Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus. It can be freely downloaded at https://people.math.wisc.edu/~hkeisler/calc.html
I have mixed feelings about this. I've been through nonstandard analysis. My response was, "We shouldn't need the axiom of choice to define the derivative."
But I think that it is extremely important to understand that the infinitesimal notation really MEANS something. Here is some Python to demonstrate.
# d is a functor. It takes a function and returns a second function.
# The second function captures the change in f over a small distance.
# The dx/2 business reduces artefacts of it being a finite distance.
def d (f, dx=0.001):
return lambda t: (f(t + dx/2) - f(t - dx/2))
# d²x / dx²
def second_derivative (f):
return lambda t: d(d(f))(t) / (d(x)(t) * d(x)(t))
def x (t):
return t
def cubed (t):
return t*t*t
print("The second derivative of cubed at 1 is near", second_derivative(cubed)(1))
You write, "we shouldn't need the axiom of choice to define the derivative."
The good news is that we don't!
Only model-theoretic approaches, which justify the infinitesimal methods by constructing a hyperreal field, require (a weak form of) the axiom of choice [2].
However, there are axioms for nonstandard analysis which are conservative over the usual choice-free set theory ZF. The three axioms of Hrbacek and Katz presented in the article "Infinitesimal analysis without the Axiom of Choice" [1] are the best recent example: these axioms allow you to do everything that is done in Keisler's book and more (including defining the derivative), and you never need to invoke the axiom of choice to justify them.
[2] Essentially, the set of properties satisfied by a fixed nonstandard hypernatural gives rise to a non-principal ultrafilter over the naturals. The axiom of choice is necessary to prove the existence of non-principal ultrafilters in (choice-free) set theory, but the existence of non-principal ultrafilters is not sufficient to prove the axiom of choice.
Most calculus students don't need the full formal power of rigorous analysis. Calculus, taken alone and with the elementary properties of the real numbers assumed and a few elementary properties of infinitesimals (0 <<< infinitesimal^2 <<< infinitesimal <<< any positive real), can get you a lot of power for very little formal work.
I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.
As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation and observe that you generally don't need L'Hôpital's rule to work them out. It is possible to produce examples that can be calculated by L'Hôpital's rule, but not by simply understanding o-notation. But it isn't easy, and you're unlikely to find them in textbooks.
It is probably true that as you go on, limits are more useful in higher mathematics than o-notation. But o-notation is far more useful in most subjects that use mathematics. Given how easy it is to master limits if you know o-notation, why not teach o-notation first?
More importantly for this audience, the idea of limits lies at the heart of numerical analysis. Explicating the quantifiers of the definition of a limit is the first step in obtaining control over any estimator of your data. This, among other things, is why I am perpetually baffled at the people in this audience who say that limits is something "they will never need". Limits, their algebra (which subsumes all of high school algebra and inequalities) and their delicate analysis is what makes a ton of numerical algorithms work; Higham's classic has it all, and is perfectly clear about it all.
> I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.
I'd say that limits in the sense that you mean (as opposed to category-theoretic limits) are precisely the domain of calculus or, if one wishes so to call it (because one is proving things!), analysis. For example, many US universities, mine included, regard the computation of infinite series as part of Calculus II.
My calculus teacher tried to talk me out of taking the AP Calculus AB exam because I wasn’t doing that well in his class. “Calculus the Easy Way” which teaches in the setting of a mythical fantasy world where applied problems are solved with calculus unlocked the secret of calculus for me. I ended up scoring a “4” which was a big triumph for me, as it placed me out of taking calculus with all the people who did not want to be taking calculus.
Douglas Downing's Trigonometry Made Easy is also a really great book for non-maths people. Same approach of situating Trigonometry in a mythical fantasy land. I remember Trigonometry as a chore of trying to memorize double-angle formulas and such, but this book really helped connect trig in an intuitive way.
I didn't know he had a calculus book, I'll have to check it out now.
> Douglas Downing's Trigonometry Made Easy is also a really great book for non-maths people.
> but this book really helped connect trig in an intuitive way.
To me, understanding math in an intuitive way is actually how "maths people" think about math. The numbers and formulas are just a means to an end to get there.
When they came to require still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth's days, they called “second minùtes” (i.e.: small quantities of the second order of minuteness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called.
A "third minute" (of 1/60 second) could be useful in talking about human reaction times. We could say that people commonly have a reaction time of 7 to 15 thirds, and that alcohol consumption may slow that reaction time by a further 8 thirds or more.
