Clarification about the concept of number

Question

I am reading a book called Numerical Notation: A Comparative History (by Stephen Chrisomalis).

The first chapter (Introduction), second and third paragraph go like this:

If you look up from this page and examine your surroundings, I am certain that you will encounter at least one instance of numerical notation, probably more. Moreover, unless you have a Roman numeral clock nearby, I am nearly certain that all of the numerals you encounter are those of the Hindu-Arabic or Western 1 system. Numerals serve a wide variety of functions: denotation – “Call George, 876 – 5000”; computation – “21.00 × 1.15 = 24.15”; valuation – “25 cents”; ordina- tion – “1. Wash dishes, 2. Sweep fl oor, 3. Finish manuscript”; and so on. Most of the thousands of numerals we see each day barely register on our conscious minds; regardless, we encounter far more written numbers in our lifetime than we do sunsets, songs, or smiles. Until the past few centuries, the opposite was true for most people.

These ten digits are so prevalent that it is easy to equate our numeral-signs with the set of abstract numbers. In this view, 62 does not merely signify the abstract concept “sixty-two” – it is the raw form of the number itself, the stuff of pure mathematics (or perhaps pure numerology). That these signs are frequently encountered and used in mathematical contexts contributes to the prevalence of such attitudes. According to this view, our numeral-signs constitute abstract number, and other systems (when recognized as such) are simply archaic devia- tions from the abstract entity comprised by these signs.

My question is about the line

In this view, 62 does not merely signify the abstract concept “sixty-two” – it is the raw form of the number itself, the stuff of pure mathematics (or perhaps pure numerology).

Aren't "the abstract concept “sixty-two” and "the raw form of the number itself, the stuff of pure mathematics" the same thing: the idea of 62: that which is common between 62 cats and 62 stones? If not what are they referring to?

Answer 1

The string $62$ is distinct from the string $50 + 12$. But the two strings represent a single number. I think the author is suggesting that people often conflate numbers with strings that represent numbers. I read the line with the following emphasis:

In this view, 62 does not merely signify the abstract concept “sixty-two” – it is the raw form of the number itself, the stuff of pure mathematics (or perhaps pure numerology).

Anecdotally, it seems true to me that people do indeed conflate numbers with strings. I suspect that this underlies the common discomfort with the fact that $0.999\dotso = 1$. The two strings $0.999\dotso$ and $1$ are not equal, and people sometimes fail to draw a distinction between a number and its decimal expansion(s).

Answer 2

As @Daniel Fisher already noted, the authors are stressing the difference between the sign and the signified.

When I sign my name at the bottom of this document, I am not putting myself there, and if you happen to delete this document, no harm would come to my physical self. Likewise, when I'm writing '62' here, it is a sign, pointing at the abstract concept (the number 62). It is not the abstract concept which is written down, only a sign referring to it.