Frustrated by how physics is generally explained/taught

In summary, despite efforts to overcome obstacles, physics often feels like a difficult and frustrating subject. The thing is, I think it's possible to overcome these obstacles by paying more attention to the conceptual aspects of the theory.
  • #1
burakumin
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I'm neither a physicist, nor working in a physics-related domain but I've tried for years to get interested in physics because of a fascination for questions like "what is the world" and "how does it works". I suppose this is a feeling shared by other people here. The thing is I have often became stuck in my attempt to understand, even sometimes about things that are said to be "simple". Maybe it takes a lifetime to get those things. Certainly I'm only clever enough to reach a given level of understanding.

However over the years I became aware that there is a recurrent obstacle in books, classes, explanations, articles that have always kept me either frustrated or confused, most of the time both. I would call it an excessive concern for computational aspect over conceptual ones.

As an example, the idea of a physical system I was taught younger was to think of a "real" piece of matter on which we could make experiments and then attach numbers. In this approach every property is a sort of "number". The piece of matter is for the intuitive physical aspect: something you can touch and imagine. Numbers are ... for calculation only. Actually those numbers are most of the times numbers-that-depends-on-other-numbers but this is only a technical detail you know. Learn the formula, learn multi-variable calculus and that's it. At the contrary, after my studies it was a real eye-opener to understand (as a self-taught) a physical system could be defined itself as a configuration/phase space and related properties as fields (or even better bundle sections) on that space. Suddenly a lot of things made sense because the physical semantics of the system could be described inside a mathematical structure.

First I must say I'm definitely a math-inclined guy. Second I deeply believe that maths can (and should) encode the semantics of theories. But as illustrated above I've always noticed maths are more often considered as a mere pragmatical tool that help us creating supposedly abstract nonrealistic "models" used for computation. On the contrary I see maths as the best language to speak about physical objects and concepts themselves. When for example I consider spacetime as a manifold, I don't think I'm creating an ideal object in my mind or in my notebook that "looks like" reality. I consider the idea that spacetime can be thought as a particular manifold itself: that somehow it makes sense to speak about the "real" spacetime using the vocabulary and the concepts of differential geometry (of course that may be partially inaccurate: I know the hole argument against manifold substantialism in this precise example, but I do not think it invalidates the general approach).

In my view, looking for a structure for a theory in physics should imply that :
  • central concepts should be present even if with a mathematical form
  • obviously distinct concepts should be distinct kinds of mathematical objects
  • non physical information (like any computational devices) should never be present as a necessary core component of the structure
But here is the kind of things we generally encounter:
  • arbitrary coordinates everywhere : space coordinates that refers to arbitrary directions and arbitrary center, the so-called canonical coordinates on the phase space that are absolutely not canonical, tensors-as-arrays, vectors-as-tuples, matrices, Christoffel symbols, Lie groups described as matrices groups, etc
  • other relative objects : frames of reference that inject privileged viewpoints called "observers", all quantities that are dependent on the frames of reference (speed, kinetic energy, work, …) position vector/angular momentum/torque/moment of inertia which all depend on an entirely non-physical geometrical point, wavefunctions that depends on both coordinate systems and frames, etc
  • wrongly-categorised objects : vectors that should be bivectors or points, "values" that are in fact fields, fields, that should be measures, gradients that should be forms, states as Hilbert space vectors, groups instead of homogeneous spaces, etc
  • completely inconsistent (but supposedly more intuitive) objects: differentials as infinitesimal quantities, eigenvectors for unbounded operators, etc
What deeply irritates me (at best) or drives me to dispair in this kind of approach is that:
  • it results in a mix between physical information and representational artifacts. In the end what is physical and what is not becomes awefully blurry
  • many distinctions are flattened to fit primitive mathematical entities so that you end up with a gigantic list of long equations between a bunch of numerical values. ##\mathbb{R}^n## is suddenly everywhere. I even came up with a name for this process: the ##\mathbb{R}^n##-ization of physics.
I don't deny that there exists approaches that privilege a more conceptual mindset but:
  • They are generally few except if you consider higher level domains of physics. If you're just trying to have a very clear and precise understanding of more basic stuff you're stuck with the usual material (show me a sound introduction to classical thermodynamics)
  • They rarely covers all aspects (AFAIK analytical mechanics cannot handle non conservative forces)
  • They generally still contain a certain degree of arbitrariness (again analytical mechanics can be coordinate-free but is still frame-dependent)
  • It's not always shown how they are related to more widely used approaches so you have to invest energy and time to create the links (because often you have no choice)
My question is basically: are there other people here with the same mindset and how do they cope with it without giving up entirely on physics ? I tried to adapt my manner of thinking. I did tried. But it feels so much unnatural.
 
