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Willie Maartens

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Holism and Reductionism: The Missing Dimension is Software
By Willie Maartens   
Rated "G" by the Author.
Last edited: Tuesday, April 10, 2007
Posted: Monday, April 09, 2007

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Holism, reductionism, and the missing dimension





Holism (from holus, a Greek word meaning all, entire, and total) is the idea that all the properties of a given system (physical, chemical, biological, social, economic, mental, linguistic, etc.) cannot be determined or explained by the sum of its [physical] component parts alone. Instead, the system as a whole determines in an important way, how the parts behave.


Holism is in essence the reintegration of the physical (body, or hardware) and spiritual (spirit, mind, or software) aspects of the universe.


The idea has ancient roots. Examples of holism can be found throughout human history and in the most diverse socio-cultural contexts, as has been confirmed by many ethnological studies.


The term holism was introduced by the South African statesman Jan Christiaan Smuts (1870-1950) in his 1926 book, Holism and Evolution. Smuts defined holism as “The tendency in nature to form wholes that are greater than the sum of the parts through creative evolution”.


Smuts recognised the dilemma in which humanity finds itself then and today. Smuts began his book by stating, “Among the great gaps in knowledge those which separate the phenomena of matter, life, and mind still remain unabridged”. (Smuts, Holism and Evolution, 1987:2)


Holism opens the door to spirituality. In holism, there is the possibility for the unification of the spiritual, religious, evolutionary, scientific, and materialistic approaches to life. It requires a revolutionary change in attitudes from all sides to fit each as part of a whole system.


The general principle of holism was concisely summarised by Aristotle (384-322 BCE) in the Metaphysics, as “The whole is more than the sum of its parts”.


Reductionism is seen as the opposite of holism. Reductionism in science says that a complex system can be explained by reduction to its fundamental parts (through analysis). Essentially then, chemistry is reducible to physics, biology is reducible to chemistry and physics, and psychology and sociology are reducible to biology, etc.


However, when compared to computer technology, we can deduce that physics and chemistry are physical (matter/energy, hardware) in nature; biology contains both hardware and software (programs, information, and data) aspects, and psychology and sociology are purely software. Where software can never be reduced to, or compared with, hardware. Software is sui generis. Therefore, biology (body plus life, mind, and spirit) can never be reduced to physics and chemistry (matter/energy).


Drawing on thinkers ranging from Plotinus (205-270 AD) and Sri Aurobindo (1872-1950) to Hegel (1770-1831) and Jean Piaget (1896-1980), and grounding his own thought in extensive meditation practices, Ken Wilber synthesises modern science and traditional spirituality to provide a progressive understanding of cosmic, biotic, human, and divine evolution. He is also a pertinent character in the story of Holism.


Ken Wilber now seeks to unite the perennial idea of The Great Chain of Being), as informed by spiritual, cultural, social, and natural scientific evolutionary concepts, with a four-fold set of distinctions allegedly capable of analysing all phenomena.


Drawing on the notion of holons developed by Jan Smuts and Arthur Koestler, Wilber maintains that virtually all phenomena are wholes from one perspective and parts from another.


A cell in an organism, for example, is a whole (a system) that includes parts, but is also a part of the organism (a part-system). Emphasising that holonic evolution generates emergent qualities; Wilber divides the ‘Kosmos into four grand domains: physiosphere, biosphere, noösphere, and theosphere.


The physiosphere (the physical universe) includes the non-biological features of the universe, including the stars and planets that arose in the billions (109) of years following the Big Bang.  


The biosphere (the breathing coat around our planet), the domain of life, depends upon the much older and much vaster physiosphere, but involves features that transcend the physiosphere.


Finally, the biosphere gives rise to the noösphere (the mental realm), which includes complex sentient life such as mammals and humans. Again, the noösphere both depends on physiosphere and biosphere, but also transcends them, by exhibiting emergent characteristics, including self-consciousness, language, and rationality.


The theosphere (the spiritual abode), which both includes and transcends the other three domains, refers to dimensions of consciousness that include what is traditionally understood by God.


“From the parts [matter/energy] it is never possible to reach the whole [matter/energy plus life, mind, and spirit], because the sum of the parts still gives us a collection of partial and external relations and not that inward totality which constitutes the whole”. (Smuts, Holism and Evolution, 1987)


Holism is sometimes described as the opposite of reductionism (or scientific reductionism). Some proponents of reductionism think rather that it is the opposite of greedy reductionism and therefore holism should be contrasted with atomism.


