Why isn't heat capacity defined as $(\frac{d\tau}{dQ})_V$ instead? In my thermo notes, $C_V$ is defined in words to be the heat needed to change the temperature of the system. Intuitively, this definition "adds heat" and "measures change in temperature." Thus, according to this logic, we should define $C_V = (\frac{d\tau}{dQ})_V$, but instead the notes proceed to define it the opposite way, as $C_V = (\frac{dQ}{d\tau})_V$, instead. Why is this? How should I intuitively think about heat capacity?
 A: So as a practical matter, if you look up a specific heat, you will get a number in units of $\text{J}/(\text{kg}\cdot\text{K})$ and you multiply that number by the mass of the substance that you want to heat up and the temperature change that you want to create, and it tells you how much energy you need to inject to do that.
You are looking at this as a pure mathematician, and that is not wrong. I have a similar thing where everyone calls dispersion relations $\omega(k)$ but I find this very opaque and prefer $v_\text{p}(\lambda)$. For us, we just have to learn the convention and how to translate to more familiar terms.
Indeed with phase changes you will frequently get a bunch of different internal energies that map to the same temperature as liquid water freezes or ice thaws at a specific temperature, and there’s no “function” there in the first place.
However when we do this the conventional way, the heat capacity becomes an extrinsic property of the object and is connected more directly to a conservation equation, so presumably that is why the community has found it more meaningful in the past.
If you prefer mathematical beauty, let me submit that the Maxwell relations between heat capacities are particularly pretty and handy in a pinch.
A: One way to think about it is that heat capacity is the amount of heat energy required to raise the temperature of a system by a certain amount. In other words, it measures how much heat you need to add to a system to increase its temperature. So the equation become as we know it:
(Cv =(dQ/dt)V).
In this we are actually measuring the change in temprature when a certain amount of heat is given because temprature is more fundamental quantity.
But if  CV= (dT/dQ)V then we would be essentially calculating the Change in heat resulting from the change in temprature.
A: This goes back to the chemist Joseph Black who introduced the concept of specific and latent heats and measured those for a variety of substances. His views of the matter were based on the so-called "caloric theory of heat" in which "heat" is conceived as an indestructible fluid that can be moved from place to another, so the characteristic quantity he measured was the ability of matter to contain that fluid. In fact, the caloric theory of heat conflated what we now call energy and entropy and try to make it a single thing.
The subsequent experiments of Rumford, Mayer, Colding, Joule showed the untenability of such view but Black's concepts of specific heat and latent heat are still needed and useful if it is augmented with the later development by Carnot, Kelvin, Clausius and Gibbs.
