Will Homotopy Type Theory Soon Replace ZFC as the New Foundation of Mathematics?

Any fundamental shift in foundational thinking will take time. I still remember my first department head who had a background in computer science basics advising, 'Curriculum changes one death at a time.' I believe this adage holds true for ZFC - Homotopy Type Theory (HoTT) or whatever follows it.

Adopting HoTT in Modern Mathematics

I have been closely following and utilizing Homotopy Type Theory (HoTT) as much as I can, even employing techniques from Idris to work within the limitations of language such as F#. It's important to note that F# does not support HoTT directly, but I try to work around these limitations where possible. However, it's crucial to be cautious with statements, as the context of meaning is significant. For the most part, ZFC will remain a fundamental cornerstone, but HoTT promises to be a more expressive foundation, and we may yet see even richer foundations in the future.

HoTT and Computability

The recent aim of HoTT is to enhance computability. Having sets or setoids as basic objects in HoTT brings us closer to our goals. The invention of Cubical Type Theory (CTT) has further solidified HoTT by making the Univalence Axiom a theorem rather than an axiom. Univalence is a vital concept toward proving function extensionality in languages like Coq, Agda, and Idris.

However, HoTT is not without its challenges. For instance, CTT does not yet handle Higher Inductive Types as seamlessly as desired. This highlights that HoTT as a project is still unfinished. HoTT is a promising field, but its current state is not yet fully optimized. There are ongoing experiments in Quantitative Type Theory aimed at reducing resource usage by proving unnecessary resources can be omitted.

The Future of HoTT

Like Haskell, which was once seen as an academic experiment in type theory, HoTT is still in its academic phase. The aim is to refine and optimize the system. Every 'crisis' leads to a richer mathematical foundation. This may not be a stressful crisis in the traditional sense, but it represents a paradigm shift moving us toward answering questions we didn't even know we were asking.

Conclusion

The transition from ZFC to HoTT is a slow and ongoing process. Both theories have their strengths, and the journey toward a new foundation is full of challenges and opportunities. As mathematicians and researchers continue to work on these systems, we can expect HoTT to become an increasingly significant player in the foundational landscape of mathematics.