The GP point is correct; we implicitly convert between all these representations naturally and quickly, but there are interesting branches of mathematics that consider those conversions explicitly and find nuances (eg, category theory).
Traditional Shadow from Sunlight: A tree casting a shadow in bright sunlight during the summer, characterized by the typical darkening on the ground.
Snow Shadow: Occurs in winter when a tree intercepts snowflakes, resulting in an absence of snow under the tree despite being surrounded by snow. This is likened to a shadow because it mimics the visual absence typical of shadows, even though caused by a different kind of blockage.
Rain Shadow: Explained with the example of the Cascade Mountains, where the mountains block rain clouds leading to dry conditions on the leeward side, effectively creating a "shadow" where rain is absent due to the geographical barrier.
Car Shadow by a Truck: Described during highway driving, where a large truck casts a "shadow" by preventing cars behind it from passing. This "shadow" is the area in front of the truck where cars tend to stay clear due to the difficulty in passing the truck.
Shadow Cast by England Due to the Gulf Stream: This is a metaphorical shadow, where England blocks the warm Gulf Stream from reaching certain areas, resulting in colder temperatures on one side, while allowing it to pass above to northern Norway, warming it. This is referred to as the "shadow of England" on the coast of Norway, influenced by the flow of water rather than light.
These examples use the concept of "shadow" in various physical and metaphorical contexts, showing disruptions in different types of flows (light, snow, rain, traffic, and ocean currents) caused by obstructions.
For example, the "2" in "2π" is not the same type of "2" as in x^2 or 2x generally. Yet, physicists (to pick a random group) will blend in these factors, resulting in nonsense. As a random example, one of the Einstein field equations has "8π" in it. Eight what!? What aspect of the universe is this counting out eight of -- a weirdly large integer constant? This actually ought to be "4(2pi)", and then "4" is the number of spacetime dimensions, which makes a lot more sense.
Similarly, in at least one place the square of the pseudoscalar (I^2) was treated as a plain -1 integer constant and accidentally "folded" into other unrelated integer constants. This causes issues when moving from 2D to 3D to 4D.
The best example is perhaps the polynomial ring R[x][y], which consists of polynomials in the variable y over the ring of polynomials in the variable x over the real numbers. Any algebraist would tell you that it is obviously just the two-variable polynomial ring R[x, y] in disguise, because you can factor out all the y-powers and then the coefficients will be polynomials in x. But the rings are very much not the same at the level of implementation, and every time you use their "equality" (canonical isomorphy), you need to keep the actual conversion map (the isomorphism) in the back of your mind.
That by itself isn't a problem. But making all the other confusions you mention is a problem.
When you construct numbers using sets under ZFC axioms or inside lambda calculus what you get is representation. But 1 is just 1.
1 ∈ Z and ASCII '1' can similarly be seen as corresponding in terms of having the same glyph when displayed. But of course, there are far fewer meaningful operations common to the two.
Saying that 0 belongs to 1 is false no matter what one uses to represent those numbers in any ZFC formalisation of numbers.
It’s a map-territory distinction.
On a more serious note, if you are of a certain philosophical bent you may believe that the natural numbers have an existence independent of and outside of the minds of humans. If so, 1 is presumably not a set, even if we don’t fully understand what it is. I certainly don’t think of it as a set on a day to day basis!
But others may deny that the territory even exists, that all we have are the maps. So in this one map, 1 is a set containing zero, but in that other map, it is something different. The fact that all the different maps correspond one-to-one is what counts in this worldview, and is what leads to the belief – whether an illusion or not – that the terrain does indeed exist. (And even the most hard nosed formalist will usually talk about the terrain as if it exists!)
But this is perhaps taking us a bit too far afield. It is fortunate that we can do mathematics without a clear understanding of what we talk about!
For that matter, how do we know what infinite sets like Z and Q 'literally are', without appealing to a system of axioms? The naive conception of sets runs headlong into Russell's paradox.
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.
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.