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Comment by P_I_Staker | Hacker News Reader

How so?

As far as I've understood so far, back when calculus was invented people would think about it using infinitesimals. But nowadays it is taught without mentioning them; in short, caluclus is now taught emphasizing the ability to use the techniques and that's it.

I think that during the 19th century when it (along with logic) was being formalized they decided to throw away the old original ways to think about it. This ends up making it really hard to understand in exchange for making it "formally sound" and "user friendly".

limits are still taught... which do the same thing as infinitesimals

I think infinitesimals are more like the partitions in a Riemann Sum. The limit is where you are heading, whereas infinitesimals are how you get there.

In the finite Riemann sum written as a limit the partition is just the limit of (x1 - x2)/m. I'd say it's the reverse, you use the limit to resolve an equation of two infinitesimals.

I don't know about you but proving various derivative rules using the limit (ie, infinitesimals) has been a recurring exam question for me, and so were differentials.

The modern approach to calculus using limits do not use infinitesimals. According to Richard Courant, "Mathematicians of first rank operated ... sometimes even with mystical associations as in references to 'infinitesimals' or 'infinitely small quantities.'" I believe we now use the "epislon delta" definition of limits.

"Introduction to Calculus and Analysis" Volume 1

Limits are conceptually equivalent to infinitesimals. Another way to view it is that a limit is a way to evaluate an equation of infinitesimals.

Epsilon Delta is not our definition of limits, it's the definition of the derivative that uses limits as the "backend". But even using infinitesimals you can use the epsilon-delta definition of the derivative.

Perhaps you are referring to modern nonstandard analysis, which I am sure has equivalent definition of limits. I will note that in standard calculus Epsilon-Delta is not only used in defining derivatives, but also used in defining function limits, which of course means used in defining integrals.

Just out of curiosity, in nonstandard analysis is there an equivalent of Lebesgue integral?

There is indeed an equivalent of the Lebesgue integral available in several textbooks, for example `Nonstandard Analysis for the Working Mathematician`.

While certainly not the usual approach, calculus in terms of infinitesimals has been placed of formal foundations[1]. To be fair to Courant, however, the approach to infinitesimals in nonstandard analysis is pretty far removed from the views of Leibniz, et al., and "mystical associations" is a fair, if not necessarily impartial, criticism of the latter.

IIRC, even Newton was at least a bit skeptical of the validity of his method of fluxions, even going so far as to formulate his Principia in terms of (nearly impenetrable IMO) arguments from classical geometry instead, and the non-rigorous application of related ideas led even the best mathematicians to questionable conclusions at times (e.g., Euler's argument that 1 + 2 + 4 + ⋯ = -1, which may or may not have influenced computer science in terms of two's complement arithmetic).

As for abstraction as a source of seemingly artificial difficulty in modern mathematics, from the introduction to Courant's Differential and Integral Calculus, the first edition of Introduction to Calculus and Analysis[2],

The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy.

For similar arguments that over-reliance on formalism begins far earlier in the modern mathematical curriculum, see Feynman's New Textbooks for the "New" Mathematics [3], in which he discusses his frustrations as a working (theoretical!) physicist reviewing grade school textbooks.

As a partial counterpoint, I personally often have an easier time understanding the more abstract treatments. For example, I struggled with the traditional presentation of multivariable calculus in terms of "physically meaningful" differential operators until working through the first couple books of Spivak's Comprehensive Introduction to Differential Geometry after-hours, at which point everything sort of clicked. Had my intro (college) calculus course not coincidentally been taught by a differential geometer with a habit of presenting some of the basic ideas as asides, I may have never made it any further in the field (in terms of learning; I work in software, not maths).

Nevertheless, I'd never propose Bourbaki as a good model for elementary education!

Revised (Introduction to…) editions of Courant's calc textbooks are particularly noteworthy in this respect; to me, at least, they strike a very good balance between classical and modern formalisms, and between applications to pure and applied mathematics.

As for the application of "general abstract nonsense" to computer science and engineering, I have no doubt Courant would be pleased, with the applications if not in their presentation, as the interplay between pure mathematics and its applications was always an important subject to him (see also the introduction to his [and, nominally, Hilbert's] Methods of Mathematical Physics [4], the address he gave on variational methods in PDEs [5], often cited as a foundational work in finite element analysis, and, well, pretty much the entirety of What is Mathematics?[6]).

[1] https://en.wikipedia.org/wiki/Nonstandard_analysis

[2] https://onlinelibrary.wiley.com/doi/pdf/10.1002/978111803324...

[3] http://calteches.library.caltech.edu/2362/1/feynman.pdf

[4] https://onlinelibrary.wiley.com/doi/pdf/10.1002/978352761721...

[5] https://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943...

[6] https://archive.org/details/WhatIsMathematics

Infinitesimals can't be used to prove anything - it's really easy to write equations like

x + y + z = 0

dx/dy * dy/dz * dz/dx = -1

that make no sense if you don't know epsilon-delta calculus.

You can use infinitesimal analysis but that's a completely different thing that you probably won't enjoy if you don't think epsilon-delta is straightforward.

https://en.m.wikipedia.org/wiki/Smooth_infinitesimal_analysi...

I would guess the sheer size of it.

For instance, I can't take a derivative without the rules in front of me, though the concept of derivatives is simple.

I mean, you probably forgot how to, but taking a derivative without using any derivation rules has always been an exam question for me whenever I take a calculus class.

Derivatives are usually quite a bit easier than integrals.

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