Geometry as a branch of physics

[pmb_phy]

A friend of mine is a GRist and once explained to me that geometry is a branch of physics because the basic knowledge of geometry comes from observing nature. He's a bright guy who explains things well. I happen to agree with him. I forgot how he so elegantly put it though. I also read it somewhere. else but I can't for the life of me recall where.

Question: Do you know of a physics text which holds geometry to be a branch of physics? Do you believe that yourself?

Thanks

Pete

 

[MathematicalPhysicist]

It's not.
You can use it in physics as every maths tool (which stand by its own virtue), but it's maths.
If you can't make the distinction between maths and using maths for physics, then i don't see why you can't see every maths theory as a branch in physics.

 

[MathematicalPhysicist]

anyway, this is a philosophical question not a mathmetical one nor physical.

 

[HallsofIvy]

Uh0h, Somebody hold me down! I am about to go off in a rage. Pretty much everything had its origins in observing nature but that doesn't make everything a branch of physics! In fact, I would be surprised at someone saying that "observing nature" is physics. Why not geology? Or biology? And, of course, mathematics very quickly leaves observation behind. The basic test of a theory in physics (or any science) is how well it corresponds to experimental evidence (observation). That is not the case in mathematics: it is consistency that is crucial in mathematics.

There- and I didn't swear or break anything!

 

[ObsessiveMathsFreak]

Some people think mathematics in general is a branch of physics. They are wrong of course. It is physics that is a branch of mathematics!

 

[DeadWolfe]

If physics is a branch of mathematics its being done very poorly.

 

[MathematicalPhysicist]

It is physics that is a branch of mathematics!

exactly what i said to some user in another forum who claimed that it's not true, cheers OMF.

 

[MathematicalPhysicist]

If physics is a branch of mathematics its being done very poorly.

spoken like a true pure maths major. (-:

 

[quetzalcoatl9]

exactly what i said to some user in another forum who claimed that it's not true, cheers OMF.

1,000 people could say the same things as you, but that doesn't make it true

to say that physics is a branch of mathematics is a bit like saying that the trade of automotive mechanics is a branch of wrenches.

 

[DeadWolfe]

Physics and math have different sets of standards, different gaols, different methods, different language, etc... The fact the two subjects often use the same symbols, many times in similar ways, does not make them the same. (Though knowledge of one can bring clarity to the other).

 

[MathematicalPhysicist]

but we do have applied maths don't we?
so why isn't physics a field in applied maths?

 

[robphy]

but we do have applied maths don't we?
so why isn't physics a field in applied maths?

interesting reading:
http://aeolist.wordpress.com/2007/03/17/physics-as-applied-mathematics-not/

 

[quetzalcoatl9]

interesting reading:
http://aeolist.wordpress.com/2007/03/17/physics-as-applied-mathematics-not/

nice article, indeed it is a different "game"

 

[MathematicalPhysicist]

I don't agree with idea being laid there, that there are no mathematicinas who are also physicists, or that mathematicians don't like labs.(yes there arent a lot of them, you can name a few such as Newton as a famous figure but i guess there are others one name is richard courant which use applications in most of his books in analysis, also it's written in mactutor that wilhelm magnus had some impact in physics theorywise).
most of the labs are about analyzing statistically the information, if they don't like labs then they don't like statistics as well.

I agree that there are stuff that you need a physical intuition for them, such as analysis by units of measure, i don't think that you impede or damage your physics education by first taking courses in maths, quite the contrary, you might understnad why you can use this theorem and why not in physics.
and it hasn't persuaded me that physics isn't applied maths, yes sure the principles arent mathematical, but that's the notion of physics as empirical science, but you still use maths in it so why not call it applied maths, if it's not applying maths then i really don't know what applying means?

 

[Hurkyl]

I didn't think it worth leaving a comment on the blog, but if people here are reading it, it is probably worth commenting here.


The hypothesis that mathematicians cannot be successful in theoretical physics is absurd. Surely any scientist should recognize the critical contributions by mathematicians such as Dirac, Hilbert, Minkowski, Riemann, and Fourier. (And that's just what immediately springs to mind)

(and I don't think Riemann even had any physical applications in mind when he invented differential geometry!)



