Step 1: We start by believing in the bare minimum : our own subjective experience exists. This is the only thing we know to exist. The existence of other things can only be inferred. And to deny your own existence, you would have to exist in the first place. So it can't be denied
Step 2: We notice that our perceptions have a distinguishability property. For instance, we notice that distinguishable feelings of colors exist. Distinguishable feelings of sounds exist. This forms the basis of the idea of "collections" of distinguishable things.
Step 3: We start studying the rules by which our perceptions behave. We first map our distinguishable perceptions to distinguishable symbols, and then we study the rules of behavior of perceptions in terms of these symbols. This is physics. To test physics, we map these symbols back to our perceptions.
Here we also note that this map isn't information preserving. For example, we may map distinguishable colors to real numbers, and distinguishable sounds to real numbers. But there is nothing about real numbers that tells us how the perception of sounds is different from the perception of colors.
Linguistic maps only preserve the information about the distinguishability of perceptions. They lose the information about what perceptions are, which is something incommunicable using language.
Step 4: We study the behavior of other people, who so far only exist in our perceptions. We conclude that their behavior is similar to us. For example, we may find a pin-prick painful and also observe a similar behavior in other people. We may also see ourselves in a mirror to get a third person perspective of ourselves like what we have of others.
Here we take a leap and conclude that other people also have subjective perceptions like us.
Step 5: We observe other people reacting to things that are outside their bodies. For example, in the pin-prick, we can see that the pin is located outside the other person's body.
Now since we are identical to other people according to Step 4., we conclude that we are also reacting to things that are located outside our bodies. Our qualia like that of pain are correlated to outside sources.
Step 6: Then the physics that we came up with in step 3 actually describes an objective world out there. The outside world has distinguishable "objects". These objects also include peoples' brains.
The outside world is mathematical. Its nature may be a computational structure like cellular automata, or a more advanced set theoretic structure like General Relativity. We don't know the final model yet. These structures also include the human brain.
Step 7: Ever since step 3, we knew that the mathematical map was not information preserving. Qualia are the only thing whose existence we can be sure about since Step 1. The existence of other things is something we've inferred using leaps of faith.
Since the map isn't information preserving, we conclude that the mathematical structure does not encompass all that we know to exist.
So, in the end, we have two fundamental aspects of reality:
Subjective experience and qualia which we know to exist since the beginning. These can't be translated into language.
The objective mathematical reality, whose existence we have inferred using tiny leaps of faith.
Which steps in my argument are controversial/wrong and why?