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Geometrical Art as an Applied Science


Clifford Singer Ó 2003

Abstracts of Papers Presented to the
Mathematical Association of America
Boulder MathFest
July 31 - August 2, 2003


 Is geometrical art an applied science? There reigns a certain pre-established harmony

in geometry, what is required in one line of thought is supplied in another line, so that

there appears to be a logical necessity, independent of our individual disposition.   It is

not in the formalist’s mode of thought to deal mainly with the skilful formal

treatment of a formal question to devise an algorithm.   But, it is rather for the

intuitionists who give particular emphasis and stress on geometrical intuition in

general and not to give a complete account of a specific subject area.  An additive

construction as a supplement to a range of mathematical views that I find prevalent

through geometrical thinking as to gain insight into the shape of curves as far as

general classification and an enumeration of all fundamental forms are concerned.  


All this suggests, is the question whether it would not be possible to create an

abridged system of mathematics that is adapted to the needs of the applied sciences,

without passing through the whole realm of abstract mathematics.   It is my general

aim to gain, in the course of time, a complete view of the whole field of mathematics

with particular regard to the intuitional or in the highest sense of the term:

geometrical standpoint.


    What is the distinction between a naïve and refined intuition?  It must be stated

that the root of the matter is that naïve intuition is not exact.  While the refining of

intuition is not properly intuition at all, however, arises through a rational development

of axioms considered as being perfectly correct.  To further describe our naïve intuition,

for example when thinking of a point we do not picture in our mind an abstract

mathematical point, but rather substitute something visual and concrete in its place. 

In imagining a line, we do not picture in our minds a length without breadth, but

however a strip of a particular width.  We can always imagine a tangent as a straight

strip having a small portion or point in common with a curved strip; similarly with

respect to an osculating circle or variable curvature.





Clifford Singer, Cut Space Composition #9, ©2002, Etching on paper, 16 x 20 inches

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