Viktor Blåsjö, a historian of mathematics, writes essays and podcast notes on the history and teaching of mathematics, including calculus and Galileo.
HN commenters recommend his Opinionated History of Mathematics podcast for its problem-solving perspective, though one dislikes the first Galileo episode as hostile and dismissive.
I do definitely agree that something is lacking in current mathematical education, but calling your own counter-program "intellectual mathematics" is a bit ... on the nose.
Judging from the rest of the website, the author appears to have some rather idiosyncratic opinions. For example, he seems to be unconvinced that rigour is an essential component of mathematics (even going as far as claiming not to understand what it means): https://intellectualmathematics.com/blog/what-is-rigour-anyway/
He also has his own take on the "two cultures" distinction often postulated for mathematicians, but also apparently assigning distinctly less value to so-called "lesser technocrats", seemingly going as far as calling Euler of all people a "technocrat" and calling into question the value of Euler's famous equation e^pi*i = 1 - why? https://intellectualmathematics.com/blog/four-types-of-mathematicians/
I see no indication that the author doesn't know the mathematics he's talking about, but I also feel like he's incredibly biased towards the particular way he likes doing mathematics without considering that there are legitimately valid different approaches to doing mathematics - probably even among students.
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyperreals.
If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.
If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.
[0] https://www.bravernewmath.com/
[1] https://intellectualmathematics.com/calculus/
[2] https://people.math.wisc.edu/~hkeisler/keislercalc-06-03-26.pdf
[3] https://people.math.harvard.edu/~knill/teaching/mathe320_2015_fall/blog15/surreal1.pdf
1. I do not think Galileo is either a characteristic example of a physicist or mathematician, or a particularly good example at all [1].
2. While I would say that most would trivially agree that mathematics is a form of language [2], it is less common to argue that mathematics is _just that_. And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3].
[1] For a criticism regarding galileo's mediocrity and historically unfounded idealisation https://intellectualmathematics.com/blog/the-case-against-galileo-s01-overview/
[2] Actually, a collection of languages, as different parts of mathematics form different languages or dialects that are not easy to communicate in between even if they refer to similar objects, eg category theory vs set theory, analysis vs probability theory.
[3] https://anthonybonato.com/2017/09/19/this-is-your-brain-on-mathematics-2/
I cannot recommend enough the podcast Opinionated History of Mathematics by Viktor Blåsjö
What's great about it is that it helps put you in the shoes of the people solving the problems at these times. So it secretly teaches you how to solve problems. He also attacks some of the claims of some historians which are a bit obtuse. The arguments are really well founded and if you're wondering how ancients solved extremely complex math while never inventing the tools you'd think are needed to solve them, then this will answer a lot of that. I think it'll make you see a lot of problems in a different light. It also is just a lot of fun and can be pretty funny.
https://intellectualmathematics.com/opinionated-history-of-mathematics/
I took a look and was really turned off by the first episode which is all about dunking on Galileo and what a bad mathematician he was, and generally showing tremendous hostility to anything that isn't math. You can read it for yourself here: https://intellectualmathematics.com/blog/galileo-bad-archimedes-good/
The syllogism appears to be:
1. All good science is mathematical
2. Anyone intelligent is very good at math
3. Galileo was not very good at math
4. Therefore Galileo was dumb and did bad science.
Perhaps this was essentially Thales's own proof: https://intellectualmathematics.com/blog/first-proofs-thales-and-the-beginnings-of-geometry/
BTW it's also quite direct using vectors: the legs of the triangle are the sum and difference of radius vectors. Take their dot product, distribute it, it's zero because radii are the same length.