Elementary Calculus: An Infinitesimal Approach (2000)

https://people.math.wisc.edu/~keisler/calc.html

by mkl • 5 years ago

153 80 5 years ago

people.math.wisc.edu

mkl6• 5 years ago

This is a first-year university calculus text book based on hyperreal numbers.

marosgrego• 5 years ago

Another approach to infinitesimal calculus which allows a presentation in a relatively clear elementary way is smooth infinitesimal analysis and related synthetic differential geometry: http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...

eqbridges• 5 years ago

Is there an online course covering this?

mkl6• 5 years ago

The key terms to search for are "nonstandard calculus" or "nonstandard analysis". I just found the following set of lectures (but I haven't watched any of them); maybe they are what you want: https://www.youtube.com/watch?v=ILDkYszP2lA&list=PLDXeoTykA-...

andi99913• 5 years ago

How to show such numbers exist? Chapter 1 states they can be constructed from the real numbers and this construction is discussed in the epilog, but (I dont have enough time probably) there seem to be only some axioms in the epilogue but not a construction (or proof of existence).

pfortuny• 5 years ago

It is not complicated but it requires some work and more.

Define “f(x)” is an infinitesimal if lim f(x)=0 when x-> 0.

which is the same as saying “f(x)” is smaller in absolute value than any real number.

Nothing more.

The problem: you cannot divide by infinitesimals. Now you need to “create a field” from these numbers and the reals.: this requires the axiom of choice. There is also the sign problem.

But in the end, “that is all you neeed”. You think of infinitesimals as “numbers smaller than any true real number”.

andi99913• 5 years ago

Thanks! I know about axiom of choice (and I am fine with it), so how do you do the field construction?

I am also a bit worried, if you have a field it is also a division algebra. And I thought we know that the only divisions algebras are R, C, H? Like https://math.stackexchange.com/questions/2020399/division-al... (The link does not 100% fit). So how do hyperreals go around that 'no-go' theorem?

mkl6• 5 years ago

Well, no numbers exist in any real sense. Integers, real numbers, complex numbers, hyperreals, etc., they're all imagined, and built from axioms. Keisler does have a companion book that goes into more of the theory of hyperreals, and its bibliography (p209) points to deeper theoretical work too: https://people.math.wisc.edu/~keisler/foundations.html

andi99913• 5 years ago

They are not all build from axioms but from definition (apart from some models of natural numbers, or you go back to ZFC). Like C is build from R^2, R from the powerset of Q, Q from integer^2, integer from natural numbers.

Otherwise you do not easily know if your axioms are actually consistent.

Koshkin8• 5 years ago

Well, the real existence of abstractions is a pretty involved topic in philosophy of knowledge, and so you can’t simply say that, for example, integers do not ‘really’ exist. Because they indeed do, in a very important (and, in fact, obvious) sense: you need two cats to have a fight.

Y_Y• 5 years ago

I agree with you (for some values of "exist") and furthermore think this is a good reason why "imaginary" is a terrible name for the number set they refer to.

(I prefer "pure complex", but that's got its issues too, of course.)

dwheeler• 5 years ago

There are various ways to construct real numbers and complex numbers. One approach is given in Landau's Foundations of Analysis (a.k.a. Grundlagen der Analysis).

You might want to look at the Metamath Proof Explorer materials on constructing numbers, a good starting point is here: http://us.metamath.org/mpeuni/mmcomplex.html Metamath is a general tool that lets you specify axioms and proofs, and verifies that the proofs only depend on axioms and previously proved proofs. The Metamath Proof Explorer (MPE) is a particular application of it that uses classical logic and the ZFC set theory axioms. MPE shows how to construct numbers using these axioms, then proves a set of number axioms, and from then on uses only those axioms so that the details of any particular construction are not important. What's usually more important is showing that you can construct them.

andi99913• 5 years ago

Cool link. I know some construction of the real and complex numbers, I was asking about the construction of the hyperreals used in the textbook. I couldnt find info about that on the page.

bikenaga3• 5 years ago

Keisler wrote a monograph to accompany the textbook:

https://people.math.wisc.edu/~keisler/foundations.html

It's a bit more than an instructor's manual. It covers the construction of the hyperreals in two ways: One (in section 1E) is relatively short, and is like the compactness theorem approach Abraham Robinson originally used in his book ("Non-standard Analysis", Princeton U. Press).

The second construction uses ultraproducts (in Chapter 1*). Keisler gives a quick introduction to the logic needed to understand the ultraproduct construction, beginning with formal languages and sentential logic. So the prerequisites are there, and explained concisely but clearly. The ultraproduct construction comes up elsewhere; for applications to other areas of math, see Martin Davis's book "Applied Nonstandard Analysis" (Dover Publications). (There's a review of the books by Keisler, Davis, and Stroyan and Luxemburg: https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978...).