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  • #2
burakumin said:
My question is basically: are there other people here with the same mindset and how do they cope with it without giving up entirely on physics ? I tried to adapt my manner of thinking. I did tried. But it feels so much unnatural.

It seems to me that you have the same central complaint that led Michael Spivak to start a textbook series "https://www.amazon.com/dp/0914098322/?tag=pfamazon01-20", currently only Mechanics has been done. But if you haven't read it, it sounds like you will be quite happy with it.
 
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  • #3
Diaz Lilahk said:
It seems to me that you have the same central complaint that led Michael Spivak to start a textbook series "https://www.amazon.com/dp/0914098322/?tag=pfamazon01-20", currently only Mechanics has been done. But if you haven't read it, it sounds like you will be quite happy with it.

Actually I had already given a quick look over it. That may more be rigorous than most of the other books on the topic but that certainly does not avoid a lot of the pitfalls I've mentioned before: coordinates, coordinates, coordinates everywhere. And structures nowhere
 
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  • #4
burakumin said:
Actually I had already given a quick look over it. That may more be rigorous than most of the other books on the topic but that certainly does not avoid a lot of the pitfalls I've mentioned before: coordinates, coordinates, coordinates everywhere. And structures nowhere
It sounds like you want to do for physics what the Bourbaki group did for math. I honestly don't know if this has been done before, but the person who probably would know would be Spivak.
 
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  • #5
I think you can already find on certain topics more math-oriented conceptual approaches. The thing is again that those "Bourbaki-like" approaches seems to be more frequent when you're considering more complex theories. I don't deny that apparently plenty of people are more comfortable with the kind of approach I reject (because the explanations rely more on intuition and a so-called "common sense" and the computational aspects are only requiring simple mathematical knowledge) I'm just fed up it is so pervasive for a huge part of physics. It just looks like everyone should think the same manner. Adapt or die.

Note that I don't necessarily want something as terse and austere as Bourbaki. I just consider that on the continuous line between austere-and-too-abstract VS informal-and-overconcerned-by-computation there is a huge bias in favor of the latter.
 
  • #6
Apparently nobody seems to be in the same situation as me :frown: My post was moved in the educational forum but I didn't intend to focus on the teaching aspect but rather on a more epistemological "what does it mean to understand physics" question.

Nevertheless, as I suppose there are many teachers around, if I can ask an additional related question: Do you try to mix several approaches when presenting physics to your audience? Have you already met students which were obviously not on the same page as you (and with who communication and teaching was problematic) because they were using different perspective, mindset, metaphors, representations than you to understand and explain the same physical notions?
 
  • #7
burakumin said:
<snip>Nevertheless, as I suppose there are many teachers around, if I can ask an additional related question: Do you try to mix several approaches when presenting physics to your audience?

Yes- that's good teaching practice.

burakumin said:
Have you already met students which were obviously not on the same page as you (and with who communication and teaching was problematic) because they were using different perspective, mindset, metaphors, representations than you to understand and explain the same physical notions?

Yes, because introductory physics is required for many non-physics majors.
 
  • #8
Andy Resnick said:
Yes, because introductory physics is required for many non-physics majors.

Sure but my question was more about students of physics themselves.
 
  • #9
burakumin said:
Sure but my question was more about students of physics themselves.

What is a 'student of physics'?
 
  • #10
Andy Resnick said:
What is a 'student of physics'?

I meant someone majoring in physics.
 
