I have started reading a book on Measurement theory, so I’m going to try to write a little bit about the little bit I understand of it.

What is Measurement Theory

In general, measurement theory deals with relating physical things and mathematical things meaningfully. That is, it is a theory of taking a physical relational structure and, via “measurement”, attaching a corresponding mathematical relational structure.

The book I am reading focuses on isomorphisms of structures exclusively. This in general bothers me, because a lot of the examples are isomorphisms of physical things with real continua without justification of why the physical stuff is actually continuous and not just dense. However, I am assured that similar results can be obtained with homomorphisms in place of isomorphisms, allowing for the math to come out substantially more messy.

Why should we care about it?

My personal take on this goes back to my view on conventionality in general. A theory such as this can be useful in explaining the details of something like¹ the correspondence rules of Reichenbach or the conventionality of Schlick with respect to the connections between mathematics and physics.

Somewhat separately, measurement theory can illuminate why we can² bring to bear totally unrelated mathematical tools to a physical problem — perhaps from quite disparate domains of mathematics — while still getting a reasonable result at the end of the calculation or transformation.

There are surely other reasons, but I am only at the beginning of the text that is introducing me to the subject. I will write more as I get further along.

¹ Read: perhaps a caricature of the ideas of

² Or, perhaps more interestingly, why we can’t in some situations.