There's no getting around the fact that physicists often use mathematics in a 'sloppy' manner. This has the effect of creating a communication barrier, which discourages many mathematicians from pursuing any interest they might have had in physics.

In other words, it's not that being a mathematician hinders you from learning physics -- but instead that many physicists don't want to talk to mathematicians.



As for the comment BL made to the blog -- the art of clever approximation is called "analysis" by mathematicians. (e.g. real analysis, asymptotic analysis, analytic number theory...)

 

[robphy]

By the way, Dirac got his first degree in Electrical Engineering.
In addition, Dirac's approach to the "delta function" is sometimes given as an example of "sloppy mathematics".

I agree the quoted professors in that blog entry were too strict by saying anecdotally "No one has..." when they probably should have said "Few have..." Certainly, one can find exceptions [to their statements] of mathematicians who have been successful in physics... as Hurkyl has done.

The reason for pointing to that [imperfect] blog entry is that
Physics is distinct from Applied Mathematics... neither is a proper subset of the other, although there is an overlap.

In my experience, my advice to a future theoretical physicist is
take as many math courses as possible on top of your physics curriculum... and try to forge the connections between them.
There is a lot of physics that one will only see in an undergraduate physics curriculum. So, one is at a disadvantage without one.Let me contribute some of my anecdotal observations:

- it is better to take a first course in calculus before the first calculus-based physics course because
it seems that the calculus-needed by the physics course is taught too late in the math course
Things might be okay if your calculus-based physics course doesn't use much calculus.

- vector calculus and E&M should be taken as close together as possible because each course provided only a piece of what I wanted...

- when I teach the intro calculus-based classes [when setting up a Newton's Law problem], I emphasize the following points:
PHY: Drawing a Free-Body Diagram (with force vectors)
PHY: Writing down Newton's Second Law
MAT: Breaking the vectors into components and doing algebra to solve for the unknown
PHY: Verifying that the solution makes physical sense. Interpreting the physical implications over the range of the variables.

(MAT means pure math... in the sense that the student can present their work to the math department without physics context and get the required mathematical solution.
PHY means that a physics understanding is essential...even though the language used might involve mathematical symbols.)

To me (but not necessarily to my colleagues), the PHY parts should carry the most weight in the problem. If they can describe completely [but not necessarily do] the required mathematics, they could get almost all of the credit.(Here are some interesting observations by some mathematical physicists:
http://www.math.oregonstate.edu/bridge/papers/calculus.pdf - The Vector Calculus Gap
http://www.math.oregonstate.edu/bridge/papers/ampere.pdf - Why is Ampere's Law so hard? )

 

[HallsofIvy]

1,000 people could say the same things as you, but that doesn't make it true

True. You have to say it LOUDLY enough!

to say that physics is a branch of mathematics is a bit like saying that the trade of automotive mechanics is a branch of wrenches.

Now, that I like!

 

[llarsen]

The hypothesis that mathematicians cannot be successful in theoretical physics is absurd. Surely any scientist should recognize the critical contributions by mathematicians such as Dirac, Hilbert, Minkowski, Riemann, and Fourier. (And that's just what immediately springs to mind)

There is no doubt that mathematicians have had a profound effect on Physics since mathematics is a fundamental tool in physics. However, the bread and butter for mathematicians is proofs, which drives one to formalism. Whereas intuition and physical reasoning play a much more important role to a good physicist. I have seen numerous mathematics texts which are composed almost exclusively of theorem - proof - theroem - proof, with the author occasionally honoring the reader with a brief explanation of the value and context of the material. Even some of the better mathematical texts (in advanced mathematics) typically provide few worked out examples (if any) and are much more concerned about 'saying things precisely' than conveying conceptual understanding, the purpose, and the value of a particular set of tools. Unfortunately, the things that are left out are often what is most important to scientists (conceptual explanations, worked out examples, context, geometric explanation and intuition).

There's no getting around the fact that physicists often use mathematics in a 'sloppy' manner. This has the effect of creating a communication barrier, which discourages many mathematicians from pursuing any interest they might have had in physics.