  • #11
If I understand your frustration correctly, your main objection is the introduction of specific frames of reference and coordinates. But aren't these constructs necessary in order to do any useful calculations? The vast majority of students learning physics will not go on to become theoretical physicists who plumb the depths of our description of nature. Virtually all of them, if they use physics at all, will use it to do specific calculations in applied fields like engineering. So the techniques necessary to do calculations in real setting are an important part of learning physics. For those few who are interested in the foundational aspects of theoretical physics, the material is out there for them to study. Misner, Thorne, and Wheeler do an excellent job of discussing GR in a coordinate-free manner. I personally like the approach taken by Doran and Lasenby in "Geometric Algebra for Physicists".
 
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  • #12
phyzguy said:
your main objection is the introduction of specific frames of reference and coordinates. But aren't these constructs necessary in order to do any useful calculations?

No my main objection is what I consider as a real obsession for them and for calculation at the detriment of conceptual understanding. I don't pretend that calculation tools are useless or should not be taught, I say they could come second after a coordinate-free formal description (you can include in "coordinates" anything that can be considered as non-physical).

phyzguy said:
The vast majority of students learning physics will not go on to become theoretical physicists who plumb the depths of our description of nature. Virtually all of them, if they use physics at all, will use it to do specific calculations in applied fields like engineering.

Maybe I don't belong to the (supposed) "vast majority of students learning physics". I don't plan to become a theoretical physicist but still this approach has slowed me down or stuck me in confusion. What's the use of learning how to use a computation tool when you don't understand what the computation semantically represents?

phyzguy said:
For those few who are interested in the foundational aspects of theoretical physics, the material is out there for them to study. Misner, Thorne, and Wheeler do an excellent job of discussing GR in a coordinate-free manner. I personally like the approach taken by Doran and Lasenby in "Geometric Algebra for Physicists".

Again I don't deny they exists but again look: you're already talking about GR. What about this kind of approach for classical thermodynamics, newtownian mechanics (including rigid body kinematics, fluid mechanics, continuum mechanics), optics, classical electromagnetic theory ... ? Additionally your remark implies that "those few who are interested in the foundational aspects of theoretical physics" are "those whose want go deeper", presupposing they have already understood the usual coordinate-based approach. What if some of them got stuck because of it?
 
  • #13
After all this, I STILL don't have an understanding what the problem here is. Can someone give me a simple, coordinate-free explanation?

@burakumin : Have you applied the approach that you want and tried to teach someone else using your preferred methodology? Has it worked consistently over the various topics that you described? Have you also considered if you are the "select few" or the only one who had trouble with the "traditional" methods?

Zz.
 
  • #14
burakumin said:
What if some of them got stuck because of it?

They'll get stuck in a coordinate-free approach as well.
It is much harder to understand when you can't picture something.

As far as I can tell the only way to describe classical (i.e. Newtonian) mechanics in a coordinate-free way is to look at manifolds and other difficult stuff.
And without previous knowledge in the coordinate approach I wouldn't even understand what we were trying to do.
Coordinate freedom comes at the cost that you need more knowledge of the deep mathematical nature of the theory.
Look e.g. at Arnolds book on classical mechanics, there's a reason it is part of the "graduate texts in mathematics" series
 
  • #15
ZapperZ said:
After all this, I STILL don't have an understanding what the problem here is. Can someone give me a simple, coordinate-free explanation?

Sorry I must be too stupid to make myself clear.

ZapperZ said:
Have you also considered if you are the "select few" or the only one who had trouble with the "traditional" methods?

Clarifying this was exactly the purpose of my questions:

are there other people here with the same mindset and how do they cope with it without giving up entirely on physics ?

Have you already met students which were obviously not on the same page as you (and with who communication and teaching was problematic) because they were using different perspective, mindset, metaphors, representations than you to understand and explain the same physical notions?

JorisL said:
They'll get stuck in a coordinate-free approach as well.

I don't. I must be a counter-example.

JorisL said:
It is much harder to understand when you can't picture something.

