I am much interested in discussions such as Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". It's quite amazing that mathematics so well applies to our universe, and this raises many interesting questions which have been discussed here already, but I am interested in the converse as well: "There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world." (Lobachevsky)
So, then, my question is thus: what is known of the limits to the relationship between mathematics and physics. It seems that no matter the subject, it finds its way into a description of our reality. Is this so because we, as denizens of this universe, cannot imagine anything due to our mental structure being at some level composed of the very physical laws we wish to comprehend? Might alternative constructions of mathematics be more applicable in other universes, and we merely neglect their study, not seeing their worth? Or is there so deep, arcane tie-in between mathematics and nature, such that nature must effect each mathematically formalizable notion in some form or another. The last point is to say, how accurate was the second quote I posted? Is there a 'use' so-to-speak, a physical or social or cognitive implementation of any given arbitrary mathematical system? And what might I read on other's thoughts of why mathematics seems to hold such power on our minds and our world?
There's the idea that mathematics merely exists because it's a useful tool, though there is a certain grand structure in many areas of mathematics which start with merely few axioms, and the complexity just falls out of it naturally. Does our reality play catch-up with mathematics, or vice versa?