But the integers are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers. Looking only at the objects and not their operations 1 (integer) = 1 (rational) = 1 (real) = 1 (complex). Moreover, when we do account for the operations, we also see that 1 + 1 = 2 and 1 * 1 = 1 in every one of those systems. This isn't just a coincidence, of course; it's by design.
However, the way you arrive at 1 + 1 = 2 is not the same (though I suppose you could short-circuit the algorithm). Rational addition requires finding a common denominator, while integer addition doesn't. They achieve the same result when the inputs are integers, and again this is by design, but the process isn't the same. Ditto real addition vs. rational and complex addition vs. real.
In higher-level mathematics, the operations on the objects become definitional. We don't look at just a set of things, we look at a set of things and the set of operations upon those things. Thus "1 with integer addition and integer multiplication" becomes the object under consideration (even if it's just contextually understood) instead of simply 1. This is why they don't satisfy higher-level notions of equivalence, even if they intentionally do satisfy simple equality as taught in grade school.
Of course, the entire point of the submitted paper is to examine this in detail.
> But the integers are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers.
It depends on definitions, and, in some sense, the point of the common approach to mathematics is not just that one does not, but that one cannot, ask such questions. One approach is to look at natural numbers set theoretically, starting with 0 = ∅; to define integers as equivalence classes of pairs of natural numbers; to define rational numbers as equivalence classes of certain pairs of integers; and to define real numbers as equivalence classes of Cauchy sequences of rational numbers. In each of these cases there is an obvious injection which we are used to regarding as inclusion, but most of mathematics is set up to make it meaningless even to ask whether the natural number 1 is the same as the integer 1 is the same as ….
That is to say, if you're working on an application where encoding details are important, then you can and will ask such questions; but if I am writing a paper about natural numbers, I do not have to worry about the fact that, for some choice of encoding, the number 2 = {∅, {∅}} is the same as the ordered pair (0, 0) = {0, {0, 0}} = {∅, {∅}}, and in fact it is meaningless to test whether 2 "equals" (0, 0). The philosophy of studiously avoiding such meaningless questions leads some to avoid even testing for equality, as opposed to isomorphism; failing to do so used to be referred to in category-theoretic circles as "evil", although, as the nLab points out if you try to go to https://ncatlab.org/nlab/show/evil , it seems common nowadays to avoid such language.
This is not the point of the article. Even at the level of the objects themselves, 1 : integer is not 1 : rational. The latter is an ordered pair (1, 1) of two coprime positive integers, or an equivalence class of ordered pairs up to cancelling. Some ugly hackery is required to really make the integers equal to their respective rationals, and its consequences aren't great either (just imagine that some rationals are pairs while others are not -- that's what you get if you forcibly replace the rational k/1 by the integer k), and no one wants to do that.
However, the way you arrive at 1 + 1 = 2 is not the same (though I suppose you could short-circuit the algorithm). Rational addition requires finding a common denominator, while integer addition doesn't. They achieve the same result when the inputs are integers, and again this is by design, but the process isn't the same. Ditto real addition vs. rational and complex addition vs. real.
In higher-level mathematics, the operations on the objects become definitional. We don't look at just a set of things, we look at a set of things and the set of operations upon those things. Thus "1 with integer addition and integer multiplication" becomes the object under consideration (even if it's just contextually understood) instead of simply 1. This is why they don't satisfy higher-level notions of equivalence, even if they intentionally do satisfy simple equality as taught in grade school.
Of course, the entire point of the submitted paper is to examine this in detail.