Or, speedrunners and professional musicians can practice motions (relative to their own motions or to an ongoing rhythm) that are accurate to about 1 third.
I assume the terminology would be too easy to confuse with ⅓, so maybe they would have to be called "terces" or "tertians" or "tertiaries" or something. (Specifically, we want to say (1/60)³ hour rather than (1/3) minute!) And maybe it's not a great thing to introduce a non-decimal unit when we're also trying to get rid of them in many other contexts.
I should follow up and say that I learned that this unit has already been referred to under the name "tierce", although it's not at all common.
It could also be useful for measuring latency of Internet connections, although these have commonly gotten faster over time and many latencies can be under a tierce. On the other hand, the round-trip latency to a satellite in geostationary orbit is about 14 tierces.
Consider Analysis 1 by Terry Tao. It walks you from the very beginning and also uses a conversational style. It's one of the most pedagogically smooth or gentle books I've ever read.
I graduated from engineering school more than 30 years ago, and learned how to apply calculus without understanding why it worked. This book helped me to finally understand.
I love projects like this, but I really wish instead of converting text resources to web sites that these projects would produce epub outputs. It's great for distribution, offline reading, and scaling to different display sizes, aspect ratios, and resolutions.
I wonder if there will be a time when textbooks will be created in digital-first format, instead of being mere replicas of what print books are. It doesn’t have to be static text and images on A4 pages.
I'm familiar with those tools, but they don't preserve the formatting of the source document (when PDF) and certain types of embedded resources don't transfer properly.
Some PDF documents don't contain the source text as digitized text, either. It's just a bundle of scanned images.
Only if they exist as poor conversions. Properly formatted epub documents are worth their weight in effort.
Epub documents _are_ HTML files in a zip archive - I'd argue that if a web site is a proper presentation of the source material, an epub is even better.
Additionally, PDF documents _are_ worse for any kind of document technical or otherwise. PDF documents are primarily Postscript instructions without any relation or hierarchy to the included elements. Epub documents/HTML provide semantic relationships and hints to the content.
Late 30s here. I keep feeling like I should learn math better, but damn, I just never need it. It’s much easier to learn stuff I need. As it is, I’ve lost everything back to about 8th grade math because I’ve never used any of it, so it’s just as gone as all the French I used to know but never found an excuse to use.
[edit] and I’m dreading my kids getting past elementary school math because they’re gonna be like “why the hell am I spending months of my life on quadratic equations?” and I’m not gonna have an answer, because IDK why we did that either. At least I have answers for calculus, even if they’re not much good (“so you can do physics stuff”, “right, but will I ever need to do physics stuff?”, “uhhh… unless you really want to, no.”)
The quadratic equation is completing the square while inexplicably avoiding all the intuition of completing the square. For example, to solve x^2 + 6x + 5 = 0, you would re-write it as (x^2 + 6x + 9) - 4 = 0, which is (x + 3)^2 - 4 = 0 and hence equivalent to (x + 3)^2 = 4, so that x + 3 = ±2 and hence x = 3 ± 2 is 1 or 5. Euclid thought of things this way, though his language is, of course, very different to modern language; see, for example, Proposition 6 of Book II (http://aleph0.clarku.edu/~djoyce/elements/bookII/propII6.htm...).
That's the same answer as the quadratic formula, but makes a lot more sense to me! Of course I've cooked the numbers so that you don't wind up with surds in the answer, but those are just complications in bookkeeping, not in concept.
Quadratics are useful-- finding dimensions of things in the plane; relating area and constrained side lengths, etc. They come up a lot if you want to solve problems.
And good luck taking on calculus without being super solid in the mathematical tools you use against quadratics -- factoring, completing squares, manipulation of binomials, pairing up like terms, etc.
The author was a Soviet nuclear physicist (who participated in the creation of the H-bomb), so his main point isn't rigor. It can be a nice change of perspective from standard American texts.
Although Soviet-era books were notoriously terse & difficult to digest, they had some very aesthetic typesetting. I have owned a few (Problems in Physics by Irodov, & another by Krotov) and they all share similar design aesthetics.
Readit News
Relevant wikipedia entry: https://en.m.wikipedia.org/wiki/Nonstandard_analysis
But I think that it is extremely important to understand that the infinitesimal notation really MEANS something. Here is some Python to demonstrate.
