It's not really finished, and it's certainly not complete, but I thought it would give you something to think about. It's not necessarily mainstream, and it may turn out that I don't believe it all myself, but I hope it's interesting.
I like how you present the developments as filling in logical gaps - almost a kind of deduction. It seems inevitable that some practical application using the basic system will eventually also need one of the logical consequences. A little bit like having different operations in a UI - eventually, each combination will be tried by someone, and it's reassuring to find that they are orthogonal, and do work as expected.
..they have an independent right to existence.; mathematical structures are not invented, they are discovered.
How far does that carry? What about algorithms? Do they also have have an independent right to existence? Can you invent an algorithm at all or do you in fact discover it?
> mathematical structures are not invented, they are discovered
This is a doorway to a pretty abstract philosophical discussion, not a statement of fact. I don't think that math is actually real in the strictest sense, so I'm on the fence about invent vs discover. While I think that mathematical constructions are idealizations similar to Platonic Ideals, I don't believe like Plato did that there's a perfect Plato-space where these idealizations live.
When an artist arranges common-place paints (or sounds or dance moves or whatever) into a novel combination, he is said to have created rather than discovered a new work of art. Is there a good reason why this should not be true when a mathematician takes common-place axioms and makes a new arrangement of them that validates as a proof?
Yeah -- I think this is an example of a place where the language breaks down thanks to the different connotations that "invent" and "discover" have. It seems like math has some properties of both, and the choice of word says more about the person using the word than it does about the phenomena being described.
What about when you come up with a buggy algorithm? Did you do a bad job at inventing an algorithm, or a great job at discovering a wonderful, previously unknown (though incorrect) one? :)
This was a good read. I wish the author had gone a little bit further and spoke about how many reals there are compared to the rationals, continuum hypothesis, etc.
Although his point about riddles and puzzles is very true I think mathematics is very much a natural science.
There I thought I'd start with fractions, basically from the ground up, but I think that's wrong. I think I want to engage people first, then "drill down" on demand. The comment about wanting more about the cardinality, the C.H. and related stuff is what I was looking for.
0 and the negative numbers turned up very late in our history
I believe it's also true that the ancients distrusted polynomials (e.g., x^3+7x^2+2x+9=0) because who would be silly enough to add a volume to an area to a length, etc.
If they had just done the mathematics without relating the numbers to real world, we might have far more advanced maths today. All the more reason to pursue topics for their own sake.
It's not really finished, and it's certainly not complete, but I thought it would give you something to think about. It's not necessarily mainstream, and it may turn out that I don't believe it all myself, but I hope it's interesting.