On the other hand, holism and reductionism can also be regarded as complementary viewpoints, in which case they both would be needed to give a proper account of a given system.


The atomist divides things up in order to know them better; the holist looks at things or systems in aggregate and argues that we can know more about them viewed as such, and better understand their nature and their purpose.


However, Holism as an idea or philosophical concept is diametrically opposed to atomism. Where the atomist believes that any whole can be broken down or analysed into its separate parts and the relationships between them, the holist maintains that the whole is primary and often greater than the sum of its parts (= synergy, i.e. 1 + 1 = 3, or 2 + 2 = 5).


Erich Schumacher pointed out in A Guide for the Perplexed (1980:19), that there was a significant withdrawal and impoverishment of the human imagination in scientific (reductionist) thinking. While conventional wisdom had always presented the world as a 3-dimensional structure (as symbolised by the cross), where it was not only significant and meaningful, but of essential importance to distinguish always and everywhere between ‘higher’ and ‘lower’ things and Levels of Being. Reductionist thinking strove with stochastic determinism, zeal, and fanaticism, to get rid of the third, the vertical dimension.


So then, if we have:


  • 1 + 1 = 2 (wholes that are equal to the sum of the parts – I will call this the reductionist view), and
  • 1 + 1 = 3 (wholes that are greater than the sum of the parts, i.e. it is synergistic – I will call this the holistic view)


Then in the second case, that with synergy, we then have something like, 2 + 2 + i = 5. However, what exactly is this something, this unidentified factor “i”?


If we firstly take “i” to be an ‘imaginary’ number, where imaginary numbers involve the concept of ‘√-1’ – i.e. the square root of minus one.


We can then see that the square root of ‘-1’ (√-1) is called ‘an imaginary number’ because it does not exist in the ‘ordinary’ number system. In 1777, the Swiss mathematician, Leonhard Euler (1707-1783), symbolised the square root of ‘-1’ as ‘i’ [for ‘imaginary’]. It is also sometimes called ‘j’. [I.e. i, j = √ (-1)]


Imaginary numbers however are not ‘imaginary’; they are very ‘real’ in Algebra – although they are not ‘real numbers’. Consider a horizontal line with a zero point located on it. Points to the right can be represented as positive numbers; points to the left are negative numbers (… -3, -2, -1, 0, 1, 2, 3…).


If we did not know better, we could have called imaginary numbers mathematical hallucinations. They are much like pink elephants (or even sentient mind)! In truth, all numbers are ‘imaginary’ because they only exist in our minds – there are no numbers in the external world around us! So, this division of ‘real’ and ‘imaginary’ numbers is used only for mathematical and scientific game playing.


A second line vertical through zero, can now represent positive and negative ‘imaginary numbers’ (…3i, 2i, 1i, 0, -1i, -2i, -3i…). The entire geometrical plane is now defined by a combination of two numbers called complex numbers [i.e. ‘x + y.i’ – ‘real number’ + ‘imaginary number’]. In fact, ‘x + y.i’ = r.(Cosθ + i.Sinθ) = r.e(i.θ). Where ‘x’ and ‘y’ are real numbers, the angle ‘θ’ is measured in radians (not in degrees), and ‘r = √(x2 + y2)’.


Complex numbers are indispensable in the mathematics of engineering and science.


Casper Wessel (1745-1818), a Norwegian Land Surveyor, in 1797 was the very first person to represent real and imaginary numbers in this way, but the theory of complex numbers was already developed by Jean le Rond d′Alembert in 1746. The vector representation of complex numbers was consequently introduced by Casper Wessel in 1798.


Argand Diagrams is the method of drawing complex numbers as vectors on a coordinate grid and is named for Jean R Argand (1768-1822), an amateur mathematician who described them in a paper in 1806. A similar method, although less complete, had been suggested as early as 120 years before that by the English mathematician John Wallis (1616-1703), and later developed extensively by Casper Wessel.


It may be that even then, the method was unknown to Johann Friedrich Carl Gauss (1777-1855) and he had to rediscover it for himself in 1831.  Although it has been suggested that Gauss may have discovered the idea as early as Wessel. Some parts of Gauss’ Demonstratio Nova (1799) would seem almost miraculously derived without knowledge of the ideas of the geometry of complex numbers.


