Re: The Meaning of Abstract

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 06/01/04 

Date: Tue, 01 Jun 2004 23:28:20 +0200

Nathan Funk <noemail@noemail.com> writes:

>> I guess I don't understand abstraction.
>>
>> Is "abstract" supposed to be a synonym for "lossy compression"? If
>> so, do you regard an mp3 ripped from a CD as an abstraction of the
>> original wave file?
>
> I would definitely understand an mp3 as an abstract representation of
> the raw waveform data contained on a CD. Do you have a better example
> for the point you are trying to make?

Most of us don't regard the relation between mp3 and wave file as the
same as the relation between the concept of triangle and *all* of the
particular three-sided figures with which we are familiar[1].

That x is created by lossy compression from a source y puts x in a
relationship with a particular y. When we abstract, we forget certain
details of each in a collection of instances, not the details of just
one particular instance.

But even this forgetting is not really the defining feature of
abstraction. Consider the operation that takes algebras to their
underlying sets by "forgetting" the algebraic structure. This
operation loses some data, just like lossy compression, but it is not
an act that most of us would call abstraction. One doesn't abstract
algebras to reach a notion of set[2]. The fact that data is lost or
forgotten doesn't capture the notion of abstraction, although it may
be a feature of abstraction (similarly, there is nothing about
compression as far as the forgetting is concerned).

Anyway, if you want to say that an mp3 is an abstraction of a
particular wave file, well, you can have that reductio ad absurdum.

Footnotes:
[1] I'm not taking the position that the notion of triangle is really
somehow abstracted from our experience with particular
three-sided-figures here.

[2] I can't say why this *isn't* a good example of abstraction, but I
think one would find that most philosophers of mathematics wouldn't
call this forgetful functor an instance of abstraction. 

-- 
"So, at this time, I'd like to assure you that I am not interested in
making sure mathematicians worldwide get fired.  I've rethought my
desire to go to Congress and try to get funding for mathematicians
cut."  -- James Harris is a reasonable man.  Whew!
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