- #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

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 :

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**

- 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

- 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.

- 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)

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