I don't understand this remark. Coordinates are precisely what kills all geometric insights with me, they are precisely what prevent me to picture anything.
 
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  • #16
burakumin said:
Coordinates are precisely what kills all geometric insights with me, they are precisely what prevent me to picture anything.

Can you give an example? Because I don't see how this works.
I'm interested though.
 
  • #17
I personally find that coordinate-free notation can sometimes be useful for calculation, and that coordinate-free notation often is useful conceptually.

Even math grad students, however, have trouble doing calculations in a coordinate-free manner. From Zee's book "Einstein Gravity in a Nutshell":
Zee said:
I am certainly not against coordinate-free notations ... Coordinate-free notations are great for proving general theorems, but not so good for calculating ...chatting at lunch with two leading young researchers ... During grad school, to deepen his understanding of Einstein gravity, he enrolled in a course taught by a famous mathematician. As it happened, he was the only student able to do the problems in the final exam involving actual calculations: he did them by first using old-fashioned indices and then translating back into the abstract notation used in the course.

This was a physics student enrolled in a pure math grad course. :biggrin:
 
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  • #18
burakumin said:
I meant someone majoring in physics.

Oh, ok. Then my first answer remains the same (use several pedagogical approaches) but my second answer is now 'no, not really', because by the time I get my hands on physics majors, which is second semester junior year and beyond, they are largely assimilated. To be sure, different students learn differently and are comfortable with different approaches, but as a whole they are familiar with the same subset of metaphors, representations, etc.
 
  • #19
As a follow-up to what Andy said, I'm going to be tacky and refer to something I wrote elsewhere quite a while ago, and which has some relevancy here.

I think the OP needs to be aware that we ALL have our own ways of learning things. Part of being a student, and part of going through an academic program, is for one to figure out for oneself how one learns, or what method is the most effective for one to acquire a knowledge or an understanding efficiently. We all had to go through this on our own terms. And this is why I claim that learning how to learn is the most important thing that I learned about being a physicist.

Zz.
 
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  • #20
Hey burakumin.

What you are trying to ask is whether a different language exists (with certain properties/constraints) to describe physics.

I guess what you should ask yourself is based on your understanding of physics (which is something you will have to elaborate on), what are the differences between how you would represent information to define physics and how the existing "body" of knowledge represents information.

You mention things like co-ordinate systems and mechanics and your "dis-satisfaction" of how things are organized (which is the purpose of language) but perhaps you could outline what sort of language you think would be appropriate to use in place of what already exists and why you think it would be more appropriate.

I'm sure if you did that you'd get a lot more specific/directed "responses" for your question - particularly by those with significant education/training in the "field".
 
  • #21
Sorry to have been so long to answer.

ZapperZ said:
I'm talking about knowing WHERE to look, WHO to ask, WHAT do I need to do to understand that, HOW do *I* understand something? We all work in different ways. Knowing how I, personally, comprehend something is very important, because I have consciously tried to discover when I can make something click in my head, and when it can't.

So in the end it seems we totally agree. It's just that precisely the traditional manner of explaining physics seldom "makes something click in my head". And the fact that many teachers apparently follow Resnik's path does not help me :
Andy Resnick said:
my second answer is now 'no, not really', […]. To be sure, different students learn differently and are comfortable with different approaches, but as a whole they are familiar with the same subset of metaphors, representations, etc.

Now about these questions:
JorisL said:
Can you give an example? Because I don't see how this works.
I'm interested though.
chiro said:
I guess what you should ask yourself is based on your understanding of physics (which is something you will have to elaborate on), what are the differences between how you would represent information to define physics and how the existing "body" of knowledge represents information.

