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Thanks, very informative. > The "real" numbers, "complex" numbers and quaternions are all obtained as the results of some division operations applied to certai

Thanks, very informative. > The "real" numbers, "complex" numbers and quaternions are all obtained as the results of some division operations applied to certain kinds of vector pairs. Affine spaces (i.e. spaces of points) can be defined axiomatically, vector spaces can be defined based on affine spaces (a vector is the difference between a pair of points) and scalars are the quotients of pairs of collinear vectors. "Complex" numbers and quaternions appear naturally as quotients of certain kinds of non-collinear vectors. With a proper set of axioms for affine spaces all the properties of the derived notions, i.e. vectors, scalars (i.e. "real" numbers), "complex" numbers and quaternions result automatically. Do you have any source detailing this approach to recommend?

adrian_b 2y ago
Unfortunately I am not aware of any source that would present a coherent, consistent and complete theory of the physical quantities. The theories of the vector spaces, real numbers, complex numbers, quaternions and tensors are byproducts of the complete theory of the physical quantities. In 1873, Maxwell has written a rather complete theory of the physical quantities for that time, but his exposition is obviously obsolete now. Later authors of physics manuals have only very seldom taken care to present a logically consistent axiomatic description of the mathematical models assumed by their physics theories, while the authors of mathematics texts have preferred to present systems of axioms like they would have been the result of random choices, even if the real origin of most useful axioms has been in the work of creating mathematical models for the physical behavior. All the required axioms and theorems are dispersed in various texts of mathematics and physics. I do not remember now a concrete list of titles, but some texts about the axioms of geometric algebras and about the axioms of affine spaces may contain useful information and the most important are some texts that explain which are the minimal sets of axioms that must be satisfied by a measurable physical quantity (i.e. the set of its values must have an algebraic structure of Archimedean group, which means that it must be possible to add and compare the values) and how the sets of axioms of more complex algebraic structures, like vector spaces a.k.a. linear spaces can be actually derived from the minimum set of axioms (for any Archimedean group you can define a division operation that yields a rational number, and through a process of passing to the limit you can define the "real" numbers [passing to the limit requires a completitude axiom for the Archimedean group], thus obtaining the field of scalars associated with the original Archimedean group, which will then have a structure of vector space over the field of scalars).
- mbivert 2y ago
  Many thanks, it's more than enough for me to dig around. > while the authors of mathematics texts have preferred to present systems of axioms like they would have been the result of random choices This is a frustrating situation, and a source of confusion in my experience, for example when we jump from the high-school "column of numbers" definition of vectors to the "element of a vector space" one. Or, how confusing the axioms for topologies are.
- vinnyvichy 2y ago
  Please pardon the non-accidental facetiousness (because you and I may have a surprisingly common set of interests),-- But: you can first extend the boundary of what is meant by "physical quantity" to the Non-Archimedean https://en.wikipedia.org/wiki/Ultrametric_space#Applications https://en.wikipedia.org/wiki/Ultrametric_space#Applications and then think about going even further https://en.wikipedia.org/wiki/Self-averaging#Non_self-averaging_systems https://en.wikipedia.org/wiki/Self-averaging#Non_self-averag...

adrian_b 2y ago

Unfortunately I am not aware of any source that would present a coherent, consistent and complete theory of the physical quantities. The theories of the vector spaces, real numbers, complex numbers, quaternions and tensors are byproducts of the complete theory of the physical quantities. In 1873, Maxwell has written a rather complete theory of the physical quantities for that time, but his exposition is obviously obsolete now. Later authors of physics manuals have only very seldom taken care to present a logically consistent axiomatic description of the mathematical models assumed by their physics theories, while the authors of mathematics texts have preferred to present systems of axioms like they would have been the result of random choices, even if the real origin of most useful axioms has been in the work of creating mathematical models for the physical behavior. All the required axioms and theorems are dispersed in various texts of mathematics and physics. I do not remember now a concrete list of titles, but some texts about the axioms of geometric algebras and about the axioms of affine spaces may contain useful information and the most important are some texts that explain which are the minimal sets of axioms that must be satisfied by a measurable physical quantity (i.e. the set of its values must have an algebraic structure of Archimedean group, which means that it must be possible to add and compare the values) and how the sets of axioms of more complex algebraic structures, like vector spaces a.k.a. linear spaces can be actually derived from the minimum set of axioms (for any Archimedean group you can define a division operation that yields a rational number, and through a process of passing to the limit you can define the "real" numbers [passing to the limit requires a completitude axiom for the Archimedean group], thus obtaining the field of scalars associated with the original Archimedean group, which will then have a structure of vector space over the field of scalars).

mbivert 2y ago
Many thanks, it's more than enough for me to dig around. > while the authors of mathematics texts have preferred to present systems of axioms like they would have been the result of random choices This is a frustrating situation, and a source of confusion in my experience, for example when we jump from the high-school "column of numbers" definition of vectors to the "element of a vector space" one. Or, how confusing the axioms for topologies are.
vinnyvichy 2y ago
Please pardon the non-accidental facetiousness (because you and I may have a surprisingly common set of interests),-- But: you can first extend the boundary of what is meant by "physical quantity" to the Non-Archimedean https://en.wikipedia.org/wiki/Ultrametric_space#Applications https://en.wikipedia.org/wiki/Ultrametric_space#Applications and then think about going even further https://en.wikipedia.org/wiki/Self-averaging#Non_self-averaging_systems https://en.wikipedia.org/wiki/Self-averaging#Non_self-averag...