From: Chris Lofting (firstname.lastname@example.org)
Date: Fri 19 May 2006 - 15:33:37 GMT
The subject line is the title of a paper by Physicist Eugene Wigner:
The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in
Communications in Pure and Applied Mathematics, vol. 13, No. I (February
1960). New York: John Wiley & Sons, Inc. Copyright C 1960 by John Wiley &
On-line at http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
In this paper he "argues that the capacity of mathematics to successfully
predict events in physics cannot be a coincidence, but must reflect some
larger or deeper or simpler truth in both."
Wigner writes: "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."
It is now 2006 and we are in a far better position to understand the content
of what Wigner was musing over.
Through the analysis of research in the neurosciences, cognitive science,
and psychology, we can demonstrate the derivation of specialist perspectives
from the general perspective of the species where the latter is determined
by our genetics to a GENERAL degree.
Given this genetic template or 'bedrock', exposure to local context,
together with the development post-birth of a refined sense of 'self', gives
us a methodology of taking the general perspective and relabelling the
associated categories of that perspective to fit local contexts and so
elicit unique, particular, singular perspectives.
This relabelling allows us to represent differences with the ONE set of
categories. The relabelling means that each specialist perspective comes
with its own lexicon to represent reality, but, due to the use of the
'universal' perspective of the species, the specialisation can also be used as a source of analogy/metaphor to describe any other specialisation.
Thus the set of 'universal' categories of the species can elicit 'resonance'
such that the language of one specialisation can link to that of the other
specialisation through what both languages are tied-to in the context of
As such, we all have a sense of 'wholeness' but to what we associate that
sense is determined by local context - and so the infinite number of labels
indicating some 'whole'. By being able to associate whole-X in context A
with whole-Y in context B we form analogies by then using their formal
What this methodology enables us to do is quickly describe some novel
context by reference to properties and methods of some known context and
that includes predicting elements of the new from knowledge of the existing.
As differences are discovered so we add new labels that eventually allow the
new context to develop its own language with which to describe itself.
Included in the set of specialist perspectives are Mathematics and Physics,
the main two disciplines identified in Wigner's paper.
To analyse this particular relationship, first we need to flesh-out some
core categories in Mathematics but from the position of the neurological,
the cognitive, and the emotional to give us a 'sense' of number.
An observed methodology of the brain in the processing of information is
through the use of the WHAT/WHERE dichotomy. The properties of this
dichotomy are isomorphic to the properties of the more abstract dichotomy of
differentiating/integrating and this abstract dichotomy has elicited much
thought by such researchers as psychologist, and originator of "Personal
Construct Psychology", George Kelly who wrote:
"Our psychological geometry is a geometry of dichotomies rather than the
geometry of areas envisioned by the classical logic of concepts, or the
geometry of lines envisioned by classical mathematical geometries. Each of
our dichotomies has both a differentiating and an integrating function. That
is to say it is the generalized form of the differentiating and integrating
act by which man intervenes in his world. By such an act he interposes a
difference between incidents -- incidents that would otherwise be
imperceptible to him because they are infinitely homogeneous. But also, by
such an intervening act, he ascribes integrity to incidents that are
otherwise imperceptible because they are infinitesimally fragmented. In this
kind of geometrically structured world there are no distances. Each axis of
reference represents not a line or continuum, as in analytic geometry, but
one, and only one, distinction. However, there are angles. These are
represented by contingencies or overlapping frequencies of incidents.
Moreover, these angles of relationship between personal constructs change
with the context of incidents to which the constructs are applied. Thus our
psychological space is a space without distance, and, as in the case of
non-Euclidian geometries, the relationships between directions change with
the context." (Kelly, 1969)
We note that, in Mathematics, the differentiating/integrating dichotomy
plays a major part in the setting the foundations of a mathematical
specialisation, Calculus. Furthermore, when we focus attention on the
general properties of differentiating/integrating so we can derive the
concepts of 'thingness', of 'objectness', from differentiating as we can
derive the concept of 'relatedness' from integrating.
If we formally identify these properties we have:
A sense of WHOLE
A sense of PART
A sense of Static relatedness (sharing space)
A sense of Dynamic relatedness (sharing time)
Cognitive analysis of types of numbers, detecting their qualitative nature,
identifies four basic qualities and so categories isomorphic to the above
A sense of wholeness - whole numbers (within which are primes (objects) and
A sense of partness - rational numbers (harmonics series)
A sense of static relatedness - irrational numbers (share space with
A sense of dynamic relatedness - imaginary numbers (share time with
Thus we can associate mathematical terms with qualities derived from the
neurology and our cognition. We can extend this into generic feelings and so
draw-in 'emotions' where the above properties can be described using feeling
A sense of wholeness - blending
A sense of partness - bounding
A sense of static relatedness - bonding
A sense of dynamic relatedness - binding
Composite terms are derived by mixing the above ('blending in a bonding
Thus the types of numbers in Mathematics correlate with the basic categories
sourced from 'mindless' differentiating/integrating that is fundamental to
our neurology abstracting sensory data. This SAME neurology applies to ALL
other specialisations and so the specialisation of Mathematics will be
'isomorphic' to all other specialisations derived from the activities of our neurology and that includes Physics. As such, all specialisations are metaphors for what the brain deals with - patterns of differentiating/integrating.
Since Mathematics, and other specialisations, have arisen 'ad hoc', so it
comes with perspectives that can be 'misleading' or lack clear
definition/understanding in their relation to neurological, cognitive, and
emotional processes and their representations. This lack of understanding
has allowed Mathematics to take on a life of its own and be treated as if
something 'magical', independent of our species.