There is a good reason that mathematics is often 'sloppy' in physics. A good physicist needs strong physical intuition. Mathematics is the primary tool used to express this intuition and formalize ideas. BUT it is the physical phenomenon that is fundamental and the intuition to understand the phenonenon that is of primary importance in physics. Formalism that detracts from developing physical intuition is often left out because it proves to be a distraction. The student needs to understand how the mathematics relates to underlying physical concepts. Often an intuitive description, diagram, manipulation rules, worked out examples, and other 'imprecise' tools are much more important to developing physical intuition, than formal definitions and proofs. Later a more formal study of the mathematical tools can prove useful. Proofs are much more productive when one understanding how to USE the tools along with physical interpretation to draw on, in my experience.

V.I. Arnold (a prominent Russian mathematician) gave one of the best critiques of current trends in mathematics I have seen in an adress in Paris in 1997. (The talk can be found at http://pauli.uni-muenster.de/~munsteg/arnold.html) He claims that the separation of mathematics and physics has been 'catastrophic' for mathematics. It used to be that most mathematicians studied Physics and their problems were motivated by Physics. This led to powerful mathematical tools which had a context that provided motivation and a variety of examples to give insight. Mathematicians often had a firm grasp on Physics (and in some sense would have been considered too 'sloppy' to be considered a mathematician today and my have been labeled as merely 'theoretical physicists' - not his comment in this case). Arnold strongly criticizes the method of developing a mathematical topic based on abstract 'axioms' without providing clear understanding of the use and context of the mathematical tools since there are often geometric or physical exmples that explain the importance of a particular tool. 'It is impossible to understand an unmotivated defintion but this does not stop the criminal algebraists-axiomatisators... It is obvious that such definitions and such proofs can only harm the teaching and practical work... For what sins must a student try and tind their way through all these twists and turns [of focusing on abstraction, defintion, and proof]?'

In other words, it's not that being a mathematician hinders you from learning physics -- but instead that many physicists don't want to talk to mathematicians.

A physicist is less interested in proving a mathematical result than in using the mathematics as a tool to understand some physical phenomenon. It is common to come out of a graduate mathematics course having done numerous proofs, but not having any feel for how to apply the mathematical tools to an actual problem.

I have run into a communication barrier numerous times as I have gone to talk to mathematics professors. Typically as I have approached a new mathematical topic, I have tried to develop some context and understanding of what the tool is doing. Often, in my case, this has involved developing some geometric intuition into the tool. Commonly one has to look through many texts and do a great deal of thinking to develop a strong geometric intuition. (In many cases, as I have developed clear geometric intuition, I have been astounded that so few books present a clear geometric picture.) I have gone at various times to talk to one professor of mathematics or another to talk about a concept, geometric interpretation, or possible physical application of a particular tool. Almost invariably, rather than focusing on understanding the conceptual picture, they have wanted to focus on definitions and details. Having an informal description and picture (whether it is correct or not) brings distain. I think it is this unwillingness to handle informal conceptual thinking (which has proved a powerful motivator of some of the most prominent areas of mathematics and physics over time) that hinders mathematicians in pursiut of physics, and discourages communication between Physicists and Mathematicians in many cases, which is unfortunate because I think there is a lot to be gained by collaboration.

 

[Hurkyl]

I apologize in advance for not having the energy for a full response.



You mention "formalism" as a negative, and "manipulation rules" as a positive -- but this is somewhat contradictory. By its very definition, formalism considers "manipulation rules" to be of primary importance.

And 'formalism' and 'intuition' are not exclusive. Knowing the rules is not the same as being able to apply the rules effectively; even a formalist considers intuition to be important. (I am a staunch formalist) But intuition doesn't override the rules of the game -- if there is a conflict, then either you must change your intuition or play a different game.



One of the major driving forces behind abstraction is that a fledgling theory will often have some central important ideas and concepts, and a lot of generally irrelevant scaffolding that was used to construct the theory. One goal of abstraction is to revise the theory so that the important ideas are brought to the foreground, and the irrelevant details are pushed to the background, or even eliminated.

