Towards reframing the implications of the mathematics of remainder

Reintegration of a Remaindered World (Part #4)

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Remainder: The nature of a remainder is an early feature of the teaching of mathematics to children. It is the amount "left over" when dividing two integers, namely whole numbers (whether positive or negative). Thus in dividing 12 by 8, the amount left over is 4. The result of the division can also be expressed as 1.5 -- using the decimal point as an indicator of what proportion of 8 "does not fit" into its division into 12.

When approximating a value by a mathematical series, the remainder is the error (the amount "left over") of an approximation, such that true value = series approximation + remainder term. It is assumed that the divisor is non-zero (although that case could be interpreted as being of larger significance, as discussed below). Also of relevance is that the value of so many irrational numbers (in principle all such numbers) can be approximated in rational terms through a dynamic iterative process which always implies a remainder. This gets progressively smaller as the number is better approximated.

Understanding of "remainder" becomes of wider relevance in the light of the manner in which conceptual models and strategies are typically articulated into elements (notably as "bullet points" in slide presentations). These may be numbered by integers -- implying distinct "wholes" into which the totality is broken down. Put otherwise, the consequence is that when a systemic "model" is applied to "reality", there is every probability that a range of (unnumbered) factors may be omitted or neglected as irrelevant -- whether deliberately or unconsciously. These then constitute the "remainder" -- as notably illustrated by the recent treatment by the Intergovernmental Panel on Climate Change of the population factor in the Kaya Identity.

Unknown remainders: The issue is especially significant when the implication is that the framework is in some way "comprehensive" or "global", such that "everything" is somehow subsumed by it (as a "summation", if not a "consummation"). It is potentially even more significant in the case of any assumption that a framework is "universal". What might be held to be omitted from any Theory of Everything? Clearly it is unlikely that the neglected factors would be explicitly quantified -- rather than being understood as "numberless" . In this sense such a "remainder" is essentially qualitative, although the (im)possibility of numbering may be partially recognized in the notorious strategic recognition by Donald Rumsfeld of the known unknowns -- potentially conflated with his "unknown unknowns":

There are known knowns; there are things we know we know.
We also know there are known unknowns;
that is to say we know there are some things we do not know.
But there are also unknown unknowns -
the ones we don't know we don't know.

(discussed in Unknown Undoing, 2008)

It could then be said that it is the combination of "known unknowns" and "unknown unknowns" which constitutes the "remainder". However, being unquantified, they are strangely to be understood (through their conflation) as potentially both "less" than the "known" (incorporated into the model) and "larger" in implication than the knowledge offered by the model (if essentially partial). The situation calls for a more complex form of mathematics.

Modulus: The mathematical concept of modulus suggests further implications. Mathematically A is said to be congruent to B modulus C, if A divided by C and B divided by C have the same remainder. C is called the modulus of congruence. Stated otherwise, for a positive integer C, two integers A and B are said to be congruent modulo C, if their difference A ? B is an integer multiple of C. The number C is thus called the modulus of the congruence. Modular arithmetic enables different integers (such as A and B) to be handled mathematically by introducing such a congruence relation.

This suggests the possibility  of exploring some form of fruitful "modular relationship" between different integrative conceptual frameworks (as "integers"), each with distinct claims to be "comprehensive" -- with each necessarily "failing" in that presumption and thereby engendering a "remainder". Is there then the possibility of a "modular arithmetic" between models of reality, identifying a form of "congruence relation" between them? This would offer a powerful means of interrelating frameworks otherwise considered as incompatible and incommensurable.

