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> You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars.

I mostly agree with your explanation, but would like to emphasize that this is just a convention from mathematics which mostly carries over into physics and engineering. We like to define functions that are R -> R and similar, instead of defining special sets like R° = { r * 360° | r \in R }, corresponding to "real numbers with unit degrees", and then defining functions like sin: R° -> R. It’s just simpler to define and analyze most functions from R -> R and so we mostly do that.

But if you look up physics papers, it’s not uncommon to define functions that require unitful inputs as well. For example, the wave function in the Schrödinger equation maps a position r (3D vector with unit meter) and time t (scalar with unit seconds), to a probability amplitude (complex number with unit m^-3/2), so that \int |ψ(r,t)|^2 d3r becomes a scalar (a probability). Up wave function is still considered a function by all physicists.

The reason for preferring functions defined over domains like R is that it’s a field, and so I can do things like multiply and divide and add and subtract inside it.

If instead we start defining ‘amounts of distance’ as some set D and ‘amounts of time’ as some set T, I have all sorts of extra work to do to make it so that products of amounts of distance are ‘amounts of area’ and amounts of distance over amounts of time are ‘amounts of speed’.

‘Dimension’ is the mathematical tool that lets us bundle all that up into something that we can deal with separately, alongside a real number. And of course you can totally make functions that are dimensional - but it affects what you can do with your functions, like composition and differentiation.

You can apply functions to anything. That's the only thing "function" means. They transform values into other values, and there is no limit on what kind of values you might want to talk about.

I basically agree, at least for standard functions like sin, cos, tan, exp etc. It is even possible to see mistakes in equations just by checking that all the units to standard functions cancel out making the arguments dimensionless.

On the other hand I am still unhappy with calling the ratio of two quantities, that happen to have the same units, "dimensionless". This way, you could create any two "dimensionless" quantities and try to compare them or use one in place of the other which might be meaningless.

Yeah, "dimensionless" would mean they have equal dimension, which would mean they are comparable, which isn't necessarily the case. E.g. both radians and degrees are called "dimensionless".

Edit: Apparently "same dimension" doesn't imply "same unit".

> This way, you could create any two "dimensionless" quantities and try to compare them or use one in place of the other which might be meaningless.

Case in point:

https://trac.ffmpeg.org/ticket/11279

https://trac.ffmpeg.org/ticket/11284

You can also create meaningless dimensionless quantities by blindly mashing the number keys on your keyboard. Should we stop using keyboards?

This is a great paper from NIST that gets into some of the problems with the limit of ‘dimensionlessness’ in metrology and the SI, and in particular issues like the fact that Hertz is considered a coherent SI unit but radian isn’t.

https://www.nist.gov/publications/dimensionless-units-si

A key takeaway:

> one conclusion that is not optional is that the unit hertz cannot be regarded as a coherent unit of the SI, in contrast to its designation in the current form of the SI, where cycles are ignored and Hz may be replaced by s^−1

Really interesting read, thank you!

One thing I found especially interesting: They argue that you can express the (complex) exponential function exp(x) as a power series with powers x^k. They do not say it explicitly, but if we assume the power series comes from a Taylor series, then the k-th factor 1/(k!) is the derivative evaluated at x=0. And the k-th derivative has exactly the unit needed to cancel the unit of x^k. So, all summands of the series are unitless and hence the exponential function's argument is unitless.

This argument would hold for any function which can be written as a series like this. I am wondering whether this is actually a "problem" of the derivative operator.

> You generally can’t apply functions to dimensional units.

Perhaps not in mathematics, but in programming that's clearly possible. I guess programming is more general than mathematics.

This is precisely why (programming language) types are poor model of physics units, despite often being touted for this exact use case. 3m is not the same thing as "the value 3 of type meter". It is the multiplication of the dimensionless scalar 3 with the special "m" constant for meters.

That's why pow(3m, 2) = 9 m^2, and not `the value 9 of type meter`. Of course, you can define the type `square meter` as well, and define `pow -> meters -> positive integer -> square meters`. However this quickly becomes overwhelming once you start doing more complex expressions with multiple types. What is the type of `pow (3kg^2 * m/s, 3/2)`?

Edit to add: also, there is a simple fact that "sin(pi/2 kg)" is just not defined, in programming or math or physics or any other useful system. It's definitely not 1kg, just like sin ( (pi/2) * 2) is not sin (pi/2) * sin (2).

> with the special "m" constant for meters.

Isn't the "special constant" exactly "value 1, type meters", defined as equal to "value <...very large number...> type atoms" etc?

If not, then what would be the result of the multiplication of 3 with "m"?

> Of course, you can define the type `square meter` as well, and define `pow -> meters -> positive integer -> square meters`

As long as your power is an integer, you can reduce it to multiplication. So what you'd really want to define is the result of "<value1 of type meter> * <value2 of type meter>", "(<value1 of type meter> * <value2 of type meter>) * <value3 of type meter>" etc.

What this gets you in the end is a type algebra, but that is also not exactly a new concept.

> 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.

> 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.

No, it's definitely possible in mathematics, they've left out some details as to what the units are doing that makes them unable to be assigned to functions. I mean a regular ODE that you get from newtons laws is a set of functions that take position and time as inputs, which all have units. What they mean should be "dimensionless functions cannot be applied to dimensional variables". These are commonly functions like sin cos exp log and so on.

Well you can also square root etc.