I'm thinking about a historical approach. For example what exactly is Ptolomy system? What's the math involved? Why did the phases of Venus rebutes the model? How did Tycho make his observations? Can we repeat it for say a few months? How did Newton come up his ideas? What are Einstein's thought experiments.
Basically I want to put myself in the shoes of great minds, learn what math tools were available at the time, do their observations, know how their thought processes work.
edit Would appreciate if someone could recommend a textbook that follows such approach.
Though I lack personal experience with a historical approach to studying mathematics, a starting point could be the reading list by St. John's College as part of the materials the faculty uses to teach mathematics to their undergraduates: https://www.sjc.edu/academic-programs/undergraduate/subjects...
Readings include Archimedes's "On the Equilibrium of Planes" and "On Floating Bodies" and Nicomachus's "Arithmetic." However, I've heard of some mixed reviews on the effectiveness of learning math [1] [2] via reading historical texts at St. John's, versus more traditional approaches today in good mathematics programs.
A bit separately but as a related idea, I've spoken to a couple people who said they preferred older mathematics books from the 1900s to the 1960s for their studies over modern books. I personally prefer modern books with recent editions that are well-recommended, as they usually have good visualizations of ideas and shouldn't frequently have errors (due to their reputation). If you're studying as a hobby, though, the most effective mathematics book, assuming it's reasonably reputable and heard-of, is the one that interests you enough to stick with it for consistent study in the long-term.
If you go through for the "great books" approach, just be aware that it's not the same curriculum as a major in mathematics:
> Mathematics is one of the many subjects studied in the college’s interdisciplinary great books curriculum. There are no majors at St. John’s.
It may be fulfilling from a humanistic/personal development perspective, but you won't really have the tools to do anything useful like cryptography, algorithms, physics, statistical inference, data analysis, etc., which, in my mind are the really cool things you can understand by learning math.
An interesting book (2 volumes) would be: 2019, June Barrow-Green et. al., The History of Mathematics. A Source-Based Approach, American Mathematical Society.
People who want to study physics want to understand such theories, so would be heavily incentivised by the market.
The reason it isn't taught this way is because it's generally not seen as an efficient route, and you can understand such theories by studying them directly and in a logical progression.
What's important about the old books and theories is that they teach you that science is invention and discovery and being wrong, not memorizing rules and solving already -solved puzzles. If you only learn from modern textbooks, you get the wrong impression that these biblical tomes are what science is. Science is about finding patterns and creating order out of chaos, not memorizing the order someome else creaymtes.
They don't teach you that any more than current books do.
You might gain that because you compare the two and notice the old books are obviously wrong against new data ... something the current books happily tell you too.
Basically I want to put myself in the shoes of great minds, learn what math tools were available at the time, do their observations, know how their thought processes work.
edit Would appreciate if someone could recommend a textbook that follows such approach.