Month: October 2014

This page is dedicated to a poster session that was given at the 2014 Mind and Life conference in Boston, Massachusetts entitled “Mathematics of Enlightenment”.  The slides corresponding to the presentation are available as a PDF: emath.pdf .

There are two pages on this blog with related content:

• Svalaksana
• Mathematical primitives

There is also a post about the tetralemma (or catuskoti), and how that might be brought into accord with modern logic (or vice versa):

• A Buddhist Critique of Boolean Logic 

Svalaksana literally means “the real”.  The ancient Indian debate concerning svalaksana essentially asks, what is real?  When we identify an object, is it that we identify that object by first identifying all of its constituent atoms, and then assembling them together? Or do we (also) identify an object in virtue of its larger context? Are points svalaksana? Is the whole a svalaksana? What about regions in between? Is the requirement that objects are simple applicable to their unanalyzed appearance to the mind or their ultimate (psychophysical) unanalyzability?

In the Western philosophical tradition, this debate has focused on the reality of particulars (or concrete things) vs universals (or abstract things).  In early theories of both Buddhism and Mathematics, spatial regions are identified by the points (or “partless particles”) they contain, where those points are taken as primitive (either as axiomatic within geometry, or as the atoms of physical reality). In other words, particulars (as opposed to abstract things) are real, and the smaller the better.

Later Buddhist theories arrive at the conclusion that atoms are no more real than larger objects, in that both are imputed (or nominal) entities.  Several recent Mathematical theories similarly abandon points in favor of point-free topologies. So both ontologically and mathematically, there has been movement away from their respective reductionist positions (which define objects in terms of their parts, and therefore ontologically depend of partless entities).  Since Cognitive Set Theory similarly adopts a more holistic (instead of reductionistic) view, we present a few more details about the evolution of Buddhist psychology, and how that might map onto mathematics.

A simplified version of the evolution of the theory of svalaksana proceeds as follows:

1. According to Pramana theorists, the real is simple, or singular (“eka”): we might say more precisely that what is real is spatiotemporally contiguous: a thing which is whole is not distributed.  Mathematically, we will call it a well-connected region in N-space.
2. According to the Sautrantika view (or the “external realism” of Dunne), there are “real” external things: momentary, partless particles.  This view has been expounded in depth by Tscherbatsky [Buddhist Logic Vols 1&2], and it is compatible with the mathematical theory of points (which cannot be divided).
3. According to Chittamatra and Madhyamaka views, the objects of awareness are of the nature of mind, and only imputed.  In other words, there is no a priori reason to divide reality into a certain set of objects as opposed to another.  In other words, space is no more composed of points than it is composed of bricks: tiny  and large particles are equally real (or equally unreal). 

To establish the spatial analogue to the later Buddhist schools, we summarize the divergence from point-space as follows:

1. Regions are well-connected
2. Space is not composed of points
3. Regions are imputed

The first and second tenets are easily established in mereotopology, where space is divided into regions that are not (inherently) composed of points.  

We alter the third tenet by saying that an object only exists in virtue of an imputed boundary.  Since that boundary does not exist as an object within the original space, we say that it is an imputed boundary.  The details of how an object (the boundary) may demarcate events within a space without being an actual object within that space is taken up in more detail in another blog post.

 

The mathematical equivalent of the Madhyamaka (and especially Gelukpa) view of svalaksana (“the real”) requires an alternative to point-sets as the mathematical basis for space: these are known as “point-free” topologies. Mathematically, these theories avoid several paradoxes associated with point sets (which involve the distinction between open/closed intervals). We nominate the following tenets for a psychologically-valid, mathematical theory of conceptual and non-conceptual mind:

• Space is represented as an open-dimensional, or multi-dimensional, manifold.
• Objects are demarcated by boundaries. A boundary is an imputed object which has zero width (it is a thing of dimensionality N-1 in an N-dimensional space).
• The division of space into objects is not set-like: it has the relationship of parts to wholes, and is transitive. The complete definition of an object entails both its larger context and its boundary (a separating hyperplane or hyperspace).
• The boundary of an object may be characterized by a function.  That function is not necessarily characterized intrinsically (i.e. by the contents of that object), since it can also be specified extrinsically.
• Objects, whether they are regarded as real or merely perceptual, serve as the elements of sets (or the ultimate contents of concepts).
• Sets may be labeled, and collected into higher-rank sets.