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In order to accomplish interesting things with a given discrete collection of symbols (standing for abstract identities), we need to consider arrangements of those symbols. People who have thought about formal systems in the past have shown a very strong tendency to conceive of symbols as visual characters, and arrangements as geometric arrays. Many logicians and computer scientists have limited their notions of arrangements entirely to discrete linear sequence of symbols. This prejudice toward linear sequences is highly unfortunate, and probably comes from a tendency to think of texts produced by typewriters. In principle, though, arrangements are much more general.
Essentially, arrangements introduce additional abstract identities besides the symbols' referents, to serve as structural elements in the arrangements, along with abstract relations among structural elements, and between structural elements and symbols. For example, a linear sequence has a structural element for every place in the sequence, a relation that specifies the unique first element, a relation that specifies the unique last element (which may be the same as the first), a relation associating each element but the last with its immediate successor, and finally a relation associating each element with the symbol that occurs at its position in the sequence. The system of addition in the section on examples used linear sequences of symbols.
Haskell B. Curry realized that most of the formal systems in mathematics are best defined in terms of arrangements into branching structures, called terms. A term, like a linear sequence, has abstract identities for places in the term, and a relation specifying the unique first element, called the root. But a term has no unique last element---rather it has one or more leaves. Instead of a successor relation, there is a relation associating each element that is not a leaf with a linear sequence of one or more other elements, called its children (or by some people, its arguments). Each element except the root must be the child of exactly one other element. The root must not be a child of any element. Finally, there is a relation associating each element with the symbol that occurs at its position in the term.
Gottlob Frege, a logician we might think of as the grandfather of modern symbolic logic, proposed a system of mathematical logic based explicitly on tree-shaped terms. He had a very difficult time convincing publishers to print nice pictures of these terms. After Frege, almost all logicians surrendered to typographers, and encoded arrangements that made more sense as terms into linear sequences, using parentheses and commas and similar punctuation symbols to indicate the tree structure. Haskell B. Curry compromised by providing sequential notations, but explaining carefully that they always referred to tree-shaped terms.
Terms are almost good enough for the arrangements of symbols in most of the studies of formal systems up to the present, but we should keep our minds open to more general sorts of arrangements in the future. To introduce new sorts of arrangements, we need only explain carefully what sorts of elements they contain, and what sorts of relations may be given among elements, and between elements and symbols. The combinator calculus described in the examples section used arrangements of the symbols S, K, and I, into terms.
Although we are highly confident that all sensible formal systems can be described in terms of finite discrete sets of symbols, allowing for infinitely many arrangements, we seem to be stuck with an infinite number of potential structural elements in arrangements. We might decide to generate the structural elements systematically as arrangements of some more primitive elements. But there appears to be no completely secure place to stop at a truly fixed finite collection of identities. We appear to have a true infinite regress of principles here, just as in Lewis Carroll's story, ``What the Tortoise Said to Achilles.''
The bad news about the need for infinitely many structural elements for arrangements is softened by the fact that we do not need to produce them all in advance. Whenever we want to study a larger arrangement than we have ever considered before, we need to somehow generate one or more new identities to serve as structural elements.
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