The good news is that we don't!
Only model-theoretic approaches, which justify the infinitesimal methods by constructing a hyperreal field, require (a weak form of) the axiom of choice [2].
However, there are axioms for nonstandard analysis which are conservative over the usual choice-free set theory ZF. The three axioms of Hrbacek and Katz presented in the article "Infinitesimal analysis without the Axiom of Choice" [1] are the best recent example: these axioms allow you to do everything that is done in Keisler's book and more (including defining the derivative), and you never need to invoke the axiom of choice to justify them.
[1] https://arxiv.org/abs/2009.04980
[2] Essentially, the set of properties satisfied by a fixed nonstandard hypernatural gives rise to a non-principal ultrafilter over the naturals. The axiom of choice is necessary to prove the existence of non-principal ultrafilters in (choice-free) set theory, but the existence of non-principal ultrafilters is not sufficient to prove the axiom of choice.
The Python is correct, but the comment is not. The formula for the second derivative should be, of course:
Deleted Comment
Dead Comment
As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation and observe that you generally don't need L'Hôpital's rule to work them out. It is possible to produce examples that can be calculated by L'Hôpital's rule, but not by simply understanding o-notation. But it isn't easy, and you're unlikely to find them in textbooks.
It is probably true that as you go on, limits are more useful in higher mathematics than o-notation. But o-notation is far more useful in most subjects that use mathematics. Given how easy it is to master limits if you know o-notation, why not teach o-notation first?
https://discreteanalysisjournal.com/article/87772-a-simple-c...
I'd say that limits in the sense that you mean (as opposed to category-theoretic limits) are precisely the domain of calculus or, if one wishes so to call it (because one is proving things!), analysis. For example, many US universities, mine included, regard the computation of infinite series as part of Calculus II.
https://www.thriftbooks.com/w/calculus-the-easy-way-easy-way...
I didn't know he had a calculus book, I'll have to check it out now.
> but this book really helped connect trig in an intuitive way.
To me, understanding math in an intuitive way is actually how "maths people" think about math. The numbers and formulas are just a means to an end to get there.
Or, speedrunners and professional musicians can practice motions (relative to their own motions or to an ongoing rhythm) that are accurate to about 1 third.
I assume the terminology would be too easy to confuse with ⅓, so maybe they would have to be called "terces" or "tertians" or "tertiaries" or something. (Specifically, we want to say (1/60)³ hour rather than (1/3) minute!) And maybe it's not a great thing to introduce a non-decimal unit when we're also trying to get rid of them in many other contexts.
It could also be useful for measuring latency of Internet connections, although these have commonly gotten faster over time and many latencies can be under a tierce. On the other hand, the round-trip latency to a satellite in geostationary orbit is about 14 tierces.
[0] https://www.gutenberg.org/ebooks/33283
[1] https://calibre-ebook.com/
Some PDF documents don't contain the source text as digitized text, either. It's just a bundle of scanned images.
Epub documents _are_ HTML files in a zip archive - I'd argue that if a web site is a proper presentation of the source material, an epub is even better.
Additionally, PDF documents _are_ worse for any kind of document technical or otherwise. PDF documents are primarily Postscript instructions without any relation or hierarchy to the included elements. Epub documents/HTML provide semantic relationships and hints to the content.
[edit] and I’m dreading my kids getting past elementary school math because they’re gonna be like “why the hell am I spending months of my life on quadratic equations?” and I’m not gonna have an answer, because IDK why we did that either. At least I have answers for calculus, even if they’re not much good (“so you can do physics stuff”, “right, but will I ever need to do physics stuff?”, “uhhh… unless you really want to, no.”)
That's the same answer as the quadratic formula, but makes a lot more sense to me! Of course I've cooked the numbers so that you don't wind up with surds in the answer, but those are just complications in bookkeeping, not in concept.
And good luck taking on calculus without being super solid in the mathematical tools you use against quadratics -- factoring, completing squares, manipulation of binomials, pairing up like terms, etc.
For example, the way to solve a quadratic is to reduce it to a form one knows how to solve (via competing the square).
The specifics of the mathematics are not the prize, the methods of thinking are.
Edit: I can recommend this book for a self-guided study
https://archive.org/details/zeldovich-higher-mathematics-for...
The author was a Soviet nuclear physicist (who participated in the creation of the H-bomb), so his main point isn't rigor. It can be a nice change of perspective from standard American texts.