The Empty Set, the Origins of Mathematics and the Buddhist Concept of Sunyata.
What is the origin of numbers? In what way do numbers exist? Have they always been present as 'Platonic' abstractions, or do they require a mind to bring them into existence? Can numbers exist in the absence of matter or things to count?
Buddhist philosophy claims that all things arise out of emptiness (Sanskrit sunyata or shunyata)
According to David Loy, the English word emptiness has a more nihilistic connotation than the original Sanskrit. The Sanskrit root su conveys the concept of being swollen with possibility [LOY 1996].
The Kadampa school of Buddhist philosophy claims that all phenomena are ultimately empty of inherent existence and do not exist as things in themselves. All phenomena exist solely in dependence on other phenomena, which are themselves empty and dependently related to other phenomena and so on. No matter how deeply or far back we search, no phenomenon can ever be found which is fundamental or a 'thing-in-itself'. Neither the observer nor any observed phenomenon exist independently, but are inextricably intertwined. This viewpoint is known as dependent relationship.
Geshe Kelsang Gyatso [KELSANG GYATSO 1995] states that there are three levels of dependent relationship:
(1) Gross dependent relationship - causality - the dependence of phenomena on their causes.
(2) Subtle dependent relationship - structure - the dependence of phenomena on their perceived parts (including aspects, divisions and directions).
(3) Very subtle dependent relationship - the dependence of phenomena on imputation by mind.
These ideas are remarkably similar to the theory of the origins of mathematics first proposed by the mathematician John von Neumann, who was one of the founders of computer science. The theory relies on simple maipulations of sets.
A set is a collection of things. An empty set is a collection of nothing at all. An empty set can be thought of as nothing with the potential to become something (that is to be become a set with at least one member).
Von Neumann [VON NEUMANN 1923] proposed that all numbers could be bootstrapped out of the empty set by the operations of the mind.
The mind observes the empty set. The mind's act of observation causes the appearance another set - the set of empty sets. The set of empty sets is not empty, because it contains one non-thing - the empty set. The mind has thus generated the number 1 by producing the set containing the empty set.
Now the mind perceives the empty set and the set containing the empty set, so there are two non-things. The mind has generated the number 2 out of emptiness. And so it goes on all the way up.
So, the three levels of dependent relationship postulated by Kadampa Buddhist philosophy are apparent even at the very deepest level of mathematics.
- Numbers have causes - the algorithms that perform the operations on the sets.
- Numbers have parts and aspects. The number 1 is defined as the set which contains the empty set and so on.
- And in the final analysis the entire number system has been generated by the play of mind on emptiness, in the complete absence of the need to refer to any material thing, or things, which are being counted.
Numbers are non-physical phenomena and need make no reference to physical systems for their existence. But neither are they inherently-existent entities from the 'Platonic realms'. Numbers are dependently-related manifestations of the working of the mind.
- Sean Robsville
The unreasonable effectiveness of mathematics in science and engineering
'So we are left with something of a mystery. According to the physicalist worldview, the mind (including mathematicians' minds) is an epiphenomenon of matter which has evolved solely to ensure the survival of the selfish genes which code for it. Why should this 'top-level' phenomenon have such intimate access to the 'bottom level' phenomena such as quantum physics? After all, the two levels are supposedly separated by less well-understood (in some cases) explanatory layers such as evolutionary psychology, neurology, cell biology, genetics, molecular biology, and chemistry.'
'One interesting aspect of emergent phenomena is the different causal and organisational relationships which appear at different levels of investigation. For example, ecology emerges out of biology, which emerges out of chemistry, which emerges out of physics, which emerges out of mathematics, which emerges out of the mind contemplating the empty set. Each level of investigation has its own explanatory relationships, yet if we check carefully there is no 'added extra' coming from the side of the objects. (Everything is algorithmically compressible without remainder, there are no mysterious ingredients added as we progress from lower levels to higher levels).'
Meditating on emptiness
'....For a qualified meditator single-pointedly absorbed in emptiness, there is no difference between production and disintegration, impermanence and permanence, going and coming, singularity and plurality - everything is equal in emptiness and all problems of attachment, anger, and self-grasping ignorance are solved...'
on the mind, personal relationships, meditation and the spiritual path.
Mathematics: The Bridge to an Integral Science of Experience
The scientific method is generally thought to require that scientific verification draw only from outer experience, thus excluding all inner experience as a valid basis for scientific verification. Consequently, the scientific study of consciousness today is largely limited to the physical sciences and the externally observable correlates of consciousness. Mathematics, however, provides an example of a rigorous science based on inner experience that is nonetheless verifiable. Contrary to widespread belief, it is possible, therefore, for an authentic science to be based upon inner experience.
Arguments against Buddhism
The best way to understand the strengths of a philosophy is to attempt to refute it.
[KELSANG GYATSO 1995] Gyatso, Geshe Kelsang, Joyful Path of Good Fortune, 2nd Edition - page 349, (London: Tharpa Publications, 1995, ISBN 0 948006 46 3)
[LOY 1996] Loy, David in the afterword to Swedenborg, Buddha of the North , page 104, (Swedenborg Foundation, West Chester Pennsylvania, 1996, ISBN 0-87785-184-0)
[VON NEUMANN 1923] cited by Robert Matthews in the Sunday Telegraph 15th October 2000, page 16.
If we regard Buddhism as a combination of a philosophy, psychology and religion, then how much mileage can we get from the first two aspects before we have to start invoking religious faith?