This is pretty good, and it will exercise your mind for a minute.
The distinction between mathematical notation and its underlying reality is a crucial one. The first is an invention of humans, the second is universal. In fact, I’ve increasingly become convinced that the second actually is the universe, and mathematics is just us recognizing reality’s fundamental patterns, and devising mechanisms to describe and to model, to extrapolate, to make predictions, based on those patterns.
Of course, many of those predictions have no correlation in observed reality, at least none that has been observed yet. Many mathematicians take delight in pointing out how useless many of their endeavors are. Yet, despite this, many mathematical structures initially thought to be purely abstract do eventually end up being useful to model some aspect of nature. The ones that don’t could be thought of as either untested or falsified scientific theories.
Another way to describe what I’m saying is that mathematics is the universe. This is similar to but the reverse of the Mathematical Universe Hypothesis, which posits that the universe is a part of mathematics. Both of these ideas see an equivalence between underlying mathematical realities and the universe, but with opposite ideas of which is the more primal reality.
Which one is true? Like all metaphysical conundrums, I can’t see any way to know for sure. But my personal judgment is that mathematics being the universe is simpler. The universe being a subset of mathematics requires us to assume a trans-universe reality that we can’t observe, an assumption mathematics being the universe doesn’t require.
Of course, depending on exactly what we mean by “mathematics”, even if there is no trans-universe reality, the universe could still be thought of as a part of mathematics, but only in the same sense that it is a subset of all scientific theories, including both true and false ones.
Unless I’m missing something?