That depends on what you think math is.

From an intuitionist point of view, math is the study of human idealizations. And there is no good reason to be surprised that we would, given millions of years of trial and error, evolve a really strong intuition that would allow us to understand a lot of the natural world. Nor that once we had language and adequate time to dwell inwards that we should not be able to unwind those intuitions into precise language forms over thousands of years.

The economy of mathematics is sometimes striking, to me: that so many parts of it are really just other parts in variant forms. But I would blame that on the fact we are in a very orderly corner of the universe, compared to what might be.

If you think math is somehow independent of human psychology, and not the collective set of modelling tools at its disposal, then the consistent meeting up of fact and form becomes much more mystical. But then that big mystery becomes a good reason to question that independence.

The conventions you find so bizarre, are largely just that, conventions, if ones we are pretty much born into making. The idea that we can think of multiplication on the complex numbers as scaling and rotation has a lot to do with our own relative paucity of our own simple models of motion, and not so much to do with reality. After all, when we want to model waves, we try hard to make sure they get expressed in terms of the components of a rotation. Then when we decided to model particles, we 'found' they have rotational inertia, despite that their rotation has to be 720 degrees, and acts relatively little like actual rotation. Once you let the real awkwardness of that notion sink it, it seems to me like polar coordinates are a solution in search of a problem, not something that just happens to crawl out of so many niches.

That depends on what you think math is.

From an intuitionist point of view, math is the study of human idealizations. And there is no good reason to be surprised that we would, given millions of years of trial and error, evolve a really strong intuition that would allow us to understand a lot of the natural world. Nor that once we had language and adequate time to dwell inwards that we should not be able to unwind those intuitions into precise language forms over thousands of years.

The economy of mathematics is sometimes striking, to me: that so many parts of it are really just other parts in variant forms. But I would blame that on the fact we are in a very orderly corner of the universe, compared to what might be.

If you think math is somehow independent of human psychology, and not the collective set of modelling tools at its disposal, then the consistent meeting up of fact and form becomes much more mystical. But then that big mystery becomes a good reason to question that independence.

From that angle the conventions you find so bizarre, are largely just that, conventions, if ones we worked out over generations, and are pretty much born into making. The idea that we can think of multiplication on the complex numbers as scaling and rotation has a lot to do with the relative paucity of our own simple models of motion, and not so much to do with independent reality. After all, we really wanted circular planetary orbits. When we want to model waves, we try hard to make sure they get expressed in terms of the components of a rotation. And when we decided to model particles, we 'found' they have rotational inertia, despite that their rotation has to be 720 degrees, and acts relatively little like actual rotation. Once you let the real awkwardness of that notion sink it, it seems to me like polar coordinates are a solution in search of a problem, not something that just happens to crawl out of so many niches.

That depends on what you think math is.

From an intuitionist point of view, math is the study of human idealizations. And there is no good reason to be surprised that we would, given millions of years of trial and error, evolve a really strong intuition that would allow us to understand a lot of the natural world. Nor that once we had language and adequate time to dwell inwards that we should not be able to unwind those intuitions into precise language forms over thousands of years.

The economy of mathematics is sometimes striking, to me: that so many parts of it are really just other parts in variant forms. But I would blame that on the fact we are in a very orderly corner of the universe, compared to what might be.

If you think math is somehow independent of human psychology, and not the collective set of modelling tools at its disposal, then the consistent meeting up of fact and form becomes much more mystical. But then that big mystery becomes a good reason to question that independence.

That depends on what you think math is.

From an intuitionist point of view, math is the study of human idealizations. And there is no good reason to be surprised that we would, given millions of years of trial and error, evolve a really strong intuition that would allow us to understand a lot of the natural world. Nor that once we had language and adequate time to dwell inwards that we should not be able to unwind those intuitions into precise language forms over thousands of years.

The economy of mathematics is sometimes striking, to me: that so many parts of it are really just other parts in variant forms. But I would blame that on the fact we are in a very orderly corner of the universe, compared to what might be.

If you think math is somehow independent of human psychology, and not the collective set of modelling tools at its disposal, then the consistent meeting up of fact and form becomes much more mystical. But then that big mystery becomes a good reason to question that independence.

The conventions you find so bizarre, are largely just that, conventions, if ones we are pretty much born into making. The idea that we can think of multiplication on the complex numbers as scaling and rotation has a lot to do with our own relative paucity of our own simple models of motion, and not so much to do with reality. After all, when we want to model waves, we try hard to make sure they get expressed in terms of the components of a rotation. Then when we decided to model particles, we 'found' they have rotational inertia, despite that their rotation has to be 720 degrees, and acts relatively little like actual rotation. Once you let the real awkwardness of that notion sink it, it seems to me like polar coordinates are a solution in search of a problem, not something that just happens to crawl out of so many niches.