2 ms·

> If not, then what would be the result of the multiplication of 3 with "m"? The answer is not, and the result of 3 multiplied by m is 3m. Just like 3 multipli

> If not, then what would be the result of the multiplication of 3 with "m"? The answer is not, and the result of 3 multiplied by m is 3m. Just like 3 multiplied by pi is 3pi; or, perhaps more accurately, you can view m as a kind of vector unit, and 3m as the scalar product. Of course, none of this is exactly matching - dimensions are different from irrationals, vectors, complex numbers, etc, they are mostly a thing of their own. > What this gets you in the end is a type algebra, but that is also not exactly a new concept. Sure, that's why I said specifically programming language types. I am aware that type theory has way more complex operations on types. I think some of these may even be expressible in Idris or Haskell + some appropriate extension. But in almost all programming languages, even ones like OCaml, SML, plain Haskell, Rust, C++ with template magic, Scala, F# and what have you, there is no way to specify that the result of multiplying two values of type A is of type "A * A", especially not in a way that then allows you specify that the division of a value of type "A * A" by A has type A. So types as exposed in any of the common programming languages are horrible for modelling dimensions as used in even high school physics.

jameshart 2mo ago
> dimensions are different from irrationals, vectors, complex numbers, etc, they are mostly a thing of their own. I harbor a terrible internal mental model of dimensions which I have never really validated or explored fully, where I like to think they might be vector exponents, or something vaguely similar. If we assign each dimension to be a dimension of a vector - (length, mass, time, etc…) then a ‘distance’ might be e^((1,0,0,…)); a ‘duration’ e^((0,0,1,…)). These have the requisite properties that when we multiply and divide them, we end up adding and subtracting these vectors. So a distance times a distance is e^((2,0,0,…)) and a distance over a duration (a speed) is e^((1,0,-1,…)) They have the right basic algebraic behavior but who knows what terrible consequences they would have.

jameshart 2mo ago

> dimensions are different from irrationals, vectors, complex numbers, etc, they are mostly a thing of their own. I harbor a terrible internal mental model of dimensions which I have never really validated or explored fully, where I like to think they might be vector exponents, or something vaguely similar. If we assign each dimension to be a dimension of a vector - (length, mass, time, etc…) then a ‘distance’ might be e^((1,0,0,…)); a ‘duration’ e^((0,0,1,…)). These have the requisite properties that when we multiply and divide them, we end up adding and subtracting these vectors. So a distance times a distance is e^((2,0,0,…)) and a distance over a duration (a speed) is e^((1,0,-1,…)) They have the right basic algebraic behavior but who knows what terrible consequences they would have.