In the HoTT book issue 460 a question by gluttonousGrandma (where do people get these nicknames?) once more exposed a common misunderstanding that we tried to explain in section 5.8 of the book (many thanks to Bas Spitters for putting the book into Google Books so now we can link to particular pages). Apparently the following belief is widely spread, and I admit to holding it a couple of years ago:
An inductive type contains exactly those elements that we obtain by repeatedly using the constructors.
If you believe the above statement you should keep reading. I am going to convince you that the statement is unfounded, or that at the very least it is preventing you from understanding type theory.
Recently I reviewed a paper in which most proofs were done in a proof assistant. Yes, the machine guaranteed that the proofs were correct, but I still had to make sure that the authors correctly formulated their definitions and theorems, that the code did not contain hidden assumptions, that there were no unfinished proofs, and so on.
In a typical situation an author submits a paper accompanied with some source code which contains the formalized parts of the work. Sometimes the code is enclosed with the paper, and sometimes it is available for download somewhere. It is easy to ignore the code! The journal finds it difficult to archive the code, the editor naturally focuses on the paper itself, the reviewer trusts the authors and the proof assistant, and the authors are tempted not to mention dirty little secrets about their code. If the proponents of formalized mathematics want to avert a disaster that could destroy their efforts in a single blow, they must adopt a set of rules that will ensure high standards. There is much more to trusting a piece of formalized mathematics than just running it through a proof checker.
The HoTT book is finished!
Since spring, and even before that, I have participated in a great collaborative effort on writing a book on Homotopy Type Theory. It is finally finished and ready for public consumption. You can get the book freely at http://homotopytypetheory.org/book/. Mike Shulman has written about the contents of the book, so I am not going to repeat that here. Instead, I would like to comment on the socio-technological aspects of making the book, and in particular about what we learned from open-source community about collaborative research.
Mathematicians are often confused about the meaning of variables. I hear them say “a free variable is implicitly universally quantified”, by which they mean that it is ok to equate a formula $\phi$ with a free variable $x$ with its universal closure $\forall x \,.\, \phi$. I am addressing this post to those who share this opinion.
I spent a week trying to implement higher-order pattern unification. I looked at couple of PhD dissertations, talked to lots of smart people, and failed because the substitutions were just getting in the way all the time. So today we are going to bite the bullet and implement de Bruijn indices and explicit substitutions.
The code is available on Github in the repository andrejbauer/tt (the