# Geometry as a branch of physics

by pmb_phy
Tags: branch, geometry, physics
P: 20
 Quote by 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
That is exactly what I have thought. Physics studies space, time, matter, energy, and the forces; geometry studies just space. Therefore, geometry is part of physics and physics is an extension of geometry. Euclid's theory was the first theory of physics, as it was the physics of space. Newton's theory joined space with time, motion, matter, and gravity. Coulomb added electricity. Faraday unified electricity and magnetism, and Einstein and Lorenz figured out more accurately the relation between space and time.

As a theory of physics, geometry at some point lost touch with experiment and observation. Geometry came to be more about the math than the physics, so that geometers could eventually care less about whether the theory was true of physical space. Even now, although some discrepancy with space is clearly implied by general relativity, Euclidean geometry remains a useful branch of mathematics, since it motivates trigonometry, whose functions are necessary even in those applications of mathematics where Euclid's axioms are no longer valid. For example, hyperbolic geometry requires both trigonometric and hyperbolic functions to describe its trigonometry!

Thus geometry has become more of an interesting mathematical theory and less of a theory of physics, although as the latter it is extremely accurate if not exact. But it bears pointing out that other physical theories have undergone a similar process. String theory comes to mind. In fact, string theory has lost so much touch with reality that it is now regarded as merely an amusing mathematical toy with hardly any insight about physics (and not even wrong).

And actually, the above applies to Euclidean geometry. Since Euclid, there have been other geometries. Physical space does in fact have a geometry even if it is not (exactly) Euclidean. Space-time has a geometry, according to GR. Any theory of space or of the unified space-time construct is a geometry, and there is geometry as a branch of physics, just as there are many nonphysical geometries.
 P: 4,577 Math is like any other language out there: its simply an optimized way of representing a class of things as well as providing a framework for decomposing and analyzing these things. Languages in general are optimized for a particular purpose. Physics uses math because it is the optimal choice of language for analysis. If it weren't the optimal choice, then something else would be used.
 Sci Advisor P: 1,810 Geometry is a part of mathematics merely because how it is used. It has a formal axiomatic basis, and the reasoning is mathematical (formal, rule governed, not experimental). This is essential to geometry and any part of mathematics. Geometry is not about the real world, it is merely utilized as a representation of physical states of affairs. Geometry is not a study of (physical) space, it can be a representation of space. There is a lot of mathematical reasoning in physics, but the conclusions are never physical facts by virtue of being formally deduced, (physical) facts are always experimentally inferred, which naturally constitutes the distinction between the two fields.
 P: 9 There is a famous lecture by V.I.Arnold concerning the subject. http://neon7.110mb.com/On%20teaching%20mathematics.pdf
P: 1,810
 Quote by simeonsen_bg There is a famous lecture by V.I.Arnold concerning the subject. http://neon7.110mb.com/On%20teaching%20mathematics.pdf
That was quite horrible to read, he doesn't seem to be treating this issue justly. In my opinion, the massive exaggerations and amount of strawmen virtually destroyed his argument. He calls the formal group axioms easily forgettable and questions the need of them to a sensible person. As if we are better off starting out with some particular case of groups like geometrical transformations. That is not something I buy. It doesn't help that he is quoting long dead respective mathematicians as backup for his zeal either.

These types of arguments usually follow the lines of criticizing the physical and geometrical pedagogy of how mathematics is taught. But it is not increasing formality and generality which blinds us from intuition and applications, it is treating the special cases as canonical that blinds us for the generality we can achieve, and of course, apply. One generalizes not for its own sake, but to draw out the essential content, which is not reflected in all of the special cases physics can provide.
P: 1,719
 Quote by 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
We can never prove that the empirical world can be understood using geometry or any other intellectual tool. We can only test theories and hope for the best. Mathematics, while certainly an empirical subject where hypotheses are tested , also has the possibility of absolute proof. This is not true of any other science. So Geometry can not be a branch of Physics or Physics a branch of Geometry

Nevertheless, it does seem that there is a lawfulness to the empirical world which is well framed in mathematical terms. The connection between Mathematics and Physics is so strong that purely mathematical theorems can lead to new physical discoveries and physical theory can suggest mathematical theorems.

I think the symbiosis of Mathematics and Physics creates a new subject that subsumes each. Each is a branch of a larger field of inquiry.

