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Required fields*

What is it to write something "formally"?

Note: this question seems apt for flagging as a duplicate, but I haven't found a duplicate of it yet. Apologies in advance if I haven't done my full due diligence regarding this matter.

Now, I was reading Homotopy Type Theory: Univalent Foundations of Mathematics, and after hundreds of pages of dense symbolism,^§ of presumably (I presumed!) formalized text, this passage appears therein (in an appendix):

... in this book we have developed mathematics in univalent foundations without explicitly referring to a formal system of homotopy type theory.

The authors said much earlier (near the beginning) of the text that:

Univalent foundations is closely tied to the idea of a foundation of mathematics that can be implemented in a computer proof assistant. Although such a formalization is not part of this book, much of the material presented here was actually done first in the fully formalized setting inside a proof assistant, and only later “unformalized” to arrive at the presentation you find before you — a remarkable inversion of the usual state of affairs in formalized mathematics.

So now I'm not sure what "formalization" means. I thought it meant writing things out in the way that so many things are written out in that book. Per that book, the authors seem to identify formalization in part with a translation into a computer proof system, like COQ. But didn't we talk of formalization before modern digital computing and proof systems? Is formalization a vague property of a system? But if formalization is meant to dampen ambiguity, and if ambiguity is the handmaiden of vagueness, well...?

^§For a not-entirely-representative sample:

Answer*

You are correct in pointing out that using the term "formalisation" to
refer to "implementing in a computer proof assistant" is a novel use
of the term.  But such use is increasingly common.  At any rate there
have been *several levels* to "formalisation" even before computer
proof assistants came on the scene.  "Formalisation" does not merely
involve "writing everything in formulas".

For example, a typical calculus textbook certainly looks very
formalized in the sense of being full of formulas, but it typically
does not delve into a set-theoretic grounding of the subject.  Such
foundation is assumed as the background but is usually not made explicit,
analogously to the procedure in the book you cited (though
"formalisation" does not mean the same thing in that case).

To take a more advanced example of "several levels of formalisation",
John L. Bell develops Smooth Infinitesimal Analysis in a thoroughly
formalized fashion in his books, but the foundations of the subject in
topos/category terms is only summarized briefly in an appendix.  Such
foundations are treated in more detail in the books by Anders Kock and
others.

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