'Homotopy Type Theory for Philosophers'
Wed 1st February 2017 16:15
Room 104, Edgecliffe
Professor James Ladyman
Homotopy Type Theory for Philosophers
James Ladyman, University of Bristol
Homotopy Type Theory (HoTT) is a research programme in mathematics that connects algebraic topology with logic, computer science, and category theory. Its name derives from the way it integrates homotopy theory (which concerns spaces, points and paths) and formal type theory (as pioneered by Russell, Church, and Goedel, and developed in computer science) by interpreting types as spaces and terms of them as points in those spaces. Much of the interest in HoTT is due to the fact that it may be regarded as a `programming language for mathematics', and it is formulated in a way that facilitates automated computer proof checking. HoTT is a `foundation for mathematics' in the sense of a framework or language for mathematical practice. Within the language of HoTT it is possible to characterise mathematical structures such as natural numbers, real numbers, and groups, and to formalise proofs in homotopy theory.
However, philosophers often mean something stronger by `foundation for mathematics'. They require a foundation to provide not just a language but also a conceptual and epistemological basis for mathematics, and moreover one that can be formulated without relying upon any other existing foundation. HoTT is interesting for philosophers because of its putative foundational status in this sense. However, there are many other philosophically rich aspects to HoTT. One is the way identity is treated in the theory, which is what makes the theory in one sense 'intensional'. Another is that the Univalence Axiom in HoTT arguably (and according to Steve Awodey) expresses mathematical structuralism. Further matters of interest are that the theory is constructive and the way this interacts with its other features. Finally, type theory in general, and dependent types in particular provide a different formal approach to functions and quantification, and a different way of dealing with predication.
In this talk I will present the basic ideas of HoTT paying particular attention to its novel conceptual foundations, and discussing a selection of philosophical issues in lesser and greater depth. I will draw upon joint work with Stuart Presnell on our three year Leverhulme Project on HoTT. I will assume no prior knowledge and will aim at accessibility, but I will also be sure to provide some technical details for the logicians.
A free to download introduction to HoTT is available here:
• Ladyman, J. and Presnell, S. (2015) ‘Identity in Homotopy Type Theory, Part I: The Justification of Path Induction’, Philosophia Mathematica 23 (3): 386-406.
• Ladyman, J. and Presnell, S. (2014) ‘A Primer on Homotopy Type Theory: Part I’
• Ladyman, J. and Presnell, S. (forthcoming) ‘Does Homotopy Type Theory Provide a Foundation for Mathematics?’, to appear in The British Journal for the Philosophy of Science
• Ladyman, J. and Presnell, S. ‘Identity in Homotopy Type Theory Part 2’, to appear in Philosophia Mathematica
• Ladyman, J. and Presnell, S. ‘Universes and Univalence’, to appear in Review of Symbolic Logic
• Ladyman, J. and Presnell, S. (under review), ‘Symmetry and Representation in Physics’, Philosophy of Science
• Ladyman J. and Presnell, S. (in preparation), ‘The Hole Argument in Homotopy Type Theory’
Philosophy Club event
Index of archived news
February 2017 | January 2017 | December 2016 | November 2016 | October 2016 | September 2016 | August 2016 | July 2016 | June 2016 | May 2016 | April 2016 | March 2016 | February 2016 | January 2016 | December 2015 | November 2015 | October 2015 | September 2015 | June 2015 | May 2015 | April 2015 | March 2015 | February 2015 | January 2015 | December 2014 | November 2014 | October 2014 | September 2014 | July 2014 | June 2014 | May 2014 | April 2014 | March 2014 | February 2014 | January 2014 | December 2013 | November 2013 | October 2013 | September 2013 | July 2013 | June 2013 | May 2013 | April 2013 | March 2013 | February 2013 | January 2013 | December 2012 | November 2012 | October 2012 | September 2012 | July 2012 | June 2012 | May 2012 | April 2012 | March 2012 | February 2012 | January 2012 | December 2011 | November 2011 | October 2011 | September 2011 | July 2011 | June 2011 | May 2011 | April 2011 | March 2011 | January 2011 | December 2010 | November 2010 | October 2010 | September 2010 | August 2010 | July 2010 | June 2010 | May 2010 | April 2010 | March 2010 | February 2010 | January 2010 | December 2009 | November 2009 | October 2009 | September 2009 | July 2009 | June 2009 | May 2009 | April 2009 | March 2009 | February 2009 | January 2009 | December 2008 | November 2008 | October 2008 | September 2008 | May 2008