When for example I'm asked to think about the point at the top of the pyramid of Cheops I can perfectly imagine it without any notion of coordinate. Now if I start to refer to this location only by ##(x, y, z)## I'm loosing precious information on the actual structure I'm actually considering. And ##\mathbb{R}^3## can be many many things: a commutative group, a vector space, an homogeneous space, an affine space, a (commutative or Lie) algebra, a topological space, a metric space, a hilbert space, a (topological, differential, riemmanian or pseudo-riemmanian) manifold, an ordered set, a measured space, a probability space, etc etc. Nobody can tell me that taking the euclidean norm of ##(x, y, z) ## is mathematically unsound. But does that norm tell me anything about the pyramid? I can compare ##(x, y, z) ## to any other triple using the lexicographic order. Does this comparison have the least physical importance ? If I'm talking about a circle everybody knows what I'm talking about and everybody must understand that in this context the notion of distance must make sense otherwise the concept of circle is meaningless. Even if the argument may seem contrieved technically speaking one cannot be sure that the numerical equation ##x^2 + y^2 = 1 ## represents a circle in absense of any additional information. I can give you a coordinate system where this describes a square. Mathematics can encompass this kind of differences because in maths not everything has to be a number.

chiro said:
but perhaps you could outline what sort of language you think would be appropriate to use in place of what already exists and why you think it would be more appropriate.

Example of the kind of approaches I like:
By opposition to these ones:

Now like said before there are many other fields where it's very difficult to find references with the approach I expect.
 
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  • #22
If you are talking about information not being a number then you have to describe it's structure and algebra.

The most basic structures in mathematics are numbers and sets and if you have another structure you should mention it.

Co-ordinate systems are just bases in mathematics meaning that they organize information in a very particular constrained way.

Numbers order information in a particular way consistent with arithmetic and vector/geometric algebra. If you don't want numbers then you need to specify the states, how they are organized, the algebras used on them and why you would prefer an alternative structure as opposed to what exists within science/engineering.

If you can do that then what you're trying to say would make a lot more sense.
 
  • #23
chiro said:
If you are talking about information not being a number then you have to describe it's structure and algebra.

The most basic structures in mathematics are numbers and sets and if you have another structure you should mention it.

Co-ordinate systems are just bases in mathematics meaning that they organize information in a very particular constrained way.

Numbers order information in a particular way consistent with arithmetic and vector/geometric algebra.

Sorry Chiro but I don't understand the purpose of your answer. I already know what are structures and coordinates.

chiro said:
If you don't want numbers then you need to specify the states, how they are organized, the algebras used on them and why you would prefer an alternative structure as opposed to what exists within science/engineering.

If you can do that then what you're trying to say would make a lot more sense.

Haven't I already given examples of that? Have you seen the references I added?
 
  • #24
burakumin said:
<snip>

Example of the kind of approaches I like:
By opposition to these ones:

Now like said before there are many other fields where it's very difficult to find references with the approach I expect.

Ah! Now I think I understand what you mean. I'm not a mathematician, so my language may be wrong, but I think you are asking why functional analysis/algebraic topology is not used by most pedagogical materials. Yes?
 
  • #25
Andy Resnick said:
Ah! Now I think I understand what you mean. I'm not a mathematician, so my language may be wrong, but I think you are asking why functional analysis/algebraic topology is not used by most pedagogical materials. Yes?

General topology, differential geometry, general algebraic structures (monoids, groups, rings, fields, algebras, modules, vector spaces, ...), order theory, measure theory, set theory, graph theory, etc, etc. Plenty of already existing concepts could be used in physics. Another concrete example for @chiro is this thread.

To avoid any misunderstanding, I'm pretty aware that some may answer "because we don't want to wait for students to learn too much maths before starting to teach physics". But again I don't pretend that this kind of approach should always be used and for everybody. I would just like that there also exists this kind of alternative approach for many subfields of physics. And often it doesn't.
 
  • #26
burakumin said:
<snip> I would just like that there also exists this kind of alternative approach for many subfields of physics. And often it doesn't.

On the contrary, there is:

General introduction- https://www.amazon.com/dp/1107602602/?tag=pfamazon01-20
Fluid Mechanics- https://www.amazon.com/dp/0199679215/?tag=pfamazon01-20 (2 volumes)
Electromagnetism and optics- http://www.worldscientific.com/worldscibooks/10.1142/9251

These are just the few I have within reach. I'm sure if you search arXiv you can find monographs as well.

