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Buddhism and Mathematics
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Six aspects of mathematics which are relevant to Buddhist Philosophy
(i) The foundations of mathematics have parallels (and differences) with the Buddhist concept of emptiness (sunyata)
(ii) The bootstrapping of the integers out of the empty set provides a simple illustration of how causes and components can be expressed as algorithms and datastructures.
(iii) The further development of mathematics from its empty foundations creates computational algorithms and datastructures that can simulate all physical phenomena.
(iv) These simulations epitomize the 'unreasonable effectiveness' of mathematics in physics and engineering, which suggests that there may be aspects of the mind which are not explainable as the products of evolution.
(v) The concept of 'algorithmic compression', which is an aspect of the unreasonable effectiveness of mathematics, leads on to the ChurchTuringDeutsch principle, which gives us a workable philosophical demarcation between physical and nonphysical phenomena, including physical and nonphysical aspects of the mind.
(vi) The deep interconnection between the mind contemplating emptiness, and the workings of the physical world, suggests that mathematics may provide a bridge between ultimate and conventional truths.
1) What are the origins 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?
The mathematician John von Neumann, who was one of the founders of computer science, demonstrated that the whole numbers could be bootstrapped out of the Empty Set by mental operations without reference to any physical entities.
A set is a collection of things. The empty set is a collection of nothing at all. The 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).
The procedure is as follows:
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 nonthing  the empty set. The mind has thus generated the number 1 by producing the set containing the empty set.
Next the mind perceives the empty set and the set containing the empty set, so generating a new set encapsulating two nonthings. The mind has generated the number 2 out of emptiness. And so it goes on, all the way up through the integers, to 42 and beyond.
So, the three modes of existential dependence (causes, components and mental designation) postulated by Kadampa Buddhist philosophy are apparent even at the very deepest level of mathematics.
 Numbers are dependent upon causes  the operations on the sets.
 Numbers are dependent upon components. 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 is dependent upon the play of mind on an empty datastructure, in the complete absence of the need to refer to any material thing, or things, which are being counted.
Hence numbers are nonphysical phenomena that make no reference to physical systems for their existence. But they are not inherentlyexistent entities from the 'Platonic realms'. Numbers are dependentlyrelated manifestations of the working of the mind.
Note  the Empty Set is not synonymous with Emptiness (Sunyata) in the Buddhist sense, though there are parallels. Buddhist emptiness refers to the ultimate unfindability of all objects existing 'from their own side' . The subjective experience of Emptiness results from the exhaustion of all procedures to find an object (eg the classic meditations which try to find the real 'self').
Buddhist Emptiness is a philosophical conclusion, not a 'thing', state or attribute. The Emptiness of phenomena should not be confused with the delusion of a selfexistent void.
Emptiness is the logical conclusion of the unfindability of a 'thing in itself' once every cause and component that is 'not the thing' has been removed from its basis of imputation. Buddhist Emptiness always refers to an object whose 'inherent existence' has been refuted, for example the emptiness of a chariot. Emptiness too is empty, and Nagarjuna warned against reifying it.
The empty set, on the other hand, is what is left when everything has been removed, regardless of any original starting object. This emptiness is then 'encapsulated', and in a sense it is reified, to make it logically and mathematically manipulable.
What the empty set represents is an 'encapsulation' of emptiness which renders it logically and mathematically manipulable for certain defined purposes. According to Wiki "While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians. The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists."
Just as the defining characteristic of a chariot is its chariotness, which can't be found within the chariot itself, so the defining feature of the empty set is its emptiness, which similarly can't be found within the empty set; for if the empty set contained anything whatsoever it would no longer be empty. So the emptiness of the empty set must itself be empty.
As with the failure to find the definitive chariot, the failures of ones attempts to find a 'thing' is not an attribute of that thing. Nothing is to be found in the empty set beyond the conclusion that there is nothing to be found.
