I am trying to make my way into Homotopy Type Theory(HoTT) where a mathematician may view proofs as paths. Intuitively, this leads me to the idea of a metric on the space of mathematical propositions. Has this been developed?


Specifically, is there a way to analyse short proofs as geodesics within the space of
mathematical propositions from the perspective of HoTT? If so, might
this metric be formulated using Kolmogorov Complexity?



natural metrics for proof length
Inspired by the informal notion of *Cognitive distance*, in 2010 Charles Bennett, Peter Gács, Ming Li, Paul Vitanyí and Wojcech Zurech introduced the notion of *Information Distance* which was used in the seminal paper *Clustering by Compression*:


\begin{equation}
ID(x,y)=\min\{|p|:p(x)=y \land p(y)=x\} \tag{1}
\end{equation}


with $p$ a finite binary program for the fixed universal computer $U$ which takes the finite binary strings $x,y$ as inputs. They prove that:


\begin{equation}
ID(x,y)=E(x,y) + \mathcal{O}(\log E(x,y)) \tag{2}
\end{equation}


where the key notion:


\begin{equation}
E(x,y) = \max \{K\_U(x|y),K\_U(y|x) \} \tag{3}
\end{equation}


satisfies the criteria of a metric up to the additive term $\mathcal{O}(\log E(x,y))$.


References:
-----------


1. *Bennett, Charles H.; Gács, Péter; Li, Ming; Vitányi, Paul M. B.; Zurek, Wojciech H.*, [**Information distance**](https://doi.org/10.1109/18.681318), IEEE Trans. Inf. Theory 44, No. 4, 1407-1423 (1998). [ZBL0964.94010](https://zbmath.org/?q=an:0964.94010).
2. *Cilibrasi, Rudi; Vitányi, Paul M. B.*, [**Clustering by compression**](https://doi.org/10.1109/TIT.2005.844059), IEEE Trans. Inf. Theory 51, No. 4, 1523-1545 (2005). [ZBL1297.68097](https://zbmath.org/?q=an:1297.68097). [ArXiv:cs:0312044](https://arxiv.org/abs/cs/0312044)


