Philosophy and Transitional Mathematics Books
The math books I've been reading lately have been a bit more philosophical. Some of them have been "transitional" books. That is, books dealing with moving away from the more intuitive practical mathematics that uses both geometry and algebra to the upper year formal foundational pure mathematics that tries to rigorously justify claims from set theory and axioms and rejects the intuitive geometric ideas such as the infinitesimal.
All of my intuitions are on the practical side. I tend to think that "the unreasonable effectiveness" of mathematics is due to thousands of years of practical use of math that slowly built up geometry and algebra. It was developed because it was effective. It's only when math became so abstract in the last 150 years that the usefulness suddenly seemed mysterious.
Truth to me has to do with well established practical use. I'm deeply skeptical of mathematical objects and a priori reasoning. Truth will change with paradign shifts through history. The mathematics of an alien species would have a very different developmental history and would look nothing like our mathematics. Math is a useful tool that has instrumental truth.
I'm not suggesting that we should only do practical applied mathematics. Things like non-euclidean geometry were abstract exercises with no practical application at first. But I would say that it is more true after general relativity found a use for it.
I haven't really delved into pure math side beyond these transitional books, but there must be some practical reason why math needed to throw everything away and start from scratch. I haven't quite seen it yet. I don't quite understand the need for the paradigm shift. It is not like Calculus suddenly became true because of the epsilon-delta definition of a limit.
The book "Where Mathematics Comes From" should have been a better match for me than it was. Philosophically it seems like a good match. I was on board with the cognitive science bits that dealt with the developmental history of how human minds/brains are good at arithmetic due to subitizing (grasping groups of up to 4 without counting). It's a long book and most of dealt with justifying abstract mathematical concepts based on a foundation of metaphors and I'm not sure why they did spent all that time on it.
All of my intuitions are on the practical side. I tend to think that "the unreasonable effectiveness" of mathematics is due to thousands of years of practical use of math that slowly built up geometry and algebra. It was developed because it was effective. It's only when math became so abstract in the last 150 years that the usefulness suddenly seemed mysterious.
Truth to me has to do with well established practical use. I'm deeply skeptical of mathematical objects and a priori reasoning. Truth will change with paradign shifts through history. The mathematics of an alien species would have a very different developmental history and would look nothing like our mathematics. Math is a useful tool that has instrumental truth.
I'm not suggesting that we should only do practical applied mathematics. Things like non-euclidean geometry were abstract exercises with no practical application at first. But I would say that it is more true after general relativity found a use for it.
I haven't really delved into pure math side beyond these transitional books, but there must be some practical reason why math needed to throw everything away and start from scratch. I haven't quite seen it yet. I don't quite understand the need for the paradigm shift. It is not like Calculus suddenly became true because of the epsilon-delta definition of a limit.
The book "Where Mathematics Comes From" should have been a better match for me than it was. Philosophically it seems like a good match. I was on board with the cognitive science bits that dealt with the developmental history of how human minds/brains are good at arithmetic due to subitizing (grasping groups of up to 4 without counting). It's a long book and most of dealt with justifying abstract mathematical concepts based on a foundation of metaphors and I'm not sure why they did spent all that time on it.