Perhaps this is an appropriate time to define the different ‘traditional’ numbers that are used in Algebra. The traditional algebraic number system comprises, like everything else in the universe, a hierarchy of different types of numbers, namely:


·        Natural numbers (1, 2, 3, …) + ‘Zero’ (0) + Negative whole numbers (-1, -2, -3, …)  = Integers

·        Integers + Fractions (1/2, 1/3, ¼ …) = Rational numbers.

·        Rational numbers + Irrational numbers = Real numbers.

·        Real numbers + Imaginary numbers = Complex numbers.


Irrational numbers are any real number that cannot be expressed as the quotient of two integers (e.g. a ratio like 1/2). For example, there is ‘no’ number (none) among integers and fractions that solve the problem of finding the square root of two (√2 = 1.414 213 562…). A counterpart problem in measurement would be to find the length of the diagonal of a square whose side is one unit long (√[12 + 12] = √2); there is no subdivision of the unit length that will divide evenly into the length of the diagonal.


Algebraists of the 16th and 17th centuries were hard put to know what to think of the number ‘√ (-1)’ except that it was probably an irrational, inconsequential accident. Rene Descartes called ‘√(-1)’ and other square roots of negative numbers ‘imaginary’, and Gottfried Leibniz (1646-1716) referred to them as ‘a wonderful flight of God’s Spirit’.


These mysterious appellations were the only descriptions that algebraists at the time could manage, because they did not have any logical way of thinking about imaginary numbers. Without some conceptual model for imaginary numbers, algebraists were unable to comprehend what it meant to add, subtract, divide, and multiply these indescribable quantities. From that perspective, there was no reason to believe that these imaginary numbers were a rational part of algebra.


As we have already seen, it was not until 1797, when Casper Wessel discovered a way to conceptualise them that imaginary numbers were finally recognised by algebraists to have a proper place alongside the real numbers (as the positive and negative numbers together had come to be called). The key was in thinking of imaginary numbers as existing mathematically in a different dimension than that of the real numbers – in other words, if the real numbers were imagined to be coordinates of longitude, then the imaginary numbers would be the coordinates of latitude [or the x- and y-axes on a graph].


With this simple map in mind, Wessel was able to see just how to extend the meaning of addition, subtraction, multiplication, and division to include the imaginary numbers in Algebra. For instance, combining a real number with an imaginary number is not like adding two real numbers (i.e. ‘x + y = z, or 5¼ + 7½ = 12¾’). It is similar to combining a longitudinal coordinate with a latitudinal coordinate. The result is not a numerical sum, but instead is like the coordinates of a point on a surface (i.e. ‘(x1 + i.y1) + (x2 + i.y2) = (x1 + x2) + i.(y1 + y2) = u + i.v’, or [2 + 3i] + [5 + 2i] = [7 + 5i]). Where ‘x’, ‘y’, ‘u’, and ‘v’ are real numbers, and ‘i’ = √(-1).


Each point on the earth’s surface or on a map, a globe, or physical reality is codified by convention in terms of its longitudinal and latitudinal coordinates. Thus, according to Wessel’s conceptualisation, the algebraic relationships between real and imaginary numbers can be understood entirely in terms of a hypothetical map-like surface whose every point is labelled by a pair of numbers, with the imaginary number being measured along a latitudinal (‘y’ or vertical) scale and the real number measured along a longitudinal (‘x’ or horizontal) scale. Wessel named this mathematical surface the ‘complex plane’. The horizontal line of real numbers is just a special case, the set of numbers whose imaginary part equals zero (i.e. ‘x + 0.y’ … where ‘x’ and ‘y’ are real numbers).


If we now take “i” to be the missing dimension in reductionism’s view of reality (namely 2 + 2 = 4), as compared to the holistic view where, 2 + 2 = 5 (or 2 + 2 + “i” = 5)*; then we can call this missing factor (“i”) life, spirit, mind, or whatever because it is something from another dimension as opposed to the physical dimension of matter/energy.


The software component [programs, algorithms, or code] in computer technology is equally not comparable in any way whatsoever to the hardware component [matter/energy]; and life, spirit, and mind is likewise also not in any physical way related to matter/energy – they are from completely different dimensions altogether.


* … For example, 2 + 2 + i4 = 5 [i4 = “i” to the power 4]



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Reviewed by Tinka Boukes 4/12/2007
Maths never been my favorite subject in school!!


Love Tinka

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