The procedure for mentally generating numbers by bootstrapping out of the empty set can be set out symbolically:
0 = {} (empty set)
1 = {0} = { {} }
2 = {0,1} = { {}, { {} } }
3 = {0,1,2} = {{}, { {} }, { {}, { {} } }}
4 = {0,1,2,3} = { {}, { {} }, { {}, { {} } }, {{}, { {} }, { {}, { {} } }} }
See also http://www.mathacademy.com/pr/prime/articles/vnh/index.asp and http://www.thebigview.com/buddhism/emptiness.html
2) The simplest algorithm acting on the simplest datastructure.
The bootstrapping of the integers is one of the simplest illustrations of an algorithm acting on a datastructure. There can't be any simpler datastructure than an empty set, and the iterative procedure of nesting the sets is one of the simplest algorithms one can envisage. However neither the datastructures nor the algorithms contain any 'meaning'.
Their meaning, as it appears in terms of named integers (one, two, three etc used for counting objects) has to be designated by the mind of the observer. There is no concept of number, indeed no concept of singularity or plurality existing 'from its own side'. Singularity and plurality have to be designated by the mind of the observer over some chosen object(s) or subdivisions of objects, eg leaves, branches, trees, forests.
Counting is a mental projection, things do not count themselves, see 'The Emptiness of the Eight Extremes' in Modern Buddhism.
3) "God gave us the integers"
"God gave us the integers, all else is the work of man" is a famous statement attributed to the mathematician Leopold Kronecker. I'll let the theologians decide whether it was actually God or John von Neumann who gave us the integers, but what the statement is saying is that once you have the foundation of the integers, you can derive the rest of mathematics using a very small repertoire of operations.
In fact, the integers (represented as binaries) plus the instruction set of a computer are capable of representing any other kind of numbers and simulating and modelling any physical system (ChurchTuringDeutsch principle)
It's quite remarkable just how few operations are required to equip a general purpose computer with universal physical modelling capabilities  fewer than twenty: SET, MOVE, READ, WRITE, ADD, SUBTRACT, MULTIPLY, DIVIDE, AND, OR, XOR, NOT, SHIFT, ROTATE, COMPARE, JUMP, JUMPCONDITIONALLY, RETURN
4) The unreasonable effectiveness of mathematics in science and engineering.
The remarkable efficiency of mathematics in predicting the behavior of physical systems has fascinated many scientists.
Einstein is said to have remarked that "The most incomprehensible thing about the universe is that it is comprehensible." This observation was elaborated by Eugene Wigner in his famous paper in Pure Mathematics (Volume 13, Number 1, February 1960) entitled 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'
The theme was further developed in 'The Unreasonable Effectiveness of Mathematics' by R. W. Hamming in The American Mathematical Monthly (Volume 87, Number 2, 1980), which considered the predictive, as well as descriptive powers, of mathematics in relation to engineering.
Two surprising conclusions appear from these papers:
(i) Although it is a product of the human mind, mathematics is also involved in some strange metaphysical way at the deepest levels of physical existence. To quote Wigner:
"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."
(ii) There is no Darwinian explanation for the presence of mathematical abilities within the mind. The ability to understand physics could not have arisen by evolution. Although our bodies may well be the product of random mutation and selection all the way from amoeba to man, our minds have some 'unevolved' dimension. To quote Hamming:
"But it is hard for me to see how simple Darwinian survival of the fittest would select for the ability to do the long chains that mathematics and science seem to require".
"If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics."
Or Wigner again:
"Certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess."
So we are left with something of a mystery. According to the materialist 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. So why should this 'toplevel' phenomenon have such intimate access to the 'bottom level' phenomena such as quantum physics? After all, the two levels are supposedly separated by less wellunderstood (in some cases) explanatory layers such as evolutionary psychology, neurology, cell biology, genetics, molecular biology, and chemistry.
