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I am uncomfortable with the following statement in the draft of my paper:

[T]he unary system of representing $n$ by $n$ contiguous dots uses a singleton alphabet and requires counting rather than addition [emphasis mine] for evaluation of its words.

I am concerned readers might object to the italicized phrase, saying: "Hey, wait a second... how can you say that, when counting implies successive additions?"

For my own part, I see counting as iterated application not of addition but of the successor function, which as we know from Peano's axioms can be regarded as the foundation of addition.

Will the quote above from my paper get me into trouble with my readers? If so, what might an appropriate fix be?

JustAsking
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    I think a lot more context is needed. Is this philosophy, math, popular science writing, blog entry, homework assignment, etc.? And what level --- if popular science writing, then almost no one will have any idea what the Peano axioms are. – Dave L. Renfro Jan 05 '21 at 18:39
  • @DaveL.Renfro, I am writing a mathematical paper principally concerning what we might loosely call "the crossroads of formal languages and arithmetic." I have no aspirations beyond getting it published on arXiv, so presumably the audience will have some mathematical sophistication. But I want to make it as accessible as possible, so that it can be profitably read by an undergraduate student majoring in math or computer science. – JustAsking Jan 05 '21 at 18:50
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    At a purely formal level I think there is a distinction between adding $1$ (results from the application of a defined operation) and applying the successor operation (results from the application of a postulated operation), so maybe what you've written is OK. I'm thinking of "counting" as successive applications of the successor operation, although I'm not sure for some contexts (even when very formal) whether this is entirely accurate. I'm thinking that "counting" might be a meta-level action that doesn't really exist in the formal system's object language. – Dave L. Renfro Jan 05 '21 at 19:02

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Multiplication is repeated addition; that's the first indication you're wrong. In fact, addition is repeated counting. You'll know this if you've ever seen a small child who's playing a board game advance $5$ steps one at a time, rather than just moving straight from $12$ to $17$.

J.G.
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  • Well, I can certainly see where you stand on the question brought up in https://en.wikipedia.org/wiki/Multiplication_and_repeated_addition. – JustAsking Jan 05 '21 at 20:03
  • @JustAsking Well, multiplying by a non-negative integer, or adding a non-negative integer, admits my description; the latter is all that my answer needs. – J.G. Jan 05 '21 at 20:18