Re: Re: Timelines for category theory: a response to comments

[Patrik Eklund <peklund@cs.umu.se>]

Can we do history of category theory without considering history leading 
to category theory? Is history an attachment of subhistories, or are 
there paths that can be followed, and how do we teach these things to 
young researchers? What I say is I like the '... and foundations of 
mathematics'.

I always wondered if the term 'categorical this-and-that' every once 
in a while should be considered with its counterpart 'this-and-thatical 
category theory'. It's also about meta and object languages, I believe.
And sometimes categorists 'internalize', so that the roles of meta and 
object ar blurred. Isn't a topos basically a 'logical category', where 
'categorical logic' is something else? 'Topological category' is not the 
same as 'categorical topology', and so on.

Also, 'categories in computer science' is too general. Still, most of 
categories, used in recognized areas in computer science, relate to logic 
in one way or the other, and to logic in a broad sense. Sometimes we also 
say computer science has given many interesting problems for category 
theory. I don't think this is really true in such a phrasing. What has 
happened is that computer scientist in their work to formalize computable 
logic and computability has been forced to go back to foundations of 
mathematics in order to understand what is really going on. Computer 
scientists, however, usually don't bother to formalize something they 
already 'understand' (type theory is a good example), where a
mathematicians refuses to understand before it's formalized (that's why 
there isn't any mathematical type theory).

Yes, we can. We can do history of category theory without considering 
history leading to category theory. But why should we? And how does it 
help to bring out all flavours of things we are still working on? Will 
this history writing provide me with those utensils I need for things I 
need to do. Or do I have to go elsewhere to look for it?

History writing is also a bit dangerous as it almost says this is now the 
state-of-the-art, and if you don't play your etudes properly you are not 
allowed to play structure and provide interpration. Talent is 
thereby often surpressed, and mostly by teachers who really never 
understood counterpoint anyway.

So what I really say is I like the '... and foundations of mathematics'. 
Keeping meta and object apart, and category theory taught me how to 
do that, is important for logic and foundations, as we know e.g. from 
Gödel numbering and creating sentences about it. To which logic these 
sentences belong, nobody ever told me, so please do.

Best regards,

Patrik



On Tue, 12 Jul 2011, Graham White wrote:

> I think, judging by comments so far, that there are basically two
> goals concealed within "this project". One is to write an outline of
> category theory as it seems to us now; the other is to write a history
> of category theory, and, specifically, a history of who influenced whom.
> Both of these are very worth doing, but the second is much more
> difficult.
>
> It's difficult mainly because it entails recovering a consistent history
> from people's reminiscences, and these will not be consistent with
> each other: they will be inconsistent not just because people's memories
> are not accurate, but because everyone has remained active in the field
> and they alter their memories according to what they think now. This is
> probably especially true of mathematicians, because mathematicians
> always rephrase other people's stuff in their own terms: it's how they
> come to understand it. (Remember Goethe's remark, "Mathematicians are
> like Frenchmen: if you tell them something, they rephrase it in their
> own language, and you cannot understand it any more"? Well,
> mathematicians do that to each other as well as to non-mathematicians).
>
> The history is hard to do, but also potentially very valuable: it would
> show how a revolution in mathematics took place. Hard work, though.
>
> And *not* in the form of a Wiki, because Wikis deal with contradictions
> between documents by erasing one document in favour of the other. (I
> know, you can always look back in edit history, but it still relegates
> one of the testimonies to the sidelines: you might well be in a
> situation where you just have more than one testimony, and where it
> would not be sensible to prefer one to the other).
>
> Graham
>

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