The author is criticizing calculus courses. The level of formalism in calculus courses is relatively low, and none of it has anything to do with quantum mechanics or contradictions far down. It is only enough formalism to give a definition of derivative and integral that actually makes sense, that's all.
it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)
Now any hard-working student of university calculus can without problems understand and reproduce the results of those treatises. I think this would not be possible without the modern systematic methods, including the epsilon-delta approach.
you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.
I mentioned fourier in another comments. Those weird functions were postulated in the discussion that arised somewhere in the same time period. I don't know what the author is referring to, because he is irritatingly vague, especially for a mathematician.
it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.
If you are familiar with the concept of a limit, you can notice that Zeno considers a limiting process of two related quantities, time and the difference in the position of Achilles and the Tortoise. Since in this limiting process time get arbitrarly close to some definite value (the meeting time of Achilles and Tortoise), but never gets equal to it, it stops being so surprising that the distance between them never reaches zero, though it gets arbitrarly close. The formalism clarifies what seems parodoxical when described in natural language. I find this quite convincing, and I never found a better explanation, altough I know philosophers still dispute this.
I never said rigour takes precedence over intuition. I just think it's the inherent difficulty of calculus that stops students from understanding it, and not the epsilon-delta stuff.
99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. Infinitesimals are just fine for that.
There were logical contradictions even in Newtons and Leibniz works, far before anyone considered continuous functions without derivatives etc., they were basically making decisions about when a given operation or transformation can be applied based on intuition alone and not any logical deductions, and it wasn't rare they arrived at incorrect conclusions. Also, Newton himself did epsilon-delta reasonings, he just did not notice their generality:
Again, you are confusing the work of Cauchy and Weierstrass and the epsilon-delta stuff with all the latter even more formal approaches to treat more complicated functions, which by the way were developed in response to Fourier examining heat transfer and trying to describe it mathematically (so it's actually rooted in physics).
People were saying the same things you are just saying about limits about the geometry of Euclid, in fact Newton at one time was of the opinion all the formal development of geometry is useless. He reconsidered after obtaining nonsense geometrical results a few times...
The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.
I wish you luck doing quantum mechanics with Newton-style calculus.
I find the logically sound version of the infinitesimals approach to be much more difficult to understand than the approach using limits. For example, you have to introduce hyperreals to make it work (most common approach):
Introductory calculus classes are hardly ever rigorous. The only formalism you see is functions and limits, and that's a very useful one, far from unnecessary, it might just not always be motivated appropriately by poor teachers who themselves have little understanding of its usefulness. Your interpretation is also very far from what he has actually written. I edited my parent comment to make what I mean more clear.
Please read the article with a critical eye, some of it is complete non-sense, for example:
CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it
under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be
mentioned.
"Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox:
The ideas of Newton and Leibniz were hardly simple, they had some valid intuitions and managed to do formal manipulations that led to correct results, but in their day it was impossible to at all logically understand why what they are doing works, and not for some god knows how complicated things, but even for most elementary ones. You don't even have to go back to writings of Newton or Leibniz, just have a look at a 19th century textbook of calculus to see how noticeably strange and illogical the exposition of the subject was even then, with "infinitely small quantities" and an air of mysticism about it:
This kind of approach simply doesn't make sense, even though it happens to apparently produce correct results sometimes. Now, "function-based approach" is a weird phrase, but I guess he means the common modern exposition of elementary calculus using limits. This however wasn't developed in response to "various analytic crises". The only explanation of this statement I see is that he knows history of mathematics poorly and confuses the latter developments by Lebesgue, Jordan etc. that led to what we now call real analysis (inspired by considerations of nowhere continuous functions, continuous but nowhere differentiable functions etc.) with the earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" and satisfactorily resolved things like the before-mentioned Zeno's paradox.
If it is only motivated appropriately, the concept of a limit is actually very interesting and powerful. There is a ladder of granularity with which you can treat computational problems, with the most elementary approach being always trying to get the exact answer. However, the class of problems that can be solved this way is very narrow. You can jump over this severe restriction by getting a bound, with inequalities for example, or you could try to get an equality in the limit (when n approaches y, the sought thing x approaches w*z). Unfortunately in school people almost exclusively learn to look for the exact answer, while in mathematics proper and in real world it is much more common to look for approximations and limiting behaviour. Furthermore, since the limit concept so powerfully extends the range of problems for which we are able to state anything interesting, there are lots of mathematical disciplines that rely on it to a great extent, for example probability theory (laws of large numbers, central limit theorem, ...). You won't understand almost any higher mathematics without learning limits first!
One can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in Courant's "What is mathematics?" in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc. The problem is not the formalism, but the teachers who can't motivate the material well enough both mathematically and physically and students who are not always mature enough to put in the amount of work necessary to understand calculus, which for most of them will be by far the most difficult thing they ever attempted to learn.
https://github.com/git/git/blob/master/strbuf.h
https://github.com/git/git/blob/master/strbuf.c
See this for the story of why strncpy/strncat are insecure:
https://en.wikipedia.org/wiki/C_string_handling#Replacements