Geometry and logic

by Neil Rickert

While preparing to write this post, I did a google search for “geometric method.”  Many of the results of the search were about logic rather than about geometry as, for example, in Spinoza’s geometric method.  In a way that makes sense, for Euclid’s geometry is famous for its use of self-evident axioms.  However, that is not at all what I think of when I think about geometry and geometric method.

The way logic is used in geometry is very different from the way it is used in ordinary propositional logic.  And that is what I mainly want to discuss here.
In ordinary logic, the starting point is a set of entities, and some relations between those entities.  What are those entities is said to be determined by ontology, which is taken to be part of metaphysics.  Our knowledge about the relations between these entities is said to come from observation and induction.  How induction could work is never adequately explained.  A particular application of logic begins with premises which normally are the relational statements obtained using observation and induction.  The weakest point of ordinary logic, is in the uncertainty of the premises that come from observation and induction.  Disagreements over logic arguments often turn out to be disagreements about the premises assumed for those arguments.

With geometry, by contrast, we start with nothing.  Or, more strictly, in plane geometry we start with an empty ideal plane.  Our relational rules or axioms are about lines and triangles.  But we start without any lines or triangles.  So there is no assumed ontology.  When we do encounter lines or triangles, they are lines or triangles that we ourselves have constructed.  Our premises (or axioms) do not come from observation or induction, but from the methods that we should ideally have followed in making that construction.  This is why we can consider the axioms to be self-evident.  We are certain of them, in a way that we could not be certain of the results of mere observation or induction.

In a future post, I shall discuss what I mean by “geometric method” and why I see that as important.

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One Comment to “Geometry and logic”

  1. Geometry starts with quite a bit. The fifth postulate is not obvious. topology starts with even less, logically speaking.

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