Re: Foundations Of Mathematics Absconditus

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FOMA.  Note 3 : Mathematical Operations : Visual And Symbolic

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Re: http://www.cs.nyu.edu/pipermail/fom/2008-December/013248.html

Whether we operate in visual or symbolic terms probably depends
on our definitions of "visual" and "symbolic".  There are many
helpful suggestions to be found in the array of definitions that
C.S. Peirce developed in his theory of signs.  For example, he
defined an "icon" as a sign that denotes its objects by virtue
of a property (or a structural similarity) that it shares with
its objects.  He defined a "symbol" as a sign that denotes its
objects simply by virtue of the fact that it will be interpreted
as denoting those objects.

Icons are to signs as analogies are to inferences --
and you might even say as arrows are to categories --
so this opens up a very wide field, going beyond
the purely visual sense modality.

Symbols are what Peirce called "genuine" signs, using the word
in just the same sense that we call something "generic" in math.

Springing from an Aristotelian-Kantian tradition, Peirce tended to
form his categories in threes, and so he recognized a sign modality
that he called "indices".  An index denotes its objects by virtue of
an existential connection with them.  These might seem to occur more
in empirical than mathematical reasoning, but that would open up the
question of how empirical "math in practice" really is.

Jon Awbrey

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