The particular focus here, in the context of understanding what we are
dealing with, is on precision and on 'difference' where things 'orthogonal'
are considered to be statistically independent. A major emphasis is that of
'orthogonality' expressed in the use of Cartesian coordinates ( derived from consideration of measurement etc using vision). The most common format is the establishment of the X, Y, and Z dimensions. These dimensions, being determined in Cartesian format, are in fact representations of the dichotomy of +/-.
The establishment of the number 'line' is the setting-down of a basic
context and so we set the BASE context as the X axis and so a +/- dichotomy.
The implicit qualities of 'positive/negative' set a boundary to this
dimension and WITHIN this base context we introduce the Y axis and WITHIN
that XY context we introduce the Z axis. This focus manifests
self-referencing where it is the LABELS that differentiate +/- dichotomies.
(through use of those labels I don't need to 'see' anything)
This derivation of XYZ gives us the standard logic relationships of Z <= Y
<= X. In logic operators this represents the IMP (implies) operator where we work backwards - given Z it IMPLIES the presence of XY. Given Y it implies the presence of X. Given X we have X. There is hierarchy here reflected in the improvement in PRECISION as we move from X to XY to XYZ and onwards.
The three dichotomies will give us eight categories (2^3) where the
establishment of a 'clear' perspective means we distance the categories from
each other but retain the 'whole' that they all represent. This is done
VISUALLY through the formation of a meaning space that is a cube/octahedron
etc. Keep adding dimensions (aka +/- dichotomies) and our precision goes up
as a power of 2 - thus with 12 dimensions, when maximised in distance from
each other, form a dodecahedron (2^12 = 4096 possible categories)
The addition of SCALES to each dichotomy is equivalent to RECURSING the
dimension such that the '+/`' sense is quantified where the categories
derived are in the form of 'numbers' WITHIN the particular dimension
category (2Y is not the same as 2Z etc but we can also 'cut' the dimension
to give us finer and finer levels of precision within the dimension)
The DEGREE of scaling shows us recursion at work - this is the issue in
number line representation where at the level of WHOLE numbers we have
-4,-3,-2,-1 0 1,2,3,4... Add in PART numbers (and so rationals) and we have a new line with finer distinctions. ADD in RELATIONAL numbers, irrationals, and we have a new line with even finer distinctions. This is all examples of self-referencing.
For the representation of cyclic/morphic dynamics we have to add a PLANE
perspective through the use of imaginary numbers that allow us to
'transform' numeric representations over the plane as whatever we are representing is changing form/spatial-orientation etc.
Thus the XYZ nature of dimensions as lines shifts levels to that of
'dimensions as planes' etc (thus we go from real to complex to quaternions to octonions - the REPRESENTATION of these perspectives are in REAL number formats that develop along powers of 2 - and so that recursive focus:
REAL - (a,0)
COMPLEX - (a,b)
QUTERNION - ((a,b),(a,b))
OCTONION - (((a,b),(a,b)),((a,b),(a,b)))
EACH level repeats the previous but with finer differentiations and that
allows for 3D+ representations where the maximising of distance between all
categories gives us a lot of room to derive meaning from!
There is a focus on discretisation of Mathematics where the discretisation
of Mathematics is in the conversion from geometric perspectives to algebraic
perspectives - and so a shift from the dimensional to the dimensionless,
from an integrating perspective to a differentiating perspective.
At the level of the POINT we have high precision where, for example, a
tangent is the sharing of a point by line/circle. This is all representative
of information encoding by PHASE (and so constructive/destructive
interferences and so holograms) as compared to AMPLITUDE (photographs), as
it is a focus on use of positive and negative feedback.
All of these QUALITATIVE mappings, links to blending, bonding, bounding, and
binding (and their composite forms), allow one to FEEL meaning in the
quantitative and so build-up an intuitive experience. Since the same
qualities apply to specialist perspectives within Mathematics - e.g. algebra
vs geometry - so develops an intuitive sense of which specialist perspective
is the 'best suited' to describe some situation.
Thus all of the above 'grounds' Mathematics in us; Mathematics being a
specialisation of our neurological/cognitive focus in dealing with patterns
of differentiating/integrating. That said, the development of the neuron is
a development of a cell adapting to the environment and the indication here
is our neurology has 'copied' the main dynamics of that environment -
differentiating/integrating - and it is this adaptation that allows for our
maps to be so good in mapping both the local and the universal.
A quote at the above link is of Bertrand Russell:
"Mathematics, rightly viewed, possesses not only truth, but supreme beautya
beauty cold and austere, like that of sculpture, without appeal to any part
of our weaker nature, without the gorgeous trappings of painting or music,
yet sublimely pure, and capable of a stern perfection such as only the
greatest art can show. The true spirit of delight, the exaltation, the sense
of being more than Man, which is the touchstone of the highest excellence,
is to be found in mathematics as surely as in poetry.
--BERTRAND RUSSELL, Study of Mathematics"
How does this relate to our discussion? It so happens that another
specialisation used to interpret reality is that of emotion and emotion maps
to the SAME general, universal, categories derived above (for more on
emotions and blend, bond, bound, bind, see
http://members.iimetro.com.au/~lofting/myweb/emote.html ). Thus the
isomorphism of the various specialisations to the one general perspective of
the species means that each specialisation can be described from an
emotional perspective as it can a mathematical perspective.
I think that all of above goes towards answering Wigner's musing upon the
"unreasonable effectiveness of Mathematics" and brings out the " larger or deeper or simpler truth in both [mathematics and physics]" where we can see it is all perfectly reasonable and so understandable; there is no 'magic', there is just 'us' where we still have some way to go in getting a 'full spectrum' understanding of our nature, but we have advanced a lot since 1960.
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