Abstract finite-dimensional vectors spaces are a wonderful elementary example of this process. (And the very closely related notions of abstract tensor fields and abstract Hilbert spaces) Coordinates are a very convenient way to represent spatial vectors and to do rote calculation, but they carry a lot of mental baggage: before you can use them, you have to specify coordinate axes, and all of your manipulations are done through coordinates, rather than working directly with the underlying concepts. But if you appeal to the vector space formalism, you now get to work directly with the objects of interest, without the added baggage of bases and coordinates.

Unfortunately, and ironically, vector spaces are one of the examples of staunch resistance to 'abstraction'; some people will swear up and down that studying anything but vector spaces of n-tuples is pointless abstraction and obscures the real meaning of vectors.



As for definitions versus conceptual picture, you have to realize that people understand things differently. For example, for me, this Wikipedia page paints a very vivid picture. But, alas, I find a "physics"-style introduction to QFT to be entirely impenetrable. The end result is that I can't compute a darned thing, but I can easily understand some of the high level concepts, such as how locality fits into the picture. On a more elementary note, I once took a short course in quantum computing, which explicitly avoided doing any real 'physics', instead using purely 'abstract' linear algebra. (We didn't even talk about the Bloch sphere!) But I learned far more about quantum entanglement than I did in all of my many hours of self-study of more traditional sources.

This is another example of what I was saying about abstraction. These 'abstract' presentations were much closer to the concepts in which I was interested, and thus I was much better able to understand them. Whereas I gather little to no understanding from more 'concrete' presentations that have a lot of obscuring details.

 

[llarsen]

I think my post was a bit more like fighting works than I intended, and I apologize for that. I have deep respect for mathematics and the mathematicians that have come to master so many useful topics. The tools of modern mathematics that have been developed are awe inspiring. On the other hand, I have found more often than not that a physicist, rather than providing a clear understanding of what mathematical tool does, resorts to 'hand waving' and comes up with results that the student does not understand because no appropriate mathematical foundation was provided. Sergei Winitzki gives a justified critique of the physicists all too common tendency to neglect important mathematical details that could add much clarity to their expositions (See http://www.theorie.physik.uni-muenchen.de/~serge/why_physics_is_hard.pdf ). At the end of the paper he notes 'Finally, I would like to note that physicists require a different approach to teaching mathematics than mathematicians, mainly because physicists are focused on computational issues rather than on creating and proving theorems about abstract mathematical constructions. Therefore there seems to be no solution except to offer lecture courses of mathematics specially tailored for physicists.' The paper suggests that Physicists often fail to provide a proper mathematical foundation to student. If you have not read the talk sighted in my post above by V.I. Arnold I would highly recommend it because it does an excellent job of explaining why mathematical education often fails to provide physicists with the skills they need for studying physical phenomenon. The focus of the majority of advanced mathematical texts and courses I have encountered are primarily focused around theorem and proof. One comes away from a course able to prove various results, but with little proficiency at applying the tools to actual problems.

There seems to be a great need for physicists and scientists that is too often left unmet by the physicists or the mathematicians. For scientists, proofs, for the most part, are only important to the degree that they provide insight into how to effectively apply a mathematical tool. Unfortunately little time is usually spent at the end of a proof applying the result to problems. Typically one may just apply the result to another proof! Many scientists could certainly benefit from a more solid mathematical foundation that allow them to more effectively tackle problems. Mathematicians are usually not primarily concerned with teaching one how to use a tool to tackle problems. This seems to be the great divide between the mathematicians approach to mathematics and the needs of a scientist, and in my experience, there are few who have done an adequate job bridging the gap (i.e. provided a solid foundation for understanding mathematical tools with a particular emphasis on understanding the geometric or physical interpretations along with a solid ability to use the tools in calculations.) I personally feel that mathematicians would make much stronger contributions directly to physics, and physicists would show much more proficiency with mathematics, if mathematics courses and text would focus on motivating examples and developing a students intuition, as suggested by Arnold, as well as developing a students proficiency in applying the tools to various practical problems, rather than a focus primarily on proof. Unfortunately, the culture does not seem to lean that direction at present.

 

[Hurkyl]

If you have not read the talk sighted in my post above by V.I. Arnold I would highly recommend it because it does an excellent job of explaining why mathematical education often fails to provide physicists with the skills they need for studying physical phenomenon.

