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While in general I agree with this paper, there are various details in its arguments that I consider wrong or incomplete. For instance, he lists 3 main domains

While in general I agree with this paper, there are various details in its arguments that I consider wrong or incomplete. For instance, he lists 3 main domains of applications that have driven the development of various branches of mathematics. While one of the application domains has indeed started as hydrodynamics (later also as aerodynamics), despite the remaining importance of fluid dynamics, I believe that nowadays the most important application in this domain is the simulation of semiconductor devices, which is required in the development of new manufacturing processes, and which certainly dwarfs even the simulations of military watercraft or aircraft. Another example is that he claims that quaternions cannot be obtained in a straightforward way as a generalization of "real" numbers or "complex" numbers. He is right about this only because he attempts to derive such numbers in the reverse direction in comparison with how they must be derived logically and how they have been discovered historically. Unfortunately this wrong direction of derivation from some arbitrary axioms of "real" numbers is contained in almost all modern mathematics manuals. Both logically and historically, vectors are the primary concept and scalars are a secondary concept that is derived from vectors. 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. What are now called "real numbers", have been called for millennia "measures", where a scalar, i.e. a real number, was explicitly derived from a pair of physical quantities, e.g. lengths, by dividing a quantity to be measured to a measurement standard unit. In the past the fact that the scalars are derived from vectors and not vice-versa was obvious. (When "real" numbers were called "measures", the term "number" was restricted to integers and rational numbers.) The modern teaching method where a set of axioms for the "real" numbers is chosen arbitrarily and other mathematical objects are derived by arbitrary rules from the "real" numbers obscures the meaning of the "real" numbers and the historical development of mathematics. The "real" numbers are not the result of arbitrary choices, but they result automatically from attempting to model the properties of the space-time, which is the primitive concept. While the "real" numbers and all related continuous quantities are derived from the space-time, the "numbers" in the old sense, i.e. the integers and the rational numbers and other related discrete quantities are derived from the finite sets.

mbivert 2y ago
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

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).