You can also look at this as being undefined for the point, but
defined for an interval on the curve which is arbitrarily close to
that point.
--joshua
On Jul 9, 2008, at 10:37 AM, Frank Wimberly wrote:
In differential geometry a curve with a given parameterization has a
velocity at a point. This is not a category error; it’s a definition.
Frank
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
On Behalf Of Robert Holmes
Sent: Wednesday, July 09, 2008 9:49 AM
To: [EMAIL PROTECTED]; The Friday Morning Applied
Complexity Coffee Group
Subject: Re: [FRIAM] Mentalism and Calculus
This is based on nothing more than reading the entry on categories athttp://plato.stanford.edu/entries/categories/
so please take with a pinch of salt...
It seems that the tools necessary to construct category systems are
severely broken. Specifically, there is no generally accepted method
for distinguishing between categories. For example, the Ryle/Husserl
method boils down to a highly subjective notion of whether a
statement is absurd or not. That means it's perfectly possible for
Nick to see a category error ("it's crazy to say that a point can
have position and velocity") and me not to see one ("nothing wrong
with a point having position and velocity") and we can both be right.
IMHO, this means that category theory really can't tell us very much
about calculus.
Robert
On 7/8/08, Nicholas Thompson <[EMAIL PROTECTED]> wrote:
All who have patience,
Once of the classic critiques of mentalism .... the belief that
behavior is caused by events in some "inner" space called the
mind ... is that it involves a category error. The term "category
error" arises from ordinary language philosophy (I think). You made
a category error when you start talking about some thing as if it
were a different sort of thing altogether. In other words, our
language is full of conventions concerning the way we talk about
things, and when we violate those conventions, we start to talk
silly. To an anti-mentalist a "feeling" is something that arises
when one palpates the world and to talk about our "inner feelings",
say, is to doom ourselves to silliness. Feelings are inherently "of"
other things and to talk of "feeling our own feelings" is, well, in
a word, nutty.
As many of you know, I have been engaged in a geriatric attempt to
recover what slipped by me in my youth, the chance to understand
the Calculus. As I read more and more, it became clear to me that
the differential calculus was based on a huge "category error." To
speak of a point as having velocity and direction one had to speak
of it at if it were something that it essentially wasn't. And yet,
of course, the Calculus flourishes.
Now the reason I am writing is that I am not sure where to go with
this "discovery." One way is to renounce my behaviorism on the
ground that category errors ... any category errors ... are just
fine. Another way is to start to think of the mind/behavior
distinction in some way analogous to the derivative/function
distinction. That mind is just the derivative of behavior. For
instance, a motive, or an intention, is not some inner thing that
directs behavior, but rather the limit of its behavioral direction.
A third way, is to wonder about how the inventors of calculus
thought about these issues. They, presumably, were steeped in
mentalism and it cannot have escaped their notice that they were
attributing to points qualities that points just cannot have. Many
of the texts have been reading have alluded to the idea that some
contemporaries ... perhaps Newton himself ... attributed to the
Calculus some sort of mystic properties. I really would like to know
more about that. Any intellectual historians out there????
So, I am hoping somebody will help me go in any, or all, of these
directions.
--Nthompson 04:14, 9 July 2008 (GMT)
This noodle, and perhaps some subsequent revisions and commentary,
may be found at http://www.sfcomplex.org/wiki/MentalismAndCalculus
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([EMAIL PROTECTED])
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
