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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
See also:
Arguments against Buddhism
The best way to understand the strengths of a philosophy is to attempt to refute
it.
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.'
Emergence
'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...'
Buddhist Teachings
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.
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References
[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.