I confess I find it extraordinarily difficult to take him seriously, with all the vitreol and exaggeration. (At least, I hope he doesn't literally mean what he says )





There is something I wanted to say in my last post but forgot... the "theorem - proof - theorem - proof" style does have it's place: it is very easy to search for particular bits of information, and if you can follow it, the information density makes for very efficient reading. IMHO, the ideal situation (even if it isn't practical for the average student) is to have several texts on the subject available for simultaneous study.

And one other comment; I hear mathematicians bemoaning many of these same complaints about the state of mathematical education.

 

[MathematicalPhysicist]

there are books that also add historical anecdotes in maths. (few of them).

 

[llarsen]

Arnold's tone is quite acerbic isn't it. Not necessarily the way to endear yourself to your target audience . But he is a well respected mathematician with a lot of experience, and I think he does capture the frustration of many scientists who have struggled to understand advanced topics in mathematics.

It takes much more work than it should to develop a solid feel for a new mathematical topic. I have often found after studying a new mathematical subject (sometimes for months, or even years) some insight, or way of presenting the topic, and I have an AHA! THAT'S WHAT THEY WERE TALKING ABOUT! moment. Why did the author just come out and say that!

For example, I spent years working with and studying various texts related to linear algebra and determinants before I learned the direct geometric connection to parallelapiped's (or n dimensional volume elements). This helped make the connection to the chain rule. It helped me understand why degeneracy was such an issue in transformations and what it meant. This insight made so many of the theorems of linear algebra almost obvious. It has been incredibly useful in my study of Differential Geometry. It has made the purpose and importance of the jacobian much more clear to me and made it obvious why the rank of the jacobian is so important. Why is it that almost no linear algebra texts emphasized this simple geometric connection that makes things much simpler to understand?

It has been all too common for me that the concepts which clarify a subject for me are not presented in many of the texts, but come either from analogy to prior mathematical topics I have come to understand, or through thinking deeply (sometimes over a long time) about what the theorems are saying, or from some random text I run into after studying the topic for a while which does a beautiful job of presenting the key insights that help one understand the significance of the theorems. It is marvellous when the pieces start to come together - what an awesome feeling! I am incredibly greatful to the authors who take the time to paint the picture and guide your understanding and make the path much more easy for you to follow.

 

[J77]

It's one of those discussions that go round and round...

Physics vs Maths (or Chem or Bio or Geo...)

eg. I went to a talk last year from Gross about the future of physics. He highlighted 10 'hot' points, amongst them...

Biophysics, Slow-fast systems and Neuroscience.

The way he explained them was what people have been doing in dynamical systems for a long time.

Sometimes I think that people who like to make barriers between the different sciences (which typically is a barrier between physics and the rest ) are trying to hold onto something, protecting themselves through ignorance; ie. are scared to see how other people do it in other fields if it knocks their ways a bit...

I'm ranting now...

 

[CRGreathouse]

The hypothesis that mathematicians cannot be successful in theoretical physics is absurd. Surely any scientist should recognize the critical contributions by mathematicians such as Dirac, Hilbert, Minkowski, Riemann, and Fourier. (And that's just what immediately springs to mind)

Wasn't Dyson's degree in mathematics?

 

[robphy]

Wasn't Dyson's degree in mathematics?

Apparently, Freeman Dyson has a BA in Mathematics... but no Ph.D.
http://www.sns.ias.edu/~dyson/
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Dyson.htmlBy the way, this thread has seemed to move away from the initial question...
...and the initial question, although thought-provoking, itself isn't really a TA&DG question.

 

[Hurkyl]

For example, I spent years working with and studying various texts related to linear algebra and determinants before I learned the direct geometric connection to parallelapiped's (or n dimensional volume elements). This helped make the connection to the chain rule. It helped me understand why degeneracy was such an issue in transformations and what it meant. This insight made so many of the theorems of linear algebra almost obvious. It has been incredibly useful in my study of Differential Geometry. It has made the purpose and importance of the jacobian much more clear to me and made it obvious why the rank of the jacobian is so important. Why is it that almost no linear algebra texts emphasized this simple geometric connection that makes things much simpler to understand?

