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Geometry as a branch of physics 
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#37
Jan1611, 12:55 AM

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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trigonometry, whose functions are necessary even in those applications of mathematics where Euclid's axioms are no longer valid.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. Spacetime has a geometry, according to GR. Any theory of space or of the unified spacetime construct is a geometry, and there is geometry as a branch of physics, just as there are many nonphysical geometries. 


#38
Jan1611, 01:22 AM

P: 4,572

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. 


#39
Jan1611, 02:24 AM

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P: 1,741

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.



#40
Jan1611, 04:20 AM

P: 9

There is a famous lecture by V.I.Arnold concerning the subject.
http://neon7.110mb.com/On%20teaching%20mathematics.pdf 


#41
Jan1611, 07:14 AM

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P: 1,741

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. 


#42
Jan1611, 09:02 AM

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P: 1,716

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. 


#43
Jan2311, 12:32 AM

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#44
Feb411, 04:50 PM

P: 429

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 510 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 aweinspiring 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. 


#45
Feb411, 05:01 PM

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 spoonfed 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). 


#46
Feb411, 05:28 PM

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#47
Feb411, 07:19 PM

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P: 1,716

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. 


#48
Feb411, 08:13 PM

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P: 1,965

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!"



#49
Feb511, 05:42 AM

Sci Advisor
P: 1,716




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