Definitions:
"differentiate v.
1 distinguish, discriminate, contradistinguish, separate, contrast, oppose, set off or apart, tell apart: They
must learn how to differentiate one species from another.
2 modify, specialize, change, alter, transform, transmute, convert, adapt, adjust: All
organisms possess the power to differentiate special
organs to meet special needs.
integrate v. combine, unite,
blend, bring or put together, assemble, merge, amalgamate, join, knit, mesh, consolidate, coalesce, fuse; US desegregate:
We must integrate all the parts into a coherent whole. Several cultures have been well
integrated into our community."
Oxford Dictionary.
Commutative (1)
Relating to, involving, or characterized by substitution, interchange, or exchange. (2) Independent of order.
Used of a logical or mathematical operation that combines objects or sets of objects two at a time. If a ×
b = b × a, the operation indicated by × is commutative.
The Basics
The concept of commutative symmetry reflects a symmetry of natural laws where this symmetry
"...dictates that all particles in nature are split into two categories that behave differently in an ensamble. One category obeys the rule that particules of the same type, for example electrons, must avoid one another. According to these rules, identical particles may only be alone in a state. All the particles of this category are known collectively as fermions .... The other category is governed by precisely opposite rules, which not only allow but even dictate that identical particles concentrate densely in states. These particles are known as bosons." (Tarasov, 1986)
From an abstract perspective, in particular that of properties of differentiation versus those of integration, the characteristics of fermions and bosons reflect properties of differentiation and integration respectively.
Thus the act of differentiation reflects a focus on an individual to the exclusion of all others. In other words the focus is on an exclusive OR, A XOR B is the rule. (In particle physics this exclusion is formalised into the "Pauli Exclusion Principle")
The act of integration reflects a focus on grouping individuals such that they lose their individuality, they all sum to become an integrated whole. In other words the focus is on BOTH/AND-ness, A AND B is the rule.
In the human brain, and in the brains of all other neuron-dependent lifeforms, the hemispheres of the neocortex, the most developed part of the brain, reflect the SAME characteristics as those distributed above to 'differentiations' and 'integrations'. As such the more integrating parts of the brain, in most the more right side, reflect an A AND B perspective of reality - a sense of integration with reality where interactions are in the form of 'whole' responses to stimuli through the use of instincts and habits. To complement this more integrating 'side' is a more differentiating side, in most the left side, where the A XOR B perspective rules.
To continue with the analogy to fundamental particles, "...if a state is occupied by a fermion, then no other fermion of this type may be in this state....Moreover, [for bosons] the more dense is a given state populated the higher the probability that other bosons of this type will come to it. We have thus, on the one hand, clearly expressed individualism (leptons and baryons), and, on the other hand, as clearly expressed collectivism (photons and mesons)" (Tarasov).
Tarasov goes on to make the point that there are marked differences here in that for fermions there "exist conservation laws according to which the difference between particles and anti-particles remain unchanged...whereas for [bosons] there are no such laws"
The main point here is in the nature of electrons in that they are fermions and without their exclusive nature we would not have the diversity we see in elements of the periodic table. In other words we see here a fundamental property of differentiation - the determination of DIFFERENCE as compared to integrations that reflect more the formation of SAMENESS.
We can introduce another dichotomy here that reflects differentitations/integrations, difference/sameness, fermions/bosons and that is the dichotomy of BINARY/UNARY.
In the brain, the dynamics of though processes reflects the extraction from the unary of 'something' that is then encapsulated and analysed to become idealised and as such reflect a BINARY perspective, a clear distinction of A from NOT-A. In the brain this seems to reflect the extraction of PARTS from a WHOLE and the idealisation of those parts AS IF wholes.
This pattern is reflected in the dynamics of fermions/bosons where out of bosonic states can emerge a PAIR of fermions that then divide into discrete forms, e.g. the pair of electron/positron. In other words the realm of the binary reflects the realm of particle/anti-particle that, when joined express as a flash of light - a boson.
In current models of cosmology, as you cool down the universe so fermions will form into pairs to then 'dissapear' into a boson 'soup'. Bosons too will condense into whats is called a Bose-Einstein Condensate.
The processes of fundamental particles reflect the same processes of brain function in processing information where the binary perspective reflects a high energy perspective of A XOR B, an idealisation of 'something' that in turn can 'fall back' into the pool of information of the individual/collective/species (A AND B). The moment we extract this 'A' from the pool so out will pop its 'anti-particle' form as NOT-A aka B (e.g. yang automatically elicits yin).
In Chinese cosmology we find this pattern expressed hierarchically as:
T'ai Chi / Wu Chi (exaggerted, differentiated, particular/ not exaggerated, integrated, general or extension/lack of extension) - realm of general A AND B
Yin-Yang / T'a Chi (now T'ai Chi is the particular integrated whole and yin-yang acts to differentiate) - realm of particular A AND B
Yang / Yin (the pair 'split' into yang (differentiations bias, the light, positive, male) and yin (integrations bias - the dark, negative, female) A XOR B (or, to make it asymmetric, A IMP B [a implies B])
Dichotomy Mathematics
In Quantum mechanics the differences between fermions and bosons are reflected in the use of the 'wave equation' which immediately brings out another property of fermions and bosons and, by analogy, another property of differentiations and integrations, namely the property expressed as another dichotomy in the form of asymmetry/symmetry.
