I think I finally get it – why my mathematical colleagues who teach service courses for science students don’t like to include units. For years I have been trying to convince my mathematician friends to be more careful with units.
Occasionally, I would run into an irritating example. Once I walked into the lecture hall before my class to set up a demo. On the board, left over from a calculus class in the previous hour, was an integral of “rho + 1” over an area of the plane. Next to it, the lecturer had written “rho = mass density.” If “rho” is a mass density, you can’t add a pure number to it. It’s like saying “I stayed for a certain amount of time – and, oh yes, for one more.” One more what? Hour? Minute? Second? You have to specify units, and then you wouldn’t write just the number “1”.
I’ve written about a second example  that I found on a calculus exam in a class for scientists.
The population density of trout in a stream is
where r is measured in trout per mile and x is measured in miles. x runs from 0 to 10.
(a) Write an expression for the total number of trout in the stream. Do not compute it.
If “x is measured in miles” then the “1” in the numerator of the fraction is a distance (1 mile) and the “1” in the denominator of the fraction is an area (1 square mile). Using the same symbol for two different kinds of quantities is not good scientific practice. We do it sometimes – consider all the different things that k can mean in physics – but to do it with just numbers is particularly bad -- and having two different "1"s seems particularly egregious.
For a number of years now, I’ve thought that the reason that mathematicians didn’t want to use units was because they wanted to be consistent about the level of the mathematics they used. Unit checks are in a sense “advanced” mathematics corresponding to group theory
– a course many science students never take and that mathematicians and mathematical physicists typically only take as advanced students. The idea of group theory is to classify how mathematical structures change when something else is changed. For example, what happens to geometrical objects if you look at them in a mirror? Do they stay the same? If not, can the result be rotated so it’s the same as the original? Another example is, “What happens if you rotate your coordinate axes about the origin?” This analysis of this last explains why we use vectors and what we mean by a vector. And it’s group theory that underlies the properties of angular momentum in quantum mechanics, and therefore is responsible for the structure of the periodic table.
The reason that unit checks have to do with group theory is because units are about how a quantity changes when the standard that is used to assign a number to a physical quantity changes. If I measure a length with the unit of inches, what I mean is that I see how many times the standard size of “inch” fits into my length. I get some number. If I measure it with centimeters (a different and smaller standard size), I get a different (and a bigger) number. The same thing happens with area, but the number assigned to length increases by a factor of 2.54 while the number assigned to area increases by a factor (2.54)2 = 6.45.
So making something into “not just a number” but into a quantity that may transform into something else is dealing with a kind of quantity that is more complex than might be appropriate to talk about in an intro math class.
But thanks to a discussion I had last week (see the post “Units and stoichiometry
”), I now get that it’s really even worse than that.
What I discussed there is that when we are measuring something with units, what we are doing is mapping some aspect of the physical world into a mathematical structure – the real number line (or in the case of stoichiometry, the positive integers). What I pointed out was that we were actually modifying the math by blending it with our physical concept. That although it looks like we’re mapping something into the real number line, we limit the math that we keep. With distances, if we define a fixed origin, we can add and subtract our numbers freely, getting positives and negatives – and being able to interpret them with a conceptual correlate – something that makes sense to us and we can interpret in the world in which we live. For lengths, this works for multiplying two lengths together – or even three, as we connect that result to the mental concept of an area or a volume. We can go beyond three powers – sort of – by using metaphor and analogy, talking about multiple dimensions and hypervolumes.
With masses, we have to be more circumspect. Negative numbers are meaningless for mass and we typically don’t use them. (Modulo some issues in quantum field theory.) We sometimes multiply masses – for example when considering the gravitational force between two masses. But in most other cases where the product of two masses appears, the result can be rearranged to be a mass times a function of the ratio of masses. We don’t have a conceptual correlate for a “square mass”. (Though I wonder whether we couldn’t come up with one.) “Square times” appear in physics calculations, such as in accelerations or kinetic energy, but again, we don’t have a direct conceptual correlate.
So what I’ve realized as a result of this discussion is that we use units not just to keep track of “different kinds of numbers” as I previously thought. Rather, we use them as a warning to check for “physical reasonableness” in a way that permits some kinds of legitimate mathematical calculations and forbids others in a way that might depend on physical context. So if I’m calculating a kinetic energy, I don’t care that “time squared” occurs in the denominator. If I were calculating a time interval, I would not want to carry out a calculation that resulted in a “square time.”
The use of units is therefore subtle and requires a blending of physical and mathematical knowledge in a way that constrains mathematical manipulations that the mathematics by itself does not. This puts unit analysis outside the realm of what many mathematicians want to be doing in a math class – teaching an “honorable” description of the mathematics that remains true to the math.
I now have more sympathy for their objections so I’m willing to cut them some slack, but still insist that students need to know how to deal with units to use math in science. I have two resulting messages for the two groups:
Mathematicians – Don’t try to “paste in” units into your math classes in a casual or sloppy way. If you must do it (and I would like it if you would), be careful – and realize that you may have to get students to bring in knowledge that goes beyond math, building on their understanding of length, time, and mass from their everyday experience.
Scientists – Don’t assume that units are trivial or even simple or that the mathematicians have handled it for you in their classes. Set aside some time for a careful discussion of why you care and why it’s important – and use units carefully in your lectures and lecture notes.
I have found that even when I am Draconian about units in my teaching – taking off lots of points for wrong units on exams – students still don’t believe that it’s important and that I really care -- and they seem not to understand what's going on. In my future classes, I’m certainly going to try to share this more extended discussion and justification of units with my students. I’ll let you know if it works better than what I’ve done in the past.