Andrej Bauer on Intuitionistic Logic and Constructive Mathematics

13:42 A modality is a Monad on Propositions (and the theory of a monad is decidable! See Oracle modalities by Andrew W Swan and Andrej Bauer and Ronald Brown On Monads and Groupoids and Alexander Grothendieck on his idea for a Science Fiction Novel on Motives).

To understand this idea you need to be able to understand Kleene's notion of Realizability and Hyland's Effective Topos which depends heavily upon Tripos Theory, a tripos here is "a topos-representing, indexed, pre-ordered set".  Kleene's attempts to interpret intuitionistic logic were all more than a little weird, I suspect he had dubious motives. See Markov's principle and Realizability.  Intuitionistic logic is in a sense strictly more expressive than classical logic. See Emily Riehl on Univalent Foundations.

Subscribe to Types and Topology Workshop.

Dr. Martin Escardo on Searchable sets and ordinals in system T given at a six month 2012 programme called Semantics and syntax: a legacy of Alan Turing:


Subscribe to INI Seminar Room 1.

I have been finding more and more interesting things people have written on coalgebraic types and totality:

See Mark Jago on Type Theory in Computer Science, Logic and Linguistics.

Comments

Post a Comment

Popular posts from this blog

How Could One Unify CMU and MIT

Image
I mean, how to get the best of the two different philosophies of computation. One is based on typed programming languages and the other on engineering with diagrams . This is about practical computing and the cost and feasibility of software development in general. Here is the problem: We have a lot of algorithms which can all be described abstractly using some sort of pseudocode, or perhaps using some particular language (usually Python!). These algorithms are often well-studied and a lot is known about them in terms of their computational complexity in time and space. Substantive practical software systems invariably employ many such algorithms, often implemented in libraries with more or less well-specified APIs. But very few of these libraries are capable of interoperating because they are either packages written in some specific programming language like Java or Haskell, say, or they are written in C and used as object code, or they are written in an interpreted language like Sc...
Read more

Tensor Fields and Simplicial Complexes

Image
Listening to Freya Holmér last night I started to get glimmers of an idea I had long ago about how to represent vector spaces in computational processes using this recursive abstract type :  abstype 'a point = POINT   of {getx :  'a vector,       diff :  'a point -> 'a point,       move :  'a point -> 'a point,       scale : 'a -> 'a point,       proj :  'a point -> 'a} with    fun new i (op +) (op -) (op * ) dot =    let fun self x =  POINT       {getx = x,        move = fn (POINT pr) =>           (self (x + (#getx pr))),        diff = fn (POINT pr) =>          self (x - (#getx pr)),        scale = fn i =>          (self (x * i)),        proj = fn (POINT pr) =>     ...
Read more

HTML in Blog Posts

Just testing stuff: SVG You can click on the blue circle: MathML There is not much you can do with this. See Mathematical Markup Language 1.01 Specification   7.1.5 Mixing and Linking MathML and HTML . It's the big problem  of how you compose languages.   a x 2 + b x + c = 0 You can click on the discriminant: x = − b ± b 2 − 4 ⁢ a ⁢ c 2 ⁢ a 2D Canvas Sound  Beep! WebGPU Next level Parser expression grammar compiler:  https://peggyjs.org/online.html Devine Lu Linvega's unxtal assembler/debugger:  https://wiki.xxi...
Read more