FOM: wider cultural significance, part 2
Stephen G Simpson
simpson at math.psu.edu
Tue Mar 2 02:41:58 EST 1999
This is the second in a series of FOM postings about the wider cultural
significance of f.o.m.
Jacques Dubucs 12 Feb 1999 06:02:14 wrote
> After a long period of opprobrium ... there is an increasingly
> fashionable tendancy to consider again geometric reasoning by means
> of diagrams or figures as somehow irreducible to logical
> inference. ....
and in 16 Feb 1999 09:48:52
> this train of thoughts, which emphazises rather strong precedence
> of geometry over logic, is more represented in "continental" (i.e:
> european) areas than elsewhere.
and from this arose a lively, ongoing, FOM discussion of the
relationship between geometrical reasoning and logical reasoning in
mathematics. This is an excellent discussion. I am learning a lot
from this discussion, expecially Seligman's postings on diagrammatic
reasoning and the postings of Shipman and Machover on visual proof. I
want this discussion to continue in its current mathematical
At the same time, this thread also seems to tie in with a much
broader cultural/political/philosophical issue: `picturism', or: What
is the proper role of pictures in cognition?
My reference for picturism is Leonard Peikoff, `A Picture is Not an
Argument', The Intellectual Activist, vol. 13, no. 2, February 1999,
pages 7-23. Peikoff explains:
The appeal to pictures as the standard of truth is omnipresent
in our society ... [many excellent examples omitted] ... Should
American troops enter Somalia? Let the TV networks show emaciated
children dying of hunger. Should American troops leave Somalia?
Let the networks show the naked corpse of an American soldier being
dragged through the streets by the natives. (Thanks to picturism,
we now have foreign policy decided in essence by TV producers.)
... I hold that the picture method of resolving disputes is
espistemologically corrupt; ....
I do not object to pictures used merely as illustrations, after it
has been made clear that the pictures have no evidentiary
significance. What I object to is pictures used cognitively, in an
abstract discussion, i.e., pictures used to try to solve, or even
help solve, a problem in philosophy or politics. A picture used in
such a manner represents the antithesis of thought, of logic, of
Picturism is a form of epistemological manipulation of the viewer.
He is inveigled into turning his mind passively to an isolated
segment of a complex issue, which deflects him from the total. ...
Pictures, let me say, can sometimes be a necessary part of a process
of cognition. The proper pattern here is the one laid down in our
courtrooms. Take the O. J. Simpson case .... In legal terms, the
pictures were *probative*. ....
In dramatic contrast to a murder trial, the issues which people
dispute in philosophy and politics always involve broad
generalizations or principles; .... In philosophy and politics, a
picture is *always* prejudicial, never probative. The picture
offers perceptual data only, ....
To summarize, our culture harbors a `picturist' trend that wants to
replace logical argument by graphic pictures. And Peikoff opposes
this trend by taking a strong anti-picture, pro-logic stand, at least
with respect to politics and philosophy.
Now compare this to our geometrical reasoning thread here on FOM. I
am *not* trying to say that the two issues are one. It may be
possible to eschew pictures in philosophical/political disputes yet
find appropriate roles for them in mathematics. For my part, although
I agree with Peikoff on the need to reject `picturism' in philosophy
and politics, I also think Seligman and Shipman here on FOM have made
a pretty good case for `diagrammatic reasoning' and `visual proof' in
Nevertheless, there seem to be some close parallels here. In
politics, `picturism' disintegrates cognition by causing the viewer to
function at the perceptual rather than the conceptual level. In
mathematics, pictures can mislead in a somewhat similar manner. This
seems to be more than just a vague analogy. In addition, I see a
fairly clear cultural connection. This comes out when continental
mathematicians and philosophers go overboard with claims that
geometrical reasoning is *irreducible* to logical reasoning, or that
pictorial proof can or should *displace* logical proof. It seems to
me that some of this is partly motivated by some sort of postmodern
hostility or rebellion. The target of the rebellion seems to be logic
and rational argument in mathematics. The connection to `picturism'
in politics seems evident.
Am I way off-base about this? I don't want to overstate this cultural
connection. Still, there seems to be something there ....
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