[Haskell-cafe] Category Theory woes

[Nick Rudnick]

Alexander Solla wrote:
>
> On Feb 18, 2010, at 2:08 PM, Nick Rudnick wrote:
>
>> my actual posting was about rename refactoring category theory; 
>> closed/open was just presented as an example for suboptimal 
>> terminology in maths. But of course, bordered/unbordered would be 
>> extended by e.g. «partially bordered» and the same holds.
>
> And my point was that your terminology was suboptimal for just the 
> same reasons.  The difficulty of mathematics is hardly the funny names.
:-) Criticism... Criticism is good at this place... Opens up things...
>
> Perhaps you're not familiar with the development of Category theory.  
> Hans Aberg gave a brief development.  Basically, Category theory is 
> the RESULT of the refactoring you're asking about.  Category theory's 
> beginnings are found in work on differential topology (where functors 
> and higher order constructs took on a life of their own), and the 
> unification of topology, lattice theory, and universal algebra (in 
> order to ground that higher order stuff).  Distinct models and notions 
> of computation were unified, using arrows and objects. 
>
> Now, you could have a legitimate gripe about current category theory 
> terminology.  But I am not so sure.  We can "simplify" lots of 
> things.  Morphisms can become arrows or functions.  Auto- can become 
> "self-".  "Homo-" can become "same-".  Functors can become "Category 
> arrows".  Does it help?  You tell me.
I think I understand what you mean and completely agree...

The project in my imagination is different, I read on...
>
> But if we're ever going to do anything interesting with Category 
> theory, we're going to have to go into the realm of dealing with SOME 
> kind of algebra.  We need examples, and the mathematically tractable 
> ones have names like "group", "monoid", "ring", "field", 
> "sigma-algebras", "lattices", "logics", "topologies", "geometries".  
> They are arbitrary names, grounded in history.  Any other choice is 
> just as arbitrary, if not more so.  The closest thing algebras have to 
> a unique name is their signature -- basically their axiomatization -- 
> or a long descriptive name in terms of arbitrary names and adjectives 
> ("the Cartesian product of a Cartesian closed category and a groupoid 
> with groupoid addition induced by....").  The case for Pareto 
> efficiency is here:  is changing the name of these kinds of structures 
> wholesale a win for efficiency?  The answer is "no".  Everybody would 
> have to learn the new, arbitrary names, instead of just some people 
> having to learn the old arbitrary names.
Ok...
>
> Let's compare this to the "monad fallacy".  It is said every beginner 
> Haskell programmer write a monad tutorial, and often falls into the 
> "monad fallacy" of thinking that there is only one interpretation for 
> monadism.  Monads are relatively straightforward.  Their power comes 
> from the fact that many different kinds of things are "monadic" -- 
> sequencing, state, function application.  What name should we use for 
> monads instead?  Which interpretation must we favor, despite the fact 
> that others will find it counter-intuitive?  Or should we choose to 
> not favor one, and just pick a new arbitrary name?
The short answer: If the work I imagine would be done by exchanging here 
a word and there on the quick -- it would be again maths style, with 
difference only in justifying it with naivity instead of resignation.

The idea I have is different: DEEP CONTEMPLATION stands in the 
beginning, gathering the constructive criticism of the sharpest minds 
possible, hard discussions and debates full of temperament -- all of 
this already rewarding in itself. The participants are united in the 
spirit to create a masterpiece, and to explore details in depths for 
which time was missing before. It could be great fun for everybody to 
improve one's deep intuition of category theory.

This book might be comparable to a programming language, hypertext like 
a wikibook and maybe in development forever. It will have an appendix 
(or later a special mode) with a translation of all new termini into the 
original ones.

I do believe deeply that this is possible. By all criticism on Bourbaki 
-- I was among the generation of pupils taught set theory in elementary 
school; looking back, I regard it as a rewarding effort. Why should 
category theory not be able to achieve the same, maybe with other means 
than plastic chips?

All the best,

    Nick