As to your question, why is Physics not generally taught that way, my unsatisfactory answer is that I'm not familiar enough with the math to plausibly teach the material. I can learn it (and I have been learning the material, after a fashion), but I'm not currently prepared to teach that way.

You would (or at least should) definitely appreciate this one: http://www.springer.com/us/book/9780387989921
 
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  • #27
Andy Resnick said:
As to your question, why is Physics not generally taught that way, my unsatisfactory answer is that I'm not familiar enough with the math to plausibly teach the material.
Right, in order to teach such a course effectively, the instructor needs to be well-versed in both the physics and the math. I would wager that relatively few physicists have the deep math background that would be necessary.

Also, you need students who are prepared to take such a course. At least here in the US, I think relatively few physics students (either undergraduate or graduate) would have a suitable math background. How many math students at that level would also want to learn the physics?
 
  • #28
What I was trying to ask is what sorts of structures and languages you were intending to use to convey the body of knowledge for science (in this case physics).

The thing is that without these elements, no one is going to understand what you really mean.
 
  • #29
I too don't really understand what exactly the OP is about. Since you are concerned with the teaching of physical concepts, maybe it would help to go into detail about a really basic example. Imagine you are in high school and want to teach a motivated class some physics. Can you give a tangible outline of what you would do? Which topic do you choose and how do you present it? Then we can contrast your ideas much better with more mainstream approaches.

Taking advances theories like GR as example doesn't make sense to me unless you think it would be best to start with the most advanced theory and derive the more basic theories which describe our everyday experience as special cases. But then again I would like to see an outline of how you would start teaching people who are not familiar with physics yet, so that there's a tangible thing to discuss.
 
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  • #30
I think the OP just had bad introductory physics. I still remember my introduction to vectors, probably around 9th grade - "a vector is a magnitude and a direction" - which is the definition of the geometric quantity, not the coordinates. Also, the Feynman lectures stresses the difference between real physical quantities like a step of certain size in a certain direction which is invariant, compared to the set of three numbers which depend on the reference frame.
 
  • #31
Andy Resnick said:

Thanks for the references Andy. Some of them look interesting. Unfortunately it is difficult to evaluate them as they are not free. Furthermore it seems that the first reference, while apparently very nice on the mathematical aspect, does not deal with classical mechanics but directly with einsteinian relativity and QM, what I won't call an introduction to physics.
Andy Resnick said:
These are just the few I have within reach. I'm sure if you search arXiv you can find monographs as well.

I'll try to deepen my research in that directions.

kith said:
Imagine you are in high school and want to teach a motivated class some physics. Can you give a tangible outline of what you would do? Which topic do you choose and how do you present it? [...] I would like to see an outline of how you would start teaching people who are not familiar with physics yet, so that there's a tangible thing to discuss.

First let's make it clear that my question was moved in the teacher forums but my intention was more to focus on an epistemological rather than an strictly educational problem (what sort of language do/could/should we use to explain and understand physics). Anyway that's ok if it remains here. Second I'm not a physics teacher, I don't plan to become one and as it is probable we received education in different countries I'm not sure considering "what and how I would like to teach to high school students" would provide a relevant point of comparison. And last as I already said it I'm not trying to impose a better approach for teaching physics to the layman. I'm only asking for more numerous and more frequent alternative approaches.

chiro said:
What I was trying to ask is what sorts of structures and languages you were intending to use to convey the body of knowledge for science (in this case physics).
The thing is that without these elements, no one is going to understand what you really mean.

kith said:
I too don't really understand what exactly the OP is about. Since you are concerned with the teaching of physical concepts, maybe it would help to go into detail about a really basic example. But then again I would like to see an outline of how you would start teaching people who are not familiar with physics yet, so that there's a tangible thing to discuss.

I had already mentioned this thread. Is this already too complex? Classical galilean physics can use it. To re-explain here I was suggesting that the physical concept of extensive property could be formally defined as a measure and could include many vector/tensor-valued quantities like momentum, angular momentum, inertia tensor, etc by opposition to an intensive property that could be defined as a field. To me this gives a very clear categorization of how quantities behave with respect to aggregation of systems. While fields are defined point by point, measures are defined locally and additively. Measures also naturally encompass discrete and continuous variants of the same notion. For example a mass measure can contain discrete masses, standard densities, linear mass densities and area mass densities at the same time. This categorization also naturally triggers the question of quantities that are neither extensive nor intensive and what could be their mathematical nature. This is an example of language and question that help me understand physics.
 
