A very simple question took me down the depths of mathematics: Why can’t we add two temperatures like 30 °C and 10 °C and get 40 °C?

Of course temperatures in Kelvin can be added, and of course temperature differences can be added. There is a little used scientific convention denoting temperature differences as e.g. C° while absolute temperatures are °C. This difference in notation, and the very significance of the Kelvin scale, presages some fairly fundamental underlying issues which we will explore in this article. (Or rather, we will rediscover what made physicists choose this distinction in the first place). But… wait… how are temperature differences obtained again?! Yep… we have the weird situation of the Celcius (and Fahrenheit) scale allowing subtraction while not allowing addition.

Importantly, its not even the only such physical quantity. absolute time is also unable to added – I’ll challenge you to add 1^st March 2023 and 3^rd July 2025 and tell me the result! Absolute electric potential (which gives rise to voltage) also cannot be added – absolute electric potential simply cannot be assigned a unique numeric value (although the potential at infinity is commonly fixed to be zero, its not really a privileged choice).

Why exactly do we have this problem? Before we dive in, be aware that “why?” is not always an answerable question. In fact, taken too far, it leads to the problem of the infinite epistemic regress. With that caveat, lets go however far we can.

What do we mean by temperature first of all? Apparently the degree of hotness or coldness (I didn’t like this definition in middle school and I still don’t. Edit: Parameter defining equilibrium distribution of energies in statistically independent particles and parameter characterising objects in thermal equilibrium feel much better.). It is written with a number and a unit, in this case Celcius but would work with any scale. We imagine we could put any number in there, and that typically means we use real numbers (in this case with the restriction that T ≥ −273.16°C), and hence it looks like you could apply all arithmetic operations to it. But clearly addition and also multiplication aren’t fitting in for temperature (if you recall that multiplication can be viewed as repeated addition).

No physics envy here

It turns out that the most famous theoretical justification comes not from physics or mathematics, but from psychology (with statistics having a claim to it). And although a few formidable mathematicians such as Tukey (a co-inventor of the extremely consquential Fast Fourier Transform and also apparently the origin of the word “bit” and “software”!) have contributed to the field, I am not alone in feeling like there’s been a conspicuous lack of mathematical attention (to put it mildly).

The contribution from psychology are the “scales of measurement” first by Stevens. He divides measurements into nominal scale (just names like colour), ordinal scale (names with ranks like college grades), interval scale (numbers with difference but no ratios), and ratio scale (number with a true zero and hence all arithmetic operations are allowed, edit: also called an absolute scale). If you think he was secretly an algebraist LARPing as a psychologist I’m with you, dear reader.

Stevens’ classification answers our original question saying that the lack of a true-zero disallows the meaning of “temperature is a quantity of something” from the Celcius scale, which is thus an interval scale. Since temperature is not quite a quantity of heat, adding them doesn’t give us a real quantity of any kind. The same applies to time (because the Gregorian calendar isn’t counting time from the Big Bang), and electric potential (because the physical theory simply doesn’t allow a real zero).

And – this will be interesting some of you who aren’t into the natural sciences (unnatural scientists?) – pointers to memory in languages such as C or C++ also can’t be added. However there is a much subtler and more interesting mathematics that underlies pointer arithmetic, which I will be covering in one or more followups to this article. Stay tuned!

Postscript: Measurement theory

Stevens’ contributions attracted only a little attention from natural scientists for a long time. Tukey’s additions frankly don’t feel like a substantial upgrade to this framework. Chrisman’s further expanded framework too doesn’t add much, with a very important exception for cyclic quantities like months or times within a day which were sorely needed (more in the upcoming articles in this series!). Only recently has this field gotten the rigour it should have, now called Measurement Theory, and is best accessed via a good review, given that it doesn’t have a wiki article yet.