This one I can explain -- linear algebra is highly ubiquitous. You certainly learn the properties directly related to linear algebra, such as the image of a rank 2 map is two-dimensional, but you simply can't expect every possible application of linear algebra to appear in a linear algebra text!

e.g. it certainly seems reasonable to me that "if the derivative of a map^* is rank 2, then the image is locally a plane" would be first shown in a differential geometry lecture.


I don't think it's fair to complain that the applications of linear algebra to differential geometry wait until your differential geometry class.

But I do think it's probably fair to complain that there isn't a proper analytic geometry course anywhere in the usual cirriculum. (I'm not convinced it would be correct to stuff an entire analytic geometric course into Linear Algebra 1)


*: i.e. the Jacobian

 

[llarsen]

For example, I spent years working with and studying various texts related to linear algebra and determinants before I learned the direct geometric connection to parallelapiped's (or n dimensional volume elements). This helped make the connection to the chain rule. It helped me understand why degeneracy was such an issue in transformations and what it meant. This insight made so many of the theorems of linear algebra almost obvious. It has been incredibly useful in my study of Differential Geometry. It has made the purpose and importance of the jacobian much more clear to me and made it obvious why the rank of the jacobian is so important.

I don't mean to imply that all the ideas above belong in a linear algebra course. But the insight that the determinant is the volume of a parallelepiped formed by the vectors that make up the columns of the matrix makes properties of the determinant much more transparent. Maybe more useful is the idea that matrix rows can be related to the a choice of linear coordinates in a Euclidean space and that the determinant is a volume element that can be represented by drawing a constant surfaces of equation associated with each row of the matrix. Since various linear algebra texts already present the application to solving a linear set of algebraic equations, this concept seems compatible with what is presented in many courses. It gives one a mental picture of what a determinant represents, and allows one to almost guess at some of the properties of determinants prior to proving results. This insight can guide one in proving results, but helps one see the point better. The key is that a simple insight can lead one to understand results much better. Until one gains this type of key insight, the relevance of proofs is often lost on the individual.

 

[shoehorn]

I didn't think it worth leaving a comment on the blog, but if people here are reading it, it is probably worth commenting here.


The hypothesis that mathematicians cannot be successful in theoretical physics is absurd. Surely any scientist should recognize the critical contributions by mathematicians such as Dirac, Hilbert, Minkowski, Riemann, and Fourier. (And that's just what immediately springs to mind)

I have little to contribute on this apart from saying that the claim that Dirac was a mathematician is, frankly, laughable.

 

[robphy]

I have little to contribute on this apart from saying that the claim that Dirac was a mathematician is, frankly, laughable.

I guess one may look back and make such statements using today's standards and viewpoints [possibly calling him more of a "mathematical physicist"]... but at least some of his contemporaries regarded him as a mathematician...

from http://nobelprize.org/nobel_prizes/physics/laureates/1933/dirac-bio.html

Here, he studied electrical engineering, obtaining the B.Sc. (Engineering) degree in 1921. He then studied mathematics for two years at Bristol University, later going on to St.John's College, Cambridge, as a research student in mathematics. He received his Ph.D. degree in 1926. The following year he became a Fellow of St.John's College and, in 1932, Lucasian Professor of Mathematics at Cambridge.

 

[shoehorn]

I guess one may look back and make such statements using today's standards and viewpoints [possibly calling him more of a "mathematical physicist"]... but at least some of his contemporaries regarded him as a mathematician...

from http://nobelprize.org/nobel_prizes/physics/laureates/1933/dirac-bio.html

At the time, if you graduated with a PhD from Cambridge and your dissertation centred on what we now regard as "theoretical physics", they packed you off with a PhD in mathematics.

As usual with Cambridge, nothing here is ever (or was ever) straightforward.

 

[MathematicalPhysicist]

i guess your'e from cambridge.
why isn't it clear, as i see it they have two departments there one for pure maths and the other for applied maths and theoretical physics, i guess the phd refers to both departments.

 

[J77]

DAMPT would be the place to do both theoretical physics and applied maths.

 

[MathematicalPhysicist]

DAMPT would be the place to do both theoretical physics and applied maths.

yes, so?

 

[mathwonk]

why are these philosophy arguments taking space here?