To partially quote and partially paraphrase Tarasov's comments on identifying the different equations for fermions and bosons, and so by analogy equations representative of the dynamics of the differentiate/integrate dichotomy, and by extension all dichotomies used by the brain to derieve meaning, we have:
y1 (I) as the equation of a wave function of particle I in state 1 and y2 (II) as the wave function of particle II in state 2.
What Tarsov labels as a microobject is a system of particles I and II. "The wave function of the microobject Y (I, II) can be expressed as the product of the wave functions of the constituent particles [and so a sense of BOTH/AND - A AND B]. Since the particles are assumed to be identical, it is then unknown which of them is really in state 1 and which in state 2 [and so "A" or "B" are undifferentiated as yet]. We will then have to take into account both y1(I)y2(II) and y2(I)y1(II)
(as if the particles in the microobject were continually changing their places [oscillation]). Commutative symmetry requires that the wave function Y (I, II) of the object meet the condition
|Y (I, II)|2 = | Y (II, I)|2 [this is the familar wave equation format where the square of the absolute value of A is the same as the square of the absolute value of B.]
Out of the combinations of the above products of single-particle wave functions we can construct two functions that meet this condition
YS (I, II) = y1(I)y2(II) + y2(I)y1(II)
and
YA (I, II) = y1(I)y2(II) - y2(I)y1(II)
The first of these is symmetrical, it does not change its sign under commutation : YS (I, II) = YS (II, I). This function describes a system of bosons. The second function is antisymmetric, it changes its sign under commutation of particles: YA (I, II) = -YA (II, I). This function describes a system of fermions. [And so reflects A/NOT-A dynamics]. This can be readily verfied. If we assume that both particles are in the same state, for example state 1, then it follows from the expression of YA that thus function vanishes. Hence this situation is impossible." (Tarasov).
By use of analogy we can note that (a) YS reflects properties of integration, the realm of the unary, and so of superpositions, whereas (b) YA reflects properties of differentiations, the realm of the binary, the differentiating.
We can conjecture that, since analysis of the differentiated shows us tight integration WITHIN the differentiation so from the brain perspective, our species nature is YS reflecting the integrated whole of species instincts and the universe. Emerging out of that, through the development of an ability to differentiate parts has come YA and contained within YA is another YS that could be considered to reflect consciousness in that consciousness allows for superpositions.
Overall what we see here is oscillations, brought down to the concrete of WHAT (YA ) and WHERE ( YS)
The final 'analogy' is to the development of fermions from bosons where this development process is of fermions emerging from a 'boson soup' into pairs and then into individuals. This process reflects the same development of ideas, of 'dot' precision, of 'object' thinking in our minds - we see the IDM-identified 'dimension of precision' at work where we move from the unary to the binary. Beyond the binary, the trans-binary, is the realm of complexity/chaos and so the possible source of consciousness, reflecting the oscillations in 'what-where' mentioned earlier.
The 'conflicts' we experience of consciousness nature vs species nature reflects the conflict of bosonic vs fermionic characteristics in different levels of the hierarchy that is our neurology. Thus our high precision experiments reflect the use of YA perspectives up against the species' perspective of YS This can cause paradox.
Note how in all of this the increase in energy levels ensures an increase in 'fermionic' processes, as reflected in the increase in analysis leading to an increase in differentiations. These in turn create more and more 'cuts', more and more borders and it is on borders that complexity/chaos rules.
In this page I have used the term 'analogy' here rather than 'simile' in there there is more here than just coincidence in that I believe we are seeing the adaptation of life to the universe through the recruitment and abstraction of universal principles in the brain and on into consciousness. As such, the dynamics of 'left brain-right brain' down to the level of 'axon-dendrites' reflects the same patterns as those of fermions and bosons, due either to:
(a) The structure of our neurons determine the structure of all information we can process and as such their adaptation to a LOCAL context means all we can ever 'see' in our models are 1:many formated dichotomies (differentiate precision is to ONE, integrate precision is to a PAIR) applied LOCALLY such that viewing beyond the local will still see the same formats regardless of any other possibility where that possiblity we will see as paradox.
OR
(b) The structure of the neuron reflects the recruitment and abstraction of properties of the Universe as a whole such that we can trace a development path from the 'big bang' to consciousness through the fundamental template dichotomy of differentiate/integrate.
The IDM material aims to flesh-out all of the possible interpretations of reality we can make using differentiate/integrate and so aid in developing a better understanding of ourselves and our Universe.
ref:
TARASOV, L., (1986) "This Amazingly Symmetrical World" Mir Publishers