autonomous terminology

autonomous terminology

[John Baez]

Dusko wrote:

colin mclarty's cryptic comment is very interesting to me, and it seems to
> strike at the heart of some matters of interest.
>

Colin's comment didn't seem cryptic to me - let me guess what he meant.


> On May 9, 2010, at 3:41 PM, Colin McLarty wrote:
>
>  Dusko Pavlovic Asks
>>
>>  is there any reason why words should be taken seriously?
>>>
>>
>> That just depends on whether or not you want to be understood by people
>> who do not already know everything you are going to say.
>>
>
> there are at least two ways to interpret this.
>
> 1) "you can only say something new if you declare what your words mean.
> otherwise, people will interpret them in their own way, and understand only
> what they already know."
>
> 2) "you can only say something new if you contribute to the evolution of
> language. otherwise, everything you say are just words that people already
> know, mostly in combinations that they already tried."
>

I thought he meant:

3) If you don't take the prevailing meaning of words seriously, you're
likely to talk in ways that people won't understand, unless they happen to
already know everything you're trying to say.

I worry about this point a lot, because I often want to "fix" standard
mathematical terminology that I dislike, and I have to weigh my desire to do
that against my desire to be understood by people who are unwilling to learn
new ways of talking.

For example: do I use "n-category" to mean "weak n-category", which will
eventually be the most sensible course of action, but may be premature, or
do I use it to mean "strict n-category", as tradition dictates?

Best,
jb


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Re: autonomous terminology

[Colin McLarty]

2010/5/12 John Baez <baez@math.ucr.edu>:

wrote

> Colin's comment didn't seem cryptic to me - let me guess what he meant.

Thanks. I did not mean it as cryptic.

> I thought he meant:
>
> 3) If you don't take the prevailing meaning of words seriously, you're
> likely to talk in ways that people won't understand, unless they happen to
> already know everything you're trying to say.

Yes, and also taking the choice of new or revised terminology
seriously when it seems called for, just as this thread has been
doing.

But the problem is not only to entice people unwilling to learn new
terminology.  There is also the problem of telling where change truly
is needed and what change.  I do not offer this as a new thought but
just to keep it in the discussion.

I will say, for my part, I have no objection to learning a new system
of terminology for higher categories, or even a couple of new systems
of terminology.  But as an outsider with some curiosity, I have the
impression that as of this time I'd need to learn quite few systems to
get much into the field.  That can happen, or course, but I am glad to
see people taking these choices of words seriously.

best, Colin


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Words, sets, categories, and graphs

[Vaughan Pratt]

Dusko asked:
  > is there any reason why words should be taken seriously?

Colin answered:
  > That just depends on whether or not you want to be understood by
  > people who do not already know everything you are going to say.

I took Dusko's question to be about particular words rather than words
in general.  However I've become rather attached to the latter in
connection with foundations of mathematics.

It's fashionable in certain circles to consider mathematics to be
founded on the concept of set, customarily defined by ZFC, a first order
language with a single nonlogical symbol, membership, a binary relation.
   After considerable development one can eventually arrive at the notion
of free monoid on an alphabet, whose elements can be identified with the
words over that alphabet.  On that basis it is argued that sets are
conceptually prior to words.

Throughout this development however the exposition is mediated by words.
   On that basis it would appear that words must be conceptually prior to
sets, since without words how can sets be explained?

Category theory is as much organized around the binary operation of
composition as ZFC set theory is around the binary relation of
membership.  Composition of morphisms works like concatenation of words
with the following two refinements.

1.  Multiple objects.  Whereas any two words can be concatenated,
morphisms have a domain and codomain which composition must respect.
This creates a form of phrase structure grammar in which each phrase has
not one type but two, one at each end.  Words thereby form the edges of
a graph G, the finite paths in which constitute the composites, forming
the free category G* on G.

(William Wood's notion of Augmented Transition Network, ATN, as
developed in his 1968 Harvard Ph.D. thesis, see also CACM 13:10 (Oct.
1970), works roughly that way.  The idea is to replace the notion of
nonterminal vocabulary V_N with that of state set Q of an automaton,
with the transitions labeled with either terminals from the terminal
vocabulary V_T or subroutine calls to other graphs.  An Augmented Syntax
Diagram, ASD, is similar but with the edge labels moved to the vertices.
   It would be nice if the categorial grammars of Ajdukiewicz (1935),
Bar-Hillel (1953), and Lambek (1958) were vertex-oriented along those
lines but I haven't seen linguists explicate them that way.)

2.  Commutative diagrams.  A congruence T on, or typed equational theory
of, the paths in such a graph.

A category C = (G,T) consists of a graph G whose finite paths, as words
with typed endpoints, are understood modulo a given congruence T on G*,
namely as the quotient C = G*/T.  Every category can be presented in
this way, just as every group can be presented by generators and relators.

(Like many computer scientists I'm a big fan of both graph theory and
equational logic and have no objection to inferring from the above that
graphs and equational theories should be considered conceptually prior
to categories.  And for that matter prior to sets, since a set can be
defined as simply a graph with no edges, which is how I generally
picture them.)

Had Russell and Whitehead developed Principia Mathematica by starting
with words, they could have defined numbers as words on a one-letter
alphabet, with addition defined simply as concatenation.  Their proof
that 1+1 = 2, which with their approach they were able to complete by
Volume II, could have been shortened to the reasoning I+I = II, which
could surely be dealt with by page (ii) of the forward.  By extending
this approach to numbers to provide for larger alphabets, multiple
objects, and commutative diagrams, they could have compressed their
three volumes into one chapter, freeing up time and space for further
principles.

That's with a century of hindsight, mind you.  That much hindsight
applied to heavier-than-air flight would have been harder because
gasoline engines were relatively new in 1903, cf.
http://en.wikipedia.org/wiki/Steam_aircraft .  Graphs and congruences
are not at all new.

Vaughan




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