Occurrences of numerals

Numerals—strings of characters called digits—are types not tokens, using Pierce's type-token distinction. Written tokens of numerals are visible and can be destroyed: erased, eradicated, incinerated, and so on. In senses used in this article, there are exactly ten digits—0,1, …, 9—, each a one-digit numeral. All longer numerals are concatenations of shorter numerals, where ‘concatenation’ is used in the sense of concatenation theory or string theory. The digit 1 occurs in infinitely many numerals including 1, 12, 123, and so on, each having exactly one occurrence of 1. The numeral 12 occurs in infinitely many numerals including 12, 123, and so on, each having exactly one occurrence of 12.

In this article, we follow the common practice of using each numeral autonymously to denote itself: as 12 is a two-digit numeral. Of course, it is equally correct, and in some contexts preferable, to use the numerals heteronymously to denote numbers: as 12 is a number, namely twelve. Then one makes heteronymous names for the numerals—italic-names, quote-names, or phoneticals: 12, ‘12’, and wun-tu, respectively. Here the numerals are never used heteronymously to denote numbers.
According to standard terminology, one given numeral occurs in another iff the latter contains an occurrence of the former. No occurrence of 1 occurs in any numeral: the digit 1 itself is what occurs. The expression ‘an occurrence of an occurrence’ is incoherent. Also, one given numeral is a sub-numeral of another iff the former occurs in the latter. The occurrence relation—expressed by the relational verb ‘to occur in’—is reflexive, anti-symmetric, and transitive.
12 occurs twice in 12312: the first occurrence of 12 in 12312 precedes the occurrence of 3; the second occurrence of 12 in 12312 follows the occurrence of 3.
Every numeral has a unique positive length—the number of its digit occurrences—and each occurrence of a numeral has the length of its numeral. The length of an occurrence is the number of places it occupies in its numeral. Further, every occurrence is the nth occurrence of a certain numeral in another certain numeral, for a certain positive integer n.
What is “an occurrence”? As detailed in various places including the article type-token distinction, some authors have accepted as answer The Prefix Proposal that a given occurrence of a numeral in a given numeral be identified with the initial sub-numeral ending with the given occurrence. This means that the first occurrence of 12 in 12312 is 12 itself and the second occurrence of 12 in 12312 is 12312 itself. It further means the first occurrence of 12 in 12312 occurs in the second occurrence of 12 in 12312. Finally, it means that the length of an occurrence of a numeral depends on the numeral it occurs in and its order of occurrence: the first occurrence of 12 in 12312 is a two-digit numeral whereas the second occurrence of 12 in 12312 is a five digit-numeral.
There are other awkward consequences of the Prefix Proposal. The above and other awkward consequence of the Prefix Proposal are eschewed by The Prefix-suffix Proposal that a given occurrence of a numeral in a given numeral be identified with the ordered pair consisting of the initial sub-numeral ending with the given occurrence plus the final sub-numeral beginning with the given occurrence. This means that the first occurrence of 12 in 12312 is <12:12312> and the second occurrence of 12 in 12312 is <12312: 12>. It further means that it would be wrong to say that the first occurrence of 12 in 12312 occurs in the second occurrence of 12 in 12312. Nevertheless, the Prefix-suffix Proposal has its own awkward consequences.

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