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 Neuman [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 to 42, and maybe even beyond.
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.
But 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 therefore fundamentally devoid of inherent 'Platonic' existence.
The philosophical implication of these findings is that mind ontologically precedes number. In other words in the foundations of Being, mind is more fundamental than number. Asking 'How many minds are there?' is equivalent to asking 'What is the color of electromagnetism?' This sheds a new light on the age-old confrontation between monotheism and polytheism - does God have one mind or many? Is this a silly question?
The sociology of science and unreasonable effectiveness of mathematics
During the late 1980s there emerged a branch of sociology known as 'Science and Technology Studies' or STS, which had the objective of showing that the results of scientific findings did not represent any underlying reality, but were purely the ideology of dominant groups within society.
[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.