According to the BIPM and Wikipedia, "amount of substance" (as measured in moles) is one of the base quantities in our system of weights and measures. Why?
I get why the mole is useful as a unit. In fact, my question isn't really about the mole at all; I just mention it because what little information I could find generally talked about moles, not about "amount of substance". Nor am I asking about why it's chosen as a base quantity and not a derived quantity. I get that any particular choice of bases is more or less arbitrary.
I don't understand why it's a dimensional quantity at all. It is, after all, just a count of things; every student is taught to think of it as "like 'a dozen', only more sciencey". Can't we just call it a dimensionless number?
No, says SI; molar mass doesn't just have dimensions of $\mathsf{M}$, it has dimensions of $\mathsf{M}\cdot\mathsf{N}^{-1}$; and Avogadro's number isn't just a number, it's got units of "per mole" (or dimensions of $\mathsf{N}^{-1}$).
Contrast this with an "actual" dimensionless quantity, plane angle (and its unit the radian). Now, you might say that it's dimensionless because radians are defined as arc length over radius, and so plane angle is just $\mathsf{L}\cdot\mathsf{L}^{-1}$; cancel out and you have no dimensions. That strikes me as arbitrary. We could just as easily argue that arc length is "really" a quantity of $\mathsf{a}\cdot \mathsf{L}$ (where $\mathsf{a}$ is plane angle), because it's the measurement of a quantity that subtends $\mathsf{a}$ at distance $\mathsf{L}$.
But this isn't needed; plane angle isn't even a derived quantity, it's a non-quantity. Plane angle is accepted as dimensionless. Why isn't amount of substance?
As I said, I've found very little on this question. From the Wikipedia article on the mole, I found a PDF of an interesting IUPAC article on atomic weight. It acknowledges the argument (as does the Wikipedia article), but dismisses it out of hand by saying (essentially) "of course counting things is a way of measuring things, so of course we need a unit of measurement for it".
Other than that, Wikipedia (as far as I can tell) touches on eliminating the mole only in the context of eliminating other units (as for example in natural systems of units). The Unified Code for Units of Measure blithely cuts moles from the base units as being "just a count of things", but doesn't go into why SI says it is necessary.
Is there any official rationale for the inclusion of "amount of substance" as a dimension? Failing that, can anyone provide, or point me to, some good reasons why it's so special?
EDIT: Thank you all for your input. The more I've thought about it, the more I've come to feel that there's no reason why "count of stuff" shouldn't be a dimension (it's clearly different from, say, a dimensionless number included as a scale factor), and that my unease with the idea comes from simple habit: in any case not involving moles, it tends to get left out. Really, I'm now more wondering why angles are considered dimensionless...
Reading before coming here:
Answer
So, here's the thing. The chemistry that underlies molar mass ratios dates back at least to 1805. We've known that if you divide by a certain "relative mass" number you can get whole-number ratios for atoms in a pile of stuff, for that long. It took us about 60 more years to get a handle on how large atoms were with the estimations of Loschmidt, who worked out that atoms are much smaller than the wavelengths of visible light -- too small to ever "see". This gave a rough count of how many atoms there were in a confined space, too -- but we weren't able to connect these two different quantities (atomic relative masses, count of atoms) together to figure out the mass of a single atom until some work done by Einstein on diffusion in Brownian motion (1905) and some concrete numbers could finally be rolled in with Millikan's oil-drop experiment (1910).
So due to history and convenience, the chemists are basically at the level of saying, "okay, we have N grams of this stuff, our mass spectrometer says that it's M grams per mole, so we've got N/M moles, that includes N/M moles of nitrogen and 15 N/M moles of hydrogen due to the known atomic composition, ..." and so on. You never have to add the uncertainty in Avogadro's number to these calculations; the "size" of a mole isn't important. It's only important when you start to want to know things that are "beyond" historical chemistry approaches, like counting actual numbers of atoms.
With all that said, you'll be heart-warmed to know that there is a unit revision being considered by the SI organization, and one of the proposals is to fix the number of atoms in 1 mole. But of course they will still use as a guideline that "1 mole of carbon-12 has exactly 12 grams of mass"; it will just transition from what is now "exactly" to what will be "almost exactly."
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