Buddhists don't dispute that some aspects of the mind have been determined by evolution of the brain, but there also seem to be highly functional attributes of the mind that have no evolutionary explanation. The unusual effectiveness of mathematics is one of these.
5) Algorithmic compression.
A very powerful aspect of the unusual effectiveness of mathematics in the physical sciences, is what is known Algorithmic compression'.
The laws of physics may be regarded as being analogous ('isomorphic') to algorithms, with the physical objects being analogous to the datastructures the algorithms act upon. All physical systems can thus be simulated by a general purpose computer. This is known as the ChurchTuringDeutch principle
Algorithmic compression is explained in this excerpt from an article by Gregory Chaitin :
'My story begins in 1686 with Gottfried W. Leibniz's philosophical essay Discours de métaphysique (Discourse on Metaphysics), in which he discusses how one can distinguish between facts that can be described by some law and those that are lawless, irregular facts. Leibniz's very simple and profound idea appears in section VI of the Discours, in which he essentially states that a theory has to be simpler than the data it explains, otherwise it does not explain anything. The concept of a law becomes vacuous if arbitrarily high mathematical complexity is permitted, because then one can always construct a law no matter how random and patternless the data really are. Conversely, if the only law that describes some data is an extremely complicated one, then the data are actually lawless.
Today the notions of complexity and simplicity are put in precise quantitative terms by a modern branch of mathematics called algorithmic information theory. Ordinary information theory quantifies information by asking how many bits are needed to encode the information. For example, it takes one bit to encode a single yes/no answer. Algorithmic information, in contrast, is defined by asking what size computer program is necessary to generate the data. The minimum number of bitswhat size string of zeros and onesneeded to store the program is called the algorithmic information content of the data. Thus, the infinite sequence of numbers 1, 2, 3, ... has very little algorithmic information; a very short computer program can generate all those numbers. It does not matter how long the program must take to do the computation or how much memory it must usejust the length of the program in bits counts...
...How do such ideas relate to scientific laws and facts? The basic insight is a software view of science: a scientific theory is like a computer program that predicts our observations, the experimental data. Two fundamental principles inform this viewpoint. First, as William of Occam noted, given two theories that explain the data, the simpler theory is to be preferred (Occam's razor). That is, the smallest program that calculates the observations is the best theory. Second is Leibniz's insight, cast in modern termsif a theory is the same size in bits as the data it explains, then it is worthless, because even the most random of data has a theory of that size. A useful theory is a compression of the data; comprehension is compression. You compress things into computer programs, into concise algorithmic descriptions. The simpler the theory, the better you understand something'
In summary: If a computer program or algorithm is simpler than the system it describes, or the data set that it generates, then the system or data set is said to be 'algorithmically compressible'.
This concept of algorithmic simplicity/complexity can be extended from the realms of mathematics into physical systems. The complexity of a physical system is the length of the minimal algorithm than can simulate or describe it. Thus the orbits of the planets, which seemed so complex to the ancients, were shown by Newton to be algorithmically compressible into a few short equations.
Related to algorithmic compression is the fact mentioned earlier, that the small repertoire of opcodes acting on binary encoded integers is capable of describing and predicting the behavior of all physical systems.
The computer model of the three levels of dependency
So causal dependency can be modelled as algorithms, and compositional/structural dependency can be modelled as datastructures, but where does that leave conceptual dependency?
According to Buddhist philosophy, the function of the mind cannot be reduced to physical or quasiphysical processes.
In Buddhist psychology, the mind is clear, formless, and knows its object. Its knowing the object constitutes the conceptual dependency, which is fundamental, axiomatic and cannot be explained in terms of other phenomena, including algorithms and datastructures.
So the issue that separates the Materialist from the Buddhist is whether there is anything left to explain about reality once algorithms and datastructures have been factored out.
The Materialist would answer that algorithms and datastructures offer a complete explanation of the universe, without any remainder. The mind is included in the overall scope of this 'computationalist' explanation.
