Re: Explanations

["F. William Lawvere" <wlawvere@hotmail.com>]

Dear Peter
We thought of "other" of course. But that word has no agreed on
mathematical definition. Students who think it means "distinct"
will be confused when told that it is a special application of the UMP
that yields the graph of a map as a section of a projection.
(Perhaps best if "self" is a special case of "other" ?)

Sammy always scolded Jon, Fred, Myles, and me that such "helpful"
explanations make difficult the digestion and mathematical use
of simple clear definitions. (I don't think this excludes explanation
in a separate paragraph or footnote).
Bill

> From: peasthope@shaw.ca
> Date: Fri, 29 Apr 2011 11:56:48 -0800
> To: categories@mta.ca
> CC: peasthope@shaw.ca
> Subject: categories: Re: Explanations
> 
> Charles & everyone,
> 
> Earlier peasthope wrote,
> "...changing a few words of a sentence can make a concept obvious rather
> than nebulous".  Revise that to "obvious rather than difficult".
> 
> From:	Charles Wells <charles@abstractmath.org>
> Date:	Fri, 22 Apr 2011 09:37:44 -0500
>> Can you give specific examples?  I suspect that in most cases the change
>> introduces a useful metaphor that was hidden before.
> 
> Here is a small example from the _Conceptual Mathematics_ of
> Lawvere and Schanuel.  No offense to the authors or the book.
> It's an indispensible and invaluable resource.
> 
> L&S page 292, "Definition ... equalizer ... and for each x:T-->X ... there is
> exactly one e:T-->E ... ."    "For all T" is implicit.
> 
> http://en.wikipedia.org/wiki/Equalizer_(Mathematics) , "In category theory
> ... defined by a universal property, ... object E and morphism eq ... such that,
> given any other object O and morphism m ... ."
> 
> For me, the reference to "any other object O" helps.  The definition in  the
> Wikipedia seems to reveal the "universality" of the equalizer better.  The
> diagram also helps.
> 
> A trivial issue for most readers but a small detail can make a difference  for
> a student.
> 
> Regards,                     ... Peter E.
> 
> -- 
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> 
> 

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