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  • #32
You're going to have to define it in terms of numeric structures or sets.

The description you have given is way too vague and you will need to specify the constraints mathematically - again based on set structures or those synthesized by ordinary numbers (including complex numbers).

If you can't do this then it means you don't actually know what you are saying and trying to do - the only way to know what you are doing is to be precise about it.
 
  • #33
chiro said:
You're going to have to define it in terms of numeric structures or sets.
What is "it" in this sentence?
chiro said:
The description you have given is way too vague and you will need to specify the constraints mathematically - again based on set structures or those synthesized by ordinary numbers (including complex numbers).
What does mean "to specify the constraints mathematically" here? From my point of view, you're the one a bit too vague. My purpose was not present an entire mathematical framework in this single post, just to give an idea of the language I'd like to find more often. But let's put it this way: let ##\mathcal{A}## a 3-dimensional euclidean affine space (3 dimensions are in fact not very satisfying: a 4D affine space would be better even for newtownian mechanics or even a 4D affine manifold, but let's put that point aside in this post). Its natural separated topology generates a single Borel ##\sigma##-algebra ##\sigma_\mathcal{A}##. A massic object in that space could be described as a finite measure ##m##. We can equip it with a vector field ##v## (its speed) whose support is included in the support of ##m## and that takes values in ##V##, the vector space associated to ##\mathcal{A}##. We could of course discuss the regularity of ##v##. I generally consider that a measurable piecewice-analytical function is enough to describe all situation of Newtonian dynamics I know about but I would be delighted to be contradicted. Now you can define the integral/product:
$$p = v \cdot m = A \in \sigma_\mathcal{A} \mapsto \int_A v \cdot \textrm{d}m \in V$$
which is a vector-valued measure called momentum of our system...
Etc.

Is this still too vague for you?
 
  • #34
It means you define the structures, algebras, relations, constraints and anything that reduces the uncertainty of what it is that you are trying to say.

A set has a very good definition which is why many mathematicians use them.

A number (from naturals to reals) also has a good definition.

Structures are in terms of sets and numbers with things synthesized from those quantities.

Sets have the set algebra (union and intersection) while numbers are based on arithmetic (including modulus) and things synthesized from these (like exponentiation as an example).

Forget all the high level stuff for the moment and get this stuff sorted first.

If you can produce all the information needed to represent what you are meant to convey then it will be easy to understand you.

Don't bother about buzz-words at the moment - use the language in mathematics as it exists and allow us to combine it so that we get an idea of what you are trying to say so that we can differentiate it from what already exists and then comment on the difference.

At the moment, you are too vague because you are not using the terms that are largely standardized. Start small if you need to and avoid buzz-words.

Also - realize symbols have no meaning unless they map to specific states of information. When you deal with mathematics you deal with constraint under consistency. You are going to have to highlight the constraint and how you make it consistent.

Finally - when you talk about spaces you should be telling us how you organize the information differently from what exists and why.

In physics things are organized with continuity and geometry because that is how humans sense this information in the real world. It doesn't have to be organized this way but it makes sense with visual and natural intuition being suitable. If you have an alternative then specify how you would organize it, why you would do so, what the advantages are with this new organization is and how you would make it all consistent and analytical.

The states will always be the same regardless of the approach since they map to exactly the same things - but the organization will be different and from what you are saying the thing you disagree with is how the information is organized and yet you haven't addressed these questions at all in any significant capacity.
 
  • #35
burakumin said:
Furthermore it seems that the first reference, while apparently very nice on the mathematical aspect, does not deal with classical mechanics but directly with einsteinian relativity and QM, what I won't call an introduction to physics.

That book ("Geometry of Physics") not an introduction to physics, it's an introduction to using algebraic geometry in physics.
 

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