In contrast, the Buddhist would claim that the mind is a third fundamental factor irreducible to algorithms and datastructures. Hence the mind itself is not algorithmically compressible, but is responsible for carrying out algorithmic compression.
Supporting this view one may point out that algorithms, as executed, do not contain within themselves any meaning. For example, the following two statements reduce to exactly the same algorithm within the memory of a computer
(i) IF RoomLength * RoomWidth > CarpetArea THEN NeedMoreCarpet = TRUE
(ii) IF Audience * TicketPrice > HireOfVenue THEN AvoidedBankruptcy = TRUE
Neither do datastructures as stored in machine format contain any meaning.
Such considerations have led critics of philosophical computationalism to claim that algorithms can only contain syntax, not semantics. Hence computers can never understand their subject matter. All assignments of meaning to their inputs, internal states and outputs have to be defined from 'outside the system'.
This may explain why the process of writing algorithms does not in itself appear to be algorithmic. The real test of computationalism would be to produce a general purpose algorithmwriting algorithm. A convincing example would be an algorithm that could simulate the mind of a programmer sufficiently to be able to write algorithms to perform such disparate activities as controlling an automatic train, regulating a distillation column, and optimising traffic flows through interlinked sets of lights.
According to the computationalist view this 'Mother of all Algorithms' must exist as an algorithm in the programmer's brain, though why and how such a thing evolved is rather difficult to imagine. It would certainly have conferred no survival advantage to our ancestors until the present generation (even so, do programmers outreproduce normal people?).
The Mother of all Algorithms
The proof of computationalism would be to program the Mother of all Algorithms on a computer. At present no one has the slightest clue of how to even start to go about producing such a thing.
According to Buddhist philosophy this is hardly surprising, as the Mother of all Algorithms is itself NOT an algorithm and never could be programmed. The mother of all algorithms is the formless mind projecting meaning onto its objects (i.e. conceptually designating meaning on to the sequential and structural components of the algorithm as it is being written).
The nonalgorithmic dimension of mind, of understanding of meaning, is needed to turn the programmer's (semantically expressed) requirements and specifications into the purely syntactic structural and causal relationships of the algorithmic flowchart or code.
[For people with a scientific education, the Turing Machine provides one of the most easily understood refutations of materialism, physicalism and the mechanistic model of the mind. The argument is as follows:
 The behavior of all machines, computers and physical systems is reducible without remainder to the operations of a Turing machine.
 The behavior of the mind shows at least two functions  'aboutness' (intentionality) and qualitative experience (qualia)  that cannot in principle be reduced to the operations of a Turing machine.
 Therefore, there are some aspects of the mind that are nonmechanistic and nonphysical.
See Mind and Mechanism – Buddhism and the Turing Machine for a full explanation.]
6) Ultimate and conventional mathematics
The question remains of how mathematics, which is derived from the mind contemplating emptiness, gives such an excellent predictive description of conventional physical reality. It may be that mathematics, in some strange way, acts as a bridge between ultimate and conventional truths. Or maybe the bridge actually is just part of a loop, and what goes around comes around...
42) Conclusion: The ultimate, fundamental nature of life the universe and everything
So let's combine reductionism with the fundamentals of mathematics: The structures and operations of mathematics are reducible to the structures and operations of the mind.
 The structures and operations of the mind are reducible to the structures and operations of biological macromolecules.
 The structures and operations of biological macromolecules are reducible to the structures and operations of organic chemicals.
 The structures and operations of organic chemicals are reducible to the structures and operations of atoms.
 The structures and operations of atoms are reducible to the structures and operations of mathematics.
 The structures and operations of mathematics are reducible to the structures and operations of the mind.
 which is where we came in!
This allows us to draw the following equally valid conclusions:
(i) Matter is fundamental
(ii) Mind is fundamental
(iii) Both mind and matter are fundamental
(iv) Neither mind nor matter are fundamental
(v) Nothing is fundamental
(vi) Forty two
 Sean Robsville