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This page is under construction, June 2013.
Symbolic and Iconic Formal Systems
Iconic arithmetic incorporates the Additive Principle. Additive numeric systems have been in use since the dawn of history. An additive system is one in which the representation of a sum is the same as the representation of the parts. When we place coins in our pocket, we use an additive system; the value of the collection of coins in our pocket (the sum) is equivalent to the value of the coins taken individually (the parts). Additive systems usually have a physical interpretation, however they can also be abstract — an example is the tally system, which uses identical marks to indicate how many. In contrast, a symbolic system requires rote memorization of both representations and algorithms, since by design the representation of concepts is independent of their meaning. Additive systems are graphic and intuitive, while symbolic systems are typographic and counterintuitive. While the intent of a symbolic system is to completely separate semantics from syntax, the intent of an additive system is to maintain a close connection between visceral understanding and representation.
The above figure illustrates how representation approaches reality as we move from symbolic to spatial models. The word “house” bears no resemblance to what it describes. Even a skeleton sketch of a house conveys far more information and is far easier to understand. As we move to the right past a cartoon sketch, an idyllic painting, a three-dimensional model and a photograph of an actual house, what we mean by a house becomes clearer and easier to follow. We can’t show an actual house on the page, that is where meaning meets actuality.
Even within symbolic syntax, some representations intentionally carry a visual meaning within their structure, while others intentionally do not. For example, some of the different ways people write the concept three include:
Some of these representations are iconic since they embody “threeness”, others are abstract since we would have to be told that the particular representation is meant to identify three. Toward the right we can see that mathematics itself provides an unlimited number of representations for three. The last example is vonNeumann’s construction of whole numbers from the empty set.
Symbolic systems are one-dimensional textual representations that rely on the sequence of symbols or tokens for meaning. Iconic systems rely on appearance rather than sequential ordering. Decimal digits lose the additive property when they stop looking like the quantity they represent. Place-value numeration loses the additive property by focusing on the maintenance of place-value alignment and carrying (digit overflow). Conventional algebra is completely symbolic, incorporating spatial and additive systems only anecdotally. Symbolic algebra includes rules that permit transformation of the sequence of symbols and that permit replacement of some sequences by specific symbols. For example, the sequence
For example, the rule
The axioms and transformation rules for additive iconic systems differ substantively from those of group theoretic symbolic systems. This difference in conceptual foundations has been largely obscured by placing additive and iconic math within the domain of specific concrete manipulations, while placing symbolic math within the domain of general abstract concepts. Iconic systems are thought not to be formal and rigorous. This opinion is incorrect. Iconic math provides a formal basis for number systems that is based on the additive principle. The principles of iconic math cover all operations in arithmetic and algebra, and have been extended into several other areas of mathematics such as knot theory, elementary differential calculus, and propositional and predicate logic. Work on iconic systems has led to the development of tools that permit side-by-side comparison of iconic and symbolic representations and that can identify and evaluate sources of errors and misunderstandings. Quite possibly, combining both symbolic and iconic math may improve the math performance of novices. But there exists a very substantial obstacle: the implicit rules that the mathematical community uses to define mathematics exclude iconic systems.