[mbivert]

> 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.

https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378...

[mindcrime]

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:

  The Calculus: A Genetic Approach by Otto Toeplitz

[jaymzcampbell]

Stillwell's book is incredible, it hits the right balance between a textbook and a (advanced) lay person introduction to a huge range of topics.

[nyssos]

You're conflating a few things here.

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.

Wildberger is an ultrafinitist.

[kthielen]

> 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.

[nyssos]

Sorry, yes, typo.

[mbivert]

> You're conflating a few things here.

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.

https://youtu.be/I0JozyxM1M0?si=IFdWcEWNeNKDid7t&t=39

[Tazerenix]

Wildberger is a crank

[mbivert]

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…