PEP-308 a "simplicity-first" alternative

[Gareth McCaughan]

Erik Max Francis wrote:

[... that in mathematics -> is the standard symbol for implication
and => typically means something else]

>  Michael Hudson wrote:
>  
> > In which context?  In my world, => is implication.  It's even
> > $\implies$ in LaTeX, I think. -> is often used to denote a map.
>  
>  In a logical context.  _VNR_ for instance uses the single horizontal
>  line -> arrow for logical implication (section 15.1).

I just checked 20 pure mathematics books, somewhat at random.
(I picked a lot of set theory and logic ones because they
have more need of a symbol for implication.)

Among books on logic and set theory (6 books):

  - 2 use -> for "implies" throughout.
  - 2 use -> in the formal theory and "implies" in the metatheory.[1]
  - 1 uses => for "implies" throughout.
  - 1 uses the reverse of the "proper subset" symbol for implication,
    following the older practice of, e.g., Russell and Whitehead.

Among other books (14 books):
  - 5 use => for "implies" throughout.
  - 9 don't use any symbol for implication.

For the most part, symbols for implication are rare outside books
on set theory and logic. (You need "therefore" much more often
than you need "implies".)

Much the commonest uses of -> are to indicate a mapping and
to mean "converges to". These are much, much commoner than
"implies" except in books on logic and set theory. Actually,
the "mapping" meaning is a pretty strong contender in some
books on set theory. :-)

Earlier, Erik wrote:

> > > In mathematics when the double line version is used,
> > > it's often used to mean something else, like "is
> > > equivalent to" or "evaluates to".

I found no instance of => being used for anything other than
"implies". I have seen it used to mean "evaluates to", though
I've seen other symbols for that just as often. I've never
seen it used to mean "is equivalent to", and I hope I never
shall.


[1] In logic and set theory, you often need to talk about
    formal systems for doing mathematics in; but you also
    need to do mathematics yourself, *about* rather than *in*
    those systems. The terms "formal theory" and "metatheory"
    denote, respectively, a formal system in which one can
    do mathematics, and whatever system (usually not formalized)
    you're using for talking about those formal systems.

-- 
Gareth McCaughan  Gareth.McCaughan at pobox.com
.sig under construc