For a number like 2 to exist, the implication is that there are two things in the universe that can be exactly equal.
Even if there are two or more of something (not a settled question in physics), the idea that you can add two such groups of two and have it be equivalent to some other group of four of that thing is only an abstraction in our heads.
In reality, if you have two apples in two pockets and I have four in mine, all we can say is that there are eight regions of the universe we like to call “apples”. 2+2=4 isn’t true in the same way that Sherlock Holmes’s address is because the former depends on a shared illusion/set of abstractions.
I'm not convinced. You are defining the number 2 as a material cardinal, and then coupling the existence of the number 2 to the existence of two completely identical material objects. But why would the existence of those objects give rise to the existence of the number 2?
Can't I similarly define that the number 2 is a materialized ordinal, and simply count the revolutions of a moving object, and state that the number 2 is instantiated by observing the periodicity of a single material object?
This has nothing to do with natural numbers. What does have to do with them is simple practical considerations like that you need two apples to treat two horses. This is why (small) natural numbers do, in fact, exist - in a very clear practical sense. And if you only have one apple and get bitten by the second horse who is now upset, you can only blame yourself for ignoring such an obvious fact.
But it is a style of thinking that is adopted because it is useful to human lives on human timescales, not because it is true.
The truth is there are no apples, no people, no objects even. There is just an immense swarm of subatomic particles interacting woth one another across the universe.
If you could observe the particles in that “apple” over a billion years instead of a more human timescale, it won’t make any sense to think of it as an object anymore. The particles started out in stars far and wide, are briefly frozen together in a fruit on earth, and will soon be spread apart widely again.
Some of the particles in the universal soup have combined into a chemical reaction that thinks it is you, and has evolved useful ways of perceiving patterns in other particles (for instance, object permanence). Because this phenomenon (you) is able to perpetuate itself by thinking in this way (perhaps by, among other things, finding and eating apples?), these thought patterns are sticky. Or in your words, practical. But not the whole picture.
Perhaps I am misinterpreting, but it seems to me like this would occur whenever you formed a coherent question. "How many apples are there?" would be insoluble, whilst "how many apples are there in this room?" or "how many red apples are there resting on that table?" would result in an answer belonging to the set of naturals.
So, given you have "one (element) and one different (element)", at that point, you have, ipso facto, 2 elements.
I can simply say there's only one's of apples. The existence of the number two is optional in this case. Its existence is completely up to my own mind (Distance? Likeness? Etc.). Prove to me in this case that the number two exists outside our own minds.