> it wasn't made during the nineteenth century

That's because people were totally focused on physics, and math was just a useful tool sometimes. Doing physics was the true goal and observation the final arbiter of truth.

Nowadays, that distinction is blurred but for the opposite reason; people think that anything conceived by sound math must be true, and observation has taken a back seat.

What does that mean? Physics is still empirical at the end of the day. Experiments decide what theories best explain the world. Math doesn't have such a requirement. It doesn't need to model natural phenomenon. Your physics lecturer sounds like a Platonist.

Well also the idea of physics as the field we currently have didn't exist much before the 17th century. Movement of bodies, astronomy, fluid dynamics, electromagnetics, optics, etc. all kind of were their own thing (if they existed at all). Fundamental developments in calculus in the late 1600s enabled these subjects to be collected under one method of study/analysis which we now call physics. As much of modern math follows from the lineage of calculus the border between the things being modeled and the tools for modeling them is naturally kind of blurry, however the distinction did still exist quite strongly throughout this entire period. Look at ex. probability or algebra, although often researchers were pursuing both physics and math, they were aware that the subjects were distinct.

> One of my physics lecturers at university made the offhand observation that the distinction between physics and mathematics is a twentieth-century idea:

It's actually a 19th century idea. The discovery or acceptance of non-euclidean geometry in the 19th century untethered math from physics or physics from math.

> and it seems to be disappearing in the twenty-first.

It can't disappear because math is no longer tied to the physical world. Math is simply theorem generation regardless of whether the axioms and theorems apply to the physical world.

The math used in physics is only a tiny subset of possible math.

Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

— V.I. Arnold: "On teaching mathematics" (1997)

Math probably split off a bit because of the attempts at formalization. That was a useful tangent though, arguably giving us computer science via the lambda calculus, Turing machines, etc.

I would argue that in the twenty-first century the distinction is even greater thanks to computer science.