BTW: In my opinion,the idea that geometry alone is a branch of physics is too narrow. For instance,analysis is largely inspired by Physical problems.
P: 2
 Quote by robphy 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. - 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.
In highschool I'm taking AP calculus and AP physics back-to-back with the same teacher (2nd to 3rd courses same teacher). Second semester when calculus learns integrals, physics is going to derive better formulas. Just as robphy emphasized, my teacher does the same; so you may take solace and that the new generation is learning.
P: 429
 Quote by llarsen 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 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. ... '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]?' 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.
I'm sorry, this is complete nonsense.

Do you think that mathematicians find reading these texts easy? I am (attempting) to do a pure mathematics PhD right now. Reading mathematics texts still takes me forever- I frequently have to spend about 5-10 minutes, if not more, reading a single page.

You see, this is expected. Do you think that the writers of these fantastic mathematical theorems have done them in complete mathematical abstraction? Such awe-inspiring theorems can not come from such pure abstraction, they always come from some sort of geometrical, or other, interpretation of what you are studying.

My point is, part of doing mathematics is this constant struggle. In my opinion, understanding a piece of mathematics is almost equivalent to visualising it in some way that you can understand. This is part of the skill (and fun!) of learning high level mathematics. Writers of mathematical texts don't expect you to read off the page and have an instant understanding of it (usually!), they expect you to read it, think about it, abstract it, ponder some more, visualise it (in you own unique way- I emphasise this, everyone thinks in different ways) and finally comprehend the work correctly.

Perhaps this is why physicists (seemingly) find some of this mathematics so daunting- perhaps they are not used to the fact (or don't realise), that studying mathematics is such a struggle, and this struggle and adeptness of visualising difficult pure mathematical concepts is part of the skill of learning maths.
 P: 429 Oh, and what I meant to add is that "Having an informal description and picture (whether it is correct or not) brings distain." is nonsense- what you said about mathematical texts not being cluttered with examples and applications is fair and true (and explained above). My mathematics supervisor, when introducing me to a problem, always does so in a reasonably loose and informal way and tries to make sure that I have some vague understanding of what is going on before he tries to give me the formal details. I'm sure others will tell you similar stories. And if they don't do this, they expect that you should do this one your own, which is often more beneficial than being spoon-fed everything. And, afterall, how do you think mathematicians solve problems? Give a mathematician any problem, and he won't be able to start until he has visualised what is going on in some way. In fact, to prove a point, try and think of some mathematical theorem or problem to solve. You will immediately try to visualise the problem in some way to get to an answer (or at least you should), unless it is a simple "plug in numbers" sort of question (obviously).
P: 429
 Quote by 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. Pete
Oh, and to not stray off topic and answer the question- the answer is, of course, no, because geometry doesn't have to be studied by observing nature and can easily be studied completely independently of nature. This is clearly demonstrated by the way it was axiomitised (is this a word?) and study of this actually lead to the biggest breakthroughs in it (i.e. the discoveries of non-Euclidean geometries), which would have got nowhere by simply studying nature. It may have started that way, but I don't know of many physicists researching geometry (only, perhaps, using it).
 Sci Advisor P: 1,719 Mathematics is an intuitive science that relies upon insight rather than deduction. It is not a tool only. It is a science. Gauss called it the queen of sciences. Mathematicians do not only prove theorems. In fact theorem proving is just a technique that they use to develop new theories. Other sciences do not have absolute proof so they use other techniques like experiments. But the goal for all is the same, to test and develop new theories. For example, the theory of characteristic classes of vector bundles is not a pile of anal deductions. It is a geometric theory. It is breathtakingly imaginative. And it is indispensible in physics. Physics is not a science only. It is a tool as well. It is a tool of engineering. And it is a tool of mathematics. Many theorems of mathematics are suggested in physical phenomena just as many physical phenomena are predicted in mathematical theorems. To say that mathematicians can't think because they reject informality is false. The truth is that mathematical thinking is incredibly profound. Now a days physicists also have to be mathematicians to think in their own field.
 HW Helper P: 1,991 The start and bits of this thread recalled to me of a comment of one character about another in a novel by Nigel Balchin (a novelist whose peak was in the 1940's and whose novels are often about scientists): "Oh he's one of these physicists who thinks Science is a branch of physics!"