SUO: 09 May 2002 -- Divergence & Coincidence
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I will pause in my study of the connections between
differential geometry and differential logic to see
if I can supply a clearer statement of my rationale
for highlighting this particular nexus in the realm
of mathematical possibilities.
I think that I first became aware of a certain disconnection
between the disciplines of mathematics and philosophy during
the time of my "post-pubescent foundational identity crisis",
as one of my mathematical mentors at the time humorously but
dismissively dubbed it, when I found that none of the elders
that I bothered with my worries had any concern with matters
of foundations, or even much interest in the entire business
that only my philosophy teachers dignified with the title of
the "philosophy of mathematics". Long time passing -- I got
over that, once or twice, but much further on, when I turned
to the work of mobilizing mathematics in computational terms,
the same set of problems recurred with a vengeance, and in a
way that I could no longer avoid. The fact is that standard
accounts of so-called "proof theory" tell us next to nothing
about how anything worthwhile in mathematics gets discovered.
More time passed, and I found myself with far more practical
issues to work on, but the very same sorts of problems arose
again, this time in the guise of communication channels that
appeared to have gone missing between the owner-operators of
different cognitive styles -- for example, the "qualitative"
or the linguistic-logical style versus the "quantitative" or
the computational-mathematical style, with all the varieties
of which I came into contact in various research communities.
Being afflicted with a backtracking, subgoaling sort of mind,
I naturally or second naturally sought to trace the symptoms
back to the roots of their sufficient reasons and vera causa.
Peirce, of course, had a few choice things to say about this
disconnect between field and laboratory science and seminary
studies of logic and philosophy, but even further back there
was Kant already complaining about the unhealthy divergences.
But I did find a nexus when the logical and the mathematical
modes of describing things, real and imaginary, were not yet
so far apart from each other, and that is the point to which
I am trying to draw your attention.
In sum, the best information that I have with respect to the
commensurability issues that we constantly face in this work,
namely, the demands of intercommunication and interoperation,
is just that these problems were largely solved in principle,
under the auspices of their quantitative analogues, at least,
in the course of the 19th Century developments of the theory
of manifolds and the differential geometry that goes with it.
Now we all know that there's a really big difference between
principle and practice, and when you take into consideration
the dimensions of that fissure compounded by the scarcity of
adequate bridges for traversing the qualitative-quantitative
divide, I do not think that anybody needs to worry about any
lack of work to do in the present time-frame.
Jon Awbrey
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Differential Geometry for Engineers
Preface
http://suo.ieee.org/ontology/msg04056.html
1. Introduction
http://suo.ieee.org/ontology/msg04057.html
http://suo.ieee.org/ontology/msg04058.html
2. Manifolds And Their Maps
2.1. Differentiable Manifolds
http://suo.ieee.org/ontology/msg04059.html
http://suo.ieee.org/ontology/msg04060.html
http://suo.ieee.org/ontology/msg04061.html
2.2. Examples
http://suo.ieee.org/ontology/msg04062.html
2.3. Manifold Maps
http://suo.ieee.org/ontology/msg04063.html
3. Tangent Spaces
http://suo.ieee.org/ontology/msg04065.html
All of the above material is excerpted from:
| Brian F. Doolin & Clyde F. Martin,
|'Introduction to Differential Geometry for Engineers',
| Marcel Dekker, New York, NY, 1990.
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