The rest of Keisler's monograph contains mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF.

Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company].

When Keisler's book came out, someone decided that it would be reviewed for the Bulletin of the AMS by Errett Bishop, who (as I recall) was a noted constructivist. You can read the review here: https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977... Having a constructivist review a calc textbook that uses nonstandard analysis was probably not a great idea. The review ends with: "Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique." I believe Keisler wrote a reply in a subsequent issue of BAMS, but I can't find it.

andi99913• 5 years ago

I do not see how 1E shows existence. The definition of the natural extension needs the hyperreals in the first place. (reading on mobile, maybe I am missing something here)

bikenaga3• 5 years ago

Addendum to my post above - I didn't notice that mkl had linked to Keisler's "Foundations" earlier - my bad!

mcguire3• 5 years ago

A proof of existence for any number?

andi99913• 5 years ago

if you assume ZF, then 1={0} and existence follows from existence of empty set.

Koshkin8• 5 years ago

While infinitesimals lie at the very heart of the classical approach to calculus, in this day and age it is important to complement a course based on what is now considered non-standard analysis with one of the more standard (limit-based) courses. (I think this is also true in other areas, e.g. if the students are taught a physics course based on, say, geometric algebra, they also should be trained in the more ‘standard’ ways of understanding things.)

mkl6• 5 years ago

I think most universities won't even have a non-standard analysis course available to complement the courses using the standard limit approaches. Your fear seems quite unwarranted to me, since standard analysis courses are usually the only option, and always outnumber nonstandard analysis courses.

rnhmjoj• 5 years ago

Why isn't this approach to calculus more common? Does it lead to more complex proofs or have any serious drawback?

I understand the ε,δ way of defining the limit is important because it extends to other context such as continuity in topological spaces etc; however, as a physicist, infinitesimal quantities reflect the way we really think about calculus intuitively, so it makes sense make them "first-class" numbers.

Smaug1233• 5 years ago

I claim that the first undergrad course in Analysis is not actually there primarily to teach you real analysis. Rather, it's there to teach you the difference between "forall exists" and "exists such that forall", and to hammer into you that intuition is not by itself a proof. Nonstandard analysis, I claim, is not as good a vehicle for those purposes as epsilon-delta analysis is.

pfdietz• 5 years ago

Terry Tao's blog has entries on nonstandard analysis, which he uses in various ways as a working mathematician.

https://terrytao.wordpress.com/tag/nonstandard-analysis/

https://terrytao.wordpress.com/2010/11/27/nonstandard-analys...

roydanroy2• 5 years ago

Recent Annals of Statistics paper that uses nonstandard analysis: https://www.e-publications.org/ims/submission/AOS/user/submi...

TeeMassive• 5 years ago

I quickly skimmed through the book, this looks great for a much needed refresh after 10 years of a non-math intensive programming career. Question: are there solutions to the problems?

kkylin• 5 years ago

People who like this may also be interested in Nelson's little book on probability using infinitesimals: https://web.math.princeton.edu/~nelson/books/rept.pdf

billytetrud2• 5 years ago

This is how all calculus should be taught.

ganzuul• 5 years ago

inf + 1 = inf, which is idempotent, so if you extend the complex numbers with infinity they become a semiring. Is the ire that infinitesmals draw related to this?

iib2• 5 years ago

From what I understand, you do not work directly with infinite, but infinitude, basically a hyperinteger that is bigger than any integer. For which H + 1 = H does not hold.

hansvm• 5 years ago

There are lots of different "infinities" floating around in mathematics, and that probably doesn't help the matter.

The extended reals or extended complex plane do behave as you describe (and infinite cardinalities have the same property), but infinitesimal treatments of calculus do not. You're instead working in an ordered field that contains the reals and some element greater than all reals.

Supposing such a thing can exist (they can exist in ZFC), the fact that you're working in an ordered field gives you a lot of infinitesimals and infinities (and also a lot of numbers "between" the ordinary reals).

rsj_hn3• 5 years ago

If you are worried about epsilons and deltas being too hard, then you can write your calculus text to not be rigorous rather than trying to make it rigorous with non-standard analysis. I think giving up on some of the formalism while still sketching the ideas of proofs is the way to go from a pedagogical point of view.

bgorman• 5 years ago

Have you compared the approaches? I learned with epsilon delta, but I would be interested in seeing data about learning outcomes with alternative approaches.

In my experience, epsilon delta understanding was pretty much unrelated to passing calculus, where tests consisted of formula/substitution crunching.

rsj_hn3• 5 years ago

In my experience, epsilon delta understanding was pretty much unrelated to passing calculus, where tests consisted of formula/substitution crunching

That's a shame. There are so many exciting things to learn in calculus that you can skip the epsilon delta stuff and still do so much more than formula/substitution crunching. Calculus is the gateway to differential geometry, topology, mathematical physics, differential equations, taylor series which are useful for numerical approximations and so many other things, analytic number theory, complex analysis, many forms of statistics. So many wonderful things you can do with it.

