Math isn't NOT intuitive as far as math itself goes. It happens to be a very nice language. I tend to think that relativity and quantum mechanics do not mesh well because of the language we use to describe them. While standard maths is NP complete, it does not seem to inherently describe the universe without getting all crazy complex. I encourage you to try and comprehend even some of the simplest M-Theory equations. (not saying I do, but if math was really the language of the universe m-theory equations would be quite simple).
You know how in science and math people describe certain things as 'elegant' ? For example, E=mc^2 is very elegant. If math were elegant, you could get from here (higgs bosons) to there (atoms) with a simple equation which has been recursed x times to produce a mathematical model of an atom.
Math isn't bad, it's quite useful in fact. It's just not the natural language of the universe like everyone thinks it is.
There is a good possibility though that the universe is inherently complex in certain aspects for our brains.
Human brains are evolved to survive in the part of the physical world observable by human organs, not to comprehend the essential forms of the universe.
E=MC^2 could be just an aspect of reality that happens to coincide with the way observable physical world works.
Maybe a different math for describing the universe written by an alien intelligence with different sensory organs could be simple for them, not still incomprehensible for us. And that alien intelligence is just a jackpot hit in the evolutionary possibility space (and the jackpot may well have never been hit).
E=MC^2 could be just an aspect of reality that happens to coincide with the way observable physical world works.
All aspects of reality coincide with the way the physical world works. Or, what does that sentence mean?
Maybe a different math for describing the universe written by an alien intelligence with different sensory organs could be simple for them, not still incomprehensible for us.
I think there is a strong argument against this. That some alien race is really good at visualizing certain kinds of N-dimensional systems of some weird kind of space is plausible, but to throw all of mathematics out, I think, is not. They would have a mere superset of the mathematics we have.
In particular, take computation. Are aliens never going to deal with strings of units of data? If they do, right there you have the notion of their length, and of natural numbers. You have notions like concatenating strings. And the homomorphism between the two. And we're off on the road to abstract algebra.
It's very unlikely for smart aliens not to develop the same study of discrete math and abstract algebra that we do, unless they were never to use discrete units of information, and were only capable of processing infinite amounts of information at a time. Would they then have no use for the idea of associativity? Of proofs that consist of a finite set of symbols in a finite alphabet? Of proofs that consist of a finite set of symbols in an uncountably infinite alphabet? With any of these they would on the road to having a superset of our mathematical knowledge, rather than something different. We don't know how such a being would be physically possible, in the first place.
I think it would be more correct, and less exciting, to predict that some aliens might be smarter than us in certain ways.
All aspects of reality coincide with the way the physical world works.
I think observable is the keyword here... we happened to be able to observe enough that we can conclude that E = mc^2
As for your second point... I don't think I agree. The fact of the matter is that as humans, we are essentially unable to reason about how beings with a different set of senses would think. With respect to cosmology, we basically react to how light moves around and that forms the basis of our science (our other senses don't really come into the picture). Its very difficult to think about an intelligence based on some other source of information and impossible to know what sort of data would be present (since we don't have access to the information).
You know how in science and math people describe certain things as 'elegant' ? For example, E=mc^2 is very elegant. If math were elegant, you could get from here (higgs bosons) to there (atoms) with a simple equation which has been recursed x times to produce a mathematical model of an atom.
Math isn't bad, it's quite useful in fact. It's just not the natural language of the universe like everyone thinks it is.