Lepton, while I do applaud your attempt to use logic in all your arguments (especially considering how scarce logical thinking is these days) I have to criticize your use of "x+x=2x" as a reason as to why mathematics doesn't accurately model the world around us.
"Apple" like all other words in every human language is a generalization, the term "apple" does not take into consideration every individual apple's characteristics, if we were to do so without having to describe indefinitely (the apple has a mass of 152 grams, with a stick that tilts 30 degrees off a straight upward line etc to ensure only a single apple is individually described) we would have to come up with a new word for every single new thing we encounter so every object (every apple for example) would have a different name that pertains solely to it - this would be impractical to say the least. Instead of get caught in all the impracticality we instead generalize and say "apple" describes all objects with defined general characteristics. Similarly in the equation x+x=2x the variable x could be given the value of 1 apple (apple defined as above), since that is the only way the equation could be used relevantly to your example, x+x=2x is as descriptive as the sentence 1 apple plus 1 apple equals to 2 apples.
In short your argument suggests that math does not take into consideration the properties of the objects involved, my answer is that neither does language - you don't describe the exact mass of an apple you want if you want to send somebody to the shops, do you? Or the exact elasticity of a rubber band you want bought? Or the exact temperature and chemical composition of the soda you want. The reason we don't describe the exact properties of an object both in language and in mathematics is simply down to the fact that it would make both language and mathematics impractical tools of communication - humans never take into account all of an objects properties simply because it would be illogical to do so. I hope you see from my argument that the term apple is no more descriptive in a sentence than the variable x is in an equation. In your example above the friend is as descriptive as the equation because his sentence doesn't take into consideration all of the properties of the apple either - we don't do that in "real life" and so math is perfectly suitable for modelling "real life".
Anyway in reference to the physics perhaps string theory is correct and there are another 7 dimensions we simply fail to experience, and some particles are capable of experiencing and travelling in and out of these dimensions. In that case is it not possible that prior to the big bang the singularity (according to the inflationary theory) that existed existed in the extra-spatial dimensions and then when it expanded it formed a positive bias towards the space-time we perceive and that explains why the other dimensions (as believed by string theorists) are infinitesimally small.
Alternatively we could take into consideration the possibility that brane- theory is right and that a our membrane of space existed prior to what we call the big bang and it was simply populated with energy (latter converted to matter) by a collision with another brane. However is must be noted that both these theories have logical flaws as they lead simply to another level of creation that we cannot explain - the question of creation is perhaps unavoidably an example of the Munchhausen trilemma because it leads either to circular regression (the universe exists because it exists exemplified by brane theory), axiomatic regression (the universe exists because of something we must take as fact ie the unquestionable existence of a god or a singularity we cannot explain - shown by religion and "non-string theory" inflation) or infinite regression (where we can search for a cause ad infinitum).
However you should note that I am not a physicist, simply a 15 year old Kenyan teenager, therefore I am hardly an informed opinion. However you do not need a formal education to question, because everybody has the right, ability and perhaps even the duty to live an examined life.