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Is it just two marks/numerals representing a singular number 1, or are they actually two instances of 1?
And what about in a set with repetition such as {1, 1, 2, 3}?
And if these are actually multiple instances of 1, how could that be, when in the domain of discourse concerning numbers, numbers are individual objects, which would mean they can't be instantiated?
One can instantiate the numeral '1' in a sequence or multiset multiple times, in a set '1' only once, and the concept of 1 cannot be instantiated more than once ever, given how the concept 1 is grounded in Peano's Axioms, for instance.
In philosophy, you are talking about the type-token distinction and the difference between the particular and the universal. At this point, there's terminology enough to make language articulate.
There is the concept of one which might be understood alternatively as "oneness" and exists compositionally in a proposition.
There is a the numeral '1' which is the grapheme that arbitrarily or amodally represents the concept.
There may be alternative ways of writing the one, such as in a different typeface, and such a '1' would be an allograph.
In other languages, there are different graphemes for one, such as in roman numerals (I) and Japanese (一). We often call these characters. Thus, '1', 'I', and '一' are distinct graphemes or symbols entirely but represent the same concept.
In philosophy, the terms reference and sense are often used to describe the difference between the label and what is labeled, resp. There are other philosophical term like symbol and referrent by Bolzano.
In linguistics and the philosophy of language, there is the distinction of syntax which is the form we represent the concept with, and the semantics which is the concept itself, and often the difference in use-mention entails using delimitters so that one and 'one' refer to the semantics and the syntax, resp.
In mathematics, a sequence or multiset means that multiple occurrence of the same numeral or grapheme represents different instances. Thus (1,2,1,3,1) might be read as (11,2,12,3,13). Each of these can be respresented as existentially distinct using what mathematicians call an index set, where the '1' in the index set is a counting number.
And if these are actually multiple instances of 1, how could that be, when in the domain of discourse concerning numbers, numbers are individual objects, which would mean they can't be instantiated?
So now, we have enough language to explain. If you have a sequence of numbers, say composed of the naturals (1,2,1,3,1), each numeral '1' is a symbol that refers to the sense of 'oneness' which mathematically is often defined these days by Peano's Axioms. The instances of numerals are not the same because the sequence is indexed as a mapping of (1,2,1,3,1)->(1,2,3,4,5). Since the first numeral one is the first element in the sequence, and the second is the third, and the last is the fifth, we say there are three instances of the same numeral that represent three different elements of the sequence and can index them using the naturals as (11,2,12,3,13). We use the Indian-Arabic numeral system, but in the same concept of "oneness" would be just as viably discussed in the Roman number system with the sequence (I,II,I,III,I), and we could even index them with Roman numerals (Ii,II,Iii,III,Iiii) and it would make no difference conceptually, because the grapheme is an arbitrary assignment of sorts (what one might call historical accident). We could use binary numbers too! Instantiation or reification in the philosophical sense means that one merely is producing multiple tokens of the same type, which is pretty close to how computer science uses the term. So, how we refer to them as duplicates, occurrences, instances, reifications, etc. recognizes the basic distinction between the syntax of the symbol and the semantics of the symbol. For instance, in a sequence, the various numerals '1' have different positions, and therefore different properties. Were they to be drawn by hand, there would be differences of each relative to each other, just as there are differences in concept between the ordinal '1' and the 'cardinal' one conceptually. Thus, a domain of discourse is nothing more than the context for talking about '1', 'I', one (as an entity), and "oneness" (as a property) in any discourse, and that domain determines precisely what instantiation means which depends on each on particular context.
This is easy. Zoom in for a closer look.
Fig. A
In the figure I have zoomed in and circled and labelled both instances of 1 in the expression. I think I got all of them. Perhaps more become visible at higher magnification.
Here's a computer scientist's view:
"1+1" in this context is a mathematical expression, consisting of the following terms:
finally, applying addition to the values 1 and 1 evaluates to the value 2, leaving "2" as the final result of the given expression.
So, how many instances of 1 are present depends on whether you are counting values or references. Two symbols represent the value 1, but the value 1 itself is unique and, since the expression does evaluate to 2, I would argue that the value 1 is not present in the expression at all, because in order to consider the value of an expression you first need to evaluate it.
If there were only one instance of one in one plus one then how would one distinguish that from the one instance of one that appears in one by itself?
