On the subject of more context in math, I've always wondered if having a grasp of the history of math would be helpful in getting better at solving mathematical problems. i.e. would learning more about how math developed over time, and how people solved important problems in the past, help me in trying to solve some other problem today?
Years ago I bought the 3-volume set "Mathematical Thought from Ancient to Modern Times", but never had the time to get past the first few chapters. I'd be interested in any recommendations for math history tomes like that.
I found it helpful in some of my University math classes when I actually took the time to read the biographies that some of the textbooks included, the classes when I skimmed past them I did not remember details of the proof formulae for. But I don't know if this says more about the aid of history to the process of remembering or about my a priori interest in the topic for those particular classes.
For a really good example of integrating the history along with the mathematics, and much more accessible than those math texts, I would recommend "Journey Through Genius" by Dunham[0]. It may be a little dated (published in 1990) and its focus is limited to algebra, geometry, number theory, and the history is perhaps too Western-biased, but it's good and it's short. Its material would make a solid foundation to build on top of because, in addition to the historical context, it shows a lot of the thought process into approaching certain landmark problems.
When I was an undergrad doing the mandatory measure theory course, I stumbled on a super old book(pre-1950 if I remember correctly, library card showed 3 people taking it out in the last 10 years) ,the name of which I forgot, that basically "re-built" the process that Lebesgue/Caratheodory/Riemann/etc followed, the problems they encountered (i.e Vitali set), why Lebesgue measure was the way it was and so on.
I really wish I could remember the name of the book, but it made so much more sense than how even something like Stein Shakarchi or Billingsley, which introduced measures by either simply dumping the Vitali set on you as the main motivation or just not really explaining why stuff like outer measure/inner measure made sense.
Boyer/Merzbach "A History of Mathematics" [0] is a tome in that vein. It spends a lot of time discussing the (mind-boggling, to a modern-mathematics-educated reader) methods that ancient peoples used to do real math (e.g. for engineering) as a way to motivate the development of the features of modern symbolic mathematics.
> I'd be interested in any recommendations for math history tomes like that.
Not a book, but FWIW, I've enjoyed a few videos from Norman Wildberger's "Math History" playlist[0]. Interestingly, he has a unconventional view of infinite processes in mathematics, a point of view that used to be common about a century ago or so.
I'm sure knowing some amount of history is useful, but there must be a limit to how much of it is practically useful though.
I've been going through that very video lecture series the last couple of weeks. Good stuff. And in the lectures he mentions a number of books. I looked a few up on Amazon, and then looked at the associated Amazon recommendations, and so far have this small list of books related to Maths history that look worth reading:
Mathematics and Its History (Undergraduate Texts in Mathematics) 3rd ed. 2010 Edition by John Stillwell
The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics) by Carl B. Boyer
A History of Mathematics by Carl B. Boyer
A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics)A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics) by Dirk J. Struik
Introduction to the Foundations of Mathematics: Second Edition (Dover Books on Mathematics) Second Edition by Raymond L. Wilder
Mathematical Thought from Ancient to Modern Times by Morris Kline (3 volume set)
The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time by Jason Socrates Bardi
There is also a "thing" in mathematics that is sometimes called the "genetic approach" where "genetic" is roughly equivalent to "historical" or maybe "developmental". IOW, a "genetic approach" book teaches a subject by tracing the development of the subject over its history. One popular book in this mold is:
Constructivists are only interested in constructive proofs: if you want to claim "forall x in X, P(x) is true" then you need to exhibit a particular element of x for which P holds. As a philosophical stance this isn't super rare but I don't know if I would say it's ever been common. As a field of study it's quite valuable.
Finitists go further and refuse to admit any infinite objects at all. This has always been pretty rare, and it's effectively dead now after the failure of Hilbert's program. It turns out you lose a ton of math this way - even statements that superficially appear to deal only with finite objects - including things as elementary as parts of arithmetic. Nonetheless there are still a few serious finitists.
Ultrafinitists refuse to admit any sufficiently large finite objects. So for instance they deny that exponentiation is always well-defined. This is completely unworkable. It's ultrafringe and always has been.
> if you want to claim "forall x in X, P(x) is true" then you need to exhibit a particular element of x for which P holds
I don’t mean to be pedantic (although it’s in keeping with constructivism) but in the case you describe, you don’t have to provide a particular x but rather you have to provide a function mapping all x in X to P(x). It may very well be that X is uninhabited but this is still a valid constructive proof (anything follows from nothing, after all).
If instead of “for all” you’d said “there exists”, then yes constructivism requires that you deliver the goods you’ve promised.
It's likely: I purposefully stayed loose about the "infinite processes" to avoid going awry. I do however remembered him justifying his views as such though: he's not going into details, but he's making that point here[0] (c. 0:40). I assumed — perhaps wrongfully — that he got those historical "facts" correct.
A crank who provides hundreds of hours of fairly decent mathematical education content free of charge; it's not because he harbours unusual/fringe opinions that he's altogether worthless…
Years ago I bought the 3-volume set "Mathematical Thought from Ancient to Modern Times", but never had the time to get past the first few chapters. I'd be interested in any recommendations for math history tomes like that.