The Unreasonable Effectiveness of Mathematics in the Natural Sciences

From: Chris Lofting (
Date: Fri 19 May 2006 - 15:33:37 GMT

  • Next message: Chris Lofting: "RE: RS"

    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 & Sons, Inc.

    On-line at

    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 eliciting 'meaning'.

    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 labels.

    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:

    Object oriented: A sense of WHOLE A sense of PART

    Relationships oriented: 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 properties:

    A sense of wholeness - whole numbers (within which are primes (objects) and composites (relationships))

    A sense of partness - rational numbers (harmonics series)

    A sense of static relatedness - irrational numbers (share space with another/others)

    A sense of dynamic relatedness - imaginary numbers (share time with another/others)

    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 terms of:

    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 context' etc)

    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 ). 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.


    =============================================================== This was distributed via the memetics list associated with the Journal of Memetics - Evolutionary Models of Information Transmission For information about the journal and the list (e.g. unsubscribing) see:

    This archive was generated by hypermail 2.1.5 : Sun 21 May 2006 - 03:39:58 GMT