I seem to have unusually low fondness for dimensional analysis, and feel it is overhyped. Some bits are useful, like having units at all, or a couple type signature like ideas—except the ones I find the most useful are ‘officially’ dimensionless! To me, ‘radians’ are a unit (which isn’t the same as length!), log(x) has the unit of ‘log-[X units]‘, etc. Z-scores are sorta unitless, but only in the way that is “X metres”, X is unitless. The version for Z-scores is, of course, “SDs of [X units]”, which is in part its own unit (but with implicit casts to [X units]). As an example, if someone is +x SDs in variable X, and variable Y correlates with correlation r with X, then the mean estimate for their Y value is r*x SDs in Y. Diopters and wavevectors both are officially in inverse length units, yet to me wavevectors actually have the units of radian(phase)/meter—and indeed I think of optical power separately from spatial frequency. I don’t usually use the unit of “moles”, but rather the unit of “moles of X”. While I am sure my way of thinking is not original, books and teachers wouldn’t use these sorts of ‘units’ except occasionally at the college level, by professors who informally reason in a similar way—a way that does not get into the course material! For some reason people think that “dimensions” must refer to “physical quantities” as if radians and decibels are not a true physical quantity.
For mathematics, basically the only useful units are those ‘officially dimensionless ones’. Thinking in terms of types, however, is extremely useful—I wouldn’t understand differential geometry otherwise. As an example, a covector field on manifold M basically has type M → (((M → R) → R) → R). Being passingly familiar with basic category theory lets you draw diagrams of various concepts, where often the main value add of the diagram is just “this function has this type, and is a composition of these other functions, and equal to this other composition of functions”.
If someone merely says that it’s important to track what each variable in a physical equation represents, then I will of course agree with them. But some of the claims dimensional analysis advocates make in e.g. physics just seem like bullshit. “Ah, you see, you can deduce that this equation is wrong because of the unit mismatch—and see, over here there should be a proportionality factor dependent of material properties!” seems to be a terrible way to reason because of how “just add a factor whose value is 1 and that has units that make it work” literally happens sometimes.
To me it is also a bad way to think of proportionality factors—in reality whatever reasoning you used to get the equation should be obviously something that is merely proportional, or alternatively you are adding a ‘fudge factor’ meant to account of deviations from your model (which you are implicitly assuming to not vary much), or you are literally taking a taylor series whose coefficients you plan to fit to data—and how could you forget the coefficients there, when you clearly don’t know the derivatives? Likewise, I do not think “figure out how it should scale, as a fermi estimate” falls under the same category—that part is great, but has almost nothing to do with units or dimensions, and everything to do with being good at making simple approximate models.
The Buckingham-pi theorem is clearly cool. Yet, I have not seen someone actually use it, instead of retroactively using it. And even just taking the exampre from the Wikipedia page of the pendulum: I find it worrying that the analysis doesn’t alert you that it should fail for large angles. As another famous example, I don’t think it provides a good explanation of the Reynolds number: it’s much better to think of comparing inertial effects to viscous ones. In practice to come up with dimensionless characterizing quantities I find it easier to just divide each term in an equation by a ‘scale value’, basically taking the scale as my units. See for example the van der Waals equation, which is usefully written with variables taken relative to the critical point.
Importantly, the reason why those work, and are ways that people actually reason, is because they are about physics. They require thinking about the physical system, and give you analyses of it—and thus you can notice things like “ah, the angle has to be small”, can understand which dimensionless units have important physical interpretations, and not worrying about typecasts that shouldn’t be worried about.
I seem to have unusually low fondness for dimensional analysis, and feel it is overhyped. Some bits are useful, like having units at all, or a couple type signature like ideas—except the ones I find the most useful are ‘officially’ dimensionless! To me, ‘radians’ are a unit (which isn’t the same as length!), log(x) has the unit of ‘log-[X units]‘, etc. Z-scores are sorta unitless, but only in the way that is “X metres”, X is unitless. The version for Z-scores is, of course, “SDs of [X units]”, which is in part its own unit (but with implicit casts to [X units]). As an example, if someone is +x SDs in variable X, and variable Y correlates with correlation r with X, then the mean estimate for their Y value is r*x SDs in Y. Diopters and wavevectors both are officially in inverse length units, yet to me wavevectors actually have the units of radian(phase)/meter—and indeed I think of optical power separately from spatial frequency. I don’t usually use the unit of “moles”, but rather the unit of “moles of X”. While I am sure my way of thinking is not original, books and teachers wouldn’t use these sorts of ‘units’ except occasionally at the college level, by professors who informally reason in a similar way—a way that does not get into the course material! For some reason people think that “dimensions” must refer to “physical quantities” as if radians and decibels are not a true physical quantity.
For mathematics, basically the only useful units are those ‘officially dimensionless ones’. Thinking in terms of types, however, is extremely useful—I wouldn’t understand differential geometry otherwise. As an example, a covector field on manifold M basically has type M → (((M → R) → R) → R). Being passingly familiar with basic category theory lets you draw diagrams of various concepts, where often the main value add of the diagram is just “this function has this type, and is a composition of these other functions, and equal to this other composition of functions”.
If someone merely says that it’s important to track what each variable in a physical equation represents, then I will of course agree with them. But some of the claims dimensional analysis advocates make in e.g. physics just seem like bullshit. “Ah, you see, you can deduce that this equation is wrong because of the unit mismatch—and see, over here there should be a proportionality factor dependent of material properties!” seems to be a terrible way to reason because of how “just add a factor whose value is 1 and that has units that make it work” literally happens sometimes.
To me it is also a bad way to think of proportionality factors—in reality whatever reasoning you used to get the equation should be obviously something that is merely proportional, or alternatively you are adding a ‘fudge factor’ meant to account of deviations from your model (which you are implicitly assuming to not vary much), or you are literally taking a taylor series whose coefficients you plan to fit to data—and how could you forget the coefficients there, when you clearly don’t know the derivatives? Likewise, I do not think “figure out how it should scale, as a fermi estimate” falls under the same category—that part is great, but has almost nothing to do with units or dimensions, and everything to do with being good at making simple approximate models.
The Buckingham-pi theorem is clearly cool. Yet, I have not seen someone actually use it, instead of retroactively using it. And even just taking the exampre from the Wikipedia page of the pendulum: I find it worrying that the analysis doesn’t alert you that it should fail for large angles. As another famous example, I don’t think it provides a good explanation of the Reynolds number: it’s much better to think of comparing inertial effects to viscous ones. In practice to come up with dimensionless characterizing quantities I find it easier to just divide each term in an equation by a ‘scale value’, basically taking the scale as my units. See for example the van der Waals equation, which is usefully written with variables taken relative to the critical point.
Importantly, the reason why those work, and are ways that people actually reason, is because they are about physics. They require thinking about the physical system, and give you analyses of it—and thus you can notice things like “ah, the angle has to be small”, can understand which dimensionless units have important physical interpretations, and not worrying about typecasts that shouldn’t be worried about.