As a test, take a song that you know (preferably one that doesn't include words, so that they don't confuse things) and whistle the end of it. Now whistle the end again, but leave out the last note. Annoying as heck, right? It isn't just because you're used to how that song ends: it would be grating in the same way if I cut off the last note of almost any song and played it for you. That's because there are syntactic rules, hardwired into our brains, that govern how a song should end: in particular, it should end with the XXX of the key it's played in. This isn't a book about musicology, but there are lots of other rules like this, and generally we are as oblivious to them as we are to the rules of English syntax. Ignorant, that is, until the instant the rules are violated.
Recent studies using MRIs have shown that musical syntax, like linguistic syntax, is processed in Broca’s area. It is, moreso at least than any other area of the brain, the task master that sounds an alarm whenever we hear or say something that violates the patterns we expect. “I were walking” and the mangling of music both make the same part of the brain deeply uneasy. Talking or listening to music will involve extremely disparate parts of our mental toolkit, depending on what we’re talking about, the emotional content of the music, etc. But in both cases Broca’s area is running in the background, making sure that we are using our syntax correctly. Other parts of the brain, which understand the subject matter at hand, will sound the alarm if what is being said is grammatically-correct but ridiculous.
The relationship between music and other mental faculties has struck a chord in the public. The “Mozart effect” suggests that listening to certain classical music can improve spatial-temporal IQ and intelligence in general, and has made a lot of money for people selling “Baby Einstein” products that expose infants to Mozart. It looks like it’s all snake oil: the IQ boost is small and temporary at best. But the fact that it caught on suggests that people are aware that music resonates with our brain on a deep level. Anecdotally many thinkers have used music to calm their mind and stir their creative energies - Einstein, for example, was known to play the violin when he was stuck on a problem.
Math has its own syntactical rules, and they are no more complicated than English - in fact generally they're a *lot* simpler. The difference is that Broca's area doesn't instinctively handle calculus the way it handles English or Beethoven, and we are forced to do some of the work ourselves, on a conscious level. It looks like Broca's area lends a hand, especially with foundational skills like counting, but our conscious minds shoulder some of the burden. With training the syntax of math can become second nature too, but as far as I know the jury is out on whether it's done by Broca's area.
Working with a symbolic language is in general an extremely difficult task, and the only reason that we can do it so easily is that nature has primed us to pick up natural language like a sponge - it's an instinct. But this doesn't make us great linguists any more than spinning intricate webs makes a spider a great geometer, or flying at 200 kph makes a falcon an aeronautical engineer. Despite centuries of study, we still don't fully understand language intellectually - linguistics is a very active area of research. Most of us spend more time using language as a tool to talk about other subjects, rather than discussing it in its own right - but it's good to appreciate that we can only do that because Mother Nature has hardwired the ability into us.