Geometry and logic (2)

by Neil Rickert

In my first post for this series, I mentioned “geometric method,” but I did not explain what I mean when I use that expression.  That is what I want to discuss today.  In doing so, I widen our horizons beyond mathematics, and look at how we use geometric methods in the real world.  In particular, I shall discuss its epistemic significance.

The most basic step in geometry is to draw a line.  By drawing that line, we divide the plane (or whatever it is where we draw the line) in two parts corresponding to what is on either side of that line.  It is the use of that basic step that I consider to be at the core of geometric method.  The idea is humorously illustrated with The Bolzano-Weierstrass method of big game hunting.

Bisect the desert by a line running N-S. The lion is either in the E portion or in the W portion; let us suppose him to be in the W portion. Bisect this portion by a line running E-W. The lion is either in the N portion or in the S portion; let us suppose him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion is ultimately surrounded by a fence of arbitrarily small perimeter.

When we apply the geometric method to reality, as illustrated in that lion hunt, we divide up the world.  The various division steps divide reality into parts, so I am tentatively using the word “partition” to describe this activity.  The suggestion that we carve nature at its joints seems to go back to Plato.  However, we carve more finely than the use of natural “joints” would allow.

Let me illustrate the idea, with the example of an explorer looking to see what he can find in a newly discovered land.  As he treks through the land, he notices a river, and uses that river as a basis for partitioning.  He calls the part of the land east of the river “Territory A”, and the part of the land that is west of the river he dubs “Territory B.”  While looking around the land, he notices that there are pine trees in Territory A, and oak trees in Territory B.

As a result, he can now say:

  1. Territory A is the part of the land that is east of the river;
  2. Territory B is the part of the land that is west of the river;
  3. There are pine trees in Territory A;
  4. There are oak trees in Territory B.

If we consider those to be propositions, then the first two are analytic propositions, or propositions which are true by virtue of the meanings of terms.  The third and fourth propositions are synthetic, or propositions that are true by virtue of the way that the world is.

Here is the importance of the partitioning:  without that act of partitioning, we would only be able to say “There are pine trees and oak trees.”  By virtue of having partitioned the land, we can make the more precise assertions of propositions 3 and 4.  Whether we actually assign names to those partitions, as in propositions 1 and 2, is not as obviously important as is action of partitioning itself.

Partitioning is a vitally important part of the epistemic project (the project of being able to have knowledge of our world).  That partitioning allows more precise descriptions, as in the example of propositions 3 and 4 above, demonstrate its importance.

As for propositions 1 and 2, I believe their role to be mainly a social and cultural one.  How we partition reality is not completely fixed by nature, so depends on choices.  Naming the parts, as is done in those propositions, helps us spread the way we partition through the cultural group, so that everyone partitions in about the same way.

I am inclined to believe that use of geometric methods (i.e. partitioning) is the basis for perception and is the basis for ordinary logic.  Moreover, I am inclined to believe that scientific laws mainly arise from the use of geometric methods, rather than from induction.

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