Dear Florin,
though our both approaches to discover ultimate reality could not be much more different as they are yet, i want to mention one thing i realize as common in our both lines of reasoning. But let me say some more general things before i come to the point.
I made up myself with some (more or less) deep thoughts about the nature of maths and why it can be that our world can be so accurately described by an "abstract" system like math. For me, math is as relational as you supposed it to be. But what are the relations math deals with? Human relationships obviously deal with humans, does math therefore deal with *numbers*?
In my opinion, maths (without physics) is at first about quantities. It's about the relation between quantities and how this quantities can be transformed into one another. Therefore "rearrangement" seems quit natural for considering the nature of maths. It's about the relation between a one and the *seemingly* many. But in physics there seems to be also *qualities* in the sense of complexity and total different behaviour as consequences of total different quantities (and rearrangements) of some systems, subsystems and elementary systems. So we can surely conclude that rearrangements of quantities in some cases can abviously produce different qualities in our world, as far as human beings can conclude this out of their limited perspective that is biologically fixed. For example in chemistry, there are some fundamental entities (atoms) made out of some more fundamental entities (protons, neutrons, electrons, quarks). If you are right and nature is indeed infinite in complexity, what could this mean? Would this mean that under every "fundamental" level there exists one "more fundamental" level and so on infinitely with an infinite level of accuracy after the decimals? Well, the question is rethoric, because in yours approach the fundamentals are somewhat of math-like nature and started with bare quantities (numbers). But every number of them can theoretically be infinite in its own right. But how could this correspond to the physical world where dynamics and change govern the tide of events? More precisely: Is your view of maths, is ultimate reality just another variation of "anthropic" reasoning with respect to human observation of repeatability and lawfullness that can - for whatever reasons - be desribed mathematically?
What i want to say with that is that for me it is not quite clear how the relation between quantities and qualities in our world really looks like. And now i come to my point. A mathematical statement can tell us a (relative) truth about qualities which have to be related by an equality-sign in the abstract language of maths. So the left side is equal to the right side (and in pure maths at least vice verca too). An inequality for example of the Bell-type can only tell us how the things possibly *not* are. So mathematical transformations that quantitatively equal each other are in my opinion pure tautologies and therefore could be indeed interpreted as a change of the quality of a system from human perspective. The same is truth for my own approach with ultimate reality (see therefore my essay).
But a mathematical sentence does not express a thought in the sense of qualia, creativity and mindfullness. We only use mathematical sentences to mechanically conclude from sentences that do not belong to maths to sentences that also do not belong to maths. You maybe would say that that's not true in your framework because in your framework *all* sentences belong to maths. For the latter is your perspective on the whole issue.
Well, if your perspective would be true that maths is an infinite landscape and reality as well, i would perhaps agree. But i strongly assume that the very notion of "infinity" in all its historical and philosophical facets is the biggest contradiction of all: It defines itself by saying it has no definite borders. So i would say the very notion of infinity is a totally misleading tautological concept because it only says: infinity = undefined.
Insofar as you use the very concept of infinity for your approach i would say that you use an undefined space to define your lines of reasoning by an act of (honorable) creativity. For me, the latter could be the most important ingredient of reality that could have truly qualitative properties instead of only quantitative ones. But as you in my opinion said very clearly, only time perhaps can tell us. My personal conviction is, that the whole question about the ontological and epistemic status of quantum mechanics is indeed a deep issue of the relationship between quantity and quality, hence between defined and undefined parts of ultimate reality.