Qualitative vs Quantitative perspective: Peter Collins (Mathematical Dimensions and Psychological Development (1), Spectrum of Mathematics, 5 October 2011; Mathematical Dimensions and Psychological Development (2), Spectrum of Mathematics, 12 October 2011) argues that, properly understood, every number expression represents a dynamic interaction as between a a base quantity and a dimensional number (that is relatively of a qualitative nature):

So what we might refer to in conventional (Type 1) terms as the number quantity 2, more accurately is expressed as 2¹ (where 2 is quantitative and 1 -- relatively -- of a qualitative holistic nature). However because the very nature of linear (1-dimensional) understanding is to reduce qualitative to quantitative type meaning, from a Type 1 perspective, interpretation of numbers is invariably reduced in a mere quantitative manner. However when correctly appreciated in holistic Type 2 terms, number expressions have an intimate bearing on the interpretation of all the main stages on the spectrum of psychological development.

Amplifying on this argument (in a private communication), Collins argues (with respect to the above example of dividing 12 by 8, that the remainder is 4, that the result could also be expressed as a fraction as 1):

The very nature of the conventional quantitative approach to mathematics is that it is defined in linear (1-dimensional) terms where qualitative meaning is thereby reduced to quantitative. Now the movement from the whole notion of 2 as an integer to the inverse part notion of can be expressed as 2^-1.

So 2^-1 = 1/(2¹) = .

The fascinating feature then from a qualitative perspective is that this very process of obtaining the reciprocal part, entails the dynamic negation of rational linear understanding. And the negation of what is rational (and conscious) thereby entails the corresponding movement to what is intuitive (and unconscious). Therefore in the very dynamics of moving from whole to part notions in experience (and in reverse from part to whole), intuitive understanding is implicitly required to enable a successful transition to take place.

However because in formal terms the nature of conventional mathematics is defined merely in a linear rational manner, this entails that the qualitative intuitive dimension is ignored with subsequent interpretation taking place in a reduced quantitative manner. And such reductionism then leads to continual fragmentation of experience (in quantitative terms).

So when we view reality in a detached impersonal manner, the (part) remainder which does not readily fit in with one's rational perspective can be thereby excluded -- literally -- as wholly irrelevant. [emphasis added]

Linear vs Circular perspective: Collins then argues:

I have been long fascinated by the fact that when we raise 1 to a simple fraction (such as 1/3) that we get a switch from a linear to a circular notion (with the result lying on the circle of unit radius in the complex plane). The deeper explanation of why this is the case actually resides in the fact that here the base number 1 and the dimensional number 1/3 are actually quantitative and qualitative with respect to each other.

Therefore in a more accurate experientially refined approach to mathematics we would no longer understand numbers as static quantities but rather as dynamic interactive entities always entailing two complementary aspects that are - relatively - quantitative and qualitative with respect to each other. And the interaction of these aspects enables continual transformation in the nature of number to take place.

Sets: These concerns follow from earlier exploration of sets of preferred sizes (Representation, Comprehension and Communication of Sets: the role of number, 1978; Patterns of N-foldness; comparison of integrated multi-set concept schemes as forms of presentation, 1984).

The preoccupation relates to the issue of the "chunking" capacity of human memory as highlighted in one of the most highly cited papers in psychology (George A. Miller, The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review, 1956). Related issues have been explored from the perspective of cognitive psychology by George Lakoff and Rafael N??ez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

Topology and remainder: It is appropriate to note in passing the possibility of further insight from the preoccupation with remainder in the field of topology (A. V. Arhangel'skii,  Two Types of Remainders of Topological Groups, Commentationes Mathematicae Universitatis Carolinae, 2008; Peter Collins, Extensions of topological spaces with strongly-discrete remainder, Topology and Appl. 1999). The implication of "compactification" would appear relevant to the above argument (A. V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl., 2005).

The term "sobrification" is introduced in this connection by Rudolf-E. Hoffmann (Sobrification of partially ordered sets, Semigroup Forum, 17, 1, pp. 123-138; On the Sobrification Remainder, Pacific J. Math., 83, 1, 1979, pp. 145-156):

... the notions of "ideal" and "idealcompletion" of an (upper semi-) lattice may be considered as special cases of a construction from set-theoretic topology, the "sobrification" of a space

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