Good objections, but I think they all have answers:
Of course it would, though it's underlying reality wouldn't be what we expect. (it might be a pointer in some program, instead of a collection of atoms)
But a pointer is information. At the physical layer, it's a configuration - when you say char *p=malloc(1); you haven't created an electron to sit in &p, rather you've configured existing electrons so as to have a pattern that is meaningful to one of your higher-level maps, but as a good reductionist you know that your higher maps don't represent anything ontologically basic about the territory.
As far as I know from any given perspective, existence is a binary quantity, something either is or isn't.
That's the point - I'm showing that the belief that 'existence' results from physical manifestation reduces to the absurdity that more physical manifestations of something means it exists more, which is nonsensical.
In Object-Oriented Programming, there's Classes which are a description of the instantiated objects, and there are the objects themselves, which you can do different things with than with the classes.
String a="5"; String b=a;
Does this make String("5") exist twice as hard? No; it doesn't even change whether it exists or not. It may be an object rather than a class, but it's still just a collection of information, which can be represented once, many times, or not at all, without ever changing the value of String("5").toInteger().
Again, how do you know there's no XML tag attached to the mathematics of our universe?
I don't know it, but such a tag would be epiphenomenal, and therefore it makes sense to assume it doesn't exist. If you're so sure that physical reality is manifest/instantiated, then how do you know, and why does it happen to follow regular mathematical laws? I think that believing in manifest physical reality is like believing in p-zombies (or, which comes to the same thing, dualistic consciousness).
The Tegmark IV hypothesis is exciting but I don't see how it's supposedly so self-proving as its proponents suggest.
It's not self-proving - at least, I don't claim it so - it's just simpler (shorter message length) to say "{equations}" than to say "mass-energy and space and time, which follow {equations}", and thus in the absence of evidence for the existence of manifest reality (which evidence I do not believe to be possible) a Bayesian, taking an Occamian prior, should conclude that the former is more likely than the latter. The only reason we mostly believe the latter is that, like Copenhagen, it came first.
EDIT: also, my views are distinct from (and were formed before I read) Tegmark IV; in particular his paper makes no mention of the concept of instantiation (and seems to me to be assuming that all Tegmark worlds are physically instantiated), and he wants to limit his hypothesis to computable structures, which I consider unnecessary and unjustified - I'm happy with the existence of structures so transfinite we can't even imagine them, perhaps because I'm a mathematician rather than a physicist.
but as a good reductionist you know that your higher maps don't represent anything ontologically basic about the territory.
I know that -- because as a good reductionist I recognize that my map is a different portion of the territory, that portion of the territory which resides inside my head and which is merely correlated to the actual territory outside of my head.
But tsn't that the opposite of what you and Tegmark IV are saying: namely that the maps we call "equations" represent something ontologically basic about the whole of the territory? ...
Follow-up to: Syntacticism
I wrote:
In my experience, most people default1 to naïve physical realism: the belief that "matter and energy and stuff exist, and they follow the laws of physics". This view has two problems: how do you know stuff exists, and what makes it follow those laws?
To the first - one might point at a rock, and say "Look at that rock; see how it exists at me." But then we are relying on sensory experience; suppose the simulation hypothesis were true, then that sensory experience would be unchanged, but the rock wouldn't really exist, would it? Suppose instead that we are being simulated twice, on two different computers. Does the rock exist twice as much? Suppose that there are actually two copies of the Universe, physically existing. Is there any way this could in principle be distinguished from the case where only one copy exists? No; a manifest physical reality is observationally equivalent to N manifest physical realities, as well as to a single simulation or indeed N simulations. (This remains true if we set N=0.)
So a true description requires that the idea of instantiation should drop out of the model; we need to think in a way that treats all the above cases as identical, that justifiably puts them all in the same bucket. This we can do if we claim that that-which-exists is precisely the mathematical structure defining the physical laws and the index of our particular initial conditions (in a non-relativistic quantum universe that would be the Schrödinger equation and some particular wavefunction). Doing so then solves not only the first problem of naïve physical realism, but the second also, since trivially solutions to those laws must follow those laws.
But then why should we privilege our particular set of physical laws, when that too is just a source of indexical uncertainty? So we conclude that all possible mathematical structures have Platonic existence; there is no little XML tag attached to the mathematics of our own universe that states "this one exists, is physically manifest, is instantiated", and in this view of things such a tag is obviously superfluous; instantiation has dropped out of our model.
When an agent in universe-defined-by-structure-A simulates, or models, or thinks-about, universe-defined-by-structure-B, they do not 'cause universe B to come into existence'; there is no refcount attached to each structure, to tell the Grand Multiversal Garbage Collection Routine whether that structure is still needed. An agent in A simulating B is not a causal relation from A to B; instead it is a causal relation from B to A! B defines the fact-of-the-matter as to what the result of B's laws is, and the agent in A will (barring cosmic rays flipping bits) get the result defined by B.2
So we are left with a Platonically existing multiverse of mathematical structures and solutions thereto, which can contain conscious agents to whom there will be every appearance of a manifest instantiated physical reality, yet no such physical reality exists. In the terminology of Max Tegmark (The Mathematical Universe) this position is the acceptance of the MUH but the rejection of the ERH (although the Mathematical Universe is an external reality, it's not an external physical reality).
Reducing all of applied mathematics and theoretical physics to a syntactic formal system is left as an exercise for the reader.
1That is, when people who haven't thought about such things before do so for the first time, this is usually the first idea that suggests itself.
2I haven't yet worked out what happens if a closed loop forms, but I think we can pull the same trick that turns formalism into syntacticism; or possibly, consider the whole system as a single mathematical structure which may have several stable states (indexical uncertainty) or no stable states (which I think can be resolved by 'loop unfolding', a process similar to that which turns the complex plane into a Riemann surface - but now I'm getting beyond the size of digression that fits in a footnote; a mathematical theory of causal relations between structures needs at least its own post, and at most its own field, to be worked out properly).