Imagine using your new found calculus knowledge to show that the orbit of the planets about the sun is an ellipse -- which is what Newton used calculus to do -- OR learning all about epsilons and deltas. Which would be more fun as a student learning calculus. Make the math exciting enough and people will put up with the drudgery of calculations.

mcguire3• 5 years ago

"...you can write your calculus text to not be rigorous..."

Oooh, the comments you'll get from other mathematicians...

rsj_hn3• 5 years ago

I can take the heat. School texts don't need to be rigorous and there are different levels of rigour. I think Bourbaki did more harm than good.

ska2• 5 years ago

you can write your calculus text to not be rigorous rather than trying to make it rigorous with non-standard analysis.

e.g., many engineering calculus courses, and explicitly some of the texts.

eigenhombre• 5 years ago

This was the book my 1st sem. calculus class used in 1986, at UW-Madison (where the link is hosted and where the author presumably teaches). I remember it as providing a fairly gentle introduction to differentiation, providing a stepping-stone to more traditional notions of limits later on.

I was just thinking the other day about how enjoyable I found learning calculus, compared to what others tend to report. Perhaps this book was part of the story -- very cool that it is being provided free of charge now!

mcguire3• 5 years ago

In roughly 1987-1988, I took the required 2 semesters of calculus at UT Austin. It actually took me 4 semesters, for a number of reasons including that I was seriously unprepared for calculus.

But the worst part of the experience was the method of starting with epsilon/delta and limits, to "explain" what was going on, and then throwing that away to get on with solving problems using differentials and integrals. The lectures almost universally took the form of the professor going through a proof of the technique in question and then assigning a set of problems from the text.

bachmeier• 5 years ago

I had to retake the first calculus class. Didn't have much to do with the difficulty of the material, but rather a professor doing things that would get him pulled from the classroom in 2021. Just one of the many things he'd do is call on someone to answer a question during the class. If he didn't like the answer he'd tell them they were lazy or that they should quit college.

dkarl• 5 years ago

But the worst part of the experience was the method of starting with epsilon/delta and limits, to "explain" what was going on, and then throwing that away to get on with solving problems using differentials and integrals

I guess that was pretty normal once upon a time? I remember that was how my calculus textbooks did it. First pictures to develop some visual intuition, then a gentle foray into how to make the intuitions more rigorous using limits, sequences, and series, and then techniques for solving certain differentiation and integration problems. It makes sense to me; it shows that the techniques are based in math and not in magic, and it lays the groundwork to introduce Taylor series later. It also foreshadows the kinds of proofs that students might do in real analysis if they decide to pursue an engineering or physical science major.

I guess every math class could be very different if you knew it was the last math class that students would take, but you don't know that, and it's problematic to separate students according to which ones will study the material further before they're even exposed to it. If a student takes a class and unexpectedly finds they want (or need) to take further classes in the area, they don't want to be find out later that, "Oh, sorry, we didn't think you would pursue this subject, so we gave you the version of the class that didn't prepare you for the next one."

hilbert42• 5 years ago

This is a nice gesture by H. Jerome Keisler, every bit helps to promote a math subject that frightens off so many.

That said, it's a shame the PDFs are in image format instead of text as it makes it more difficult to copy notes etc. from the text into one's workbooks. This would save the student time in taking notes.

Another minor criticism I have is that like many texts on mathematics it lacks both detail on the background to mathematical concepts and their applications. For example, the discussion of Eula's Formula on page 879 is somewhat incomplete. In some ways the Wiki page https://en.wikipedia.org/wiki/Euler%27s_formula does a better job in that it has a diagram of the 'Three-dimensional visualization of Euler's formula' (its application to circular polarization, etc.). Including this diagram and having some discussion about the significance of the relevance of the i, e and ᴨ in this famous relationship would have put the subject into better context.

(I recall the significance was lost on me when I first studied the subject for the same reasons. I think that many mathematicians don't realize that many of us ordinary mortals just don't think the same way they do, so including more visual descriptions and diagrams, typical applications and historical background do actually help one's understanding. However, I acknowledge that including this extra material makes textbooks larger and thus more expensive. Perhaps that's the penalty we have to pay.)

chobytes• 5 years ago

This helped me a lot when I was taking calculus. We were assigned Stewart, but I basically learned the from this.

Analysis from Rudin, and later some manifold theory from Lee, have converted me to more mainstream views on the subject... However I think this is probably how calculus should be introduced at first, particularly to say highschoolers.

Elementary Calculus: An Infinitesimal Approach (2000)

Related Stories