In homotopy type theory, homotopy types can be viewed as logical types and it is possible to prove some theorems about them without using any underlying space (no simplicial set, no topological space). It is a kind of synthetic algebraic topology. Is it just a coincidence that the word "type" may have these two meanings ? What I mean is: what had in mind the one who invented this terminology of "homotopy types" ? And do we even know where that terminology of "homotopy types" comes from. I guess that the answer will be that it is a coincidence but since it is written nowhere in the book, I ask the question in order to be sure.



A (very naive) question about Homotopy Type Theory
Maybe I should repost my comment above as a genuine answer:


It's a coincidence, but a very fortunate one (for a change). 


* When people started saying "[homotopy type](http://ncatlab.org/nlab/show/homotopy%20type)" $X$ they meant "the type of $X$" as in "what kind of space is X?".
* When people said "[type](http://ncatlab.org/nlab/show/type)" $X$ in logic ([type theory](http://ncatlab.org/nlab/show/type%20theory)), they meant the type of its [terms](http://ncatlab.org/nlab/show/term), as in "what kind of thing is an $x \in X$?".


In a naive sense, these two usages of "type" are not actually the same. Remarkably, as one digs deeper and understands that [intensional type theory](http://ncatlab.org/nlab/show/intensional%20type%20theory) is essentially [homtopy type theory](http://ncatlab.org/nlab/show/homotopy%20type%20theory), it turns out by lucky coincidence that at the bottom of it indeed this is the same notion of "type" in both cases. 


I think its one of those trivialities that deserve to be regarded as "deep".


