdfdb.com/pdf/applications-of-galois-theory-in-physics
This may help to digest complex numbers:
https://www.researchgate.net/publication/236985939_The_Operator_j_and_a_Demonstration_that_costheta__jsintheta__ejtheta
Article The Operator j and a Demonstration that $\cos\theta + j\sin\...
I have voted up your claim: "this discussion is futile. Certainly there are, and have been, areas of mathematics which are not used in physics."
Nevertheless, the answer depends on what is understood as physics. If the study of human-brain logical structure is included in the term "physics", then every human-brain construction falls in the scope of physics. From this view point, any mathematical topic with no application in physics cannot be built by any human mind. Thus, the proper question would be ¿can human mind create some mathematics for which there is no application in physics?
How about an answer based on philosophy? Plato thought there was an entirely separate existant world of ideal forms; i.e., an ideal square, an ideal perfect horse, and ideal chair etc... Sir Roger Penrose has extended Plato's world of forms to include, as well as ideals of objects and geometric shapes, mathematical concepts and entities. Shadows from these ideals somehow project down into our world.
If this were true, we could assume the existence of an ideal mathematical concept/entity that has no physical application in our world, since the ideal world is independent of ours. Whatever the projection mechanism is, I think it would only project into our world ideals that have some physical relevance.
Thus, I think the answer is yes: there are mathematical concepts or entities that have no application to physics in our world. Unfortunately, we will never know about them.
Professor Juan-Esteban
you have asked a very important question: "can human mind create some mathematics for which there is no application in physics? "
We must consider how our mind can create and understand this precise mathematical logic, as Brouwer once stated ;
"One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics. (Luitzen Brouwer)"
"The mind can be considered as a relationship machine which has evolved to understand the logical consistency of the world about us and hence relate things in a systematic and logical manner.
e.g. Once eating poison fruit was related with dying, then this relationship remained true and consistent. In this way a logical mind is a natural evolutionary consequence of the logical universe (as it enhances our survival).
This is why we are able to think in terms of mathematics. Our brain is a logical relationship machine, and mathematics is a logical relationship language."
-----------------------------------------------------------------------------------------------------------
On Mathematics, Mathematical Physics, Truth and Reality, by Geoff Haselhurst
http://www.spaceandmotion.com/mathematical-physics/logic-truth-reality.htm
The above is a highly recomended reference for this and other related philosophy of mathematics issues.
I think that the mathematics of infinite cardinal numbers is useless in physics. Particularly infinities that are higher than the countable infinity don't seem to be used. Also, the real numbers that are used, are approximated with a finite number of digits. And since the set of all subsets of an infinite set has a higher infinity, there is an infinite number (at least a countable infinite number) of infinite cardinal numbers. I think this is too infinite to have any useful application.
If maths is all in the definitions (or constructs), then wherein are the definitions ('definition' meaning: of-the-finite, btw)? NOT? in reality or 'physics'? JUST all in the mind or imagination, and wherein then do the 'mind' and imagination reside?? Hhmm? From what does anything derive? From the universe or somewhere else, since the universe is finite and we have no USE for infinity? (I must have been daydreaming in some OTHER universe to have missed that lesson)!
What defines usefulness? Only finite applications? What were all those finite 'applications' before they found their way to your ipad? It is the whole cloth failure of imagination in physics, not maths, that accuses maths of wanting in applications.
Well, of course all mathematics is in the mind, all physics is in the mind, even the whole view of the universe is in the mind. Actually the only way to really understand everything is to see what's closer than the mind. We already are before we think. What we are cannot be described by the mind. But this is a spiritual discussion. In a mathematical discussion I would say that only what's finite is applicable to things outside mathematics.
What is "closer than the mind" does not mathematically necessarily have to point solely to the spiritual realm as "the only way to really understand everything." From a mathematical point of view, "closer than the mind" to really understanding everything is the very universe in which the mind (although to its own detriment not always the ego) subsists.
Yes, by "closer than the mind" I mean the source of everything, including the mind. It seem you are using the term "universe" to mean the source. Since it is "closer than the mind" it cannot be described by the mind. Describing it with the mind would be like trying to use a flashlight to see the sun. The sun can be seen by its on light, and the truth can be seen by light of our own Being.
I agree that your "closer than the mind" phraseology is incongruent with perhaps my using 'universe' to mean the source as opposed to your using the spiritual as the ONLY source or way. What better authority than the mind to describe the universe? In fact, case in point to your seeing the sun with a flashlight analogy: Just one of each and every scientist that ever discovered something new, using Copernicus who with maths, his mind, and observations saw the sun to be the center. The next generation later, Galileo using his mind concurred, and the "spiritual" Church agreed with you it is blasphemous that his little mind (flashlight) declare the sun the center and offed his head. Good thing this generation is now in the 21st century, I hope.
No, by spirituality I definitely don't mean the church with all its ideas, even violent ideas. Let's go back to a discussion about mathematics. Is there any useful application of cardinals with higher infinities than the countable infinity?
- Badges
- Science topic
- Similar topics
- Philosophy
- Philosophy Of Science