Well, one can think of, say, natural numbers as a mystical "system of symbols" but I find them extremely close to the real world when I count change in my pocket.
The way I see it, there are three separate endeavors here: understanding the world, understanding various forms of math, and understanding which forms of math are applicable to the real world and why.
Some math may not have any application that we know of in the real world. Maybe there is one but nobody has yet realized what it is. Or maybe the real world will change in the future so that there is one. Or maybe there won't be one and that's OK.
Incidentally, I think understanding how to correctly apply math to a real world problem is a separate skill from understanding the math itself. And just because you're a good at one doesn't mean you're good at the other. It's relatively easy and common to make the error of understanding a situation in a very shallow or inaccurate manner, then applying math which itself is completely valid, then coming to a wrong conclusion that you're very confident of because of how solid the math is. In "P, and P implies Q, therefore Q", it's easy to get so excited about the beauty and wonder of P implies Q that you spend too little time understanding whether P is actually the case. If you've got a hammer, and you're in love with that hammer, you can become blind to whether something actually is a nail.
But that's what they they were designed for. The first "numbers" were arguably clay tokens meant to represent goods in storage https://en.wikipedia.org/wiki/History_of_ancient_numeral_sys... Numbers were and are primarily used for accounting purposes. If they didn't accurately represent the change in your pocket then the system would be improved until it did. There's nothing magical about it.
Some math may not have any application that we know of in the real world. Maybe there is one but nobody has yet realized what it is. Or maybe the real world will change in the future so that there is one. Or maybe there won't be one and that's OK.
Incidentally, I think understanding how to correctly apply math to a real world problem is a separate skill from understanding the math itself. And just because you're a good at one doesn't mean you're good at the other. It's relatively easy and common to make the error of understanding a situation in a very shallow or inaccurate manner, then applying math which itself is completely valid, then coming to a wrong conclusion that you're very confident of because of how solid the math is. In "P, and P implies Q, therefore Q", it's easy to get so excited about the beauty and wonder of P implies Q that you spend too little time understanding whether P is actually the case. If you've got a hammer, and you're in love with that hammer, you can become blind to whether something actually is a nail.