How to express temperatures greater or less than

How do you correctly describe a 'minus' temperature when comparing to another 'minus' temperature? For example, is -20C '>' or '<' -15C? The temperature is warmer, so therefore, is it greater than -20?

Although there are contexts when it feels natural to say greater even though what we mean is greater in magnitude (i.e. greater in absolute value), everyday thermometry is not one of those contexts, and the usual mathematical convention applies.

We would say that -20 °C is lower than -15 °C or, equivalently, less than -15 °C.

                                                                   (-20 °C < -15 °C)

Conversely, -15 °C is higher than -20 °C or, equivalently, greater than -20 °C.

                                                                   (-15 °C > -20 °C)

Here are some examples:

Heterogeneous nucleation is most likely at temperatures less than -10 C (especially around -15 C)… (US National Weather Service)

Temperatures less than -30 °C (-22 °F) can occur, but typically only once in five years. (Current Results)

Wind chill temperatures less than -20F are considered low, temperatures less than -40F are extreme and temperatures less than -60F are considered dangerous. (Unisys Weather)

Their results indicated that minimum temperature was probably the factor limiting mistletoe distribution. Temperatures less than -39 °C were lethal to A. americanum, and temperatures less than -29 °C were lethal to A. douglasii. (Dwarf Mistletoes: Biology, Pathology, and Systematics)

The damage to blueberries, though, was not nearly as bad as he expected going into the season because of temperatures greater than minus-20 degrees Fahrenheit. (Farm World)

Concentrated orange juice shipped at temperatures greater than minus 18 degree Celsius would be classified as “not frozen” orange juice as opposed to “frozen.” (US Department of Argiculture)

Butter shall not be held under long storage at temperatures greater than minus 10° Celsius. (Dairy Industry Regulations 1977, Western Australia)

Discussion

As I said, there indeed are contexts when it feels natural to say greater even though what we mean is greater in magnitude. In such contexts, saying less while referring to negative numbers can cause confusion. One example: suppose we want to say that in order for an electrical spark to occur, a particular object must carry enough negative charge. If we say a spark will occur if the total charge is less than -3 C, this sounds confusing. To prevent confusion, what we may say instead is this: a spark occurs when the magnitude of the negative charge exceeds 3 C.

But for such confusion to be possible, it must be meaningful to talk about the magnitude of a quantity that can be either positive or negative.

Thermometry is not one of those contexts.

On the one hand, if one is using the Celsius or Fahrenheit scale, then the magnitude of temperature is not meaningful. (Some would say that these scales are examples of 'interval scales', although that 'typology of measurement' is controversial.)

On the other hand, if one is using the absolute temperature scale (i.e. Kelvins or Rankines), then the magnitude of temperature is meaningful (according to the aforementioned typology of measurement, these are 'ratio scales'). But in this case there can be no confusion, because the absolute temperature normally cannot be negative (but see the Appendix).

Appendix: negative absolute temperature

There do exist some special situations (which can only occur when there is an upper bound on the total amount of energy that the system can have, and when one can actually put into the system an amount of energy comparable to that upper bound) in which it is useful to define negative absolute temperature. In such cases the potential for confusion is even greater than usual, because of the peculiar way how negative temperatures 'work'. To prevent confusion, it is better to think about a quantity proportional to the inverse absolute temperature. This quantity (sometimes called the coldness) is denoted β and is, from the standpoint of fundamental physics, more meaningful than the absolute temperature itself. In particular, it is always correct to say that the lower the β, the warmer the object, even as β turns negative. Basically, β measures relative occupations of energy levels in a system: if n_A is the occupation of a level of energy E_A and n_B that of a level of energy E_B, then

                                n_A/n_B = e^{- β (E_A-E_B)}.

So for 'normal' systems, the lower-lying energy levels are more occupied than the higher-lying ones, and so β is positive (and so is the absolute temperature; the two always have the same sign). As the absolute temperature grows, β becomes smaller and smaller (while still being positive), and the higher energy levels become more and more occupied, although still less than the lower-lying ones. When β reaches zero, signifying infinite absolute temperature, all levels become equally occupied regardless of their energy. Finally, if the higher energy levels are actually more occupied than the lower ones (that's often called 'population inversion', the most famous example being lasers), then β is negative and so is the absolute temperature. The hottest temperature, corresponding to β going to minus infinity, corresponds to a situation where only the highest energy state is occupied. This is similar to how the coldest state, corresponding to zero absolute temperature (β going to positive infinity) is one where only the lowest energy state is occupied. Note that when we talk about the limit as absolute temperature goes to zero, it matters whether we approach that limit from the above (correspoding to β going to positive infinity) or from the below (correspoding to β going to negative infinity): absolute temperature scale is discontinuous at zero.

Now that we understand β, it is easier to understand the following statements: a negative absolute temperature is actually hotter than any positive one. In fact, as one increases positive temperature (β, positive but going towards zero), it is meaningful to say that one can reach positive infinity (zero β) and then go even hotter, whereby one jumps from positive to negative infinite temperature (which are one and the same temperature, both corresponding to zero β); and then as the temperature grows from negative infinity towards zero (β becoming more and more negative), correspondingly the object gets hotter and hotter. The hottest temperatures are those that are negative but very close to zero (β going to negative infinity). The moral of the story: when negative absolute temperatures are present, in order not to be driven insane, forgo the absolute temperature in favor of β.