https://medium.com/@marinbenc/whats-a-turing-machine-and-why-does-it-matter-1cd1b4606c6a

You don’t always think about mathematicians building stuff like engineers do, but the truth is mathematicians build things all the time. It’s just that the things they build exist mostly in their minds (and published papers). They’re not concrete things you can look and feel. They do this to prove theorems or properties. Turing did just that when he “built” the Turing machine.

To set the scene, imagine it’s around 1928 and no programmable computer was invented yet. But mathematicians did study algorithms and computable things. Since computers were so limited, the only way to figure out universal properties of all programs was to think really hard.